prompt
stringlengths 0
312k
| target
stringlengths 2
62.4k
|
|---|---|
\section{Introduction}
In this note we revisit Sobolev spaces on $\mathbb{R}^n$, $n \geq 1$, from the point of view of difference quotients. For $1 \leq p < \infty$, the homogeneous Sobolev space $\dot{W}^{1,p}$ on $\mathbb{R}^n$ consists of all locally integrable functions $u$ modulo constants, whose distributional gradient $\nabla u \in L^p$. It is normed by
\[
\|\nabla u\|_{L^p} = \Big( \int_{\mathbb{R}^n} |\nabla u|^p \, \mathrm{d} x \Big)^{1/p}.
\]
We will also work with the homogeneous $\BV$ space (BV stands for \emph{bounded variation}). It consists of all locally integrable functions $u$ modulo constants, whose distributional gradient $\nabla u$ is a finite Radon measure (written $\nabla u \in \mathcal{M}$).
In other words, it is the space of all $u \in L^1_{\text{loc}}$ such that
\begin{equation}
\label{eq_Aef7bohJ2pieShooDeupeeli}
\sup \left\{ \Big| \int_{\mathbb{R}^n} u(x) \operatorname{div} \phi(x) \, \mathrm{d} x \Big| \colon \phi \in C^1_c(\mathbb{R}^n;\mathbb{R}^n), \|\phi\|_{L^{\infty}(\mathbb{R}^n;\mathbb{R}^n)} \leq 1 \right\}
\end{equation}
is finite (which in particular contains $\dot{W}^{1,1}$). It is normed by
\[
\|u\|_{\BV} \coloneqq \|\nabla u\|_{\mathcal{M}}
\]
which is just the supremum in \eqref{eq_Aef7bohJ2pieShooDeupeeli}.
In \cite{Brezis_VanSchaftingen_Yung_2021}, a new formula was established for $\|\nabla u\|_{L^p(\mathbb{R}^n)}$ for $u \in C^{\infty}_c$ that involves only difference quotients and no gradients.
In \cite{BSVY}, this formula was extended to all $u \in \dot{W}^{1,p}$, or all $u \in \BV$, and in fact we found a natural one-parameter family of such formulae.
A related limiting formula has been obtained earlier by Nguyen \cite{Nguyen06} and by Brezis and Nguyen \cite{BN2018}.
The one-parameter family of formulae we found can be used to recover certain Gagliardo-Nirenberg interpolation inequalities due to Cohen, Dahmen, Daubechies and DeVore \cite{CDDD}.
It also allows us to prove some substitutes when such interpolation inequalities fail (cf.\ \cite{Brezis_VanSchaftingen_Yung_2021Lorentz}).
Somewhat underpinning all these is a certain interplay between $L^p$ and weak-$L^p$ (also known as $L^{p,\infty}$ in the scale of Lorentz spaces).
In \Cref{sect:Lorentz} we review the basics about Lorentz spaces and some related facts about real interpolation; we also explain a simple instance of how an $L^p$ norm on a lower dimensional space can be realized as a weak-$L^p$ norm in a higher dimensional space, a phenomenon that drives our main theorems.
In \Cref{sect:BBM} we further motivate our main results by looking at difference quotient characterizations of fractional Sobolev spaces (also known as diagonal Besov spaces) and the related BBM formula.
In \Cref{sect:diffquot} we state and comment on the proofs of our main results, which is a one-parameter family formulae of Sobolev / $\BV$ norms and $L^p$ norms.
In \Cref{sect:applications} we give some applications towards Gagliardo-Nirenberg interpolation, emphasizing the importance of the existence of a one-parameter family of formulae, and contrasting our formulae for $\BV$ with the one-parameter family of embeddings by Cohen, Dahmen, Daubechies and DeVore.
In \Cref{sect:further} we discuss some other related works and some possible further directions.
The current paper is based on the lecture delivered by the last named author at the award ceremony of the inaugural Antonio Ambrosetti medal.
\Cref{thm6} and some of the material in \Cref{sect:applications} are extensions of known results, which may not have appeared explicitly in the literature before.
\subsubsection*{\it Acknowledgements}
The idea of replacing strong $L^p$ by weak-$L^p$ in the context of Sobolev-Gagliardo-Nirenberg-type inequalities involving $W^{1,1}$ goes back to the work of L. Greco and R. Schiattarella \cite{MR4106822}.
We are grateful to Carlo Sbordone for calling the attention of one of us (HB) to their result.
P.-L.Y. would like to thank Qingsong Gu for sharing his insights in their collaboration on \cite{MR4249777}.
A.S. and P.-L.Y. would like to thank the Hausdorff Research Institute of Mathematics and the organizers of the trimester program ``Harmonic Analysis and Analytic Number Theory'' for a pleasant working environment in the summer of 2021.
The research was supported in part by NSF grant DMS-2054220 (A.S.) and by a Future Fellowship FT200100399 from the Australian Research Council (P.-L.Y.).
\section{Lorentz spaces and real interpolation} \label{sect:Lorentz}
Let $(X,\nu)$ be a measure space.
For $1 \leq p < \infty$, if $f \in L^p(\nu)$ for some measure $\nu$, then for every $\lambda > 0$,
\[
\|f\|_{L^p(\nu)}^p = \int_X |f|^p \, \mathrm{d}\nu \geq \lambda^p \, \nu \{x \colon |f(x)| > \lambda\}.
\]
In particular, if $f \in L^p(\nu)$, then
\begin{equation}
\label{eq_Oth0beiZau1soo7ne8Chaefo}
\sup_{\lambda > 0} \Big( \lambda \, \nu\{x \colon |f(x)| > \lambda\}^{1/p} \Big) < \infty;
\end{equation}
but the converse is not necessarily true.
If $f$ is measurable on $X$ and the supremum in \eqref{eq_Oth0beiZau1soo7ne8Chaefo} is finite, then $f$ is said to be in weak-$L^p(\nu)$.
Its weak-$L^p$ (quasi)-norm is defined as the supremum in \eqref{eq_Oth0beiZau1soo7ne8Chaefo}, and denoted by $[f]_{L^{p,\infty}(\nu)}$.
A classic example is given by $f(x) = |x|^{-n/p}$; it is in weak-$L^p(\mathrm{d} x)$ on $\mathbb{R}^n$, because
\[
\mathcal{L}^n \{x \in \mathbb{R}^n \colon |x|^{-n/p} > \lambda\} = \mathcal{L}^n \{x \in \mathbb{R}^n \colon |x| \leq \lambda^{-p/n}\} \simeq \lambda^{-p}.
\]
(Henceforth we write $\mathcal{L}^n$ for Lebesgue measure on $\mathbb{R}^n$.)
This $f$ is not in $L^p(\mathrm{d} x)$, because
\[
\int_{\mathbb{R}^n} |f|^p \, \mathrm{d} x = \int_{\mathbb{R}^n} |x|^{-n} \, \mathrm{d} x = +\infty,
\]
and this serves as a motivation of our main result in what follows.
Later on we will also need the Lorentz spaces $L^{p,r}(\nu)$, which for $1 \leq p, r < \infty$ are defined as the space of all measurable $f$ on $X$ with
\[
[f]_{L^{p,r}(\nu)} \coloneqq \Big(r \int_0^{\infty} \lambda^r \nu\{x \colon |f(x)| > \lambda\}^{r/p} \frac{\mathrm{d} \lambda}{\lambda} \Big)^{1/r} < \infty.
\]
They arise as real interpolation spaces: if $1 \leq p_0 < p_1 \leq \infty$ and $\frac{1}{p} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$ for some $0 < \theta < 1$, then for $1 \leq r \leq \infty$,
\[
L^{p,r}(\nu) = [L^{p_0}(\nu),L^{p_1}(\nu)]_{\theta,r}
\]
where for any Banach spaces $B_0$ and $B_1$, and $f \in B_0 + B_1$, the interpolation norm is defined as
\[
\|f\|_{[B_0,B_1]_{\theta,r}} \coloneqq \Big( \int_0^{\infty} t^{-\theta} \inf_{f = f_0 + f_1} (\|f_0\|_{B_0} + t \|f_1\|_{B_1} )^r \frac{\mathrm{d} t}{t} \Big)^{1/r}
\]
for $1 \leq r < \infty$, and
\[
\|f\|_{[B_0,B_1]_{\theta,\infty}} \coloneqq \sup_{0 < t < \infty} t^{-\theta} \inf_{f = f_0 + f_1} (\|f_0\|_{B_0} + t \|f_1\|_{B_1} )
\]
when $r = \infty$.
It is also well-known that
\[
[f]_{L^{p,r}(\nu)} = \|f\|_{L^p(\nu)} \quad \text{if $r = p$}.
\]
In the rest of this article, we will be exploiting a relationship between the $L^p$ norm on a space and the weak-$L^p$ quasi-norm on a higher dimensional space. The simplest instance of this might be the identity
\begin{equation} \label{eq:Tao}
\|f\|_{L^p(X, \, \nu)} = \left[ \frac{f(x)}{y^{1/p}} \right]_{L^{p,\infty}(X \times (0,\infty), \, \nu \, \mathrm{d} y)}, \quad 1 \leq p < \infty;
\end{equation}
we thank Terence Tao for communicating this comment.
\section{The Bourgain-Brezis-Mironescu formula} \label{sect:BBM}
Let's introduce now a shorthand for the first order difference operator:
\begin{equation} \label{eq:diffop}
\Delta_h u(x) \coloneqq u(x+h)-u(x) \quad \text{for $x, h \in \mathbb{R}^n$}.
\end{equation}
For $h$ small, and $u \in C^{\infty}$, we have then $|\Delta_h u(x)| \simeq |\nabla u(x) \cdot h|$. Hence roughly speaking, we might believe that $|\nabla u(x)| \simeq \frac{|\Delta_h u(x)|}{|h|}$, at least if one averages over all possible directions for $h$. As a result, to express $\|\nabla u\|_{L^p(\mathbb{R}^n)}$ using a difference quotient instead of a gradient, a naive guess might be to try
\[
\iint_{\mathbb{R}^{2n}} \frac{|\Delta_h u(x)|^p}{|h|^p} \, \mathrm{d} h \, \mathrm{d} x
\quad \text{
in place of } \quad
\int_{\mathbb{R}^n} |\nabla u(x)|^p \, \mathrm{d} x.
\]
This is not working, because if $u_{t}(x) \coloneqq u(t x)$, then
\[
\int_{\mathbb{R}^n} |\nabla u_{t}(x)|^p \, \mathrm{d} x \quad \text{scales like $t^{p-n}$}
\]
but
\[
\iint_{\mathbb{R}^n \times \mathbb{R}^n} \frac{|\Delta_h u_{t}(x)|^p}{|h|^p} \, \mathrm{d} h \, \mathrm{d} x \quad \text{scales like $t^p$}.
\]
A proper scaling will be achieved if we consider
\[
\iint_{\mathbb{R}^{2n}} \frac{|\Delta_h u(x)|^p}{|h|^p} \frac{\mathrm{d} h \, \mathrm{d} x}{|h|^n}
\]
instead; if given $b \in \mathbb{R}$ we introduce a difference quotient\footnote{Note that our notation here is different from that in \cite{BSVY}; the $Q_b u(x,y)$ in \cite{BSVY} would be written as $\mathcal{Q}_{1+b} u(x,h)$ with $h = y-x$ in the current paper. As a result, the set $E_{\lambda,b}[u]$ in \cite{BSVY} should be compared to the set $\mathcal{E}_{\lambda,1+b}[u]$ in \eqref{eq:Esetdef} below.}
\begin{equation} \label{eq:diffquot}
\mathcal{Q}_b u(x,h) \coloneqq \frac{|\Delta_h u(x)|}{|h|^b},
\end{equation}
then the above suggests that we consider the integral $\iint_{\mathbb{R}^{2n}} \mathcal{Q}_{1+\frac{n}{p}} u(x,h)^p \, \mathrm{d} h \, \mathrm{d} x$.
This idea works if we are dealing with fractional Sobolev spaces.
Indeed, for $0 < s < 1$ and $1 \leq p < \infty$, the fractional Sobolev space $\dot{W}^{s,p}$ is the space of all $u \in L^1_{\text{loc}}(\mathbb{R}^n)$ such that
\[
\begin{split}
\|u\|_{\dot{W}^{s,p}}^p &\coloneqq \iint_{\mathbb{R}^{2n}} \mathcal{Q}_{s+\frac{n}{p}} u(x,h)^p \, \mathrm{d} h \, \mathrm{d} x = \iint_{\mathbb{R}^{2n}} \frac{|u(x+h)-u(x)|^p}{|h|^{sp+n}} \, \mathrm{d} h \, \mathrm{d} x < \infty.
\end{split}
\]
When $1 < p < \infty$, it is known to be equal to the diagonal Besov space $\dot{B}^s_{p,p}$ with comparable norms.
So this suggests again that maybe $\|\nabla u\|_{L^p(\mathbb{R}^n)}$ should be compared to $\|\mathcal{Q}_{ 1+\frac{n}{p}}u\|_{L^p(\mathbb{R}^{2n}, \, \mathrm{d} x\, \mathrm{d} h)}$?
Unfortunately this does not work; as observed in \cite{Bourgain_Brezis_Mironescu_2001} (see also \cite{Brezis_2002}), even for $u \in C^{\infty}_c(\mathbb{R}^n)$, unless $u \equiv 0$, the $L^p\!$ norm on $\mathbb{R}^{2n}$ is always infinite! The issue here is that $|h|^{-n/p} \notin L^p(\mathrm{d} h)$ on $\mathbb{R}^n$.
On the other hand, we did see $|h|^{-n/p} \in L^{p,\infty}(\mathrm{d} h)$, and this will partly motivate our main result in the next section.
For now, let's recall the ``BBM-formula'', by Bourgain, Brezis and Mironescu in \cite{Bourgain_Brezis_Mironescu_2001}, which explores what happens to $\|u\|_{\dot{W}^{s,p}}$ as $s \to 1^-$.
On $\mathbb{R}^n$, one instance of this formula says for $1 \leq p < \infty$ and (say) $u \in C^1_c$, that we have\footnote{The original BBM formula was stated and proved for bounded domains in \cite{Bourgain_Brezis_Mironescu_2001}, but the result extends easily to the whole $\mathbb{R}^n$. See e.g.\ \cite{BSY_BBM}*{Appendix A}.}
\begin{equation} \label{eq:BBM}
\lim_{s \to 1^-} (1-s) \|u\|_{\dot{W}^{s,p}}^p = \lim_{s \to 1^-} (1-s) \|\mathcal{Q}_{ s+\frac{n}{p}}u\|_{L^p(\mathbb{R}^{2n}, \, \mathrm{d} x\, \mathrm{d} h)}^p = \frac{k(p,n)}{p} \|\nabla u\|_{L^p}^p
\end{equation}
where $k(p,n)$ is given explicitly by
\begin{equation} \label{eq:kpn}
k(p,n) \coloneqq \int_{\S^{n-1}} |e \cdot \omega|^p \, \mathrm{d} \omega, \quad e \in \S^{n-1}.
\end{equation}
In particular, $\|\mathcal{Q}_{s+\frac{n}{p}} u\|_{L^p(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)}$ blows up like $(1-s)^{-1/p}$ as $s \to 1^-$ unless $u$ is a constant, another indication that $\|\mathcal{Q}_{1+\frac{n}{p}} u\|_{L^p(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)}$ is not good for computing $\|\nabla u\|_{L^p}$.
Our first main result in the next section offers an alternative point of view, that does not involve varying $s$ (as in \eqref{eq:BBM}), but involves a weak-$L^p$ norm instead of the $L^p$ norm on $\mathbb{R}^{2n}$; remember $|h|^{-n/p}$ is not in $L^p(\mathrm{d} h)$, but it is in weak-$L^p(\mathrm{d} h)$.
Before we close this section though, we mention a related result of Maz'ya and Shaposhnikova \cite{MR1940355}.
They explored what happens to $\|u\|_{\dot{W}^{s,p}}$ as $s \to 0^+$ instead of $s \to 1^-$, and showed that for (say) $u \in C^1_c$ and $1 \leq p < \infty$,
\begin{equation} \label{eq:MSh}
\lim_{s \to 0^+} s \|u\|_{\dot{W}^{s,p}}^p = \lim_{s \to 0^+} s \|\mathcal{Q}_{s+\frac{n}{p}}u\|_{L^p(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)}^p = \frac{2 \sigma_{n-1}}{p} \|u\|_{L^p}^p
\end{equation}
where $\sigma_{n-1}$ is the surface area of $\S^{n-1}$.
This result will be compared to \Cref{thm6} below.
\section{Difference quotient characterizations} \label{sect:diffquot}
Our first result is a characterization of $\|\nabla u\|_{L^p}$ for $u \in C^{\infty}_c(\mathbb{R}^n)$.
We use the notations of difference and difference quotient introduced in \eqref{eq:diffop} and \eqref{eq:diffquot}.
\begin{thm}[see \cite{Brezis_VanSchaftingen_Yung_2021}] \label{thm1}
Let $n \ge 1$, $1 \leq p < \infty$ and $u \in C^{\infty}_c(\mathbb{R}^n)$.
Then
\begin{equation} \label{eq:sup_BVY}
\|\nabla u\|_{L^p} \simeq [\mathcal{Q}_{1+\frac{n}{p}} u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)} = \Big[ \frac{\Delta_h u}{|h|^{1+\frac{n}{p}}} \Big]_{L^{p,\infty}(\mathbb{R}^{2n},\, \mathrm{d} x \, \mathrm{d} h)}.
\end{equation}
In other words, for $\lambda > 0$ and $b \in \mathbb{R}$, denote by
\begin{equation} \label{eq:Esetdef}
\mathcal{E}_{\lambda, b}[u] \coloneqq \Big\{(x,h) \in \mathbb{R}^{2n} \colon \mathcal{Q}_{b} u(x,h) > \lambda \Big\}
\end{equation}
the superlevel set of $\mathcal{Q}_{b} u$ at height $\lambda$. Then
\[
\|\nabla u\|_{L^p}^p \simeq \sup_{\lambda > 0} \Big(\lambda^p \mathcal{L}^{2n}\bigl(\mathcal{E}_{\lambda, 1+\frac{n}{p}}[u]\bigr) \Big).
\]
In fact, we also have
\begin{equation} \label{eq:limit_eq_BVY}
\frac{k(p,n)}{n} \|\nabla u\|_{L^p}^p = \lim_{\lambda \to +\infty} \Big(\lambda^p \mathcal{L}^{2n}\bigl(\mathcal{E}_{\lambda, 1+\frac{n}{p}}[u]\bigr) \Big).
\end{equation}
Here $k(p,n)$ is given by \eqref{eq:kpn}.
\end{thm}
A few remarks are in order.
First, the power $1+\frac{n}{p}$ is dictated by dilation invariance: if $[\mathcal{Q}_b u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)}$ scales like $\|\nabla u\|_{L^p}$ upon replacing $u(x)$ by $u(tx)$ for $t > 0$, then $b = 1+\frac{n}{p}$.
Next, the limit equality \eqref{eq:limit_eq_BVY} can be proved using Taylor expansion, in a way somewhat reminiscent to the proof of the BBM formula.
In fact, $\mathcal{Q}_{1+\frac{n}{p}} u$ is approximately $|h|^{-n/p} |\nabla u(x) \cdot \frac{h}{|h|}|$ when $|h|$ is small, and heuristically $\mathcal{E}_{\lambda,1+\frac{n}{p}}[u]$ can be approximated by the set
\[
\tilde{\mathcal{E}}_{\lambda,1+\frac{n}{p}}[u] \coloneqq \Big\{(x,h) \in \mathbb{R}^{2n} \colon |h|^{-n/p} \Big|\nabla u(x) \cdot \frac{h}{|h|}\Big| > \lambda \Big\}
\]
when $\lambda$ is big.
But for all $\lambda > 0$,
\[
\lambda^p \mathcal{L}^{2n}\bigl(\tilde{\mathcal{E}}_{\lambda, 1+\frac{n}{p}}[u]\bigr) = \frac{k(p,n)}{n} \|\nabla u\|_{L^p}^p.
\]
This heuristic can be turned into an actual proof for \eqref{eq:limit_eq_BVY}.
Moreover, in light of the limit equality \eqref{eq:limit_eq_BVY}, to prove \eqref{eq:sup_BVY}, we only need to prove an upper bound for a weak-$L^p$ norm, namely
\[
[\mathcal{Q}_{1+\frac{n}{p}} u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)} \lesssim \|\nabla u\|_{L^p},
\]
which can be done using a Vitali covering lemma (plus the method of rotations in dimensions $n > 1$).
It is somewhat reminiscient of the proof that the Hardy-Littlewood maximal function is bounded from $L^1$ to weak-$L^1$; see also work of Dai, Lin, Yang, Yuan and Zhang \cite{DLYYZ_metricmeasure} who extended our proof to some metric-measure spaces of homogeneous type.
Finally, the weak-$L^p$ quasi-norm in the theorem cannot be replaced by any (bigger) Lorentz $L^{p,r}$ quasi-norm where $r < \infty$.
\begin{thm}[see \cite{Brezis_VanSchaftingen_Yung_2021Lorentz}] \label{thm2}
(i) If $u$ is measurable on $\mathbb{R}^n$ and
\[
[\mathcal{Q}_{1+\frac{n}{p}}u]_{L^{p,r}(\mathbb{R}^{2n}, \, \mathrm{d} x \, \mathrm{d} h)} < \infty
\]
for some $1 \leq p, r < \infty$, then $u$ is a.e.\ a constant.\\
(ii) Indeed, if $u$ is measurable on $\mathbb{R}^n$ and
\[
\lim_{\lambda \to +\infty} \lambda^p \mathcal{L}^{2n} \Big\{(x,h) \in \mathbb{R}^{2n} \colon \mathcal{Q}_{1+\frac{n}{p}} u(x,h) > \lambda \Big\} = 0
\]
for some $1 \leq p < \infty$, then $u$ is a.e.\ a constant.
\end{thm}
The difficulty here is that we only know $u$ is measurable.
If we already know $u \in C^{\infty}_c$, the conclusion of \Cref{thm2} follows from \Cref{thm1}; if we already know $u \in \dot{W}^{1,p}$, then the conclusion follows from our next two theorems.
The case $p = 1$ relies on a result of Poliakovsky \cite{poliakovsky}.
\Cref{thm1} provided a way of computing the $\dot{W}^{1,p}$ norm of a function $u \in C^{\infty}_c(\mathbb{R}^n)$ up to a multiplicative constant.
It turns out there is a natural \emph{one-parameter} family of such formulae for $\|\nabla u\|_{L^p(\mathbb{R}^n)}$, for \emph{general} $u \in \dot{W}^{1,p}$ or $u \in \BV$ (not just for $u \in C^{\infty}_c$); in applications often the question at hand determines which formula one uses within this family.
To state such formulae, let $\gamma \in \mathbb{R}$.
Define the measure
\begin{equation} \label{eq:nu_def}
\mathrm{d} \nu_{\gamma} \coloneqq |h|^{\gamma-n} \, \mathrm{d} x \, \mathrm{d} h
\end{equation}
on $\mathbb{R}^{2n}$.
(The case $\gamma = n$ corresponds to the Lebesgue measure $\mathrm{d} x \, \mathrm{d} h = \mathcal{L}^{2n}$ we used earlier.)
Then we have two theorems, the first dealing with the case $p > 1$, the second dealing with the case $p = 1$; again we use the constant $k(p,n)$ defined in \eqref{eq:kpn}.
\begin{thm}[see \cite{BSVY}] \label{thm3}
Let $n \ge 1$, $1 < p < \infty$ and $u \in \dot{W}^{1,p}(\mathbb{R}^n)$.
Then for $\gamma \ne 0$,
\begin{equation} \label{eq:W1p_sup}
\|\nabla u\|_{L^p} \simeq [\mathcal{Q}_{1+\frac{\gamma}{p}} u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})} = \Big[ \frac{\Delta_h u}{|h|^{1+\frac{\gamma}{p}}} \Big]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})}.
\end{equation}
Furthermore, if $\mathcal{E}_{\lambda,b}[u]$ is the superlevel set of $\mathcal{Q}_b u$ at height $\lambda$ given in \eqref{eq:Esetdef}, then
\begin{equation} \label{eq:W1p_lim}
\frac{k(p,n)}{|\gamma|} \|\nabla u\|_{L^p}^p =
\begin{cases} \lim_{\lambda \to +\infty} \Big(\lambda^p \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \Big) \quad &\text{if $\gamma > 0$,} \\
\lim_{\lambda \to 0^+} \Big(\lambda^p \nu_{\gamma} \bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \Big) \quad &\text{if $\gamma < 0$}.
\end{cases}
\end{equation}
\end{thm}
The case $\gamma = -p$ of the above limit equality \eqref{eq:W1p_lim} is due to Nguyen \cite{Nguyen06}.
Next, for the case $p = 1$ we have a similar theorem for $\BV$, but with a number of \emph{additional twists}!
\begin{thm}[see \cite{BSVY}] \label{thm4}
Suppose $n \ge 1$. Then for $\gamma \in \mathbb{R} \setminus [-1,0]$ and $u \in \BV(\mathbb{R}^n)$,
\begin{equation} \label{eq:sup_p=1_good}
\|u\|_{\BV} = \|\nabla u\|_{\mathcal{M}} \simeq [\mathcal{Q}_{1+\gamma} u]_{L^{1,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})} =\Big[ \frac{\Delta_h u}{|h|^{1+\gamma}} \Big]_{L^{1,\infty}(\mathbb{R}^{2n},\nu_{\gamma})}.
\end{equation}
Furthermore, if $\mathcal{E}_{\lambda,b}[u]$ is the superlevel set of $\mathcal{Q}_b u$ at height $\lambda$ given in \eqref{eq:Esetdef}, then the formula
\begin{equation} \label{eq:limit_eq_p=1_good}
\frac{k(1,n)}{|\gamma|} \|\nabla u\|_{\mathcal{M}} = \begin{cases} \lim_{\lambda \to +\infty} \Big(\lambda \nu_{\gamma}\bigl(\mathcal{E}_{\lambda,1+\gamma}[u]\bigr) \Big) \quad &\text{if $\gamma > 0$} \\
\lim_{\lambda \to 0^+} \Big(\lambda \nu_{\gamma}\bigl(\mathcal{E}_{\lambda,1+\gamma}[u]\bigr) \Big) \quad &\text{if $\gamma < -1$}
\end{cases}
\end{equation}
holds for $u \in \dot{W}^{1,1}$ but can \emph{fail} for $u \in \BV$ (e.g.\ if $u = \mathbf{1}_{\Omega}$ where $\Omega \subset \mathbb{R}^n$ is any bounded domain with smooth boundary, then the limits above exist but is equal instead to $\frac{k(1,n)}{|\gamma+1|} \|\nabla u\|_{\mathcal{M}}$).
For $\gamma \in [-1,0)$,
\begin{equation} \label{eq:sup_p=1_bad}
\sup_{u \in C^{\infty}_c(\mathbb{R}^n), \, \|\nabla u\|_{L^1(\mathbb{R}^n)} = 1} [\mathcal{Q}_{1+\gamma} u]_{L^{1,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})} = +\infty;
\end{equation}
furthermore, the formula
\begin{equation} \label{eq:limit_eq_p=1_bad}
\frac{k(1,n)}{|\gamma|} \|\nabla u\|_{L^1} =
\lim_{\lambda \to 0^+} \Big(\lambda \nu_{\gamma}\left(\mathcal{E}_{\lambda,1+\gamma}[u]\right) \Big)
\end{equation}
remains true for all $u \in C^1_c(\mathbb{R}^n)$, but \emph{fails} for $u \in \dot{W}^{1,1}(\mathbb{R}^n)$, and the failure is generic in the sense of Baire category, despite the fact that for all $u \in \dot{W}^{1,1}(\mathbb{R}^n)$ we do have
\begin{equation} \label{eq:upperbdd_by_liminf}
\frac{k(1,n)}{|\gamma|} \|\nabla u\|_{L^1} \leq
\liminf_{\lambda \to 0^+} \Big(\lambda \nu_{\gamma}\left(\mathcal{E}_{\lambda,1+\gamma}[u]\right) \Big).
\end{equation}
\end{thm}
Note that the range of $\gamma$ allowed in \eqref{eq:sup_p=1_good} in \Cref{thm4} is smaller than that in \Cref{thm3}; in fact, \eqref{eq:sup_p=1_bad} shows that \eqref{eq:sup_p=1_good} is false when $\gamma \in [-1,0)$, even if one only restricts to functions in $C^{\infty}_c$.
The case $\gamma = -1$ of the limiting formula \eqref{eq:limit_eq_p=1_bad} has already been established by Brezis and Nguyen \cite{BN2018}: they have already shown that \eqref{eq:limit_eq_p=1_bad} remains true for all $u \in C^1_c(\mathbb{R}^n)$, but \emph{fails} for general $u \in \dot{W}^{1,1}(\mathbb{R}^n)$.
On the other hand, the counterexamples in the case $\gamma \in (-1,0)$ relies on the construction of a Cantor set of dimension $1+\gamma$; in fact in this range of $\gamma$, one may choose approximations of the associated Cantor-Lebesgue function (similar to the one in the proof of \Cref{thm7}(ii) below) to establish \eqref{eq:sup_p=1_bad}, and use sums of such functions to construct counterexamples to \eqref{eq:limit_eq_p=1_bad} in $\dot{W}^{1,1}$.
We remark again once $\nu_{\gamma}$ is fixed (by fixing $\gamma$), the powers $1+\frac{\gamma}{p}$ and $1+\gamma$ in the denominators of the difference quotients in \Cref{thm3} and \Cref{thm4} are dictated by dilation invariance.
Also, the two theorems do not address what happens if $\gamma = 0$; it turns out things fail strikingly in the case $\gamma = 0$.
It can be shown \cite{BSVY}*{Theorem 1.5} that if both $u, \nabla u \in L^1_{\text{loc}}(\mathbb{R}^n)$ then
\[
\inf \{ \lambda > 0 \colon \nu_0(\mathcal{E}_{\lambda,1}[u]) < \infty\} = \|\nabla u\|_{L^{\infty}};
\]
in particular, if in addition $[\mathcal{Q}_1 u]_{L^{p,\infty}(\mathbb{R}^{2n},\nu_0)} < \infty$ for some $1 \leq p < \infty$, then $u$ is a.e.\! a constant.
It may be fitting to comment a bit on the proofs of the positive results in \Cref{thm3} and \Cref{thm4}.
Let $n \geq 1$ and $1 \leq p < \infty$.
The issue here is that $C^1_c(\mathbb{R}^n)$ is dense in $\dot{W}^{1,p}(\mathbb{R}^n)$ only when $n > 1$ and $p \geq 1$, or when $n = 1$ and $p > 1$.
In other words, $C^1_c(\mathbb{R})$ is not dense in $\dot{W}^{1,1}(\mathbb{R})$.
Fortunately it is always possible to approximate a general function in $\dot{W}^{1,p}(\mathbb{R}^n)$ in norm by $C^1$ functions whose \emph{gradients} are compactly supported.
Let $\mathfrak{C}$ denote this latter set of functions.
We have
\[
C^1_c \subsetneq \mathfrak{C} \subsetneq \dot{W}^{1,p},
\]
and it is no harder to prove, for $u \in \mathfrak{C}$ than for $u \in C^1_c$, the upper bound for $[\mathcal{Q}_{1+\frac{\gamma}{p}} u]_{L^{p,\infty}(\nu_{\gamma})}$ for all $\gamma \in \mathbb{R} \setminus \{0\}$, or the required upper bound for $\limsup_{\lambda \to +\infty} \bigl(\lambda^p \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \bigr)$ when $\gamma > 0$.
Thus we can pass to limits and conclude the same for a general function in $\dot{W}^{1,p}(\mathbb{R}^n)$ (an additional argument allows us to conclude the same upper bound for $[\mathcal{Q}_{1+\gamma} u]_{L^{1,\infty}(\nu_{\gamma})}$ for all $u \in \BV$ and all $\gamma \in \mathbb{R} \setminus \{0\}$).
On the other hand, when $\gamma < 0$, it is not too hard to establish the desired upper bound for $\limsup_{\lambda \to 0^+} \bigl(\lambda^p \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \bigr)$ for all $u \in C^1_c(\mathbb{R}^n)$. If in addition $p > 1$, or $n \geq 2$ and $\gamma < -1$, then we can use the density of $C^1_c(\mathbb{R}^n)$ in $\dot{W}^{1,p}(\mathbb{R}^n)$ mentioned above, together with the upper bound for $[\mathcal{Q}_{1+\frac{\gamma}{p}} u]_{L^{p,\infty}(\nu_{\gamma})}$ already proven, to pass to limit to obtain the same conclusion for general $u \in \dot{W}^{1,p}(\mathbb{R}^n)$. The remaining case is then $n = p = 1$ and $\gamma < -1$; in this case, one needs to first establish the desired upper bound for $\limsup_{\lambda \to 0^+} \bigl(\lambda \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\gamma}[u] \bigr) \bigr)$ for the bigger class $u \in \mathfrak{C}$ before passing to limit to a general $u \in \dot{W}^{1,1}(\mathbb{R})$.
Finally, one can establish directly, for all $u \in \dot{W}^{1,p}(\mathbb{R}^n)$, the desired lower bound for $\liminf_{\lambda \to +\infty} \bigl(\lambda^p \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \bigr)$ if $\gamma > 0$, and that for $\liminf_{\lambda \to 0^+} \bigl(\lambda^p \nu_{\gamma}\bigl( \mathcal{E}_{\lambda,1+\frac{\gamma}{p}}[u] \bigr) \bigr)$ if $\gamma < 0$.
This completes our discussion of all positive results regarding the limiting formulae in \Cref{thm3} and \Cref{thm4}.
Note that if $p > 1$, the lower bound for $[\mathcal{Q}_{1+\frac{\gamma}{p}} u]_{L^{p,\infty}(\nu_{\gamma})}$ for $u \in \dot{W}^{1,p}$ in \eqref{eq:W1p_sup} follows from the limiting formulae \eqref{eq:W1p_lim}.
On the other hand, when $p = 1$, $\gamma \in \mathbb{R} \setminus [-1,0]$, the limiting formulae \eqref{eq:limit_eq_p=1_good} can fail for $u \in \BV$.
Thus to prove the lower bound for $[\mathcal{Q}_{1+\gamma} u]_{L^{1,\infty}(\nu_{\gamma})}$ in \eqref{eq:sup_p=1_good} for all $u \in \BV$, one must proceed differently.
The BBM formula comes to our rescue; in fact, the same argument also proves the following theorem, which \emph{characterizes} $\dot{W}^{1,p}$ ($1 < p < \infty$) and $\BV$:
\begin{thm}[see \cite{BSVY}] \label{thm5}
Let $n \ge 1$, $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, $\gamma \in \mathbb{R}$. If $[\mathcal{Q}_{1+\frac{\gamma}{p}}u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})} < \infty$, then
\[
u \in \begin{cases}
\dot{W}^{1,p}(\mathbb{R}^n) \quad & \text{if $1 < p < \infty$} \\
\BV(\mathbb{R}^n) \quad & \text{if $p = 1$}.
\end{cases}
\]
\end{thm}
See also Poliakovsky \cite{poliakovsky}*{Theorem 1.3}, who proved, among other things, the same result for $\gamma = n$ under an additional hypothesis $u \in L^p(\mathbb{R}^n)$; in fact, in that case the hypothesis $[\mathcal{Q}_{1+\frac{n}{p}}u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{n})} < \infty$ can be weakened to
\[
\limsup_{\lambda \to +\infty} \Big(\lambda^p \mathcal{L}^{2n} \bigl( \mathcal{E}_{\lambda,1+\frac{n}{p}}[u] \bigr) \Big) < \infty.
\]
It may be instructive to contrast \Cref{thm5} with \Cref{thm3} and \Cref{thm4}: note that \Cref{thm3} and \Cref{thm4} do not address what happens unless $u \in \dot{W}^{1,p}$ or $u \in \BV$.
To summarize, \Cref{thm3}, \Cref{thm4} and \Cref{thm5} imply that for $u \in L^1_{\text{loc}}(\mathbb{R}^n)$, $1 < p < \infty$ and $\gamma \ne 0$,
\[
u \in \dot{W}^{1,p} \quad \Longleftrightarrow \quad \Big[ \frac{\Delta_h u}{|h|^{1+\frac{\gamma}{p}}} \Big]_{L^{p,\infty}(\mathbb{R}^{2n},\nu_{\gamma})}
= [\mathcal{Q}_{1+\frac{\gamma}{p}}u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})}
< \infty.
\]
Similarly, for $u \in L^1_{\text{loc}}(\mathbb{R}^n)$ and $\gamma \in \mathbb{R} \setminus [-1,0]$,
\[
u \in \BV \quad \Longleftrightarrow \quad
\Big[ \frac{\Delta_h u}{|h|^{1+\gamma}} \Big]_{L^{1,\infty}(\mathbb{R}^{2n},\nu_{\gamma})}
= [\mathcal{Q}_{1+\gamma}u]_{L^{1,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})}
< \infty.
\]
In a slightly different direction, in place of $\|\nabla u\|_{L^p(\mathbb{R}^n)}$, one can also obtain a similar one parameter family of formulae for $\|u\|_{L^p(\mathbb{R}^n)}$.
\begin{thm} \label{thm6}
Let $n \ge 1$, $1 \leq p < \infty$ and $u \in L^p(\mathbb{R}^n)$. Then for $\gamma \ne 0$,
\begin{equation}
|
\label{eq:Lp_sim}
\|u\|_{L^p} \simeq [\mathcal{Q}_{\frac{\gamma}{p}} u]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})} = \Big[ \frac{\Delta_h u}{|h|^{\frac{\gamma}{p}}} \Big]_{L^{p,\infty}(\mathbb{R}^{2n}, \, \nu_{\gamma})}.
\end{equation}
Furthermore, if $\mathcal{E}_{\lambda,b}[u]$ is the superlevel set of $\mathcal{Q}_b u$ at height $\lambda$ given in \eqref{eq:Esetdef}, then
\begin{equation} \label{eq:Lp_limit}
\frac{2\sigma_{n-1}}{|\gamma|} \|u\|_{L^p}^p =
\begin{cases}
\lim_{\lambda \to 0^+} \bigl(\lambda^p \nu_{\gamma}\bigl(\mathcal{E}_{\lambda,\frac{\gamma}{p}}[u]\bigr) \bigr) \quad &\text{if $\gamma > 0$} \\
\lim_{\lambda \to +\infty} \bigl(\lambda^p \nu_{\gamma}\bigl(\mathcal{E}_{\lambda,\frac{\gamma}{p}}[u]\bigr) \bigr) \quad &\text{if $\gamma < 0$}.
\end{cases}
\end{equation}
where $\sigma_{n-1}$ is the surface area of $\mathbb{S}^{n-1}$.
\end{thm}
In this limiting formula \eqref{eq:Lp_limit}, we let $\lambda \to 0^+$ if $\gamma > 0$, and let $\lambda \to +\infty$ if $\gamma < 0$, contrarily to what happened in \Cref{thm3} and \Cref{thm4}.
Also, in the limiting formulae in \Cref{thm3} and \Cref{thm4}, we had a constant $k(p,n)/|\gamma|$, and here we had a constant $2\sigma_{n-1}/|\gamma|$; these should be compared, respectively, to the constant $k(p,n)/p$ in the BBM formula \eqref{eq:BBM}, and the constant $2\sigma_{n-1}/p$ in the Maz'ya--Shaposhnikova formula \eqref{eq:MSh}.
The case $\gamma = n$ of \eqref{eq:Lp_limit} was proved in \cite{MR4249777}.
Note that we do not obtain a characterization of $L^p(\mathbb{R}^n)$, contrarily to \Cref{thm5}: the $L^{p,\infty}(\nu_{\gamma})$ norms of $\mathcal{Q}_{\frac{\gamma}{p}}u$ are finite (in fact zero) when $u$ is a non-zero constant.
We also note that the differences $\Delta_h u(x)$ in \Cref{thm6} can be replaced by other expressions, such that the sums $S_h u(x) \coloneqq u(x+h)+u(x)$, as we will see in the proof below.
\begin{proof}[Proof of \Cref{thm6}]
We consider two cases.
\textbf{Case 1: Suppose $\gamma > 0$.}
In this case, to prove the upper bound in \eqref{eq:Lp_sim}, note that
\begin{align*}
&\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}\\
\subset & \Big\{(x,h) \colon |h|^{\gamma/p} < \frac{2|u(x)|}{\lambda}\Big\} \bigcup \Big\{(x,h) \colon |h|^{\gamma/p} < \frac{2|u(x+h)|}{\lambda} \Big\}
\end{align*}
so for any $\lambda > 0$,
\[
\nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
\leq 2 \nu_{\gamma} \Big\{(x,h) \colon |h|^{\gamma/p} < \frac{2|u(x)|}{\lambda} \Big\}
\lesssim \frac{1}{\lambda^p} \int_{\mathbb{R}^n} |u(x)|^p \, \mathrm{d} x.
\]
To prove \eqref{eq:Lp_limit}, and hence the lower bound in \eqref{eq:Lp_sim}, first assume $u$ has compact support in $B_R(0)$. Then
\begin{align*}
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
=& \Big\{(x,h) \colon |x| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x)|}{\lambda}\Big\} \\
& \quad \bigcup \Big\{(x,h) \colon |x+h| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x+h)|}{\lambda}\Big\} \\
& \quad \bigcup \Big\{(x,h) \colon |h| \leq 2R, |h|^{\gamma/p} < \frac{|\Delta_h u(x)|}{\lambda}\Big\}
\end{align*}
where all three sets are disjoint. We have
\begin{align*}
& \nu_{\gamma} \Big\{(x,h) \colon |x+h| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x+h)|}{\lambda}\Big\} \\
= \, & \nu_{\gamma} \Big\{(x,h) \colon |x| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x)|}{\lambda}\Big\} \\
= \, & \frac{\sigma_{n-1}}{\gamma} \int_{|x| \leq R} \Big( \frac{|u(x)|^p}{\lambda^p} - (2R)^{\gamma} \Big)_+ \, \mathrm{d} x
\end{align*}
so
\begin{align*}
& \lim_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon |x+h| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x+h)|}{\lambda}\Big\} \\
= & \lim_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon |x| \leq R, |h| > 2R, |h|^{\gamma/p} < \frac{|u(x)|}{\lambda}\Big\} \\
= & \lim_{\lambda \to 0^+} \frac{\sigma_{n-1}}{\gamma} \int_{|x| \leq R} \Big( |u(x)|^p - \lambda^p (2R)^{\gamma} \Big)_+ \, \mathrm{d} x = \frac{\sigma_{n-1}}{\gamma} \int_{\mathbb{R}^n} |u(x)|^p \, \mathrm{d} x.
\end{align*}
We also have
\begin{align*}
& \limsup_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon |h| \leq 2R, |h|^{\gamma/p} < \frac{|\Delta_h u(x)|}{\lambda}\Big\} \\
\leq & \limsup_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon |x| \leq R, |h| \leq 2R \Big\} = 0.
\end{align*}
Together this establishes \eqref{eq:Lp_limit} when $u$ has compact support.
If now $u$ is a general $L^p$ function, we approximate by a sequence of compactly supported functions $u_j$, so that $\|u_j - u\|_{L^p(\mathbb{R}^n)} \to 0$ as $j \to +\infty$. Then for any $\varepsilon > 0$, $j \geq 1$,
\[
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \subset
\Big\{(x,h) \colon \frac{|\Delta_h u_j(x)|}{|h|^{\gamma/p}} > \lambda(1-\varepsilon) \Big\} \bigcup \Big\{(x,h) \colon \frac{|\Delta_h (u_j-u)(x)|}{|h|^{\gamma/p}} > \lambda \varepsilon \Big\}
\]
so using the previous result for $u_j$,
\[
\limsup_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \leq \frac{1}{(1-\varepsilon)^p} \int_{\mathbb{R}^n} |u_j(x)|^p \, \mathrm{d} x + \frac{C}{\varepsilon^p} \int_{\mathbb{R}^n} |u_j(x)-u(x)|^p \, \mathrm{d} x.
\]
Similarly,
\[
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \supset
\Big\{(x,h) \colon \frac{|\Delta_h u_j(x)|}{|h|^{\gamma/p}} > \lambda(1+\varepsilon) \Big\} \setminus \Big\{(x,h) \colon \frac{|\Delta_h (u_j-u)(x)|}{|h|^{\gamma/p}} > \lambda \varepsilon \Big\}
\]
so
\[
\liminf_{\lambda \to 0^+} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \geq \frac{1}{(1+\varepsilon)^p} \int_{\mathbb{R}^n} |u_j(x)|^p \, \mathrm{d} x - \frac{C}{\varepsilon^p} \int_{\mathbb{R}^n} |u_j(x)-u(x)|^p \, \mathrm{d} x.
\]
We let $j \to +\infty$ before letting $\varepsilon \to 0^+$ in these inequalities to obtain \eqref{eq:Lp_limit}.
\textbf{Case 2: Suppose $\gamma < 0$.}
To prove the upper bound in \eqref{eq:Lp_sim}, note that
\begin{align*}
&\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}\\
\subset & \Big\{(x,h) \colon |h|^{|\gamma|/p} > \frac{\lambda}{2|u(x)|}\Big\} \bigcup \Big\{(x,h) \colon |h|^{|\gamma|/p} > \frac{\lambda}{2|u(x+h)|}\Big\}
\end{align*}
so for any $\lambda > 0$,
\[
\nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
\leq 2 \nu_{\gamma} \Big\{(x,h) \colon |h|^{|\gamma|/p} > \frac{\lambda}{2|u(x)|}\Big\}
\lesssim \frac{1}{\lambda^p} \int_{\mathbb{R}^n} |u(x)|^p \, \mathrm{d} x.
\]
To prove \eqref{eq:Lp_limit}, and hence the lower bound in \eqref{eq:Lp_sim}, first assume $u \in L^{\infty}$, say $|u| \leq M$, with compact support in $B_R(0)$. Then for $\lambda > 2M (2R)^{|\gamma|/p}$, we have
\[
\frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \quad \Longrightarrow \quad 2M |h|^{|\gamma|/p} > 2M (2R)^{|\gamma|/p} \quad \Longrightarrow \quad |h| > 2R,
\]
in which case at most one of $x, x+h$ can be in $B_R(0)$. So
\begin{align*}
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
=& \Big\{(x,h) \colon |x| \leq R, |h| > 2R, |h|^{|\gamma|/p} > \frac{\lambda}{|u(x)|}\Big\} \\
& \quad \bigcup \Big\{(x,h) \colon |x+h| \leq R, |h| > 2R, |h|^{|\gamma|/p} > \frac{\lambda}{|u(x+h)|}\Big\}
\end{align*}
where the two sets in the union are disjoint. Hence
\begin{align*}
\nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
&= 2 \int_{|x| \leq R} \int_{|h| > \max\{2R, (\frac{\lambda}{|u(x)|})^{p/|\gamma|}\}} |h|^{\gamma-n} \, \mathrm{d} h \, \mathrm{d} x\\
&= \frac{2 \sigma_{n-1}}{|\gamma|} \int_{|x| \leq R} \min\Big\{(2R)^{-|\gamma|}, \frac{|u(x)|^p}{\lambda^p}\Big\} \, \mathrm{d} x
\end{align*}
which says
\[
\begin{split}
\lambda^p \nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\}
&= \frac{2 \sigma_{n-1}}{|\gamma|} \int_{|x| \leq R} \min\Big\{\lambda^p (2R)^{-|\gamma|}, |u(x)|^p\Big\} \, \mathrm{d} x \\
&\to \frac{2 \sigma_{n-1}}{|\gamma|} \int_{|x| \leq R} |u(x)|^p \, \mathrm{d} x
\end{split}
\]
as $\lambda \to +\infty$ by monotone convergence.
If now $u$ is a general $L^p$ function, we approximate by a sequence of bounded, compactly supported functions $u_j$, so that $\|u_j - u\|_{L^p(\mathbb{R}^n)} \to 0$ as $j \to +\infty$. Then for any $\varepsilon > 0$, $j \geq 1$,
\[
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \subset
\Big\{(x,h) \colon \frac{|\Delta_h u_j(x)|}{|h|^{\gamma/p}} > \lambda(1-\varepsilon) \Big\} \bigcup \Big\{(x,h) \colon \frac{|\Delta_h (u_j-u)(x)|}{|h|^{\gamma/p}} > \lambda \varepsilon \Big\}
\]
so using the previous result for $u_j$,
\[
\limsup_{\lambda \to \infty} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \leq \frac{1}{(1-\varepsilon)^p} \int_{\mathbb{R}^n} |u_j(x)|^p \, \mathrm{d} x + \frac{C}{\varepsilon^p} \int_{\mathbb{R}^n} |u_j(x)-u(x)|^p \, \mathrm{d} x.
\]
Similarly,
\[
\Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \supset
\Big\{(x,h) \colon \frac{|\Delta_h u_j(x)|}{|h|^{\gamma/p}} > \lambda(1+\varepsilon) \Big\} \setminus \Big\{(x,h) \colon \frac{|\Delta_h (u_j-u)(x)|}{|h|^{\gamma/p}} > \lambda \varepsilon \Big\}
\]
so
\[
\liminf_{\lambda \to \infty} \lambda^p \nu_{\gamma} \Big\{(x,h) \colon \frac{|\Delta_h u(x)|}{|h|^{\gamma/p}} > \lambda \Big\} \geq \frac{1}{(1+\varepsilon)^p} \int_{\mathbb{R}^n} |u_j(x)|^p \, \mathrm{d} x - \frac{C}{\varepsilon^p} \int_{\mathbb{R}^n} |u_j(x)-u(x)|^p \, \mathrm{d} x.
\]
We let $j \to +\infty$ before letting $\varepsilon \to 0^+$ in these inequalities to obtain \eqref{eq:Lp_limit}.
\end{proof}
\section{Applications to Gagliardo-Nirenberg interpolation} \label{sect:applications}
The existence of a one-parameter family of characterizations in the previous section is not just natural, but useful in applications.
For instance, Cohen, Dahmen, Daubechies and DeVore \cite{CDDD} proved that for any $0 < t < 1$ and any $1 < q < \infty$, if
\begin{equation} \label{eq:CDDDassump}
t < \tfrac{1}{q},
\end{equation}
and if
$(\frac{1}{p},s) = (1-\theta) (\frac{1}{q},t) + \theta (1,1)$
for some $0 < \theta < 1$, then for any $u \in \BV \cap \dot{W}^{t,q}$, one has the interpolation inequality
\begin{equation} \label{eq:CDDD}
\|u\|_{\dot{W}^{s,p}} \lesssim \|u\|_{\dot{W}^{t,q}}^{1-\theta} \|u\|_{\BV}^{\theta}.
\end{equation}
\begin{center}
\begin{tikzpicture} [scale = 4]
\draw[arrows=->] (0,0)--(1.1,0);
\draw[arrows=->] (0,0)--(0,1.1);
\node at (0,1.2) {{$s$}};
\node at (1.3,0) {{$1/p$}};
\fill [green!50!black] (1,1) circle[radius=0.02cm];
\node at (1.1,1.1) {{{\color{green!50!black} $\BV$}}};
\draw[black,dotted] (0,0)--(1,1);
\fill [green!50!black] (0.7,0.4) circle[radius=0.02cm];
\node at (0.7,0.25) {{{\color{green!50!black} $\dot{W}^{t,q}$}}};
\fill [blue] (0.85,0.7) circle[radius=0.02cm];
\node [anchor=west] at (0.85,0.7) {{{\color{blue} $\dot{W}^{s,p}$}}};
\draw[blue] (0.7,0.4)--(1,1);
\node at (1.5,0.5) {{{\color{blue} slope $> 1$}}};
\end{tikzpicture}
\end{center}
Their proof uses bounds for coefficients of wavelet expansions of a general function in $\BV(\mathbb{R}^n)$.
Indeed, let $\psi^0 \coloneqq \varphi$ and $\tilde{\psi}^0 \coloneqq \tilde{\varphi}$ be a pair of one-dimensional compactly supported scaling functions which are in duality:
\[
\int_{\mathbb{R}} \varphi(t) \tilde{\varphi}(t-k) \, \mathrm{d} t = \delta(k), \quad k \in \mathbb{Z},
\]
where $\delta$ is the Kronecker delta, and let $\psi^1 \coloneqq \psi$, $\tilde{\psi}^1 \coloneqq \tilde{\psi}$ be their corresponding univariate wavelets.
Define, for any $e \in E \coloneqq \{0,1\}^n \setminus \{(0,0,\dots,0)\}$,
\[
\tilde{\psi}^e(x) \coloneqq \tilde{\psi}^{e_1}(x_1) \dots \tilde{\psi}^{e_n}(x_n), \quad x = (x_1, \dots, x_n);
\]
also define, for any $e \in E$ and any dyadic cube $I = 2^{-j} (k + [0,1]^n)$,
\[
\tilde{\psi}^e_I(x) \coloneqq 2^{jn} \tilde{\psi}^e(2^j x - k).
\]
For any $\gamma \in \mathbb{R}$, one can also define a measure $\tilde{\nu}_{\gamma}$ on the product of $E$ with the set of all dyadic cubes $\{I\}$, so that
\[
\tilde{\nu}_{\gamma}(\{(e,I)\}) \coloneqq 2^{-j(\gamma+n)}
\]
if $e \in E$ and $I$ has side length $\ell(I) = 2^{-j}$.
A result in \cite{CDDD} says that if $u \in \BV(\mathbb{R}^n)$ and
\[
u^e_I \coloneqq \int_{\mathbb{R}^n} u(x) \tilde{\psi}^e_I(x) \, \mathrm{d} x,
\]
then for any $\gamma \in \mathbb{R} \setminus [-1,0]$, the sequence $(\frac{u^e_I}{\ell(I)^{1+\gamma}})$ indexed by $e$ and $I$ is in weak-$\ell^1$ with respect to $\tilde{\nu}_{\gamma}$, with
\begin{equation} \label{eq:CDDD_weakL1}
\Big[\Big(\frac{u^e_I}{\ell(I)^{1+\gamma}}\Big)\Big]_{\ell^{1,\infty}(\tilde{\nu}_{\gamma})} \lesssim \|u\|_{\BV} \lesssim \Big\|\Big(\frac{u^e_I}{\ell(I)^{1+\gamma}}\Big)\Big\|_{\ell^{1}(\tilde{\nu}_{\gamma})}.
\end{equation}
Using \eqref{eq:CDDD_weakL1}, a proof of \eqref{eq:CDDD} was given in \cite{CDDD}; indeed a stronger result was proved there, namely
\begin{equation} \label{eq:CDDD_strong}
[\dot{W}^{t,q},\BV]_{\theta,p} = \dot{W}^{s,p}.
\end{equation}
The inequality \eqref{eq:CDDD_weakL1} bears a superficial resemblance to our difference quotient characterization \eqref{eq:sup_p=1_good} for the $\BV$ norm.
Indeed even the proofs are somewhat similar: both relies on covering lemmas in the range $\gamma \in \mathbb{R} \setminus [-n,0]$, and the range $\gamma \in [-n,-1)$ for \eqref{eq:CDDD} was dealt with in \cite{CDDD} using the coarea formula, while the same range for \eqref{eq:sup_p=1_good} was dealt with in \cite{BSVY} using the method of rotation.
While we did not manage to use \eqref{eq:sup_p=1_good} to recover a proof of \eqref{eq:CDDD_strong}, the characterization \eqref{eq:sup_p=1_good} does allow us to give a simple alternative proof of \eqref{eq:CDDD}, which we describe as follows.
\begin{proof}[Proof of \eqref{eq:CDDD}]
Let $\gamma_0$ be $-1$ times the slope connecting the points $(1,1)$ and $(\tfrac{1}{q},t)$, i.e.
\begin{equation} \label{eq:ga_0def}
\gamma_0 \coloneqq -\frac{1-t}{1-\frac{1}{q}}.
\end{equation}
The assumption \eqref{eq:CDDDassump} shows that $\gamma_0 < -1$.
Let $u \in \BV \cap \dot{W}^{t,q}$. Our characterization for the $\BV$ norm (see \Cref{thm4}) shows that
\begin{equation} \label{eq:formula1}
\|u\|_{\BV} \simeq [\mathcal{Q}_{1+\gamma_0} u]_{L^{1,\infty}(\nu_{\gamma_0})}.
\end{equation}
On the other hand,
\begin{equation} \label{eq:formula2}
\|u\|_{\dot{W}^{t,q}} = \|\mathcal{Q}_{t+\frac{\gamma_0}{q}}u\|_{L^{q}(\nu_{\gamma_0})}
\end{equation}
because from \eqref{eq:nu_def}
\[
\Big( \iint_{\mathbb{R}^{2n}} \frac{|\Delta_h u|^{q}}{|h|^{t q+n}} \, \mathrm{d} x \, \mathrm{d} h \Big)^{\frac{1}{q}}
= \Big( \iint_{\mathbb{R}^{2n}} \frac{|\Delta_h u|^{q}}{|h|^{t q+\gamma_0}} \, \mathrm{d}\nu_{\gamma_0} \Big)^{\frac{1}{q}}.
\]
Similarly
\begin{equation} \label{eq:formula3}
\|u\|_{\dot{W}^{s,p}} = \|\mathcal{Q}_{s+\frac{\gamma_0}{p}}u\|_{L^{p}(\nu_{\gamma_0})}.
\end{equation}
But since $\frac{1}{p} = (1-\theta) \frac{1}{q} + \theta$, we have, for any measurable function $F$, that
\begin{equation} \label{eq:interpol_Lorentz}
\|F\|_{L^p(\nu_{\gamma_0})} \lesssim \|F\|_{L^{q}(\nu_{\gamma_0})}^{1-\theta} [F]_{L^{1,\infty}(\nu_{\gamma_0})}^{\theta};
\end{equation}
indeed, for any $\lambda > 0$,
\begin{align*}
\int |F|^p \, \mathrm{d} \nu_{\gamma_0}
&= \int_{|F| \geq \lambda} |F|^p \, \mathrm{d} \nu_{\gamma_0} + \int_{|F| < \lambda} |F|^p \, \mathrm{d} \nu_{\gamma_0} \\
&\leq \frac{1}{\lambda^{q-p}} \int |F|^q \, \mathrm{d} \nu_{\gamma_0} + \int_0^{\lambda} s^{p-1} \nu_{\gamma_0} \{|F| > s\} \, \mathrm{d} s \\
&\leq \frac{1}{\lambda^{q-p}} \|F\|_{L^{q}(\nu_{\gamma_0})}^q + \frac{\lambda^{p-1}}{p-1} [F]_{L^{1,\infty}(\nu_{\gamma_0})},
\end{align*}
so choosing $\lambda$ for which
\[
\frac{1}{\lambda^{q-p}} \|F\|_{L^{q}(\nu_{\gamma_0})}^q = \frac{\lambda^{p-1}}{p-1} [F]_{L^{1,\infty}(\nu_{\gamma_0})}
\]
we obtain \eqref{eq:interpol_Lorentz}.
We apply \eqref{eq:interpol_Lorentz} to the function $F \coloneqq \mathcal{Q}_{s+\frac{\gamma_0}{p}} u = \mathcal{Q}_{t+\frac{\gamma_0}{q}} u = \mathcal{Q}_{1+\gamma_0} u$; note that our choice of $\gamma_0$ ensures $s + \frac{\gamma_0}{p} = t + \frac{\gamma_0}{q} = 1 + \gamma_0$ (they are all equal to the $y$-intercept of the line joining $(1,1)$ and $(\tfrac{1}{q},t)$).
Using \eqref{eq:formula1}, \eqref{eq:formula2} and \eqref{eq:formula3}, we obtain \eqref{eq:CDDD}, as desired.
We note that the $\gamma_0$ we used above when invoking \Cref{thm4} is dictated by the points $(\tfrac{1}{q},t)$ and $(1,1)$, and this proof does not work if we had chosen other values of $\gamma_0$.
\end{proof}
The previous proof made crucial use of the assumption $t < \tfrac{1}{q}$ in \eqref{eq:CDDDassump}, because \eqref{eq:formula1} only holds when $\gamma_0 \notin \mathbb{R} \setminus [-1,0]$.
In fact as was shown in \cite{MR3813967}, the inequality \eqref{eq:CDDD} does \emph{not} hold when $t \geq \tfrac{1}{q}$. Nevertheless, a simple adaptation of the above proof of \eqref{eq:CDDD} yields part (i) of the following theorem:
\begin{center}
\begin{tikzpicture} [scale = 4]
\draw[arrows=->] (0,0)--(1.1,0);
\draw[arrows=->] (0,0)--(0,1.1);
\node at (0,1.2) {{$s$}};
\node at (1.3,0) {{$1/p$}};
\fill [green!50!black] (1,1) circle[radius=0.02cm];
\node at (1.1,1.1) {{{\color{green!50!black} $\BV$}}};
\draw[black,dotted] (0,0)--(1,1);
\fill [green!50!black] (0.46,0.7) circle[radius=0.02cm];
\node [anchor=east] at (0.45,0.7) {{{\color{green!50!black} $\dot{W}^{t,q}$}}};
\fill [red] (0.73,0.85) circle[radius=0.02cm];
\node [anchor=south] at (0.7,0.85) {{{\color{red} $\notin \dot{W}^{s,p}$}}};
\draw[red] (0.46,0.7)--(1,1);
\node at (1.5,0.5) {{{\color{red} $0 < $ slope $\leq 1$, i.e. $\gamma_0 \in [-1,0)$}}};
\end{tikzpicture}
\end{center}
\begin{thm} \label{thm7}
Let $n \ge 1$, $0 < t < 1$, $1 < q < \infty$. Suppose $t \geq \tfrac{1}{q}$ and $(\frac{1}{p},s) = (1-\theta) (\frac{1}{q},t) + \theta (1,1)$ for some $0 < \theta < 1$. Let $\gamma \in \mathbb{R} \setminus [-1,0]$. Then the following hold.
\begin{enumerate}[(i)]
\item Let $r = \tfrac{q}{1-\theta}$. For any $u \in \BV \cap \dot{W}^{t,q}$, one has the interpolation inequality
\begin{equation} \label{eq:BVYinterpol2}
[\mathcal{Q}_{s+\frac{\gamma}{p}}u]_{L^{p,r}(\nu_{\gamma})} \lesssim \|u\|_{\dot{W}^{t,q}}^{1-\theta} \|u\|_{\BV}^{\theta}.
\end{equation}
\item The inequality \eqref{eq:BVYinterpol2} fails for some $u \in C^{\infty}_c$ if $r < \frac{q}{1-\theta}$.
\end{enumerate}
\end{thm}
The case $\gamma = n$ was already proved in \cite{Brezis_VanSchaftingen_Yung_2021Lorentz}. The proof of the general case is similar, once \Cref{thm4} is established.
\begin{proof}[Proof of \Cref{thm7}]
(i)
Note that since $(\frac{1}{p},s) = (1-\theta) (\frac{1}{q},t) + \theta (1,1)$, we have
\[
s+\frac{\gamma}{
|
to $\incross(\pi)$ and $\incross(\hat \pi)$ are equal.
Now consider points of intersection of the forms
(\ref{eq:triples}), (\ref{eq:hattriples}) for $i \notin \{ u_j, u_{j'} \}$.
If $\pi_i \in AF \cup CD$, then all four triples are crossings;
if $\pi_i \in AD \cup CF$, then all four are noncrossing.
Moreover, if we have $\pi_i \in AD$ appearing to the left of column $t$
or $\pi_i \in CF$ appearing to the right of column $t$, then
all four noncrossings are inverted. Otherwise none of the four is inverted.
If $\pi_i \in AE \cup BD \cup BF \cup CE$, then
exactly one of the triples (\ref{eq:triples}) is a noncrossing, as is
exactly one of the triples (\ref{eq:hattriples}).
These two noncrossings are inverted if we have
$\pi_i \in AE \cup BD$ appearing to the right of column $t$,
or if we have
$\pi_i \in BF \cup CE$ appearing to the left of column $t$.
Otherwise, the two noncrossings are not inverted.
If $\pi_i \in BE$, then either
both triples (\ref{eq:triples}) are crossings
while both triples (\ref{eq:hattriples}) are not
(if $(\pi_{u_j}, \pi_{u_{j'}}, p)$ is a crossing),
or vice versa
(if $(\pi_{u_j}, \pi_{u_{j'}}, p)$ is a noncrossing).
In both cases, exactly one of the two noncrossings is inverted.
Finally, observe that neither triple in
(\ref{eq:piuj}) can be an inverted noncrossing, since all four paths
appear in column $t$.
Thus the points of intersection of the forms
(\ref{eq:triples}), (\ref{eq:hattriples}), (\ref{eq:piuj})
contribute a surplus of $b$ to $\incross(\pi)$
if $(\pi_{u_j}, \pi_{u_{j'}}, p)$ is a crossing,
and to $\cross(\hat \pi)$ otherwise.
\smallskip
(Identity (\ref{eq:cdncrossid}))
For $k < p$ and all $i, i'$, we have
that
$(\pi_i, \pi_{i'}, k)$ is a column defective noncrossing
if and only if
$(\hat \pi_i, \hat \pi_{i'}, k)$ is a column defective noncrossing.
The same is true for $k \geq p$,
provided that $i, i' \not \in \{ u_j, u_{j'} \}$.
For $k > p$ and $i \notin \{ u_j, u_{j'} \}$,
no triple $(\pi_i, \pi_{u_j}, k)$, $(\pi_i, \pi_{u_{j'}}, k)$,
$(\hat \pi_i, \hat \pi_{u_j}, k)$, $(\hat \pi_i, \hat \pi_{u_{j'}}, k)$
can be a column defective noncrossing, since the definition of $j, j'$
guarantees that no path $\pi_i$ can belong to column $t$ and intersect
paths indexed by $u_j$, $u_{j'}$ to the right of $G_{J_p}$.
So far, contributions to $\cdncross(\pi)$ and $\cdncross(\hat \pi)$ are equal.
Now consider points of intersection of the forms
(\ref{eq:triples}), (\ref{eq:hattriples}) for $i \notin \{ u_j, u_{j'} \}$.
If $\pi_i \in AF \cup BF \cup CF \cup AE \cup BE \cup CE$,
then it cannot appear in column $t$ by our choice of $j, j'$.
If $\pi_i \in CD$, then all four triples are crossings.
If $\pi_i \in AD \cup BD$, then
$(\pi_i, \pi_{u_j}, p)$
is a column defective noncrossing if and only if
$(\hat \pi_i, \hat \pi_{u_j}, p)$ is,
and
$(\pi_i, \pi_{u_{j'}}, p)$
is a column defective noncrossing if and only if
$(\hat \pi_i, \hat \pi_{u_{j'}}, p)$ is.
Finally, observe that neither triple in
(\ref{eq:piuj}) can be a column defective noncrossing if
$(\pi_{u_j}, \pi_{u_{j'}}, p)$ is proper,
while exactly one of the two a column defective noncrossing if
$(\pi_{u_j}, \pi_{u_{j'}}, p)$ is defective.
Thus the points of intersection of the forms (\ref{eq:piuj})
contribute a surplus of
$1$ to $\cdncross(\pi)$
if $(\pi_{u_j}, \pi_{u_{j'}}, p)$ is a defective noncrossing,
$1$ to $\cdncross(\hat \pi)$
if $(\pi_{u_j}, \pi_{u_{j'}}, p)$ is a defective crossing,
and $0$ to both otherwise.
\end{proof}
As a consequence, we have that the map $\zeta$ preserves
a certain linear combination of the above statistics.
\begin{cor}\label{c:adamidentity}
For $W = \delta(U(\pi, u, yu))$ and $\zeta(W) = \delta(U(\hat \pi, u, \hat yu))$ in $\mathcal T_I$, we have
\begin{equation}\label{eq:adamidentity}
\frac{\cross(\hat \pi)}2 + \cdncross(\zeta(W)) + \incross(\zeta(W))
- \frac{\ell(\hat y)}2
= \frac{\cross(\pi)}2 + \cdncross(W) + \incross(W) - \frac{\ell(y)}2.
\end{equation}
\end{cor}
\begin{proof}
If $W$ is a fixed point of $\zeta$, then the result is clear.
Suppose therefore that $\zeta(W) \neq W$, and consider the triple
$(\pi_{u_j}, \pi_{u_{j'}}, p)$ appearing in the definition of $\zeta$.
If this triple is a proper noncrossing, then by Lemma~\ref{l:statids}
the left-hand side of
(\ref{eq:adamidentity}) is
\begin{multline*}
\frac{\cross(\pi) + 1 + 2b}2 + \cdncross(W) + \incross(W) - b
- \frac{\ell(y)+1}2\\
= \frac{\cross(\pi)}2 + \cdncross(W) + \incross(W) - \frac{\ell(y)}2
+ \frac{1 + 2b}2 - b - \frac 12.
\end{multline*}
One shows similarly that the result holds when the triple is a
defective noncrossing or any crossing.
\end{proof}
Finally we can state and justify a
subtraction-free formula for
$\epsilon_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)$.
The proof of the following result is nearly identical to that of
\cite[Thm.\,5.4]{KLSBasesQMBIndSgn}.
We include it here, both for the convenience of the reader,
and because it relies upon
Theorem~\ref{t:qstem},
Proposition~\ref{p:sigmastats},
Lemmas~\ref{l:TU} -- \ref{l:lengthdiff}
in this paper which differ from
the corresponding (weaker) results
\cite[Thm.\,3.7]{KLSBasesQMBIndSgn},
\cite[Prop.\,4.1]{KLSBasesQMBIndSgn},
\cite[Lem.\,5.1 -- 5.2]{KLSBasesQMBIndSgn},
in that paper,
and upon Corollary~\ref{c:adamidentity}.
\begin{thm}\label{t:qepsilon}
Let $G = G_{J_1} \circ \cdots \circ G_{J_m}$.
Then for $\lambda \vdash n$
we have
\begin{equation}\label{eq:epsilonmain}
\epsilon_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)
=
\sum_\pi \qp{\cross(\pi)}2 \sum_W q^{\incross(W)},
\end{equation}
where the sums are over path families $\pi$ of type $e$ which cover $G$,
and
column-strict $\pi$-tableaux $W$
of
shape $\lambda^\tr$.
\end{thm}
\begin{proof}
Let $B$ be the path matrix of $G$.
By Theorem~\ref{t:qstem}
and
Equations (\ref{eq:immepsilon}) -- (\ref{eq:lambepsilon}),
the left-hand side of (\ref{eq:epsilonmain}) is
\begin{equation}\label{eq:epsilonsecond}
\begin{aligned}
\sigma_B(\imm{\epsilon_q^\lambda}(x)) &=
\sum_I
\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}))\\
&=
\sum_I
\sum_{\smash{y \in \slambda}} (-1)^{\ell(y)}\qiey
\sigma_B (x^{u(I),yu(I)}),
\end{aligned}
\end{equation}
where the first two sums are over ordered set partitions
$I = (I_1,\dotsc,I_r)$ of $[n]$ of type $\lambda$.
Fixing one such partition $I$ and writing $u = u(I)$, we may use
Proposition~\ref{p:sigmastats} and Lemma~\ref{l:TU} to
express the sum over $y \in \slambda$ as
\begin{equation}\label{eq:ypiinvuupi}
\sum_{y \in \slambda}
\sum_\pi
(-1)^{\ell(y)} \qiey
\qp{\cross(\pi)}2
q^{\incross(U(\pi,u,yu))}
=
\sum_{y \in \slambda}
\sum_\pi
(-1)^{\ell(y)} \qiey
\qp{\cross(\pi)}2 q^{\incross(W)+\cdncross(W)},
\end{equation}
where
the inner
sums are over
path families $\pi$ of type
$u^{-1}yu$
which cover $G$,
and where $W = \delta_I(U(\pi,u,yu))$.
As $y$ and $\pi$ vary in the above sums,
$U(\pi,u,yu)$
varies over all tableaux in $\mathcal U_I$,
and
$W$ varies over all tableaux in $\mathcal T_I$.
Now consider a tableau $W \in \mathcal T_I$ which satisfies $\zeta(W) \neq W$.
Let tableaux $W$ and $\zeta(W)$ contain path families
$\pi$ of type $u^{-1}yu$ and $\hat \pi$ of type $u^{-1}\hat yu$, respectively.
Then the two terms
on the right-hand side of
(\ref{eq:ypiinvuupi}) corresponding to
$\zeta(W)$ and $W$ sum to
\begin{equation*}
(-1)^{\ell(\hat y)}
\qumongous{\cross(\hat \pi)}{\incross(\zeta(W))}{\cdncross(\zeta(W))}{\ell(\hat y)}
+ (-1)^{\ell(y)}
\qumongous{\cross(\pi)}{\incross(W)}{\cd
|
ncross(W)}{\ell(y)}.
\end{equation*}
By Lemma~\ref{l:lengthdiff}
and Corollary~\ref{c:adamidentity}
this is $0$.
Thus it suffices to sum the right-hand side of (\ref{eq:ypiinvuupi})
over only the pairs $(y,\pi)$ corresponding
to tableaux $W$ satisfying $W = \zeta(W)$.
By the definition of $\zeta$, each such tableau $W$
is column-strict and therefore satisfies $\cdncross(W) = 0$.
Since all tableaux in $\mathcal T_I$ are also column-closed,
each such tableau $W$ must have type $e$. Thus we have
we have $u^{-1}yu = e$, i.e., $y=e$.
It follows that the
right-hand side of (\ref{eq:ypiinvuupi})
and the third sum in (\ref{eq:epsilonsecond}) are equal to
\begin{equation*}
\qp{\cross(\pi)}2 \sum_W q^{\incross(W)},
\end{equation*}
where the sum is over all tableau $W$ in $\mathcal T_I$ which
are column-strict of type $e$.
Since all tableaux in $\mathcal T_I$ have shape $\lambda^\tr$,
the three expressions in (\ref{eq:epsilonsecond})
are equal to the right-hand side of (\ref{eq:epsilonmain}).
\end{proof}
To illustrate the theorem, we compute
$\epsilon_q^{211}(\wtc{s_{[1,2]}\ntnsp}q \wtc{s_{[2,4]}\ntnsp}q \wtc{s_{[1,2]}\ntnsp}q)$
using the wiring diagram
\begin{equation}
G = G_{[1,2]}\circ G_{[2,4]} \circ G_{[1,2]} =
\begin{tikzpicture}[scale=.5,baseline=-10]
\draw (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
\draw (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
\draw (0,-1) -- (.5,-1.5) -- (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-1);
\draw (0,-2) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-2);
\end{tikzpicture}.
\end{equation}
There are two path families of type $e$ which cover $G$, and four
column-strict $G$-tableaux of shape $211^\tr = 31$ for each:
\begin{equation*}\label{eq:Gtableauxpi}
\begin{tikzpicture}[scale=.5,baseline=-10]
\draw[dashed, ultra thick] (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
\draw[dotted, thick] (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
\draw[dashed] (0,-1) -- (.5,-1.5) -- (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-1);
\draw[-,thick] (0,-2) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-2);
\node at (-.5,1) {$\pi_4$};
\node at (-.5,0) {$\pi_3$};
\node at (-.5,-1) {$\pi_2$};
\node at (-.5,-2) {$\pi_1$};
\end{tikzpicture}
\;, \quad
U_\pi^{(1)} = \tableau[scY]{\pi_3 | \pi_1,\pi_2,\pi_4} \;,\
U_\pi^{(2)} = \tableau[scY]{\pi_3 | \pi_1,\pi_4,\pi_2}\;,\
U_\pi^{(3)} = \tableau[scY]{\pi_4 | \pi_1,\pi_2,\pi_3}\;,\
U_\pi^{(4)} = \tableau[scY]{\pi_4 | \pi_1,\pi_3,\pi_2}\;;
\end{equation*}
\begin{equation*
\begin{tikzpicture}[scale=.5,baseline=-10]
\draw[dashed, ultra thick] (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
\draw[dotted, thick] (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
\draw[dashed] (0,-1) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-1);
\draw[-, thick] (0,-2) -- (.5,-1.5) -- (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-2);
\node at (-.5,1) {$\rho_4$};
\node at (-.5,0) {$\rho_3$};
\node at (-.5,-1) {$\rho_2$};
\node at (-.5,-2) {$\rho_1$};
\end{tikzpicture}\;, \quad
U_\rho^{(1)} = \tableau[scY]{\rho_3 | \rho_2,\rho_1,\rho_4}\;,\
U_\rho^{(2)} = \tableau[scY]{\rho_3 | \rho_2,\rho_4,\rho_1}\;,\
U_\rho^{(3)} = \tableau[scY]{\rho_4 | \rho_2,\rho_1,\rho_3}\;,\
U_\rho^{(4)} = \tableau[scY]{\rho_4 | \rho_2,\rho_3,\rho_1}\;.
\end{equation*}
Since the path family $\pi$ has no crossings,
we have $\textsc{c}(U_\pi^{(i)}) = 0$ for all $i$, and each
tableau $U_\pi^{(i)}$ therefore contributes
$q^{\textsc{invnc}(U_\pi^{(i)})}$.
We have one noncrossing for each of the pairs $(\pi_2,\pi_3)$, $(\pi_2,\pi_4)$ and $(\pi_3,\pi_4)$
and two for the pair $(\pi_1,\pi_2)$. Counting the noncrossings only for pairs where the path
which intersects the other from above appears in a column left of the other, for instance $\pi_2$ and $\pi_3$
in $U_\pi^{(1)}$, we find the contributions from $U_\pi^{(1)},\dots,U_\pi^{(4)}$ are
$q, q^2, q^2, q^3$, respectively.
Since the path family $\rho$ has two crossings,
the tableaux for the path family $\rho$ each have two crossings,
and one noncrossing for
each of the pairs $(\rho_1,\rho_3)$, $(\rho_1,\rho_4)$ and $(\rho_3,\rho_4)$.
Adding the contributions together we find the contributions for
$U_\rho^{(1)},\dots,U_\rho^{(4)}$ are $q^1q^{2/2} = q^2$, $q^2q^{2/2} = q^3$,
$q^2q^{2/2} = q^3$ and $q^3q^{2/2}= q^4$ respectively.
Hence we have
$\epsilon_q^{211}(\wtc{s_{[1,2]}\ntnsp}q \wtc{s_{[2,4]}\ntnsp}q \wtc{s_{[1,2]}\ntnsp}q)
= q + 3q^2 + 3q^3 + q^4$.
Theorem~\ref{t:qepsilon} allows one to
combinatorially interpret evaluations of $\epsilon_q^\lambda$ at
(multiples of) certain elements $\wtc wq$
of the Kazhdan-Lusztig basis of $\hnq$.
In particular, for some elements $\wtc wq$ there exists a polynomial $g(q)$
such that we have
\begin{equation}\label{eq:klfactor}
g(q) \wtc wq = \wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q
\end{equation}
for some sequence $s_{J_1},\dotsc,s_{J_m}$ of reversals.
Such permutations include all \pavoiding permutations,
all of $\mfs 4$ (even $4231$ and $3412$),
all of $\mfs 5$ except $45312$,
and all $321$-{\em hexagon-avoiding} permutations.
(See \cite{BWHex}.)
\begin{cor}\label{c:klfactor}
Suppose that $\wtc wq$ satisfies
a factorization of the form (\ref{eq:klfactor})
and define $G = G_{J_1} \circ \cdots \circ G_{J_m}$.
Then we have
\begin{equation}\label{eq:klfactoreval}
\epsilon_q^\lambda(\wtc wq) = \frac1{g(q)}\sum_U q^{\incross(U)+\cross(U)/2},
\end{equation}
where the sum is over all column-strict $G$-tableaux
of type $e$ and shape $\lambda^\tr$.
\end{cor}
It would be interesting to know for which Kazhdan-Lusztig basis
elements we have the above factorization~(\cite[Quest.\,4.5]{SkanNNDCB}).
Recall from Equations~(\ref{eq:cprodq}) -- (\ref{eq:cnoprodq})
that Haiman's result~\cite[Lem.\,1.1]{HaimanHecke} that we have
\begin{equation*}
\chi_q^\lambda(\wtc wq) \in \mathbb N[q]
\end{equation*}
for all irreducible characters $\chi_q^\lambda$ and all $w \in \sn$
implies that we also have
\begin{equation}\label{eq:prodchi}
\chi_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q) \in \mathbb N[q]
\end{equation}
for all irreducible characters $\chi_q^\lambda$ and
all sequences $(s_{J_1}, \dotsc, s_{J_m})$ of reversals.
It would therefore be interesting to extend Theorem~\ref{t:qepsilon}
to irreducible characters.
\begin{prob}\label{p:qchi}
Find a combinatorial interpretation of the polynomial
(\ref{eq:prodchi})
which holds for all irreducible charcters $\chi_q^\lambda$
and all sequences $(s_{J_1}, \dotsc, s_{J_m})$ of reversals.
\end{prob}
|
\section*{Introduction}
Let $V=\mid\!\mathcal O_{\mathbb{P}^ 2}(4)\!\mid\cong\mathbb{P}^{14}$ be the space of plane quartics and $V_0$ be the open subvariety of smooth curves.
The degree four cyclic cover of the plane branched along a curve $C\in V_0$ is a $K3$ surface equipped with an order four non-symplectic automorphism group.
This construction defines a holomorphic period map:
$$\mathcal P_0:\mathcal V_0\longrightarrow \mathcal M,$$
where $\mathcal V_0$ is the geometric quotient of $V_0$ by the
action of $PGL(3)$ and $\mathcal M$ is a moduli space of polarized
$K3$ surfaces.
In \cite{K} S. Kond\=o shows that $\mathcal P_0$ gives an
isomorphism between $\mathcal V_0$ and the complement of two
irreducible divisors $\mathcal D_n, \mathcal D_h$ in $\mathcal M$.
Moreover, he proves that the generic points in $\mathcal D_n$ and
$\mathcal D_h$ correspond to plane quartics with a node and to
smooth hyperelliptic genus three curves respectively.
The moduli space $\mathcal M$ is an arithmetic quotient of a six
dimensional complex ball, hence a natural compactification is given
by the Baily-Borel compactification $\mathcal M^*$ (see \cite{BB}).
On the other hand, geometric invariant theory provides a compact
projective variety $\mathcal V$ containing $\mathcal V_0$ as a dense
subset, given by the categorical quotient of the semistable locus in
$V$ for the natural action of $PGL(3)$. In this paper we prove that
the map $\mathcal P_0$ can be extended to a holomorphic surjective
map
$$\mathcal P:\widetilde{\mathcal V}\longrightarrow \mathcal M^{*}$$
on the blowing-up $\widetilde{\mathcal V}$ of $\mathcal V$ in the point $v_0$
corresponding to the orbit of double conics. In particular, we
show that the exceptional divisor in $\widetilde{\mathcal V}$ is a GIT moduli space of hyperelliptic genus three curves which is mapped isomorphically onto the Baily-Borel compactification of $\mathcal D_h$.
The paper is divided in three main sections related to plane
quartics, hyperelliptic genus three curves and finally genus three
curves in general.\\
The first section starts reviewing the geometric construction by
Kond\=o, which defines the period map $\mathcal P_0$ on the moduli
space of smooth plane quartics.
This construction can be easily extended to stable
quartics (i.e. having at most ordinary nodes and cusps) since in
this case the 4:1 cyclic cover of the plane branched along the curve
has at most rational double points, hence its minimal resolution is
still a K3 surface. In fact the period map can be extended to a
holomorphic map
$$\mathcal P_1:\mathcal V_s\longrightarrow \mathcal M$$
defined on the geometric quotient $\mathcal V_s$ of the locus of
stable quartics for the action of $PGL(3)$. The geometry of $K3$
surfaces associated to stable quartics is also described in detail.
The analogous construction for strictly semistable quartics gives
surfaces with elliptic singularities for quartics with tacnodes and
surfaces with significant limit singularities (see \cite{S3}) in the
case of double conics. Nevertheless, by taking the Baily-Borel
compactification $\mathcal M^*$ of the ball quotient $\mathcal M$,
we can extend the period map to a morphism
$$\mathcal P_2:\mathcal V_{ss}\backslash\{v_0\}\longrightarrow \mathcal M^*,$$
where the point $v_0$ represents the orbit of double conics and
strictly semistable quartics give a smooth rational curve (with
$v_0$ in its closure) mapped to the boundary.
In section two, we give a correspondence between hyperelliptic genus three curves and certain hyperelliptic polarized $K3$ surfaces parametrized by $\mathcal D_h$.
In fact, this relation has been studied in detail by Kond\=o in \cite{Kh}.
He defines a period map
$$\mathcal P^h:\mathcal V^h\longrightarrow \mathcal D^*_h$$
from the GIT moduli space $\mathcal V^h$ of sets of eight points in
$\mathbb{P}^ 1$ to the Baily-Borel compactification of $\mathcal D_h$ and
proves that it is an isomorphism. We recall his results giving a
different approach. In particular, we show that $\mathcal D_h$ is a
moduli space for degree four cyclic covers of a cone in $\mathbb{P}^ 5$
branched along a quadratic section.
The last section contains the main theorem. We first construct a
blowing up $\widetilde{\mathcal V}$ of $\mathcal V$ in $v_0$ such
that the exceptional divisor is isomorphic to the moduli space
$\mathcal V^h$. We then prove that the period maps $\mathcal P_2$
and $\mathcal P^h$ define a global period map
$$\mathcal P:\widetilde{\mathcal V}\longrightarrow \mathcal M^*.$$
Let $\widetilde{\mathcal V}_s$ be the locus corresponding to stable
genus three curves in $\widetilde{\mathcal V}$. The morphism
$\mathcal P$ induces an isomorphism $\widetilde{\mathcal V}_s\cong
\mathcal M$ and maps the smooth rational curve $\widetilde{\mathcal
V}\backslash\widetilde{\mathcal V}_{s}$ to the boundary of $\mathcal
M^*$ (which consists of one point).
\\
Similar descriptions for sextic double planes have been given by E.
Horikawa in \cite{H1} and by J. Shah in \cite{S2}. These two papers
and \cite{S} by H.J.M. Sterk are all important references for
this work.\\
The Appendix contains a brief review on GIT moduli spaces of genus three curves.\\
\emph{Acknowledgements.} This paper is part of my PhD thesis. I'm
grateful to my advisor, Prof. B. van Geemen, for helpful comments
and careful reading. I also would like to thank Dr. \!A. Laface and
Prof. E. Looijenga for several interesting discussions and
suggestions. I acknowledge the Mathematics Department of the
University of Milano for supporting me.
\section{Plane quartics}
\subsection{Smooth quartics}
We start recalling the geometric construction introduced by Kond\=o
in \cite{K}. This can be resumed by the following commutative diagram:
$$\xymatrix{
X_C \ar[rd]_{\pi_2} \ar@{-}[rr]^ {\pi}& \ar[r]& \mathbb{P}^ 2\\
&S_C \ar[ur]_{\pi_1}& },$$
where $C$ is a smooth quartic curve, $\pi_1$ is the
double cover of the plane branched along $C$ and $\pi$
is the 4:1 cyclic cover of the plane branched along $C$. Note that
$S_C$ is a Del Pezzo surface of degree two and $\pi_1$ is the
morphism associated to the anti-canonical linear system. Its double
cover $X_C$ is a $K3$ surface. In coordinates, if the quartic $C$ is
defined by the equation $f_4(x,y,z)=0$:
$$X_C=\{(x,y,z,t)\in \mathbb{P}^ 3:\ t^4=f(x,y,z)\}.$$
The $K3$ surface $X_C$ has a natural degree four polarization
induced by the embedding in $\mathbb{P}^ 3$ and given by $\pi^*(l)$, where
$l$ is the class of a line in the plane.
Let $G$ be the covering transformation group of $\pi$. A generator
$\sigma$ for $G$ can be chosen such that the space of holomorphic
two-forms of $X_C$ lies in the $i$-eigenspace $W$ of the isometry
$\rho=\sigma^*$ on $H^2(X_C,\mathbb C)$. In particular the Picard lattice of $X_C$ contains the pull-back $L_+$ of the Picard lattice of $S_C$ as a sublattice.
Hence the period point of $X_C$ lies in the six dimensional complex ball
$$B=\{x\in \mathbb P(W): (x,\bar x)>0\}\subset \mathbb P(L_-\otimes \mathbb C),$$
where $L_-$ is the orthogonal complement of $L_+$ in $H^2(X_C,\mathbb{Z})$.
Following the definition given in \cite{DVK}, an
$(L_+,\rho)$-\emph{polarized} $K3$ surface $X$ is an $L_+$-polarized
$K3$ surface with period point in $B$ (up to the choice of an
isometry $H^2(X,\mathbb{Z})\cong H^2(X_C,\mathbb{Z})$). As proved in \cite{K}, the
moduli space $\mathcal M$ of these polarized $K3$ surfaces is the
quotient of the six dimensional complex ball $B$ by the action of an
arithmetic group $\Gamma$ of automorphisms:
$$\mathcal M=B/\Gamma,\ \Gamma=\{\gamma\in O(L_-): \gamma\circ \rho=\rho\circ\gamma\}.$$
Projectively equivalent plane quartics give isomorphic polarized $K3$ surfaces, thus the above construction defines a holomorphic period map:
$$\mathcal P_0:\mathcal V_0\longrightarrow \mathcal M,$$
where $\mathcal V_0\cong\mathcal M_3\backslash\mathcal M_3^h,$ is
the moduli space of smooth quartics (see the Appendix).
Note that all $(L_+,\rho)$-polarized $K3$ surfaces have a degree four
polarization given by the $\rho$-invariant lattice $L^{\rho}=\langle h\rangle$ in $L_+$. The polarization is \emph{ample} if there are no
$(-2)$-curves orthogonal to $h$, i.e. a $K3$ surface is not ample polarized iff
its class belongs to the \emph{discriminant locus} $\mathcal
D$ in $\mathcal M$:
$$\mathcal D=\left(\bigcup_{r\in \Delta}H_{r}\right)/\Gamma,$$
where $\Delta=\{r\in L_-:\ r^2=-2\}$ is the set of \emph{roots} of
$L_-$ and $H_{r}=B\cap r^{\perp}.$ In fact, ample $(L_+,\rho)$-polarized $K3$ surfaces correspond to smooth plane quartics.
\begin{Thm}[Theorem 2.5, \cite{K}]
The period map gives an isomorphism:
$$\mathcal P_0:\mathcal V_0\cong \mathcal M_3\backslash\mathcal M_3^h\longrightarrow \mathcal M\backslash \mathcal D.$$
\end{Thm}
If $\Lambda_r=\langle r,\rho(r)\rangle$ and $\Lambda^{\perp}_r$ is its orthogonal lattice in $L_-$, then the roots can be divided in two classes (Lemma 3.3, \cite{K}):
$$\Delta_n=\{r\in \Delta:\ \Lambda_r^{\perp}\cong U^{\oplus 2}\oplus A_1^{\oplus 8}\},\
\Delta_h=\{r\in \Delta:\ \Lambda_r^{\perp}\cong U(2)^{\oplus 2}\oplus D_8 \}.$$
This leads to a natural decomposition of $\mathcal D$ in the union of two divisors, called \emph{mirrors}:
$$\mathcal D_n=\left(\bigcup_{r\in \Delta_n} H_r\right)/\Gamma,\ \mathcal D_h=\left(\bigcup_{r\in \Delta_h} H_r\right)/\Gamma.$$
In the following sections we will see that these two divisors have a clear interpretation in terms of genus three curves.
\subsection{Stable quartics}
A plane quartic is \emph{stable} for the action of $PGL_3$ if and
only if it has at most ordinary nodes and cusps (see the Appendix).
We show that the period map has a good behaviour on the stable
locus.
\begin{Lem}\label{res}
The minimal resolution of the degree four cyclic cover of $\mathbb{P}^ 2$ branched along a stable quartic is a $(L_+,\rho)$-polarized $K3$ surface.
\end{Lem}
\proof Let $\pi:Y\longrightarrow \mathbb{P}^ 2$ be the degree four cyclic cover branched along a stable plane quartic $C$. Local computations show that
$\pi^{-1}(p)$ is a rational double point of type $A_3$ if $p$ is a node and of type $E_6$ if $p$ is a cusp of $C$ (see Ch.III, \cite{BPV}).
The minimal resolution $r:\tilde Y\longrightarrow Y$ of $Y$ is a $K3$ surface
since $r^*(K_Y)=K_{\tilde Y}=0$ and $H^1(\mathcal O_{\tilde
Y})=H^1(\mathcal O_Y)=0$. Since these $K3$ surfaces are
degenerations of ample $(L_+,\rho)$-polarized $K3$ surfaces their
period points belong to $B$.
\qed\\
By the previous Lemma a polarized $K3$ surface $X_C$ can be
associated to any stable plane quartic $C$. This defines a natural
extension $\mathcal P_1$ of $\mathcal P_0$ to the locus $\mathcal
V_s$ representing stable quartics in $\mathcal V$:
$$\mathcal P_1:\mathcal V_s\longrightarrow \mathcal M.$$
\begin{Prop}\label{s}
The period map $\mathcal P_1$ is a holomorphic extension of
$\mathcal P_0$ and maps $\mathcal V_s\backslash \mathcal V_0$ to
$\mathcal D_n.$ Moreover, the map is generically surjective onto
$\mathcal D_n$.
\end{Prop}
\proof By a result of E. Brieskorn on simultaneous resolution of
singularities (see \cite{Br}) the period map $\mathcal P_0\circ q$
(see the Appendix) can be extended holomorphically to the open
subset of stable curves in $V$. By taking the quotient for the
action of $PGL(3)$ we get the first assertion. The generic point in
$\mathcal V_s\backslash \mathcal V_0$ corresponds to a plane quartic
with one node. This case is analyzed in detail in \cite{K}, in
particular it is proved that the polarization on $X_C$ is not an
ample since, for general $C\in \mathcal V_s\backslash \mathcal V_0$:
$$Pic(X_C)\cap L_-=\langle r,\rho(r)\rangle,\ r\in \Delta_n.$$
This implies the first assertion, the second one is proved in \cite{K}, \S4.
\qed\\
We can define the \emph{singular type} of a stable quartic to be the pair $(n,c)$ where $n$ is the number of its nodes and $c$ of its cusps. The singular type induces a natural stratification of $\mathcal D_n$, since we can associate to each pair $(n,c)$ ($(n,c=\not=(0,0)$) the variety $\mathcal D_{n,c}$ corresponding to $K3$ surfaces
$X_C$ with $C$ having at least $n$ nodes and $c$ cusps.
We now describe the geometry of the generic $K3$ surface in each stratum.
For any stable quartic $C$ of singular type $(n,c)$, the $K3$
surface $X_C$ carries a natural involution $\tau_{n,c}$ in the
covering group of the 4:1 map $X_C\longrightarrow \mathbb{P}^ 2$. Let $\tau^*_{n,c}$ be
the induced isometry on $H^2(X_C,\mathbb{Z})$.
\begin{Prop}\label{stab}
The Picard lattice of the generic K3 surface $X_C$ in $\mathcal D_{n,c}$ equals the invariant lattice of $\tau^*_{n,c}$. Moreover
$$Pic(X_C)\cap L_-\cong A^{\oplus 2n}_1\oplus D^{\oplus c}_4,$$
in particular $X_C$ has Picard number $\rho(n,c)=8+2n+4c.$
\end{Prop}
\proof The case $n=c=0$ has been described in the previuos section and the case of stable quartics with $c=0$ and $1\leq n\leq 3$ is an easy generalizaton of the
results for $n=1$ given in the proof of Proposition \ref{s}.
We study in more detail the case of a plane quartic $C$ with an ordinary cusp at a point $p$.
The minimal resolution of the 4:1 cyclic cover of $\Ps2$ branched along $C$ can be obtained in the following steps (in order to simplify the notation, at each step the name of a curve is the same as that of its proper transform) :\\
1) Let $b_1:X_1\longrightarrow \mathbb{P}^ 2$ be the blow up of $\Ps2$ in $p$ and $E$ be the exceptional divisor. Notice that $C$ is smooth and intersects $E$ in one point with multiplicity $2$.\\
2) Take the double cover $\phi_1:S_C\longrightarrow X_1$ branched along $C$. Let $\phi^{-1}(E)=E',E''$.\\
3) Let $b_2:X_2\longrightarrow S_C$ be the blow up in the point $E'\cap C=E''\cap C$. Let $F$ be the exceptional divisor.\\
4) Let $b_3:X_3\longrightarrow X_2$ be the blow up in the points $C\cap(F\cup E'\cup E'')$. Let $G,H,L$ be the exceptional divisors.\\
5) Let $\phi_2: X_C\longrightarrow X_3$ be the double cover branched
along the divisor $C\cup F\cup E'\cup E''$.
The surface $S_C$ is a nodal Del Pezzo surface of degree two (see \cite{DO}), hence the pull-back of its Picard lattice in $X_C$ is isomorphic to $L_+$. In particular $X_C$ is a $(L_+,\rho)$-polarized $K3$ surface with
$$Pic(X_C)\cap L^{\perp}_+=\langle F,G,H,L\rangle\cong D_4.$$
Moreover, $X_C$ carries a natural involution $\tau_{0,1}$ induced by the cover $\phi_2$ and
the invariant lattice $L^{0,1}_+$ of the induced isometry $\tau^*_{0,1}$ is a primitive sublattice of $Pic(X_C)$ of rank $r(L_+)+4=12$.
In fact, for the generic $X_C$ in $\mathcal D_{0,1}$ the two lattices coincide by dimension reasons.
The general case (i.e. $n,c>1$) is an easy generalization of the
previous ones.
\qed
\begin{Cor}\label{irr}
The following table lists the isomorphism classes of the Picard
lattices $Pic(n,c)$ of the generic K3 surfaces in $\mathcal
D_{n,c}$, with $0\leq n,c\leq 3$ (hence the general plane quartic
with singular type $(n,c)$ is irreducible):
$$\begin{array}{lllll}
(n,c)& Pic(n,c)\\
\ \\
(0,0)& \langle 2\rangle\oplus A_1^{\oplus 7}\\
(1,0)& U\oplus A_1^{\oplus 8}\\
(2,0)& U\oplus A_1^{\oplus 6}\oplus D_4\\
(3,0)& U\oplus A_1^{\oplus 6}\oplus D_6\\
(1,1)& U\oplus A_1^{\oplus 2}\oplus D_4\oplus D_6\\
(1,2)& U\oplus A_1^{\oplus 2}\oplus D_6\oplus E_8\\
(2,1)& U\oplus A_1^{\oplus 2}\oplus D_4\oplus D_8\\
(0,1)& U\oplus A_1^{\oplus 4}\oplus D_6\\
(0,2)& U\oplus A_1^{\oplus 2}\oplus D_4\oplus E_8\\
(0,3)& U\oplus A_1^{\oplus 2}\oplus E_8^{\oplus 2}\\
\end{array}$$
\end{Cor}
\proof We adopt the same notation of the proof of Proposition
\ref{stab}. Since $X_C$ is general and $0\leq n,c\leq 3$, we can
assume $C$ to be an irreducible plane quartic. In this case, the
curve $C$ in $X_C$ is irreducible and each double point decreases
its genus by one. Moreover, the fixed locus of the involution
$\tau_{n,c}$ contains a smooth rational curve for every node in $C$
and $3$ smooth rational curves for every cusp. Hence the fixed locus
of $\tau_{n,c}$ is given by an irreducible curve of genus $3-n-c$
and $n+3c$ smooth rational curves. Note that $L^{n,c}_+$ is a
$2$-elementary lattice, let $\ell_{n,c}$ be the minimal number of
generators of its discriminant group and $\delta_{n,c}$ be the
invariant defined in \cite{N1}. By \cite{N2}, Theorem 4.2.2, we get:
$$\rho_{n,c}+\ell_{n,c}=22-2(3-n-c),\ \ \rho_{n,c}-\ell_{n,c}= 2(n+3c).$$
Hence $\ell_{n,c}=8-2c$. Moreover, if $(n,c)\not=(1,1), (1,2)$, we
have $\delta_{n,c}=1$ by \cite{N2}, Theorem 4.3.2. In these cases,
let $x$ be the class of a $-2$-curve in $L_+$ corresponding to a
bitangent of $C$ not passing through the singular points of $Q$.
Then it is easy to see that $x/2$ belongs to the dual lattice of
$L^{n,c}_+$ and $(x/2)^2=-1/2\not\in \mathbb{Z}$. It follows that
$\delta_{n,c}=1$ also in these two cases. Theorem 4.3.2 in \cite{N2}
also gives that the isomorphism class of the lattice $L^{n,c}_+$ is
determined by the invariants $(\rho_{n,c},\ell_{n,c},\delta_{n,c})$.
Hence it is enough to compute this triple of invariants for the
lattices in the second column (see \cite{N2}, Proposition
3.2.2).\qed
\begin{Rem}
A $K3$ surface associated to a singular stable quartic $C$ carries a
natural elliptic fibration, given by the inverse image of the pencil
of lines through a singular point of $C$. If $C$ is the generic
quartic with one node, then this pencil has $8$ singular fibers of
type $III$ in the sense of Kodaira, corresponding to the two
branches of the node and to the six lines through the node and
tangent to $Q$ in smooth points. If $C$ is the generic quartic with
one ordinary cusp the fibration has $6$ fibers of type $III$ and one
of type $I_0^*$. They correspond respectively to the $6$ tangent
lines passing through the cusp and to the tangent line in the cusp.
\end{Rem}
\subsection{Semistable quartics}
Let $C$ be a plane quartic with an ordinary tacnode at $p$ and $Y$ be the 4:1 cyclic cover of the plane branched along $C$. The local equation of $Y$ over the point $p$ is
analytically isomorphic to:
$$t^4+y^2+x^4=0,$$
so $Y$ has an elliptic singularity of type $\tilde E_7$. This
suggests that there will be no $K3$ surfaces corresponding to these
curves. However, we prove that the period map can be still extended
holomorphically to strictly semistable admissible quartics if we
consider the Baily-Borel compactification of the period domain
$\mathcal M$. In particular, we show that these quartics are mapped
to the boundary and we describe the corresponding degenerations of
$K3$ surfaces.
Let $\mathcal M^*$ be the Baily-Borel compactification of $\mathcal
M$ (see \cite{BB}). It easy to prove that the $i$-eigenspace $W$ of
$\rho$ on $L_-\otimes \mathbb C$ is given by
$$W=\{x-i\rho(x): x\in L_-\otimes_{\mathbb{Z}}\mathbb R\}\cong L_-\otimes_{\mathbb{Z}}\mathbb R.$$
We say that $w\in W$ is \emph{defined over} $\mathbb{Z}$ if $w=x-i\rho(x)$
with $x\in L_-\subset L_-\otimes_{\mathbb{Z}}\mathbb R$.
\begin{Lem}\label{bou}
The boundary of $\mathcal M^*$ is a disjoint union of points
corresponding to isotropic lines in $W$ defined over $\mathbb{Z}$.
\end{Lem}
\proof The period domain of $L_+$-polarized $K3$
surfaces is given by:
$$\mathcal D_+=\{z\in \mathbb P(L_-\otimes \mathbb C):\ (z,z)=0,\ (z,\bar
z)>0\}.$$ The rational boundary components of $\mathcal D_+$ are
given by points and curves corresponding respectively to isotropic
lines and planes with generators in $L_-$ (see \cite{BB}). Hence the
rational boundary components of $B$ are given by the intersection of
these components with $W$. Notice that an isotropic line $\mathbb
Cv$, $v^2=0$, $v\in L_-$ can not be contained in $W$, since $W\cap
L_-=\{0\}$. Let $\Lambda$ be an isotropic plane generated by $x,y\in
L_-$. If $\Lambda$ contains an element in $W$, then it is of the
form:
$$ax+by=a_1x+b_1y-i(a_1\rho(x)+b_1\rho(y)),$$
where $a_1,b_1\in \mathbb R$. By intersecting both sides with $y$,
this gives that $(\rho(x),y)=0$. Hence:
$$\Lambda=\langle x,y\rangle=\langle x,\rho(x)\rangle.$$
Thus the intersection $\Lambda\cap W$ is a line generated by an
isotropic vector defined over $\mathbb{Z}$:
$$ax+by=(a_1+ib_1)(x-i\rho(x)).$$
Conversely, an isotropic vector in $W$ defined over $\mathbb{Z}$ is of the
form $x-i\rho(x)$, $x\in L_-$ and the plane generated by $x,
\rho(x)$ is an isotropic plane with generators in $L_-$.
\qed\\
Let $q:V_{ss}\longrightarrow \mathcal V$ be the quotient morphism to the
categorical quotient of $V_{ss}$ by the action of $PGL_3$ and
$V_{ss}'\subset V$ be the set of admissible semistable plane
quartics (see the Appendix). The following result and its proof are
similar to those of Theorem 4.1, \cite{H1} and Theorem 5, \cite{H2}.
Consider the map $p_1=\mathcal P_1\circ q$, then the main tool is an
extension theorem by Borel, which implies that $p_1$ can be extended
to a holomorphic map of $V_{ss}'$ into $\mathcal M^*$ if
$V_{ss}'\backslash V_s$ is locally contained in a divisor with
normal crossing singularities. The following lemma gives this
property for $p_1$.
\begin{Lem}\label{nor}
Let $C$ be a point in $V_{ss}'\backslash V_s$ i.e. $C$ has an
admissible tacnode. Then $V_{ss}'\backslash V_s$ is smooth at $C$ of
codimension $3$.
\end{Lem}
\proof We first assume that $C$ has only one tacnode in $p$. Then,
after a projective transformation, the curve $C$ can be defined by
an equation of degree four of the form (see Appendix):
$$f=\sum_{i+2j\geq 4}a_{ij}x^iy^j,\ a_{02}=1,$$
in affine coordinates $(x,y)$ centered at $p$. The quartic curves in
a neighbourhood $U$ of $C$ in $V$ can be defined by the equations:
\begin{equation}\label{effeuno}f(t)=\sum_{i+2j\geq 4}a_{ij}x^iy^j+\sum{}t_{ij}x^iy^j=0\end{equation}
where $t=\{t_{ij}\}$ is a system of parameters for $U$. We call
$C(t)$ the quartic curve defined by $f(t)$. If $U$ is sufficiently
small, then a curve belongs to $U\cap (V_{ss}'\backslash V_s)$ if
and only if it has a tacnode. We can assume that $C(t)$ has the
tacnode $p(t)$ in the intersection point of the lines:
$$x+s=0,\ y+ux+v=0$$
where $s,u,v$ depend on $t$ and $y+ux+v=0$ is the tangent line at
$p(t)$. Then $f(t)$ can be written in the form:
\begin{equation}\label{effedue}\sum_{i+2j\geq 4}(a_{ij}+b_{ij})(x+s)^i(y+ux+v)^j\end{equation}
for suitable coefficients $b_{ij}$.
We now compare the equations \ref{effeuno} and \ref{effedue} to get relations of the following types:\\
a) for indices $(i,j)$ such that $i+2j\geq 4$:
$$t_{ij}=b_{ij}+\mbox{terms divisible by } s,u \mbox{ or } v;$$
b) for indices $(i,j)$ such that $i+2j<4$:
$$t_{ij}=\mbox{polynomial in } s, u, v, b_{ij}.$$
Notice that equations of type a) can be solved in $b_{ij}$ as
functions of $s, u, v$ and $t_{ij}$, $i+2j\geq 4$.
Three equations of type b), by forgetting higher terms in $u,s,v$ and $b_{ij}$, $i+2j\geq 4$ are given by:\\
b1) $\ t_{01}=2v+...$,\\
b2) $\ t_{30}=a_{31}v+4a_{40}s+a_{21}u+...$,\\
b3) $\ t_{11}=2a_{12}v+2a_{21}s+2u+...$,\\
We prove that these equations are independent. Otherwise we would
get:
$$a_{21}^2-4a_{40}=0.$$
Let $a_{21}=2\alpha$, then $a_{40}=\alpha^2$. This would imply that
$C$ has an equation of the form:
$$f=(y+\alpha x^2)^2+\sum_{i+2j\geq 5}a_{ij}x^iy^j,$$
i.e. $p$ would be an inadmissible tacnode. Thus we can solve b1),
b2), b3) in $u, s, v$ as functions of $t_{01}, t_{30}, t_{11}$ and
$t_{ij}$, $i+2j\geq 4$. Notice that the cardinality of the set of
indices $(i,j)$ with $i+2j<4$ is equal to $6$. Hence we get $3$
independent equations and it can be easily seen that they define a
variety which is smooth at $C$. In case of a quartic curve $C$ with
two tacnodes, similar computations give that $V_{ss}'\backslash V_s$
has two smooth components of codimension $3$ in $C$. By Lemma 6 in
\cite{H2}, we can find two transversal hyperplanes $L_i$, $i=1,2$
such that $V_{ss}'\backslash V_s\subset L_1\cup L_2$ in a
neighbourhood of $C$.
\qed\
\begin{Prop}\label{ss}
The period map $p_1=\mathcal P_1\circ q$ extends to a holomorphic
map:
$$\mathcal P_2:V_{ss}'\longrightarrow \mathcal M^*.$$
\end{Prop}
\proof It follows from Lemma \ref{nor} and Borel's extension theorem
(\cite{Bo}, Theorem A and Remark 3.8).
\qed\\
In fact, the image in $\mathcal V$ of the locus of inadmissible plane
quartics is only one point.
\begin{Lem}\label{in}
The quotient morphism $q:V_{ss} \longrightarrow \mathcal V$ maps
$V_{ss}\backslash V_{ss}'$ to the point $v_0$ representing the orbit
of double conics.
\end{Lem}
\proof The proof is similar to that of Lemma 11, \cite{H1}. Let $C$
be an inadmissible plane quartic. We can assume $C$ to be defined by
an equation of the form:
$$f=(yz+\alpha x^2)^2+\sum_{i+2j\geq 5}a_{ij}x^iy^jz^{4-i-j}.$$
Let $F:\mathcal C\rightarrow\Delta$ be a one parameter family of semistable
plane quartics with $F^{-1}(0)=C$. The family $F$ can be given by:
$$f(t)=(yz+\alpha x^2)^2+\sum_{i+2j\geq 5}a_{ij}x^iy^jz^{4-i-j}+t\Psi(x,y,z,t),$$
where $t\in\Delta$ and $\Psi$ is an homogeneous form of degree 4 in
$(x,y,z)$ and holomorphic in $t$. Substituting $t^5$ for $t$ and
$(x,ty,t^{-1}z)$ for $(x,y,z)$ we get the new equivalent family:
$$g(t)=(y+\alpha x^2)^2+\sum_{i+2j\geq 5}a_{ij}t^{2i+j-4}x^iy^jz^{4-i-j}+t^5\Psi(x,ty,t^{-1}z,t^5).$$
Notice that $g$ is holomorphic and that $g(0)=(y+\alpha x^2)^2$.
\qed
\begin{Thm}\label{ss2}
The period map $\mathcal P_0$ can be extended to a holomorphic map:
$$\mathcal P_2:\mathcal V\backslash\{v_0\}\longrightarrow \mathcal M^*.$$
The subvariety $\mathcal V\backslash (\mathcal V_s\cup \{v_0\})$ is a smooth rational curve parametrizing type II degenerations of $K3$ surfaces and
it is mapped to the boundary of $\mathcal M^*$.
\end{Thm}
\proof The first assertion follows from Proposition \ref{ss} and
Lemma \ref{in}. Let $C$ be a quartic curve in $V_{ss}'\backslash
V_s$ and $\Delta$ be the open unit disc in $\mathbb C$. Consider a one
dimensional family of plane quartic curves $F:\mathcal C\rightarrow \Delta$
with smooth general fiber $C_t$, $t\not=0$ and central fiber
$C_0=C$. By \cite{M}, after passing to a ramified covering of
$\Delta$, we can assume that $C_0$ is a quartic curve in a minimal
orbit. Hence, by Lemma \ref{in} we can assume $C_0$ to be the union
of two conics tangent in two points. Notice that the set of plane
quartics which are the union of two tangent conics maps onto a
smooth rational curve in $\mathcal V$ (with $v_0$ in its closure).
Taking the 4:1 cyclic coverings of $\mathbb{P}^ 2$ branched along the curves
$C_t$ we get a family $G:\mathcal X \rightarrow \Delta$ of quartic surfaces
$X_t$ in $\mathbb{P}^ 3$. The central fiber $X_0$ has two singular points of
type $\tilde E_7$ and it is birational to a ruled variety with base
curve of genus 1. In fact, by Theorem 2.4 in \cite{S1}, this
degeneration is a surface of Type II i.e. the monodromy
transformation $N$ satisfies $N^2=0$, $N\not=0$. In
particular, the monodromy transformation has infinite order. This
implies that the class of $C$ in $\mathcal V$ is mapped to the boundary of
$\mathcal M^*$. \qed
\begin{Rem}\label{shah}
In \cite{S1}, Theorem 2.4, J. Shah gives a classification of GIT
semistable quartic surfaces. It can be easily proved that the 4:1
cyclic cover of $\mathbb{P}^ 2$ branched along a plane quartic is a stable
quartic surface if and only if the plane quartic is stable. In this
case, the quartic surface has at most rational double points. Hence,
we only have surfaces of Type I (case A, Type I in Shah's theorem).
In the case of strictly semistable and reduced plane quartics, the
corresponding quartic surface is strictly semistable with isolated
singularities. In particular, we get quartic surfaces of Type II
(case B, Type II, (i) in Shah's theorem). In the case of a double
conic, the 4:1 cyclic cover of the plane branched along this curve
is the union of two quadrics tangent to each other along the
ramification curve. In particular, this surface $X$ has significant
limit singularities (case B, Surfaces with significant limit
singularities, (ii) in Shah's theorem), this means that the order of
the monodromy transformation depends on the family of surfaces
specializing to $X$. Hence the period map can not be extended to the
point $v_0$.
\end{Rem}
\subsection{Applications}\ \\
\noindent\emph{A) Vinberg's surface.} By a result of T. Shioda and H. Inose there is a one-to-one
correspondence between isomorphism classes of singular $K3$ surfaces
and equivalence classes of positive definite even lattices with
respect to the action of $SL_2(\mathbb Z)$ (see Theorem 4,
\cite{SI}). In \cite{V}, \`E.B. Vinberg describes the geometry of the $K3$ surface with transcendental lattice
$$T\cong \langle 2\rangle^{\oplus 2}.$$
In particular, he finds two interesting projective models for this
surface as a 4:1 cover of the plane branched along a stable quartic.
We give a partial version of his result and provide an alternative
proof.
\begin{Prop}[Theorem 2.5, \cite{V}]\label{vin}
The K3 surface with transcendental lattice isomorphic to $T$ has two
non isomorphic $(L_+,\rho)$-polarizations since it can be obtained
both as the 4:1 cyclic cover of $\mathbb{P}^ 2$ branched along a stable
quartic of singular type $(6,0)$ or of type $(0,3)$. The
isomorphism class of its Picard lattice is given by:
$$U\oplus A_1^{\oplus 2}\oplus E_8^{\oplus 2}.$$
\end{Prop}
\proof Let $C$ be a plane quartic with $6$ nodes i.e. the union of
four lines in $\Ps2$. The corresponding $K3$ surface $X_C$ is the
minimal resolution of a quartic surface in $\Ps3$ with $6$ rational
double points of type $A_3$. The surface $X_C$ carries a natural
involution $\tau_{6,0}$ with fixed locus equal to the disjoint union
of $10$ smooth rational curves. By Proposition \ref{stab}, $X_C$ has
maximum Picard number and the Picard lattice of $X_C$ equals the
invariant lattice $L^{6,0}_+$. By \cite{N2}, Theorem 4.2.2 this
lattice has the invariants:
$$\rho_{6,0}=20,\ \ell_{6,0}=2.$$
Hence, by \cite{N1}, Theorem 4.3.2 the isomorphism classes of the
Picard lattice and the transcendental lattice of $X_C$ are given by:
$$Pic(X_C)\cong U\oplus A_1^{\oplus 2}\oplus E_8^{\oplus 2},$$
$$T(X_C)\cong T.$$
By Corollary \ref{irr} (see type $(0,3)$) and \cite{SI}, Theorem 4,
the $K3$ surface $X_{C'}$ with $C'$ a plane quartic with three ordinary cusps is isomorphic to $X_C$.\qed\\
\noindent\emph{B) The structure of $\mathbb{Z}[i]$-module on the
transcendental lattice.} We now give a geometric interpretation to
the action of the order four isometry $\rho$ on the lattice $L_-$.
\begin{Lem}
The isomorphism class of $L_-$ is given by:
$$L_-\cong \langle 2\rangle^{\oplus 2}\oplus D_4^{\oplus 3}.$$
\end{Lem}
\proof Since $L_-=L^{\perp}_+$ it follows that $r(L_-)=14$ and
$q_{L_+}=-q_{L_-}$ on $A_{L_+}\cong A_{L_-}$. Then Corollary
\ref{irr} gives that $\ell(L_-)=8,\ \delta(L_-)=1$. Hence the proof
follows from \cite{N1}, Theorem 4.3.2 and \cite{N2}, Proposition 3.2.2.\qed\\
\begin{Lem}\label{act}
The lattices $A^{\oplus 2}_1$ and $D_4$ have a unique structure of $\mathbb{Z}[i]$-modules up to isometries.
For suitable bases in $A^{\oplus 2}_1$ and $D_4$ this action is
given by the matrices $J_2$ and $J^{\oplus 2}_2$ respectively, where
$$J_2=\left(\begin{array}{cc}
0 &1\\
-1 & 0
\end{array}\right).$$
\end{Lem}
\proof The result for $A^{\oplus 2}_1$ follows from easy
computations. We now consider the case of $D_4$. Recall that the
root lattice $D_4$ can be defined as:
$$D_4=\{x\in\mathbb{Z}^4:\ \sum{x_i}\equiv 0\ mod\,2\},$$
with the standard euclidean inner product.
A basis of $D_4$ is given by the roots:
$$e_1-e_2,\ e_2-e_3,\ e_3-e_4,\ e_3+e_4.$$
Let $f:\mathcal C\rightarrow S$ be a family of stable plane quartics over
the interval $I=[0,1]$ with fiber $C(t)$, $t\not=0$ of type
$(1,0)$ and special fiber $C(0)$ of type $(0,1)$. Then, by Ehresmann's fibration theorem, we have a natural isomorphism:
$$H^2(X_{C(t)},\mathbb Z)\cong H^2(X_{C(0)},\mathbb Z),$$
for every $t\in I$. We fix an isomorphism $H^2(X_{C(t)},\mathbb
Z)\cong L_{K3}$. By Proposition \ref{stab}
$$Pic(X_{C(t)}) \cap L_- \cong A^{\oplus 2}_1,$$
$$Pic(X_{C(0)}) \cap L_- \cong D_4.$$
Hence the family $f$ gives an embedding $A^{\oplus 2}_1\subset D_4.$
By \cite{N1}, Proposition 1.15.1 such embedding is unique up to
isometries. Hence we can assume that $r=e_1-e_2$ and
$\rho(r)=e_3-e_4,$ so $<r,\rho(r)>^{\perp}=<e_1+e_2,e_3+e_4>\cong
A^{\oplus 2}_1.$ Notice that $\rho$ preserves both copies of
$A^{\oplus 2}_1$ and, by the result for $A^{\oplus 2}_1$, it acts on
each copy by the matrix $J_2$ with respect to the natural bases.
This gives to $D_4$ the structure of a free $\mathbb{Z}[i]$-module of rank two and a $\mathbb{Z}[i]$-basis is given by:
$$f_1=e_1-e_2,\ f_2=e_1-e_3,$$
since $\rho(f_1)=e_3-e_4$, $\rho(f_2)=e_3+e_1$.
The matrix representing $\rho$ with respect to the basis $f_1,\rho(f_1),f_2,\rho(f_2)$ is $J_2\oplus J_2$.\qed
\begin{Prop}
The isometry $\rho$ on $L_-$ preserves $\langle 2\rangle^{\oplus 2}$ and each copy of $D_4$, its action is thus described by Lemma \ref{act}.
\end{Prop}
\proof
Consider a stable plane quartic $C$ of type $(0,3)$. Each cusp
gives a lattice of type $D_4\subset Pic(X_C)\cap L_-$ (Proposition \ref{stab}). Moreover $T(X_C)\cong \langle 2\rangle^{\oplus
2}$ (Proposition \ref{vin}). Hence we have that:
$$L_-\cong T(X_C)\oplus (Pic(X_C)\cap L_-).$$
By deformation to plane quartics of singular type $(0,2)$, it is
clear that the automorphism $\rho$ preserves each copy of $D_4$ and
the lattice $T(X_C)$.\qed
\section{Hyperelliptic genus three curves}
In the previous section we proved that the period map can be
extended holomorphically to the complement of the point $v_0$
representing double conics in $\mathcal V$. In Remark \ref{shah} we
observed that the period map can not be extended to this point,
since the 4:1 cyclic cover of the plane branched along a double
conic has significant limit singularities. This is connected to the
existence of \emph{hyperelliptic} $(L_+,\rho)$-polarized $K3$
surfaces i.e.\! such that the curves in the linear system defined by
the degree four polarization introduced in section 1.1. are
hyperelliptic (see \cite{D}). In fact, there is a correspondence
between these polarized hyperelliptic $K3$ surfaces and
hyperelliptic genus three curves which is described in detail by S.
Kond\=o in \cite{K} and \cite{Kh}. In this section we recover
Kond\=o's results providing an alternative description for
hyperelliptic $(L_+,\rho)$-polarized $K3$ surfaces.
\subsection{Smooth hyperelliptic curves}\label{hg}
Let $C\subset \mathbb{P}^ 5$ be a smooth hyperelliptic genus three curve
embedded by the bicanonical map and $i$ be the hyperelliptic
involution on $C$. Consider the surface $\Sigma$ in $\mathbb{P}^ 5$ defined
as follows
$$\Sigma=\overline {\bigcup_{p\in C}\langle p,i(p)\rangle},$$
where $\langle\,,\,\rangle$ denotes the line spanned by two points.
\begin{Prop}\label{hyp}
The surface $\Sigma$ is a cone over a rational normal quartic and
$C$ is a quadratic section of $\Sigma$ not passing through the
vertex. Let $\widetilde \Sigma$ be surface obtained by blowing up of
the vertex of $\Sigma$. This is a $4$-th Hirzebruch and the class of
the curve $C$ in $Pic(\widetilde \Sigma)$ is given by:
$$C=2S_{\infty}+8F,$$
where $S_{\infty}$ is the class of the section with $S_{\infty}^2=-4$ and $F$ is the class of a fiber.
\end{Prop}
\proof
A hyperelliptic genus three curve $C$ can be given by an equation of the form
$$y^2=\prod_{i=1}^{8}(x-\lambda_i),$$
for some complex numbers $\lambda_1,\dots, \lambda_8$.
The hyperelliptic involution $i$ can be written as
$$i:C\longrightarrow C,\ \ (x,y)\longmapsto(x,-y).$$
and the bicanonical map is given by
$$\phi_{\mid 2K_C\mid}:C\longrightarrow \mathbb{P}^ 5,\ \ (x,y)\longmapsto(1, x, x^2, x^3, x^4, y).$$
Hence, if $(z_0,\dots,z_5)$ are coordinates for $\mathbb{P}^ 5$, the surface
$\Sigma$ is a cone with vertex $(0,\dots,0,1)$ over the rational
normal quartic obtained by projecting $C$ on the hyperplane $z_5=0$.
Moreover, since the curve $C$ is a quadratic section of $\Sigma$,
its inverse image in $\widetilde \Sigma$ has the intersection
numbers: $(C,S_{\infty})=0,$ $(C,F)=2.$ This determines the class of
$C$ in $Pic(\widetilde \Sigma)$.
\qed\\
Let $X_C$ be the 4:1 cyclic cover of the rational ruled surface $\widetilde \Sigma$ branched along the reducible curve:
$$C\cup 2S_{\infty}\in\mid 4S_{\infty}+8F\mid.$$
It can be easily proved that $X_C$ is a $K3$ surface.
\begin{Rem}
In \cite{K} and \cite{Kh} S. Kond\=o associates a $K3$ surface to a smooth hyperelliptic genus three curve in the following way.
The hyperelliptic curve $C$ is embedded in $\Ps1\times \Ps1$ as a divisor of bidegree $(4,2)$, in coordinates $((x_0\!:\!x_1\!),(y_0\!:\!y_1\!))$ for $\Ps1\times \mathbb{P}^ 1$:
$$y_0^2\prod_{i=1}^{4}(x_0-\lambda_ix_1)+y_1^2\prod_{i=5}^{8}(x_0-\lambda_ix_1)=0.$$
Let $L_i=\{y_i=0\}$, $i=1,2$ and $S'_C$ be the double cover of $\Ps1\times \Ps1$ branched along the divisor $B=C+L_1+L_2$ of bidegree $(4,4)$. The minimal resolution $X'_C$ of $S'_C$ is a $K3$ surface.
In fact, it easy to realize that this constructions leads to the
same $K3$ surface obtained before, i.e. $X_C=X'_C$. We give a sketch
of the proof.
Let $C$ be a hyperelliptic genus three curve embedded in a cone $\Sigma$ as in Proposition \ref{hyp}. Let $p_1,\dots,p_8$ be the intersection points of $C$ with the rational normal quartic. It is easy to see that the eight fibers $F_i$ of $\widetilde \Sigma$ through the points $p_i$ are tangent to the curve $C$.
Let $\phi:S_C\longrightarrow \widetilde \Sigma$ be the double cover branched along $C$.
The inverse image of the section $S_{\infty}$ is the union of two disjoint smooth rational curves $S_1, S_2$, while the inverse images of the fibers $F_i$ split in the union of two smooth rational curves intersecting transversally in a point on the proper transform of $C$:
$$\phi_1^{-1}(F_i)=F_{i1}\cup F_{i2},\ \ \ i=1,\dots,8.$$
This gives $16$ $(-1)$-curves on $S_C$. We can assume that:
$$F_{ij}\cap S_k\not=\emptyset \Leftrightarrow j=k,\ \ \ \ \
i=1,\dots,8;\ j,k=1,2.$$
We still denote by $F$ the inverse image in $S_C$ of a fiber of $\widetilde \Sigma$.
Let $b:S_C\longrightarrow R$ be the blowing down of the eight $(-1)$-curves $F_{11},\dots,F_{41},F_{52},\dots,F_{82}.$
The surface $R$ is a rational ruled surface with sections $L_1=b(S_1)$
and $L_2=b(F)$
Notice that ${L_1}^2=(S_1+F_{11}+\dots+F_{41})^2=0$, ${L_2}^2=0.$
Hence it follows that $R\cong \Ps1\times \Ps1$, $S_C=S'_C$ and the $K3$ surface $X_C=X'_C$ is the double cover of $S_C$ branched along the divisor $C+S_1+S_2$.
\end{Rem}
The previous construction associates a polarized $K3$ surface $
|
X_C$ to any smooth hyperelliptic genus three curve $C$.
In \cite{K} and \cite{Kh} Kond\=o proved that this defines a
holomorphic period map
$$\mathcal P_0^h:\mathcal M^h_3\longrightarrow \mathcal D_h\subset \mathcal M.$$
Moreover, if we denote with $\mathcal D'$ the discriminant locus in $\mathcal D^h$ (i.e. the union of hyperplane sections of $\mathcal D_h$ defined by roots), then:
\begin{Thm}[Theorem 5.3, \cite{K}, Theorem 3.8, \cite{Kh}]\label{h1}
The period map induces an isomorphism:
$$\mathcal P_0^h:\mathcal M_3^{h}\longrightarrow \mathcal D_h\backslash \mathcal D'.$$
\end{Thm}
\subsection{Stable curves}
As stated in the Appendix, there is an isomorphism
$$\mathcal M^h_3\cong \mathcal V^h_0=V^h_0/\!/PGL_2,$$
where $V^h_0$ denotes the space of sets of eight distinct points in
the projective line. A natural compactification for the moduli space
of smooth hyperelliptic genus three curves is given by the
categorical quotient
$$\mathcal V^h=V^h_{ss} /\!/PGL_2.$$
Recall that a collection of eight points is \emph{stable} (\emph{semistable}) if and only if at most three (four) points coincide. This clearly induces a notion of (semi)stability for hyperelliptic genus three curves.
We prove that there is an isomorphism between $\mathcal V^h$ and a GIT quotient of the space of quadratic sections of a cone which preserves the sets of (semi)stable points.
Let $\Sigma\subset \mathbb{P}^ 5$ be a cone over a rational normal quartic.
\begin{Prop}\label{sec}
The space $\mathcal V^h$ is isomorphic to a GIT quotient of the space of quadratic sections of $\Sigma$ not passing through the vertex with respect to the action of the automorphism group of $\Sigma$.
\end{Prop}
\proof Let $\mathcal G$ be the group of automorphisms
of $\Sigma$, by \cite{De1} this can be described as:
$$\mathcal G =G_0\cdot GL_2/\mu_4,$$
where $G_0\cong H^0(\mathbb{P}^ 1,\mathcal O_{\mathbb{P}^ 1}(4))$, $\mu_4=\{\alpha
I:\alpha^4=1\}$ and the action of $GL_2$ on $G_0$ is the natural
one. The action of $\mathcal G$ can be extended to an action on $\mathbb{P}^
5$ and $\mathcal O_{\mathbb{P}^ 5}(1)$ admits the following $\mathcal
G$-linearization. Let $z$ be the vertex of $\Sigma$ and
$$A=H^0(\mathbb{P}^ 5, \mathcal O_{\mathbb{P}^ 5}(1))\cong H^0(\Sigma,\mathcal O_{\Sigma}(1)),$$
$$A_1=\mbox{ sections of }A\mbox{ vanishing on }z.$$
Notice that, by restricting the proper transforms of sections to
$S_{\infty}$ in the blow up $\widetilde \Sigma$ we have:
$$ A_1\cong H^0(\Ps1,\mathcal O_{\mathbb{P}^ 1}(4)).$$
Let $\{q_0,q_1,\dots, q_5\}$ be a basis of $A$ where
$\{q_1,\dots,q_5\}$ is a basis for $A_1$.
The action of $G_0$ on $A$ is given by matrices of the form:
$$\phi(\b g)=\left( \begin{array}{cc}
1 & \b g \\
0 & I_5
\end{array}\right),$$
where $\b g$ is a 5-length row vector and $I_5$ the $5\times 5$
identity matrix. Notice that $G_0$ fixes the vertex $z$ and
preserves lines through $z$. The action of $GL_2$ is trivial on
$\mathbb Cq_0$ and acts on $A_1$ via the isomorphism $A_1\cong
H^0(\Ps1,\mathcal O_{\mathbb{P}^ 1}(4)).$ In fact, $GL_2$ fixes the section
$q_0=0$ and acts on it as the group of automorphisms of the rational
normal quartic curve. In order to define the GIT quotient we now
consider the spaces:
$$B=H^0(\mathbb{P}^ 5, \mathcal O_{\mathbb{P}^ 5}(2)),$$
$$B_1 =\mbox{ sections of }B\mbox{ vanishing on }z,$$
$$B_2=\mbox{ sections of }B\mbox{ vanishing with order two on }z.$$
As in the case of $A_1$, by restricting to $S_{\infty}$ we have an
identification :
$$B_2\cong H^0(\Ps1,\mathcal O_{\Ps1}(8)).$$
We have the following $GL_2/\mu_4$ invariant decomposition:
$$B\cong \mathbb Cq_0^2\oplus q_0A_1\oplus B_2.$$
Consider the open subset of $\mid B\mid$:
$$B^*=\mid B\mid- \mid B_1\mid.$$
A point in $B^* $ is represented uniquely by an element of the
form:
$$q_0^2+q_0a+b,$$
where $a\in A_1$ and $b\in B_2$. If $\mathcal A_1$ and $\mathcal
B_2$ are the affine spaces associated to $A_1$ and $B_2$
respectively, then $B^* $ is isomorphic to $\mathcal A_1\times
\mathcal B_2$. Now we show that, up to the action of $G_0$, every
element in $B^*$ can be represented in the form $q_0^2 +b,$ where
$b\in B_2$. Let $\phi\in G_0$ taking $q_0$ to $q_0+a_0$ with $a_0\in
A_1$. Then $\phi$ takes $f=q_0^2+q_0a+b$ to the form:
$$\phi(f)=q_0^2+(a+2a_0)+(a_0^2+a_0a+b).$$
The choice $a_0=-a/2$ gives the statement and defines a map:
$$\mathcal A_1\times \mathcal B_2\longrightarrow \mathcal B_2\ \ \
(a,b)\longmapsto b-a^2/4.$$ It can be be proved as in \cite{S1},
Lemma 4.1, that this map defines a geometric quotient by the action
of $G_0$.
Let $G'$ denote the center of $GL_2/\mu_4$ (isomorphic to the
1-dimensional multiplicative group) and $G''\cong PGL_2$ be the image of
$SL_2$ in $GL_2/\mu_4$. It can be easily seen that:
$$GL_2/\mu_4\cong G'\times G''.$$
By taking the quotient for the action of $G'$ we get
$$\mathcal B_2\backslash\{0\}/\!/G'\cong \mid \mathcal O_{\Ps1}(8)\mid.$$
Moreover, the induced action of $G''$ equals the natural action of
$PGL_2$ on $\mid\mathcal O_{\Ps1}(8)\mid$. Hence, by taking the
categorical quotient of the semistable locus $\mid \mathcal
O_{\Ps1}(8)\mid_{ss}$ for the action of $PGL_2$ we get a moduli
space isomorphic to $\mathcal V^h$.\qed\\
We now give a characterization of stability as follows:
\begin{Prop}\label{cone}
A quadratic section of $\Sigma$ not passing through the vertex
is stable if and only if it has at most ordinary nodes and cusps.
\end{Prop}
\proof Let $p\in C$ and $F$ be the class of a fiber in $\widetilde
\Sigma$. We choose a basis $\{u,v\}$ of $H^0(\widetilde{\Sigma},F)$
such that $u$ vanishes at $p$. Let $l_0=\{u=0\}$,
$l_{\infty}=\{v=0\}$, $q_{\infty}\in
H^0(\widetilde{\Sigma},S_{\infty})$ and $q_0\in
H^0(\widetilde{\Sigma}, 4F+S_{\infty})$. We consider the affine
coordinates $x=u/v$ and $y=q_0/q_{\infty}$. The divisor $C$ is
defined by an equation of the form $f=q_0^2+b=0$, where $b\in
H^0(\widetilde{\Sigma},F)$ and has multiplicity $\leq 4$ at $p$. In
the affine set $\widetilde{\Sigma}-S_{\infty}-l_{\infty}$ the
divisor $C$ is defined by an equation of the form $y^2+p_8(x)=0$
where $p_8$ is a polynomial of degree $8$ which is not divisible by
$x^5$. Notice that $C$ is reduced and has at most double points. If
$C$ is stable then $x^4$ doesn't divide $p_8$, hence $C$ has at most
a node or an ordinary cusp in $p$.
\qed\\
The minimal resolution of the 4:1 cyclic cover of
$\widetilde{\Sigma}$ branched along the curve $C\cup 2S_{\infty}$,
where $C$ is a stable point in $V^h$ is an $(L_+,\rho)$-polarized K3
surface.
An application of Brieskorn's result as in the proof of Proposition \ref{stab} gives
\begin{Lem}
The period map $\mathcal P_0^h$ extends to a holomorphic map:
$$\mathcal P_1^h:\mathcal V^h_s\longrightarrow \mathcal D_h.$$
\end{Lem}
As in the case of stable plane quartics, a natural stratification
for $\mathcal D_h$ is induced by the number of nodes and cusps of a
stable hyperelliptic curve. The Picard lattice of the generic $K3$
surface in each stratum is given in \cite{Kh}, \S4.5.
\subsection{Semistable curves}
Let $C$ be a stricly semistable hyperelliptic genus three curve in a
minimal orbit. With the notation of the proof of Proposition
\ref{cone}, a local equation for the embedding of the curve $C$ in
$\widetilde \Sigma$ is given by:
$$y^2-p_8(x)=0,$$
where $p_8(x)=(x-a)^4(x-b)^4$, $a,b\in \mathbb C$. In fact, note that the
locus $\mathcal V^h\backslash \mathcal V^h_s$ is just one point. The
curve $C$ in $\widetilde \Sigma$ has two tacnodes, this implies that
the 4:1 cyclic cover $X_C$ of $\widetilde\Sigma$ branched along
$C\cup 2S_{\infty}$ has two elliptic double points. As in the case
of semistable plane quartics, it can be proved that the period map
can be extended to $\mathcal V^h$ by considering the Baily-Borel
compactification $\mathcal D^*_h$ of the ball quotient $\mathcal
D_h$. In fact, the boundary of $\mathcal D^*_h$ is given by one
point (the \emph{cusp}) and we have the following result.
\begin{Thm}[S. Kond\=o, Theorem 4.7, \cite{Kh}]\label{hiso}
The period map can be extended to an isomorphism:
$$\mathcal P^h:\mathcal V^h\longrightarrow \mathcal D^*_h.$$
The point $\mathcal V^h\backslash \mathcal V^h_s$ is mapped to the
cusp.
\end{Thm}
\section{A period map for genus three curves}
In this section we construct a blow-up of $\mathcal V$ in $v_0$ such
that the exceptional divisor is isomorphic to $\mathcal V^h$ and we
prove that the period map $\mathcal P_2$ can be extended to this
variety. In fact, we prove that this extension coincides with the
period map $\mathcal P^h$ on the exceptional divisor.
\subsection{Blow-up}
Let $T$ be a conic in $\mathbb{P}^ 2$ and $2T$ be the corresponding double
conic. We denote by $G_3(2T)$ the orbit of $2T$ by the action of
$PGL_3$.
\begin{Prop}\label{bl}
Let $\widetilde V_{ss}$ be the blow-up of $V_{ss}$ along $G_3(2T)$.
For a proper choice of the $PGL_3$-linearization on $\widetilde V$,
the fibre of
$$b:\widetilde V_{ss}\longrightarrow V_{ss}$$
over $2T$ is the semistable locus $V^h_{ss}$
(with respect to the action of $PGL_2$).
The exceptional divisor $\mathcal E$ of the induced blowing-up
$$\widetilde V_{ss}/\!/PGL_3\longrightarrow \mathcal V=V_{ss}/\!/PGL_3$$
is isomorphic to $\mathcal V^h$.
\end{Prop}
\proof The result follows from a theorem of F. Kirwan (\S 7,
\cite{Ki}) if we prove that the normal space at $G_3(2T)$ in a point
$2T$ is isomorphic to $H^0(\mathcal O_{\Ps1}(8))$ and that, under
this isomorphism, the isotropy group of $2T$ acts on it as $PGL_2$.
Let $q\in H^0(\Ps2, \mathcal O_{\mathbb{P}^ 2}(2))$ be a section vanishing
on $T$. The isotropy group of $T$ in $PGL_3$ can be easily
identified with $PGL_2$. There exists a unique $GL_2$-invariant
decomposition:
$$H^0(\Ps2, \mathcal O_{\mathbb{P}^ 2}(2))\cong \mathbb C q\oplus \Theta,$$ where $\Theta\cong H^0(\mathbb{P}^ 1,
\mathcal O_{\Ps1} (4))$. Moreover:
$$H^0(\mathbb{P}^ 2, \mathcal O_{\mathbb{P}^ 2}(4))\cong \mathbb Cq^2\oplus q\Theta\oplus \Psi,$$
where $\Psi\cong H^0(\mathbb{P}^ 1,\mathcal O_{\Ps1}(8))$.
The normal space at $G_3(2T)$ in $2T$ is isomorphic to $\Psi$.
Besides, up to the isomorphism of $\Psi$ with $H^0(\Ps1, \mathcal
O_{\mathbb{P}^ 1}(8))$, the action of $PGL_2$ is the canonical one.
Hence, by the result of F. Kirwan, the action of $PGL_3$ can be lifted to $\widetilde V$ in such a way that the exceptional divisor over $v_0$ is isomorphic to the universal categorical quotient of $V^h_{ss}$ by the action of $PGL_2$. \qed\\
We denote with $\widetilde{\mathcal V}$ the blowing-up of $\mathcal V$ in $v_0$ given in Proposition \ref{bl}:
$$\widetilde{\mathcal V}= \widetilde V_{ss}/\!/PGL_3.$$
Let $\widetilde{\mathcal V}_s$ be the subvariety corresponding to the stable
locus in $\widetilde V_{ss}$.
\subsection{Final extension}
We can now associate the isomorphism class of a genus three curve to
each point of $\widetilde{\mathcal V}$. If the point is not on the
exceptional divisor, then the curve can be embedded in $\mathbb{P}^ 2$ as a
semistable plane quartic. Otherwise, it represents a semistable
hyperelliptic genus three curve and can be embedded in the cone
$\Sigma$ as a quadratic section not passing through the vertex.
Moreover, we defined two period maps:
$$\mathcal P_2:\widetilde{\mathcal V}\backslash \mathcal E\longrightarrow \mathcal M^*\backslash \mathcal
D_h,$$
$$ \mathcal P^h:\mathcal E\longrightarrow \mathcal D^*_h.$$
We now prove that $\mathcal P_2$ and $\mathcal P^h$ give a global
holomorphic period map on $\widetilde{\mathcal V}$.
For similar results and methods see \cite{S} and \cite{S2}.
We start considering the case of a one parameter family of smooth
quartic curves degenerating to a double conic $2T$. Let $X$ be the
4:1 cyclic cover of the Hirzebruch surface $\widetilde \Sigma$
branched along $C+2S_{\infty}$, where $C$ is a quadratic section of
the cone $\Sigma$ not passing through the vertex. We define the
\emph{4:1 cyclic cover of $\Sigma$ branched along $C$ and the
vertex} to be the surface obtained by contracting the inverse images
of $S_{\infty}$ in $X$.
\begin{Prop}\label{pa1}
Let $\mathcal C\longrightarrow \Delta$ be a family of plane quartics over the
unit complex disk $\Delta$ intersecting transversally the orbit of
double conics in a smooth
double conic over $0\in\Delta$.\\
Let $F:\mathcal X\longrightarrow \Delta$ be the family of quartic surfaces in $\mathbb{P}^ 3$ giving the 4:1 cyclic cover of $\mathbb{P}^ 2$ branched along $\mathcal C$.\\ Then there exist a cover $\xi:\Delta\longrightarrow\Delta$ ramified over $0\in \Delta$ and a family $F':\mathcal Y\longrightarrow \Delta$ such that:\\
i) the fibers of $F'$ and $\xi^*F$ are isomorphic over $t\not=0$;\\
ii) the central fiber $Y_0$ of $F'$ is the 4:1 cover of a cone
$\Sigma\subset \mathbb{P}^ 5$ over a rational normal quartic, branched
along the vertex and a quadratic section not passing through the
vertex.
\end{Prop}
\proof
Let $(x_0,x_1,x_2)$ be coordinates for $\mathbb{P}^ 2$ and $t\in\Delta$. The
equation of the family $\mathcal C$ can be written in the form:
$$\mathcal C:\ \ q^2(x_i)+t\phi(t,x_i)=0,$$
where $\phi(t)\in H^0(\Ps2, \mathcal O_{\mathbb{P}^ 2}(4))$, $\phi(0)$ is
not divisible by $q$ (from the transversality condition) and $q\in
H^0(\Ps2, \mathcal O_{\mathbb{P}^ 2}(2))$ defines a smooth conic $T$. Let
$\xi$ be the base change of order two $t\mapsto t^2$ on $\Delta$ and
call $\mathcal C'$ the family of curves over $\Delta $ obtained as
pull-back of $\mathcal C$ by $\xi$. The 4:1 cyclic cover $\mathcal
X'$ of $\Delta\times \Ps2$ branched along the family $\mathcal C'$
has a natural embedding in $\Delta\times\mathbb{P}^ 3$, in coordinates
$(t;x_0,x_1,x_2,w)$:
$$\mathcal X':\ \ w^4=q^2(x_i)+t^2\phi(t^2,x_i).$$
Consider the embedding:
$$\Delta\times \mathbb{P}^ 3\longrightarrow \Delta\times\mathbb{P}^ 9$$
given by the identity on the first factor and by the $2$-nd Veronese
embedding on the second one. We consider the coordinates on $\mathbb{P}^ 9$:
$$z_{ij}=x_ix_j,\ s=w^2,\ y_k=tx_k,\ \ \ 0\leq i,j,k\leq 2.$$
The Veronese embedding $V(\mathbb{P}^ 3)$ of $\mathbb{P}^ 3$ is given by the
equations:
$$\left\{ \begin{array}{lll}
z_{ij}z_{kl}-z_{ik}z_{jl}&=&0,\\
sz_{ij}-y_iy_j&=&0,\ \ 0 \leq i,j \leq 2.
\end{array}\right.$$
Then the embedding of $\mathcal X'$ in $\Delta\times\mathbb{P}^ 9$ is the
intersection of $V(\mathbb{P}^ 3)$ with the quadratic section:
$$s^2=q^2(z_{ij})+t^2\phi(z_{ij},t^2),$$
where we think $q\in H^0(\mathbb{P}^ 9, \mathcal O_{\mathbb{P}^ 9}(1))$ and
$\phi(t^2)\in H^0(\mathbb{P}^ 9, \mathcal O_{\mathbb{P}^ 9}(2))$. We still denote by
$T$ the hyperplane section defined by $q$ in $\mathbb{P}^ 9$ and consider
the blowing-up of $\Delta\times \mathbb{P}^ 9$ along $\{0\}\times T$. This
gives a family with general fiber isomorphic to $\mathbb{P}^ 9$ and central
fiber given by the union of a copy of $\mathbb{P}^ 9$ and the exceptional
divisor:
$$E\cong\mathbb P(\mathcal O_T\oplus \mathcal O_T(1)).$$
By Grauert's contraction criterion (see \cite{Gr}), the $\mathbb{P}^ 9$
component in the central fiber can be contracted to a point. The
proper transform $\mathcal Y$ of $\mathcal X'$ in this new variety
is the intersection of the cone over $V(\mathbb{P}^ 3)$ with the quadratic
sections:
$$\left\{\begin{array}{ll}
s^2=\epsilon^2+t^2\phi,\\
q-\epsilon t=0.
\end{array}\right.$$
Let $\mathcal S$ be the projection of $\mathcal Y$ on the linear
subspace defined by:
$$y_i=0,\ \ 0\leq i\leq 2.$$
The surface $S_t$ is defined by:
$$\left\{\begin{array}{ll}
s^2=\epsilon^2+t^2\phi,\\
z_{ij}z_{kl}-z_{ik}z_{jl}=0,\ \ 0\leq i,j\leq 2,\\
q-\epsilon t=0,
\end{array}\right.$$
We denote by $\pi_{2,t}:\mathcal Y\longrightarrow\mathcal S$ the projection.\\
Consider now the projection $\mathcal W$ of $\mathcal S$ on the
subspace defined by:
$$s=y_i=0,\ \ 0\leq i \leq 2.$$
The surface $W_t$ is defined by:
$$\left\{\begin{array}{ll}
z_{ij}z_{kl}-z_{ik}z_{jl}=0,\ \ 0\leq i,j\leq 2,\\
q-\epsilon t=0.
\end{array}\right.$$
We denote by $\pi_{1,t}:\mathcal S\longrightarrow \mathcal W$ the projection
and $\pi_t=\pi_{2,t}\circ\pi_{1,t}$.
Then we have the following two cases:\\
i) $t\not=0$:\\
The surface $W_t$ is a linear section of the cone over the Veronese
surface not passing through the vertex, hence it is isomorphic to
the Veronese surface. The projection $\pi_{1,t}$ is the double cover
of $W_t$ branched along the section $B_t$ defined by:
$$\epsilon^2+\phi(z_{ij},t^2)=0.$$
In particular $S_t$ is a Del Pezzo surface of degree two. The
projection $\pi_{2,t}$ is the double cover of $S_t$ branched along
$\pi_{1,t}^{-1}(B_t)$. Hence $\pi_t$ is the 4:1 cyclic cover of
$W_t$ branched along $B_t$. The surface $Y_t$ is isomorphic to the
quartic surface $X_t$.
\\
ii) $t=0$:\\
The central fiber $W_0$ is a cone over the rational normal quartic
given by the intersection of the Veronese surface with the
hyperplane $q=0$. Let $\tilde W_0$ be the $4$-th Hirzebruch surface
obtained by blowing up the vertex of $W_0$, $S_{\infty}$ be the
exceptional divisor and $\tilde B_0$ be the proper transform of
$B_0$. The projection $\pi_{1,0}$ is the double cover of $W_0$
branched along the section $B_0$:
$$\epsilon^2+\phi(z_{ij},0)=0.$$
In particular $S_0$ has two singular points $P_{\pm}$ over the
vertex of $W_0$. The projection $\pi_{2,0}$ is the double cover of
$S_0$ branched along $\pi_{1,0}^{-1}(B_0)$ and the points $P_{\pm}$.
Hence $\pi_0$ is the 4:1 cyclic cover of $W_0$ branched along $B_0$
and the vertex of the cone i.e. the blowing up $\tilde Y_0$ of $Y_0$
in the points $\pi_{2,0}^{-1}(P_{\pm})$ is the 4:1 cyclic cover of
$\tilde W_0$ branched along $\tilde B_0+2S_{\infty}$ .\qed
\begin{Cor}
Let $\mathcal F$ be a one-parameter family in $\widetilde{\mathcal
V}$ intersecting transversally the exceptional divisor $\mathcal E$.
Then the period maps $\mathcal P_2$ and $\mathcal P^h$ glue
holomorphically on $\mathcal F$.
\end{Cor}
\proof By \cite{S2}, Proposition 2.1, up to base change, there
exists a family $\widetilde {\mathcal F}$ in $\widetilde{V}_{ss}$
representing $\mathcal F$ (such that the central fiber belongs to a
minimal orbit). Let $w$ be the intersection of $\widetilde{\mathcal
F}$ with the exceptional divisor. Let $\mathcal C$ be the projection
of $\widetilde{\mathcal F}$ to $V_{ss}$. This is a family of plane
quartics $\mathcal C$ with central fiber equal to a double conic
$2T$. By Proposition \ref{pa1}, after a base change of order two, we
can associate to $\mathcal C$ a family of surfaces such that the
general fiber is isomorphic to the 4:1 cyclic cover of $\mathbb{P}^ 2$
branched along $C_t$ and the central fiber is the 4:1 cyclic cover
of a cone $\Sigma$ branched along a quadratic section $B_0$ and the
vertex. In fact, it follows from the proof of Proposition \ref{pa1}
that the quadratic section $B_0$ is the section corresponding to $w$
as given in Proposition \ref{sec}. Since the period map is invariant
by the action of the Galois group of the double cover, by taking the
minimal resolution of surfaces in the family, we get a period map
$\mathcal P_{\mathcal F}:\mathcal F\rightarrow \mathcal M^*$ which is the
gluing of $\mathcal P_2$ and $\mathcal P^h$.
\qed\\
The existence of a global extension follows from the following
version of Hartogs' Theorem (see \cite{Ho}):
\begin{Lem}[Theorem 2.2.8, \cite{Ho}]\label{hart}
Let $f:U\rightarrow \mathbb C$ be a function defined in the open set
$U\subset\mathbb C^n$. Assume that $f$ is holomorphic in each variable
$z_j$ when the other variables $z_k$, $k\not=j$ are fixed. Then $f$
is holomorphic in $U$.
\end{Lem}
\begin{Thm}\label{end} The period
map $\mathcal P_2$ can be extended holomorphically to a period map:
$$\mathcal P:\widetilde{\mathcal V}\longrightarrow \mathcal M^*$$
such that its restriction to the exceptional divisor $\mathcal E$
coincides with the period map $\mathcal P^h$.
Moreover:\\
i) the locus $\widetilde{\mathcal V}\backslash \widetilde{\mathcal V}_{s}$ is a smooth rational curve mapped to the boundary of $\mathcal M^*$;\\
ii) $\mathcal P$ induces an isomorphism $\mathcal
P_{\mid\widetilde{\mathcal V}_s}:\widetilde{\mathcal
V}_s\longrightarrow\mathcal M.$
\end{Thm}
\proof Let $\mathcal P$ be the gluing of $\mathcal P_2$ and
$\mathcal P^h$ on $\widetilde{\mathcal V}$.
The map $\mathcal P$ is holomorphic on $\widetilde{\mathcal
V}\backslash\mathcal E$ (by Theorem \ref{ss}) and on $\mathcal E$
(by Theorem \ref{hiso}). Moreover, by Proposition \ref{pa1}, it is
holomorphic on the generic one-dimensional family intersecting
$\mathcal E$ transversally. Hence, by Lemma \ref{hart}, the period
map $\mathcal P$ is holomorphic. Since $\widetilde{\mathcal V}$ is a
compact variety (see \cite{M}) we also have that $\mathcal P$ is
surjective.
Let $B$ be the rational curve given by the image of strictly
semistable quartics in $\mathcal V-\{v_0\}$ and $b$ be the point in
$\mathcal E$ corresponding to effective divisors of the form
$4p_1+4p_2$ in $\mid \mathcal O_{\mathbb{P}^ 1}(8)\mid_{ss} $. We prove that
the point $b$ lies in the closure of the curve $B$. In fact, a point
in $B$ can be given by:
$$(q+tz^2)(q-tz^2)=q^2-t^2z^4$$
where $q=xy-z^2$ and $t\in\mathbb C$. From the proof of Proposition
\ref{pa1}, it follows that the corresponding quadratic section on
the cone is given by the union of two hyperplane sections:
$$q^2-z^4=(q+z^2)(q-z^2)=0.$$
The equation $z^4=0$ intersects the conic $q=0$ in two points with
multiplicity $4$, hence this quadratic section corresponds to the
point $b$.
Moreover, note that the injectivity of the period map on $\mathcal
V_s$ follows as in the generic case. Hence assertions $i)$ and ii)
follow from the surjectivity of the period map and theorems
\ref{ss2} and \ref{hiso}.
\qed\\
Two easy corollaries are the following:
\begin{Cor}
The boundary of the Baily-Borel compactification $\mathcal M^*$ is
given by a unique point (the cusp).
\end{Cor}
\proof This follows from Lemma \ref{bou} and Theorem \ref{end}.
\qed
\begin{Cor}
Every $(L^+,\rho)$-polarized K3 surface $X$ carries an order four
automorphism $\phi$ such that $\phi^*=\pm iI$ on $H^{2,0}(X)$.
\end{Cor}
\proof By Theorem \ref{end} an $(L_+,\rho)$-polarized $K3$ surface
is either the 4:1 cyclic cover of $\mathbb{P}^ 2$ branched along a stable
quartic or the 4:1 cyclic cover of the cone $\Sigma$ branched along
a quadratic section. In the first case, the $K3$ surface carries an
order four covering automorphism $\sigma_q$ such that the quotient
of $X$ by the involution $\tau_q=\sigma^2_q$ is a rational surface
(the blow up of a Del Pezzo surface of degree two). Hence,
$\tau^*_q$ acts as minus the identity on the transcendental lattice
$T(X)$, as in the generic case. Similarly, in the second case we
have an order four covering automorphism $\sigma_h$ and the quotient
of $X$ by the action of the involution $\tau_h=\sigma_h^2$ is a
rational surface (a blow up of $\Ps1\times\Ps1$).
\qed
\begin{Rem}
In \cite{Lo} E. Looijenga defines a special compactification of the
moduli space $\mathcal M$ which is isomorphic to the GIT
compactification $\mathcal V=V_{ss}/\!/PGL_3$ of the space of plane
quartics. This is obtained by considering a small blowing up and a
blowing down of the Baily-Borel compactification. These correspond
to the blowing up of the cusp and the blowing down of the mirror
$\mathcal D_h$.
Moreover, the same author proved in \cite{Lo2} that the algebra of
invariants of plane quartics can be identified with an algebra of
meromorphic automorphic forms on the complex 6-ball.
\end{Rem}
|
\section{Introduction}\label{sec:Intro}
Spins in GaAs and Si lateral quantum dots are promising candidates for implementing a quantum computer due to their scalability ~\cite{Loss1998,Hanson2007,Zwanenburg2013}, long coherence times ~\cite{Bluhm2010,Maune2012,Veldhorst2014,Malinowski2016}, and rapid gate operations ~\cite{Petta2005,Foletti2009,Medford2013a}. A qubit can be encoded in different ways using the spin states of one or more electrons trapped in one or more quantum dots, among these the single-spin ~\cite{Takeda2016a,Kawakami2016,Nowack2011,Veldhorst2014,Veldhorst2015a,Zajac2017}, singlet-triplet ~\cite{Petta2005,Maune2012,Foletti2009,Cerfontaine2014,Wu2014,Shulman2012,Nichol2016}, resonant exchange ~\cite{Laird2010,Medford2013a,Medford2013,Eng2015}, and hybrid spin ~\cite{Shi2012,Koh2012,Kim2014,Kim2015b,Cao2016} qubits have been successfully implemented in the laboratory. In many of these systems qubits are coupled to each other by means of the tunneling-based effective exchange interaction, which has the advantage of producing fast gates controlled electrically with gate voltages.~\cite{Petta2005,Maune2012,Brunner2011}. Furthermore, it has been recently demonstrated that symmetric exchange pulses substantially reduce the sensitivity of qubit gates to charge noise ~\cite{Hu2006,Reed2016,Martins2016,Barnes2016,Zhang2017,Yang2017,Yang2017a,Shim2017}.
Notwithstanding these advantages, the short-ranged nature of the exchange coupling ~\cite{Burkard1999,Li2009} is a potential hindrance towards scalability. However, this limitation can be circumvented by using an intermediate quantum system as a mediator ~\cite{Mehl2014c,Baart2016a,Mi2017}, for example a multielectron quantum dot ~\cite{Srinivasa2015,Croot2017,Malinowski_Thesis}. In this line, Refs.~\onlinecite{Martins2017,Malinowski2017a} study the spin properties of a multielectron GaAs quantum dot (with an estimated number of electrons between 50 and 100) exchange coupled to a single-electron quantum dot, which in turn is coupled to another single-electron quantum dot. This linear three-dot system is studied under magnetic fields both parallel and perpendicular to the two-dimensional electron gas (2DEG). In particular, the aforementioned works show that, at the transition between odd and even occupation number, the multielectron ground state is singlet-like for small hybridization and becomes triplet-like once the central electron has totally moved to the multielectron dot. As a result, the usually positive exchange energy becomes negative, even at zero magnetic field. This finding is not only important for understanding the properties of multielectron quantum dots, but also for performing dynamical decoupling on exchange-coupled spins. If the exchange coupling is restricted to be nonnegative, then special techniques are needed to dynamically correct for noise errors during gate operations ~\cite{Wang2012,Kestner2013,Wang2014}, which generally leads to longer gate times. Instead, if the exchange coupling can be tuned to both positive and negative values, then standard decoupling techniques can be used ~\cite{Goelman1989}, and this issue is avoided.
It has been demonstrated that negative exchange energy in a quantum dot with just two-electrons can be induced by a non-zero out-of plane magnetic field ~\cite{Wagner1992,Baruffa2010a,Zumbuhl2004,Mehl2014a}. Here, the out-of-plane magnetic field leads to a compression of the orbital wave functions and a larger electron-electron repulsion, which makes triplets energetically favorable. However, an in-plane or zero magnetic field does not create a wave-function compression, and thus it does not induce a negative exchange energy in a doubly occupied quantum dot, i.e. the ground state is always the singlet. In fact, there is a two-electron ground state theorem ~\cite{Lieb1962,2electron_theorem}, which states that in the absence of spin or velocity-dependent forces (the force exerted by the in-plane magnetic field is negligible since it is along the strong confinement perpendicular to the 2DEG) the state of lowest energy must be non-degenerate. An extension of the two-particle theorem to an arbitrary number of particles is given in Ref.~\onlinecite{Lieb1962}. This theorem correctly predicts the ground state of many electrons in a linear array, as shown in Ref.~\onlinecite{Riiser1993a}. Nonetheless, the multielectron ground state theorem does not apply to electrons interacting with central forces ~\cite{Lieb1962} and, for the multielectron quantum dot, the lower full orbitals do exert an effective central force onto higher orbitals. Therefore, there is no fundamental theorem or principle that prevents a triplet-like eigenstate from being the ground state of a multielectron quantum dot, regardless of the magnitude and direction of the magnetic field, as demonstrated in recent ~\cite{Martins2017,Malinowski2017a} and earlier ~\cite{Folk2001,Lindemann2002} experiments with multielectron quantum dots.
\indent In this work we demonstrate that a quantum dot does not require tens of electrons to exhibit negative exchange energy, instead, as few as 4 electrons are enough to have a triplet-like ground state in a quantum dot with zero magnetic field. We do this by performing a detailed numerical analysis employing the configuration interaction method with up to 14 electrons in both GaAs and Si quantum dots. Moreover, we use the full configuration interaction to determine the ground state of an elliptically shaped four-electron quantum dot with different eccentricities and, in doing so, we identify a threshold, in both GaAs and Si quantum dots, at which the exchange energy flips sign.
\indent The paper is divided in four sections. In Sec.~\ref{sec:Hubbard_model_Hund's_rule}, we use a simple Hubbard model to study and give a general picture of the system presented in Ref.~\onlinecite{Martins2017}. Then, in Sec.~\ref{sec:configuration_interaction}, we use a configuration interaction method to determine the ground state of a multielectron quantum dot with parabolic potential, where we consider different number of electrons and dot sizes. Moreover, a full configuration interaction calculation shows that for four electrons the ground state is triplet-like for all the dot sizes we considered. Finally, in Sec.~\ref{sec:elliptical_potential}, we use the full configuration interaction method to calculate the exact eigenenergies of four electrons confined in an elliptically shaped quantum dot, showing the effect of the dot asymmetry in the occurrence of triplet-like ground states.
\section{Hubbard model and Hund's rule}\label{sec:Hubbard_model_Hund's_rule}
We start our analysis with a simple Hubbard model that describes the system studied in Ref. \onlinecite{Martins2017}, i.e. a multielectron quantum dot (rightmost) with $2N+1$ electrons ($N=50$) tunnel-coupled to a double quantum dot containing two electrons, see Fig.~\ref{Fig.0}. We keep as many orbitals (single-particle energy levels) as necessary in the right dot and only one orbital in each of the other two quantum dots (see Fig.~\hyperref[fig:1]{\ref*{fig:1}(a)}). The system's Hamiltonian is
\begin{equation}\label{eq:Hubbard_model}
H=H_0+H_z+H_A+H_U,
\end{equation}
where
\begin{align}
H_0=&\sum_\sigma[\sum_l\epsilon_{R,l}n_{R,l\sigma}+\epsilon_Mn_{M,\sigma}+ \epsilon_Ln_{L,\sigma}\nonumber\\
&-\sum_l t_{MR,l}(c_{R,l\sigma}^\dagger c_{M,\sigma}+c_{M,\sigma}^\dagger c_{R,l\sigma})\nonumber\\
&-t_{LM}(c_{M,\sigma}^\dagger c_{L,\sigma}+c_{L,\sigma}^\dagger c_{M,\sigma})],\\
H_Z=&\frac{E_B}{2}[\sum_l(n_{R,l\uparrow}-n_{R,l\downarrow})+n_{M,\uparrow}-n_{M,\downarrow}+n_{L,\uparrow}-n_{L,\downarrow}],\\
H_A=&A_R\sum_l(c_{R,l\uparrow}^\dagger c_{R,l\downarrow}+c_{R,l\downarrow}^\dagger c_{R,l\uparrow})+A_{L,M}(c_{M,\uparrow}^\dagger c_{M,\downarrow}\nonumber\\
&+c_{M,\downarrow}^\dagger c_{M,\uparrow}+c_{L,\uparrow}^\dagger c_{L,\downarrow}+c_{L,\downarrow}^\dagger c_{L,\uparrow}),\\
H_U=&\sum_lU_{R,l}n_{R,l\uparrow}n_{R,l\downarrow}+\sum_{l_1\ne l_2, \sigma,\sigma'}U_{R,l_1l_2}n_{R,l_1\sigma}n_{R,l_2\sigma'}\nonumber\\
&+U_Mn_{M,\uparrow}n_{M,\downarrow}+U_Ln_{L,\uparrow}n_{L,\downarrow}\nonumber\\
&+\sum_{l_1\ne l_2, \sigma,\sigma'}U_{R,l_1l_2 l_2 l_1}c_{R,l_1\sigma}^\dagger c_{R,l_2\sigma'}^\dagger c_{R,l_1\sigma'}c_{R,l_2\sigma}.\label{eq:Coulomb_onsite_interaction_exchange}
\end{align}
Here, $n_{\alpha,l \sigma}=c^{\dagger}_{\alpha,l \sigma}c_{\alpha,l \sigma}$ is the number operator for the single-particle states in the left ($L$), middle ($M$), and right ($R$) quantum dot ($\alpha=L,M,R$; $\sigma=\uparrow,\downarrow$; and $l$ is the right dot's $l$-th single-particle state), $\epsilon_{\alpha,l}$ denotes the single-particle energies, $t_{\alpha\beta} $ is the tunneling amplitude between dots ($\alpha,\beta=L,M,R$), $E_B$ is the Zeeman energy, $A_\alpha$ is the hyperfine interaction (proportional to the dot size) between electrons and the nuclear spin bath, $U_\alpha$ is the ``on-site'' Coulomb interaction, and finally, $U_{R,l_1 l_2} $ and $U_{R,l_1l_2 l_2 l_1}$ are the Coulomb interaction and exchange term between orbitals $l_1$ and $l_2$ in the right dot, respectively. The large number of electrons in the right dot makes the numerical calculation of the eigenenergies too difficult to carry out without any sort of approximation. Accordingly, we make use of the so-called ``frozen-core'' approximation (FCA), where we keep the right dot's $2N$ core electrons in the lowest non-interacting states and only allow a valence electron to occupy higher energy levels. It is worth mentioning that, due to the large number of core electrons, we are only considering the direct Coulomb interaction between the core and the valence electrons, which causes a general energy shift. In the following sections we will consider cores with fewer electrons, and thus the FCA will also take into account the Coulomb exchange interaction between core and valence electrons.\\
\indent In atomic physics, it is well known that in certain configurations a combination of electron-electron repulsion and electron-nucleus attraction makes high-spin states energetically more favorable than any other lower-spin state arising from the same configuration ~\cite{Boyd1984}; this is commonly known as Hund's multiplicity rule. Similarly, for the multielectron quantum dot the exchange term, $U_{R,l_1l_2 l_2 l_1}$, in Eq.~\eqref{eq:Coulomb_onsite_interaction_exchange} induces magnetic correlations among the electron spins and, as a result, it lowers the energy of the eigenstates with spin 1. This is more evident if we set $J_{R,l_1l_2 }^F \equiv U_{R,l_1l_2 l_2 l_1}$ and, using Pauli matrix identities, we rewrite the exchange energy in Eq.~\eqref{eq:Coulomb_onsite_interaction_exchange} as~~\cite{CMFT_Alexander}
\begin{align}
&\sum_{l_1\ne l_2, \sigma ,\sigma'}U_{R,l_1l_2 l_2 l_1}c_{R,l_1\sigma}^\dagger c_{R,l_2\sigma'}^\dagger c_{R,l_1\sigma'}c_{R,l_2\sigma}\nonumber\\
=&-2\sum_{l_1\ne l_2}J_{R,l_1l_2}^F (\mathbf{S}_{R,l_1}\cdot\mathbf{S}_{R,l_2}+\frac{1}{4}n_{Rl_1} n_{Rl_2}),
\end{align}
where $\mathbf{S}_{R,l_i}$ is the spin operator acting on the $l_i$-th single-particle state.\\
\begin{figure}[!tbp]
\centering
\includegraphics[trim=0cm 10cm 0cm 9cm, clip=true,width=8.5cm, angle=0]{figure1.pdf}
\caption{Illustration of the three dot system, where the $L$ and $M$ dots form a two-electron double quantum dot and $R$ is the multielectron quantum dot. In the main text, we analyze the effective exchange interaction, $J$, between the middle ($M$) and right ($R$) quantum dots.}\label{Fig.0}
\end{figure}
\indent Since we assume that the $2N$ core electrons in the right dot are ``frozen'' (FCA), we are effectively dealing with a three-electron system. The spin Hamiltonian for three electrons coupled by nearest-neighbor exchange interactions and subject to a magnetic field is $H'=J_{LM}\left(\mathbf{S}_L\cdot\mathbf{S}_M-1/4\right)+J_{MR}\left(\mathbf{S}_M\cdot\mathbf{S}_R-1/4\right)-E_B(S_{z,L}+S_{z,M}+S_{z,R})$, where $J_{\alpha\beta}$ acts as an effective exchange interaction and $E_B$ is the Zeeman energy. The eight spin eigenstates of this Hamiltonian form a quadruplet $Q$,
\begin{align}
\ket{Q_{+3/2}}=&\ket{\uparrow\uparrow\uparrow},\\
\ket{Q_{+1/2}}=&\frac{1}{\sqrt{3}}(\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}),\\
\ket{Q_{-1/2}}=&\frac{1}{\sqrt{3}}(\ket{\uparrow\downarrow\downarrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\downarrow\uparrow}),\\
\ket{Q_{-3/2}}=&\ket{\downarrow\downarrow\downarrow},
\end{align}
and high- and low-energy doublets, which, in the absence of tunneling between left and middle dots, have the following simple form
\begin{align}
\ket{D_{+1/2}}=&\frac{1}{\sqrt{6}}(-2\ket{\downarrow\uparrow\uparrow}+\ket{\uparrow\downarrow\uparrow}+\ket{\uparrow\uparrow\downarrow}),\\
\ket{D_{-1/2}}=&\frac{1}{\sqrt{6}}(-2\ket{\uparrow\downarrow\downarrow}+\ket{\downarrow\uparrow\downarrow}+\ket{\downarrow\downarrow\uparrow}),\\
\ket{D'_{+1/2}}=&\frac{1}{\sqrt{2}}(\ket{\uparrow\uparrow\downarrow}-\ket{\uparrow\downarrow\uparrow}),\\
\ket{D'_{-1/2}}=&\frac{1}{\sqrt{2}}(\ket{\downarrow\downarrow\uparrow}-\ket{\downarrow\uparrow\downarrow}).
\end{align}
Here, the spin eigenstates $\ket{D_{+1/2}}$ ($\ket{D_{-1/2}}$) and $\ket{Q_{+1/2}}$ ($\ket{Q_{-1/2}}$) are almost degenerate, with an energy $E_{D_{\pm 1/2}}=\mp E_B/2$, whereas the low-energy doublets have an energy $E_{D'_{\pm 1/2}}=-J_{MR}\mp E_B/2$. Therefore, the effective exchange energy between the middle and right dots is given by
\begin{equation}
J_{MR}=E_{D_{+ 1/2}}-E_{D'_{+ 1/2}},
\end{equation}
where $\ket{D_{+1/2}}$ and $\ket{D'_{+1/2}}$ are the lowest spin triplet-like and singlet-like eigenstates, respectively.\\
\begin{figure}[!tbp]
\centering
\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true,width=8.7cm, angle=0]{figure2.pdf}
\caption{(a) Schematic of the energy levels and charge configuration for the three-dot system. Here, $\Delta E_{i}$ is the difference between two orbitals on the multielectron quantum dot. The schematic only shows the tunnel coupling between the middle and right dot, where $t_{MR,N+1}$ and $t_{MR,N+2}$ are the tunneling amplitudes between the middle dot's single-orbital and the lowest two orbitals above the frozen core in the multielectron quantum dot. (b) Eigenenergy spectrum, calculated as a function of the detuning $\epsilon$, for the three-dot system at the transition between $(1,1,2N+1)$ and $(1,0,2N+2)$ charge configurations. The eigenstates $\ket{D_{+1/2}}$ ($\ket{D_{-1/2}}$) and $\ket{Q_{+1/2}}$ ($\ket{Q_{-1/2}}$) are almost degenerate and the exchange energy $J$ is the difference between the triplet-like state $\ket{D_{\pm1/2}}$ and the singlet-like state $\ket{D'_{\pm1/2}}$.}\label{fig:1}
\end{figure}
\indent We use the Hubbard model, Eq.~\eqref{eq:Hubbard_model}, to calculate the energies of the aforementioned spin eigenstates. To that end, we choose a set of parameters such that the resulting energy spectrum resembles the one reported in Ref.~\onlinecite{Martins2017}. Accordingly, we set the magnitude of the energy level splittings as $\Delta E_{N+1}=0.4\mathrm{meV}$, $\Delta E_{N+2}=0.05\mathrm{meV}$, $\Delta E_{N+3}=0.4\mathrm{meV}$, where $\Delta E_{l}=\epsilon_{R,l}-\epsilon_{R,l-1}$ and $\epsilon_{R,l}$ is the right dot's $l$-th single-particle energy level (see Fig.~\hyperref[fig:1]{\ref*{fig:1}(a)}). The tunneling amplitudes are $t_{MR,N+1}=0.15\mathrm{meV}$, $t_{MR,N+2}= 0.1\mathrm{meV}$, and $t_{LM}=0.02\mathrm{meV}$. The hyperfine interactions for the multielectron quantum dot and double quantum dot are $A_R=0.4\mathrm{neV}$ and $A_{LM}=4\mathrm{neV}$, respectively. Finally, the ferromagnetic exchange term
|
in the right dot is set equal to $J_{R,l_1l_2 }^F=0.1\mathrm{meV}$ and the in-plane magnetic field is $B=500\mathrm{mT}$. With these parameters we plot in Fig.~\hyperref[fig:1]{\ref*{fig:1}(b)} the eigenenergies as a function of the detuning between left and right dot, $\epsilon\equiv(\epsilon_M-\epsilon_{R,N+1})/2$. Notice that in the $(1,1,2N+1)$ configuration, where $(n_L,n_M,n_R)$ represents the number of electrons in the left, middle, and right dots, the singlet-like eigenstate $\ket{D'_{+1/2}}$ has lower energy than $\ket{D_{+1/2}}$ ($J_{MR}>0$), but as soon as the middle dot's electron tunnels into the right dot the exchange energy becomes negative, i.e. $E_{D_{+1/2}}<E_{D'_{+1/2}}$. The latter is caused by the right dot's ferromagnetic exchange term $J_{R,l_1l_2 }^F$, which lowers the energies of all two-electron states in the right dot with total spin equal to 1. This is analogous to Hund's rule in atomic physics.\\
\indent Thus far, we have shown that, by choosing the appropriate parameters, the simple Hubbard model qualitatively reproduces the experimental results presented in Ref.~\onlinecite{Martins2017}. Moreover, the model shows that the negative exchange energy is caused by the multielectron quantum dot exhibiting a spin triplet-like rather than singlet-like ground state. In this line, Ref.~\onlinecite{Malinowski2017a} presents comparable results using a Hubbard model similar to ours, with the difference that ours takes into account the hyperfine interaction (between electrons and the nuclear spin bath) and the Coulomb interaction between orbitals in the right dot. Nonetheless, the Hubbard model does not provide enough insight into the characteristics (minimum number of electrons, dot shape, etc.) a system must have in order to display a negative exchange energy under zero magnetic field. This is addressed below using a microscopic description of the multielectron quantum dot.
\section{Configuration Interaction for a multielectron quantum dot}\label{sec:configuration_interaction}
\subsection{ Frozen-core approximation}
Here we determine the ground state of a multielectron quantum dot using a configuration interaction (CI) approach. The large number of electrons forces us to use, once again, the ``frozen-core'' approximation, which now also takes into account the Coulomb exchange interaction between core and valence electrons.\\
\indent The system can be described by the valence effective Hamiltonian ~\cite{Durand1975,Malrieu2014}:
\begin{align}\label{eq:valence_effective_Hamiltonian}
H_v=&[E_v+\sum_i^{core}h_{ii}+\sum_{i<j}^{core}(ii|jj)-(ij|ji)]\nonumber\\
&+\sum_{r,s}^{val}\tilde{h}_{rs}c_r^\dagger c_s+\frac{1}{2}\sum_{p,q,r,s}^{val}{(pq|rs)c_p^\dagger c_r^\dagger c_s c_q},
\end{align}
with
\begin{equation}\label{eq:molecular_orbital_interaction}
(kl|mn)=\int{\phi_k^*(r_1)\phi_l(r_1)\frac{1}{r_{12}}\phi_m^*(r_2)\phi_n(r_2)}dr_1 dr_2,
\end{equation}
where the single-particle orbitals are denoted by $\phi_\alpha$, and the summations marked ``core'' and ``val'' are over orbitals occupied by core or valence electrons, respectively. The terms inside the bracket in Eq.~\eqref{eq:valence_effective_Hamiltonian}, which comprises the total single-particle energy of the valence electrons ($E_v$) and the core's energy, add up to a constant, and thus they only shift the energy scale. The energy term $\tilde{h}_{rs}$ is defined as
\begin{equation}\label{eq:hopping_exchange}
\tilde{h}_{rs}=t_{rs}+\sum_i^{core}{[(rs|ii)-(ri|is)]},
\end{equation}
where $t_{rs}$ is the electron hopping between valence orbitals, and $(rs|ii)$ and $(ri|is)$ are the Coulomb interaction and exchange coupling between the valence and core electrons, respectively.\\
\indent In our numerical analysis we consider $2N+2$ electrons ($2N$ core electrons and 2 valence electrons, $0\leq N\leq6$) living in 12 orbitals. We model the lateral gate confinement of the multielectron quantum dot with a symmetric parabolic potential. Thus, the appropriate single-particle orbitals are the eigenstates of the Fock-Darwin Hamiltonian
\begin{equation}\label{eq:Fock-Darwin_Hamiltonian}
H=\frac{1}{2m^*}(-i\hbar\nabla+\frac{e}{c}\mathbf{A})^2+\frac{1}{2}m^*\omega^2r^2,
\end{equation}
where $m^*$ is the effective electron mass, $\hbar\omega$ is the quantum dot confinement energy, and $r=\sqrt{x^2+y^2}$ is the radius. Following a numerical method developed in a previous work ~\cite{Barnes2011}, and setting the external magnetic field to zero, we determine the eigenenergies and ground eigenstate of the valence effective Hamiltonian. We perform this calculation for both GaAs and Si quantum dots (see Table~\ref{Tab:1}); using for GaAs (Si) the effective electron mass $m^*=0.067m_e$ ($m^*=0.19m_e$), where $m_e$ is the electron mass, and the dielectric constant of the host material $\kappa=13.1\epsilon_0$ ($\kappa=11.68\epsilon_0$). In contrast to GaAs, Si quantum dots present a two-fold degenerate ground state. This valley degeneracy can be lifted and finely tuned by an out-of-plane electric field ~\cite{Yang2013,Veldhorst2014}. Here, we assume that such techniques have been employed to achieve a sufficiently large valley splitting, and thus we do not include a valley coupling parameter in our calculations. A comprehensive analysis including valley effects would require a detailed microscopic understanding of the intervalley coupling, which is likely device specific and is beyond the scope of this paper.
Before discussing our results, it is important to note that the multielectron quantum dot has shells at $\eta=2, 6, 12, 20, \ldots, (n^2+3n+2)$, where $\eta$ is the total number of electrons in the dot and $n=0, 1, 2, 3, \ldots$ is the principal quantum number. This is a consequence of the ($n+1$)-fold degeneracy of the quantum dot's eigenenergies, which stems from the dot's effective confinement having a symmetry very close to circular ~\cite{Reimann2002}. We are primarily interested in situations where the multielectron quantum dot can be used as a spin qubit, i.e., it initially contains an odd number of electrons where all but one electron completely fill a number of shells and form a spin singlet-like state, leaving a net spin $1/2$ from the remaining unpaired electron ~\cite{Barnes2011,Nielsen2013a,Higginbotham2014}. To examine the sign of the exchange interaction with a neighboring single-electron quantum dot we consider that the neighboring electron tunnels into the large quantum dot, giving now two valence electrons in that dot. We therefore consider first the simplest case involving just a pair of electrons, and then for larger number of electrons we focus on electron numbers $\eta= 4, 8, 14, \ldots, (n_{max}^2 +3n_{max} +4)$, which correspond to two valence electrons and where $n_{max}$ is the principal quantum number of the highest full shell in the core.
Our results, presented in Table~\ref{Tab:1}, show that when the multielectron quantum dot contains only two electrons, only a singlet ground state can be realized; this is in accordance with the two-electron ground state theorem ~\cite{Lieb1962,2electron_theorem}. Incidentally, in the case of valley degeneracy or near-degeneracy, a pair of electrons in a Si quantum dot would not necessarily follow the aforementioned theorem since electrons in different valleys could be treated as different species ~\cite{Hada2003}, which would violate the theorem's assumption that the potential is symmetric under permutations ~\cite{Lieb1962}. In Table~\ref{Tab:1} we also see that, for more than two electrons, triplet states are possible depending on the size of the dot. This indicates that having a core of electrons completely occupying lower energy orbitals is important for creating a triplet ground state. The fact that, at least for $\eta>4$, whether the ground state is a singlet or a triplet depends on the size of the dot suggests that the orbital spacing plays an important role. We know that the energy difference between shells is inversely proportional to the square radius of the quantum dot, so that small dots present well defined energy gaps between shells. Consequently, for a pair of valence electrons above a full shell and for a sufficiently large energy gap between shells (small dot), Coulomb interactions between valence and core electrons are likely reduced since excitations from core to valence orbitals are suppressed. This picture is consistent with what we observe in Table~\ref{Tab:1}, where for sufficiently small dots ($\omega/\omega_0\geq4$ for GaAs and $\omega/\omega_0\geq16$ for Si, where $\hbar\omega_0=1.0 \mathrm{meV}$) the ground state is always triplet-like, while for bigger dots ($\omega/\omega_0\leq2$ for GaAs and $\omega/\omega_0\leq8$ for Si) the comparatively larger Coulomb interactions between valence and core electrons increases the likelihood of singlet-like ground states.
\begin{table}[tbp]
\centering
\caption{Ground states (S=Singlet and T=Triplet) for different dot sizes and number of electrons. Here, $\hbar\omega_0=1.0 \mathrm{meV}$ for both GaAs and Si quantum dots.}\label{Tab:1}
\subfloat[Ground states table for GaAs.]{\begin{tabular}{@{}|P{2.5cm}||@{} P{0.5cm}| @{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}}
\hhline{-||-------}
\diagbox[width=\dimexpr\eqboxwidth{wd} + 8.8\tabcolsep\relax, height=1cm]{{\scriptsize \# of electrons}}{\raisebox{-1ex}{$\omega/\omega_0$}}
&~ 0.25 & 0.5 & 1 & 2 & 4 &8 &16\\
\hhline{=::=======}
\eqmakebox[wd]{2} & ~~ S &S & S & S & S & S & S \\
\hhline{-||-------}
\eqmakebox[wd]{4} & ~~ T &T & T & T & T & T & T \\
\hhline{-||-------}
\eqmakebox[wd]{8} & ~~ S &T & T & T & T & T & T\\
\hhline{-||-------}
\eqmakebox[wd]{14} & ~~ S &S & S & S & T & T & T \\
\hhline{-||-------}
\end{tabular}}
\bigskip
\\
\subfloat[Ground states table for Si.]{\begin{tabular}{@{}|P{2.5cm}||@{} P{0.5cm}| @{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}P{0.7cm}@{}|@{}}
\hhline{-||-------}
\diagbox[width=\dimexpr\eqboxwidth{wd} + 8.8\tabcolsep\relax, height=1cm]{{\scriptsize \# of electrons}}{\raisebox{-1ex}{$\omega/\omega_0$}}
&~ 0.25 & 0.5 & 1 & 2 & 4 &8 &16\\
\hhline{=::=======}
\eqmakebox[wd]{2} & ~~ S &S & S & S & S & S & S \\
\hhline{-||-------}
\eqmakebox[wd]{4} & ~~ T &T & T & T & T & T & T \\
\hhline{-||-------}
\eqmakebox[wd]{8} & ~~ S &S & S & T & T & T & T\\
\hhline{-||-------}
\eqmakebox[wd]{14} & ~~ S &S & S & S & S & S & T \\
\hhline{-||-------}
\end{tabular}}
\end{table}
\subsection{Full Configuration interaction for a four-electron dot}
The ``frozen-core'' approximation (FCA) was instrumental in the calculation of the results presented in Table \ref{Tab:1} and, therefore, it is important to probe the accuracy of this approximation. To that end, we use the full configuration interaction (full CI) method to calculate the exact eigenenergies of a four-electron dot with variable size and zero magnetic field. In the numerical calculation we consider 4 electrons living in the 10 lowest orbitals of a parabolic potential. Our results show that the ground state of this system is triplet-like regardless of the dot size, in accordance with the results obtained through the FCA. We also notice that the higher-orbital-content of the ground state increases proportionally to the dot size. This is due to the reduction in the energy gap between orbitals when the size of the dot increases, which allows the mixing with higher orbitals. In this regard, the FCA only provides an estimation of the ground state's orbital content, and thus the FCA's accuracy is expected to diminish for large-size dots. Nonetheless, the FCA remains a good approximation within the dot size range considered in this work.
\section{Full Configuration interaction with Elliptical potential}\label{sec:elliptical_potential}
\begin{figure}[!tbp]
\centering
\includegraphics[trim=0cm 5cm 0cm 5cm, clip=true,width=8.5cm, angle=0]{figure3.pdf}
\caption{Exchange energy vs. frequency difference of the elliptical potential.}\label{Fig.3}
\end{figure}
Apart from the dot size and number of electrons confined in a quantum dot, here we show that the shape of the dot also determines the occurrence of a triplet-like ground state. To that end we use, once again, the full CI method to calculate the exact eigenenergies of a four-electron quantum dot with elliptical potential and, in doing so, we show the effect of asymmetry on the exchange energy's magnitude and sign. In our calculation we consider 4 electrons residing in 10 orbitals, with a single-particle Hamiltonian given by
\begin{equation}
H=\frac{1}{2m^*}(-i\hbar\nabla+\frac{e}{c}\mathbf{A})^2+\frac{1}{2}m^*(\omega_1^2x^2+\omega_2^2y^2),
\end{equation}
where $\omega_1$ and $\omega_2$ are the frequencies of the harmonic oscillators for the $x$ and $y$ directions, respectively. In the absence of an external magnetic field, $\mathbf{B}=0$, the single-particle energies are the eigenvalues of the anisotropic two-dimensional harmonic oscillator:
\begin{equation}
\epsilon_{n_x,n_y}=\frac{1}{2}(\hbar\omega_1+\hbar\omega_2)+n_x \hbar\omega_1+n_y\hbar\omega_2,
\end{equation}
where we define $\tilde{n}=n_x+n_y$. Here, it is evident that the splitting between single-particle energies can be tuned by the frequency difference
\begin{equation}
\delta\omega=\omega_2-\omega_1,
\end{equation}
which effectively changes the eccentricity of the dot's elliptical potential. Accordingly, we calculate the eigenenergies of the four electrons (in both GaAs and Si quantum dots with zero magnetic field) using different magnitudes for $\delta\omega$, where, for convenience, we set $\frac{\hbar(\omega_2+\omega_1)}{2}=8\hbar\omega_0=8 meV$ and $\omega_2\geq\omega_1$. Our results are summarized in Fig.~\ref{Fig.3}, which shows the exchange energy $J$ (given by the difference in energy between the lowest triplet-like and singlet-like eigenstates) as a function of the frequency difference $\delta\omega$. Notice that when $\delta\omega=0$, i.e. the potential is parabolic, $\tilde{n}$ becomes the principal quantum number and the principal energy levels (electron shells) corresponding to $\tilde{n}>0$ are degenerate. Here, the electron distribution is such that the lowest shell is full and the next degenerate shell contains two electrons. In this configuration, as shown in the previous section, the ground state is triplet-like. However, for non-zero $\delta\omega$ the degeneracy is lifted and, in our particular case, when $\delta\omega$ is greater than a certain threshold $\delta\tilde{\omega}$ ($\delta\tilde{\omega}\approx1.65\omega_0$ for GaAs and $\delta\tilde{\omega}\approx2.8\omega_0$ for Si) the split between the formerly degenerate orbitals with $\tilde{n}=1$ is large enough to favor a singlet-like ground state and, therefore, a positive exchange energy. A similar effect is observed with larger numbers of electrons and/or larger dot sizes (see Table \ref{Tab:1}).
\section{Conclusions}
In this work we have studied the conditions under which negative exchange interactions can occur in coupled few-electron quantum dots. The negative exchange interaction between a multielectron quantum dot (with odd occupation number) and a single-electron quantum dot (which in turn is coupled to a second single-electron quantum dot) has its roots in the larger quantum dot exhibiting a spin triplet-like ground state, which occurs once the smaller dot's electron has tunneled into the larger dot. This was demonstrated using a Hubbard model for a linear three-dot system ~\cite{Note} that reproduces the experimental results presented in Ref.~\onlinecite{Martins2017}, where negative exchange was observed. The larger quantum dot with an even number of electrons and zero magnetic field was further studied using a microscopic model based on the configuration interaction (CI) method with which we determined the ground state of the multielectron quantum dot. In this CI calculation we considered different combinations of total number of electrons and dot sizes (parabolic potential), showing that the occurrence of both triplet-like and singlet-like ground states depend on those parameters and that 4 electrons is the minimum needed to have a triplet-like ground state in both Si and GaAs quantum dots. Moreover, the effect of dot asymmetry on the exchange energy is also addressed via a full CI calculation of the energy spectrum for a four-electron quantum dot with elliptical potential. The full CI calculation is repeated for different eccentricities, revealing a threshold, in both GaAs and Si dots, at which the exchange energy flips signs. Future work will explore the equally interesting three-dot system where a multielectron quantum dot acts as a quantum mediator between two single-electron quantum dots. For now, the results presented in this work show that negative exchange interactions are robust in few-electron double quantum dots, and that all the potential advantages a tunable exchange interaction can provide are accessible with as few as 4 electrons in a double quantum dot. This is fundamental for scalability purposes since it avoids the need of large quantum dots, it prevents unwanted capacitive coupling between remote dots, and it enables simpler and faster dynamically corrected gate operations.
\section{Acknowledgments}
We thank Ferdinand Kuemmeth for helpful discussions. This work is supported by the Army Research Office (W911NF-17-0287).
\bibliographystyle{apsrev4-1}
|
\section{Introduction}
Hilbert geometries, introduced by David Hilbert to illustrate the fourth of his twenty-three problems, are among the most simple and studied examples of Finsler geometries. They can be considered as a generalization of hyperbolic geometry in the context of metric geometry, and a general and now well studied question is to understand if they inherit the same geometric or analytic properties as the hyperbolic space; see for instance \cite{HandbookHilbert} for a good overview.\\
In \cite{moi:natural_finsler_laplace}, the first author introduced and began to study a new generalization of the Laplace operator to Finsler geometry. It thus gives another analytical tool to understand the differences between Hilbert geometries and the hyperbolic space. For the $n$-dimensional hyperbolic space, the spectrum of the Laplace operator is known to be the interval $[(n-1)^2/4,\infty)$. In particular, it consists only of its essential part, and there is no eigenvalue below $(n-1)^2/4$ (see for example \cite{Donnelly_essential_spectrum}). In this article, we will see that the bottom of the essential spectrum of a regular $n$-dimensional Hilbert geometry is also $(n-1)^2/4$, but that, in contrast with hyperbolic geometry, a lot of arbitrarily small eigenvalues could appear under the essential spectrum.
\subsection{Finsler and Hilbert metrics}
\begin{defin} \label{def:finsler_metric}
Let $M$ be a manifold. A \emph{Finsler metric} on $M$\ is a continuous function ${F \colon TM \rightarrow \R^+}$ that is:
\begin{enumerate}
\item $C^{2}$, except on the zero section;
\item positively homogeneous, that is, $F(x,\lambda v)=\lambda F(x,v)$\ for any $\lambda\geqslant 0$;
\item positive-definite, that is, $F(x,v)\geq0$\ with equality iff $v=0$;
\item strongly convex, that is, $ \left(\dfrac{\partial^2 F^2}{\partial v_i \partial v_j}\right)_{i,j}$ is positive-definite.
\end{enumerate}
\end{defin}
A \emph{Hilbert geometry} is a metric space $(\mathcal{C},\d)$ where
\begin{itemize}
\item $\mathcal{C}$ is a properly convex open subset of the projective space $\mathbb{RP}^n$; \emph{properly convex} means that $\mathcal{C}$ contains no affine line; in other words, it appears as a relatively compact open set in some affine chart.
\item $\d$ is a metric on $\mathcal{C}$ is defined in the following way (see Figure \ref{fig_hilbert_distance}): for $x,y \in \mathcal{C}$, let $a$ and $b$ be the intersection points of the line $(xy)$ with $\partial \mathcal{C}$; then
\begin{equation*}
d_{\mathcal{C}}(x,y) = \frac{1}{2} |\ln [a:b:x:y]|,
\end{equation*}
where $[a:b:x:y]$ is the cross-ratio of the four points; if we identify the line $(xy)$ with $\R\cup\{\infty\}$, it is defined by $[a:b:x:y]=\frac{|ax|/|bx|}{|ay|/|by|}$ .
\end{itemize}
When $\mathcal{C}$ is an ellipsoid, the Hilbert geometry of $\mathcal{C}$ gives the Klein--Beltrami model of hyperbolic space.\\
The Hilbert metric $\d$ is generated by a field of norms $F_{\mathcal{C}}$ on $\mathcal{C}$, i.e., $\d(x,y)=\inf \int_{0}^{1} F(c(t), c'(t))\ dt$, where the infimum is taken over all $C^1$ curves $c\colon [0,1]\longrightarrow \mathcal{C}$ from $x$ to $y$. In an affine chart containing $\mathcal{C}$ as a relatively compact subset, the norm $F(x,u)$ of a tangent vector $u\in T_x\mathcal{C}$ is given by the formula
\begin{equation*}
F_{\mathcal{C}}(x,u) = \frac{|u|}{2} \left(\frac{1}{|u^-x|} +\frac{1}{|xu^+|}\right),
\end{equation*}
where $|\ \cdot\ |$ is an arbitrary Euclidean metric on the affine chart, and $u^+$ and $u^-$ are the intersection points of the line $x+\R.u$ with the boundary $\partial \mathcal{C}$ (see Figure \ref{fig_finsler_metric}).
\begin{figure}[h]
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{hilbertdistance}
\caption{${d_{\mathcal{C}} (x,y) = \left|\ln [a:x:y:b]\right|/2 }$}
\label{fig_hilbert_distance}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{finsler}
\caption{${F_{\mathcal{C}} (x,u) =\left(1/|u^-x| + 1/|xu^+|\right)|u|/2}$}
\label{fig_finsler_metric}
\end{subfigure}
\caption{}
\end{figure}
In general, a Hilbert geometry fails to be a Finsler space due to regularity issues: the regularity of $F_{\mathcal{C}}$ depends on the boundary of $\mathcal{C}$, so $F_{\mathcal{C}}$ does not necessarily satisfy the first and fourth points of Definition \ref{def:finsler_metric}. However, when $\mathcal{C}$ has a $C^2$ boundary with positive definite Hessian (see section \ref{sec_hessian}), $F_{\mathcal{C}}$ is a Finsler metric. In this case, the Hilbert geometry is called \emph{regular} and we can prove that its flag-curvature is constant equal to $-1$ (\cite{Fou:EquaDiff}).
\subsection{Main results}
The definition of the Finsler--Laplace operator is recalled in section \ref{sec:operator}. As for the Riemannian one, it is an unbounded elliptic operator on a Sobolev space contained in the $L^2$ functions. As such, the Finsler Laplacian admits a spectrum which splits into a discrete part, which, if non-empty, consists only of eigenvalues of finite multiplicity, and the essential spectrum. In the case at hand, there will always be an essential spectrum as we are considering non-compact manifolds.
In hyperbolic space, the spectrum of the Laplace--Beltrami operator is the interval $\left[(n-1)^2/4, +\infty \right)$ and, therefore, has no discrete part. In the case of regular Hilbert geometries, we prove the following:
\begin{thmintro} \label{thmintro_bottom}
Let $\lambda_1(\mathcal{C})$ be the bottom of the spectrum of the Finsler Laplacian of a regular Hilbert geometry $(\mathcal{C},\d)$. Then
\begin{equation*}
0 < \lambda_1(\mathcal{C}) \leqslant \frac{(n-1)^2}{4}.
\end{equation*}
\end{thmintro}
Let us make some remarks about this theorem.
\begin{itemize}
\item A study of spectral gaps in (regular and non-regular) Hilbert geometries was already launched by the second author and C.\ Vernicos \cite{ColboisVernicos_bas_du_spectre,Vernicos:spectral_radius_hilbert}. The spectral gap they were considering turns out to be associated, in the regular case, to the \emph{non-linear} Laplacian introduced by Z.\ Shen \cite{Shen:non-linear_Laplacian}, and their techniques and difficulties differ from ours. In particular, in \cite{Vernicos:spectral_radius_hilbert}, Vernicos proves that the spectral gap he considers is also less than $(n-1)^2/4$, but the difficulties for his proof appear only when considering non-regular Hilbert metrics, contrarily to us.
\item For regular Hilbert metrics the volume entropy is always equal to $n-1$ \cite{ColboisVerovic:hilbert_geometry}. So, Theorem \ref{thmintro_bottom} in particular tells us that the inequality $4\lambda_1 \leqslant h^2$, which is true for all simply connected non-positively curved Riemannian manifolds, still holds for regular Hilbert geometries.
\item In \cite{moi:these} the first author proved that, for negatively curved Finsler manifolds, the inequality $4\lambda_1 \leqslant n h^2$ holds, where $n$ is the dimension of the manifold. For general non-compact negatively curved Finsler manifolds, it is far from clear that the factor $n$ can be removed. In this article, we prove it for what we call \emph{asymptotically Riemannian Finsler metrics} of which Hilbert metrics are a nice example. This means that the Finsler metric gets infinitely close to Riemannian outside sufficiently big compact sets (see Section \ref{sec_bottom_of_spectrum}).
\end{itemize}
Our second result shows that the difference between regular Hilbert geometry and hyperbolic geometry does not appear in the essential spectrum (or, at least, not in its infimum):
\begin{thmintro} \label{thmintro_essential_bottom}
The bottom of the essential spectrum $\inf \sigma_{\textrm{ess}}(\mathcal{C})$ of the Finsler Laplacian of a regular Hilbert geometry $(\mathcal{C},\d)$ satisfies
\begin{equation*}
\inf \sigma_{\textrm{ess}}(\mathcal{C}) = \frac{(n-1)^2}{4}.
\end{equation*}
\end{thmintro}
Below $\frac{(n-1)^2}{4}$, the spectrum of the Laplace operator is thus entirely discrete. It is then natural to ask if there is always an eigenvalue below $\frac{(n-1)^2}{4}$. We know that this does not happen in the hyperbolic space and we make the following
\begin{conjecture*}
Let $(\mathcal{C},\d)$ be a regular Hilbert geometry. The equality $\lambda_1(\mathcal{C}) = \frac{(n-1)^2}{4}$ holds if and only if $\mathcal{C}$ is an ellipsoid.
\end{conjecture*}
We are not yet able to prove this conjecture, but we show the following:
\begin{thmintro} \label{thmintro_small_eigenvalues}
Let $\eps >0$ and $N \in \N$. There exists a regular Hilbert geometry whose first $N$ eigenvalues are below $\eps$.
\end{thmintro}
In particular, we can find a regular Hilbert geometry with as many eigenvalues below the essential spectrum as we want. As the flag curvature of regular Hilbert metrics is always equal to $-1$, this gives examples of Finsler metrics of constant negative curvature with eigenvalues as small as we want.
\subsection*{Structure of this paper}
In the preliminaries, we recall the construction of the Finsler Laplacian and its basic properties. We also introduce the Legendre transform that will be an important tool all along the article.\\
In Section \ref{sec_behavior_at_infinity_regular_hilbert}, we prove that regular Hilbert geometries are asymptotically Riemannian, which is an interesting result in itself. \\
In Section \ref{sec_bottom_of_spectrum}, we prove Theorem \ref{thmintro_bottom} by showing that the inequality $\lambda_1 \leqslant h^2/4$ holds for asymptotically Riemannian metrics.\\
After recalling a few results about the essential spectrum of weighted Laplacians, we prove Theorem \ref{thmintro_essential_bottom} in Section \ref{sec_bottom_of_essential_spectrum}.\\
We finally construct Hilbert metrics with arbitrarily many small eigenvalues in Section \ref{sec_small_eigenvalues}.
\section{Preliminaries}
\subsection{Topology on the set of Finsler metrics on a manifold}\label{sec:topology}
In all the text, we will use the topology of uniform convergence on compact sets for Finsler metrics. Let $M$ be a smooth manifold. We say that a sequence of Finsler metrics $(F_n)$ on $M$ converges to the Finsler metric $F$ if, for any compact subset $K$ of $M$,
$$
\lim_{n\to +\infty} \sup_{(x,u)\in TM|_K} \left|\ln \frac{F(x,u)}{F_n(x,u)}\right| = 0,
$$
where $TM|_K$ is the restriction of the tangent bundle to $K$.
This induces a topology on the set of Finsler metrics on $M$, which is metrizable: a distance between $F$ and $F'$ can be defined as
$$
d(F,F') = \sum_n \frac{1}{2^n} \min\left\{1, \sup_{(x,u)\in TM|_{K_n}} \left|\ln \frac{F(x,u)}{F'(x,u)}\right|\right\},
$$
where $(K_n)$ is an exhausting family of compact subsets of $M$.
\subsection{Finsler Laplacian}\label{sec:operator}
In this section, we quickly recall the definition of the Finsler Laplacian we consider, which uses the formalism introduced by Foulon \cite{Fou:EquaDiff}. All the proofs and details can be found in \cite{moi:these,moi:natural_finsler_laplace}.
Let $M$ be an $n$-dimensional smooth manifold. Let $HM$\ be the \emph{homogeneous bundle} or direction bundle, that is,
$$HM := \left(TM \smallsetminus \{0\} \right) / \R^+.$$
A point of $HM$ is a pair consisting in a point $x\in M$ and a tangent direction $\xi$ at $x$. We denote by $r \colon TM\smallsetminus \{0\} \rightarrow HM$ and $\pi \colon HM \rightarrow M$\ the canonical projections. The bundle $VHM = \Ker d\pi \subset THM$ is called the \emph{vertical bundle}.
Let $F$ be a Finsler metric on $M$. As for a Riemannian space, the metric space $(M,F)$ is locally uniquely geodesic, the geodesics being defined through a second-order differential equation. We assume in the sequel that the Finsler metric is complete. In this case, its geodesic flow is well defined on the homogeneous bundle: given a point $(x,\xi)$ in $HM$, there is a unique unit-speed geodesic $c:\R\longrightarrow M$ passing at $x$ with tangent direction $\xi$ at time $0$.
The \emph{Hilbert form} of $F$ is the $1$-form $A$ on $HM$ defined, for $(x,\xi) \in HM$, $Z \in T_{(x,\xi)}HM$, by
\begin{equation}
A_{(x,\xi)}(Z) := \lim_{\eps \rightarrow 0} \frac{F\left(x, v + \eps d\pi(Z) \right) - F\left( x,v \right)}{\eps},
\end{equation}
where $v \in T_xM$ is any vector such that $r(x,v) = (x,\xi)$. The Hilbert form contains all the necessary information about the dynamics of the Finsler metric:
\begin{thm}
The form $A$ is a contact form: $\ada$ is a volume form on $HM$.
Let $X \colon HM \rightarrow THM$ be the Reeb field of $A$, that is, the only solution of
\begin{equation}
\label{eq:Reeb_field}
\left\{
\begin{aligned}
A(X) &= 1 \\
i_X dA &= 0 \, .
\end{aligned}
\right.
\end{equation}
The vector field $X$ generates the geodesic flow of $F$.
\end{thm}
We can now define the Finsler Laplacian. First we split the canonical volume $\ada$ into a volume form on the manifold $M$ and an angle form:
\begin{prop}
\label{prop:construction}
There exists a unique volume form $\Omega^F$\ on $M$\ and an $(n-1)$-form $\alpha^F$\ on $HM$, never zero on $VHM$, such that
\begin{equation}
\label{eq:alpha_wedge_omega}
\alpha^{F} \wedge \pi^{\ast}\Omega^F = A\wedge dA^{n-1},
\end{equation}
and, for all $x\in M$,
\begin{equation}
\label{eq:longueur_fibre}
\int_{H_xM} \alpha^F = \voleucl(\mathbb S^{n-1})\, .
\end{equation}
\end{prop}
\begin{rem}
The volume form $\frac{1}{(n-1)!} \Omega^F$ is the Holmes--Thompson volume form (see for instance \cite{BuragoBuragoIvanov} or \cite{Alvarez:Survey} for the definition). However, we will not need in this article any specific knowledge about the Holmes--Thompson volume.
\end{rem}
The Finsler Laplacian of a function is then obtained as an average with respect to $\alpha^F$ of the second derivative in every direction:
\begin{defin}
\label{def:delta}
For $f \in C^2(M)$, the \emph{Finsler--Laplace operator} $\Delta^F$\ is defined by
$$
\Delta^F f (x) = \frac{n}{\voleucl \left(\mathbb{S}^{n-1}\right) }\int_{H_xM} L_X ^2 (\pi^{\ast} f ) \alpha^F,\ x \in M,
$$
where $L_X$ denotes the Lie derivative in the direction $X$.
\end{defin}
This definition gives a second order elliptic differential operator, which is symmetric with respect to the Holmes--Thompson volume $\Omega^F$.
The constant in front of the operator is there in order to get back the usual Laplace--Beltrami operator when $F$ is Riemannian.
The \emph{symbol} of a second-order differential operator $\Delta$ is a symmetric bilinear form on the co-tangent bundle that can be defined in the following way: Let $\xi \in T^{\ast}_xM$, then the symbol of the operator $\Delta$ at $(x,\xi)$ is
$$
\sigma_x(\xi,\xi) = \Delta(\varphi^2)(x),
$$
where $\varphi \colon M \rightarrow \R$ is a $C^2$ function such that $\varphi(x) =0$ and $d\varphi_x= \xi$.
When the operator is elliptic, that is, when $\sigma_x(\xi,\xi) >0$ for all non-zero $\xi$, the symbol defines a dual Riemannian metric. Note that in local coordinates, the symbol is given by the matrix of the coefficients in front of the second order derivatives.
We denote by $\sigma^F$ the symbol of $\Delta^F$, as $\Delta^F$ is elliptic, $\sigma^F$ is a dual Riemannian metric. In our case, we can express $\sigma^F$ using the form $\alpha^F$:
For $\xi_1,\xi_2 \in T^{\ast}_x M$, we have
\begin{equation*}
\sigma^F_x(\xi_1,\xi_2) = \frac{n}{\voleucl \left(\mathbb{S}^{n-1}\right) } \int_{H_xM} L_X(\pi^{\ast} \varphi_1) L_X(\pi^{\ast}\varphi_2)\, \alpha^F,
\end{equation*}
where $\varphi_i \in C^{\infty}(M)$ such that $\varphi_i(x)=0$ and $\left.d\varphi_i\right._x = \xi_i$. Note that, if we identify $HM$ with $S^FM$, the unitary tangent bundle for $F$, and that we consider $\alpha^F$ as a volume form on $S^FM$ (instead of $HM$), we have this visually more agreeable formula:
\begin{equation*}
\sigma^F_x(\xi_1,\xi_2) = \frac{n}{\voleucl \left(\mathbb{S}^{n-1}\right) } \int_{v\in S^F_xM} \xi_1(v) \xi_2(v) \, \alpha^F(v).
\end{equation*}
Note that we can also see $\Delta^F$ as a weighted Laplacian (introduced in \cite{ChavelFeldman:Isoperimetric_constants,Davies:Heat_kernel_bounds}), with symbol $\sigma^F$ and weight given by the ratio between $\Omega^F$ and the Riemannian volume associated with $\sigma^F$. Indeed, we have that, if $a\in C^{\infty}(M)$ is such that $\Omega^F = a^2 \Omega^{\sigma^F}$, where $\Omega^{\sigma^F}$ is the Riemannian volume associated with $\sigma^F$, then for $\varphi \in C^{\infty}(M)$:
\begin{equation*}
\Delta^F \varphi = \Delta^{\sigma^F} \varphi - \frac{1}{a^2} \langle \nabla \varphi , \nabla a^2 \rangle.
\end{equation*}
The description of $\Delta^F$ in terms of a weighted Laplacian will come very handy for the study of the essential spectrum in Section \ref{sec_bottom_of_essential_spectrum}.
\subsection{Energy and bottom of the spectrum} \label{subsec_energy_and_spectrum}
The Finsler Laplacian has a naturally associated \emph{energy functional} defined by
\begin{equation}
E^F(f) := \frac{n}{\voleucl \left(\mathbb S^{n-1}\right) } \int_{HM} \left|L_X\left(\pi^{\ast}f \right)\right|^2 \ada. \label{eq_energy_finsler}
\end{equation}
The \emph{Rayleigh quotient} for $F$ is then defined by
\begin{equation}
R^F(f) := \frac{E^F(f)}{\int_M f^2\, \Omega^F}.
\end{equation}
Let $H^1(M)$ be the Sobolev space defined as the completion of $C^{\infty}_0(M)$, the space of smooth functions with compact support, under the norm $\lVert f \rVert^2_1 = \int_M f^2\, \Omega^F + E^F(f)$.
The bottom of the spectrum of $-\Delta^F$, considered as a symmetric unbounded operator on $H^1(M)$, is given by:
\begin{equation*}
\lambda_1 = \inf_{f \in H^1(M)} R^F(f),
\end{equation*}
Note that, as the manifolds we are interested in in this article are not compact, the spectrum has no reason to be discrete. However, if there is a discrete spectrum below the essential one, then the eigenvalues can be obtained from the Rayleigh quotient via the Min-Max principle:
\begin{thm}[Min-Max principle] \label{thm_min_max}
Suppose that $\lambda_1$, \dots, $\lambda_k$ are the first $k$ eigenvalues (counted with multiplicity) of $-\Delta^F$ and are all below the essential spectrum, then
\begin{equation*}
\lambda_i = \inf_{V_i} \sup \left\{ R^F(u) \mid u \in V_i \right\}
\end{equation*}
where $V_i$ runs over all the $i$-dimensional subspaces of $H^1(M)$.
\end{thm}
\subsection{Cotangent point of view}
We finish this preliminaries with the cotangent point of view for Finsler metrics. This is fairly well-known and we refer to \cite{moi:these} for a more detailed presentation.
\subsubsection{Dual metric}
\begin{defin}
Let $F$ be a Finsler metric on a manifold $M$. The \emph{dual Finsler (co)metric} ${F^{\ast} \colon T^{\ast}M \rightarrow \R}$ is defined, for $(x,p) \in T^{\ast}M$, by
\begin{equation*}
F^{\ast}(x,p) = \sup \lbrace p(v) \mid v\in T_xM \; \text{\rm such that } F(x,v)=1 \rbrace.
\end{equation*}
\end{defin}
\subsubsection{Legendre transform}
The tool that allows us to switch from the tangent bundle to the cotangent bundle is the \emph{Legendre transform} associated with $F$.
\begin{defin}
The \emph{Legendre transform} $\mathcal{L}_F : TM \rightarrow T^{\ast}M$ associated with $F$ is defined by $\mathcal{L}_F(x,0) = (x,0)$ and, for $(x,v) \in TM \smallsetminus \{0\}$ and $u \in T_xM$,
\begin{equation*}
\mathcal{L}_F ( x,v) (u) := \frac{1}{2} \left. \frac{d}{dt} F^2(x,v + tu)\right|_{t=0}.
\end{equation*}
\end{defin}
As $F^2$ is positively $2$-homogeneous, we have that $\mathcal{L}_F$ is positively $1$-homogeneous, that is, $\mathcal{L}_F (x,\l v) (u) = \l \mathcal{L}_F (x,v) (u) $ for $\l\geqslant 0$. So we can project $\mathcal{L}_F$ to the homogeneous bundle. Set $H^{\ast}M := T^{\ast}M \smallsetminus \{0\} / \R^+_{\ast}$ and write $\ell_F\colon HM \rightarrow H^{\ast}M$ for the projection. Considering directly $\ell_F$, instead of $\mathcal{L}_F$, can sometimes be quite helpful.
The Legendre transform $\mathcal{L}_F$ links the Finsler metric $F$ with its dual metric $F^{\ast}$
\[
F = F^{\ast} \circ \mathcal{L}_F.
\]
So, in particular $\mathcal{L}_F$ maps the unit tangent bundle of $F$ to the unit cotangent bundle of $F^{\ast}$.
Moreover, the Legendre transform $\mathcal{L}_F$ is a diffeomorphism and the following diagram commutes (see for instance \cite{moi:these}):
\[
\xymatrix{
& T^{\ast}M \smallsetminus \{0\} \ar[r]^{\hat{r}} \ar[ld]_{\hat{p}} & H^{\ast}M \ar[rd]^{\hat{\pi}} & \\
M & & & M \\
& TM \smallsetminus \{0\} \ar[uu]^{\mathcal{L}_F} \ar[r]_r \ar[ul]^{p} & HM \ar[uu]_{\ell_F} \ar[ur]_{\pi} & }
\]
For strongly convex smooth Finsler metrics, the Legendre transform can also be described using convex geometry. The Legendre transform associated with a convex $\mathcal{C} \subset \R^n$ sends a point $x$ of $\mathcal{C}$ to the hyperplane supporting $\mathcal{C}$ at $x$, or equivalently, to the linear map $p \in \left(\R^n\right)^{\ast}$ such that $p(x)=1$ and $\ker p$ is parallel to the supporting hyperplane.
\subsubsection{Continuity of the Legendre transform}
Let $V$ be a $n$-dimensional real vector space\footnote{In this section, we should think of a Finsler manifold $(M,F)$ with a fixed point $x\in M$. We look at the tangent space $T_xM$ as an $n$-dimensional real vector space, provided with a non-necessarily symmetric norm $F(x,\cdot)$.}, with a fixed Euclidean structure whose norm we denote by $F_0$ and see as a translation-invariant Finsler metric on $V$.\\
Let $\mathcal{N}$ denote the set of translation-invariant Finsler metrics on $V$. This is the same as looking at the set of non-necessarily symmetric norms on $V$, whose unit sphere is $C^2$ with positive definite Hessian.\\
The topology defined in section \ref{sec:topology} induces a topology on ${\mathcal{N}}$ which can be metrized in the following easy way: Let $HV = V\smallsetminus\{0\}/\R^+ \simeq \mathbb S^{n-1}$ be the set of rays from the origin. If $F,F'\in{\mathcal{N}}$, the ratio $\frac{F}{F'}$ is a well defined function of $HV$: if $\xi\in HV$, we have $\frac{F}{F'}(\xi)=\frac{F(u)}{F'(u)}$, where $u$ is any vector of $V$ that projects to $\xi$. Define a metric on $\mathcal{N}$ by
$$d_{{\mathcal{N}}}(F,F')= \sup_{\xi\in HV} |\ln \frac{F}{F'}(\xi)|.$$
We define a metric $D$ on the set $\mathrm{Homeo}^0(V)$ of positively $1$-homogeneous homeomorphisms of $V$ by:
$$D(H,H') = \sup_{u\in V,\ F_0(u)=1} F_0(H(u)-H'(u)).$$
Identifying $HV$ with the unit Euclidean sphere $\mathbb S^{n-1}$, we define a metric $d$ on the set $\mathrm{Homeo}(HV)$ of homeomorphisms of $HV$:
$$d(h,h') = \sup_{\xi\in HV} d_{\mathbb S^{n-1}}(h(\xi),h'(\xi)).$$
This distance is just the maximal Euclidean angle between the images.\\
For each $F\in \mathcal{N}$, the Legendre transform $\mathcal{L}_F$ is a positively $1$-homogeneous homeomorphism of $V$ and its ``projection'' $\ell_F$ a $C^1$-diffeomorphism of $HV$. We thus have applications $\mathcal{L}:F \longmapsto \mathcal{L}_F$ from ${\mathcal{N}}$ to $\mathrm{Homeo}^0(V)$ and $\ell:F \longmapsto \ell_F$ from ${\mathcal{N}}$ to $\mathrm{Homeo}(HV)$. The following lemma is immediate if we use the geometrical interpretation of the Legendre transform that we recalled at the end of the previous section.
\begin{lem}\label{lem_continuity_l}
The application $\mathcal{L}$ is a continuous bijection from $({\mathcal{N}},d_{{\mathcal{N}}})$ to $(\mathrm{Homeo}^0(V),D)$. The application $\ell$ is continuous from $({\mathcal{N}},d_{{\mathcal{N}}})$ to $(\mathrm{Homeo}(HV),d)$ but is not injective: $\ell_F=\ell_{F'}$ if and only if $F = \lambda F'$ for some $\lambda>0$.
\end{lem}
\begin{proof}
Let us explicit the continuity of $\ell$ at $F_0$ because this is all we need in this article; the continuity elsewhere follows the exact same lines.\\
Let $F\in\mathcal{N}$ such that $d_{\mathcal{N}}(F_0,F) \leqslant \ln C$ for some $C>1$. We can see that $d(\ell_{F}^{-1}\circ \ell_{F_0}, \id) \leqslant \arccos C^{-2}$.\\
Indeed, as $d_{\mathcal{N}}(F_0,F) \leqslant \ln C$, the unit sphere $S_F(1)$ for $F$ in $V$ is in between the spheres of radius $C^{-1}$ and $C$ for $F_0$, that we denote by $S_0(C^{-1})$ and $S_0(C)$. Let $\xi \in HV$. The map $\ell_{F}^{-1}\circ \ell_{F_0}$ sends $\xi$ to a point $\xi'$, such that the tangent space of $S_F(1)$ at $\xi'$ is parallel to the tangent space of $S_0(C)$ at $\xi$. Figure \ref{fig_the_control_legendre_transform} and simple trigonometry then yield the result.
\begin{figure}[h]
\begin{center}
\begin{pspicture}(0,-2.7)(6.62,3)
\pscircle[linewidth=0.04,dimen=outer](2.3,-0.4){2.3}
\pscircle[linewidth=0.04,dimen=outer](2.3,-0.4){1.7}
\psline[linewidth=0.04cm](1.2,2.3)(5.1,-0.2)
\psline[linewidth=0.04cm](2.4,-0.5)(3.8,2.0)
\psline[linewidth=0.04cm,linestyle=dashed,dash=0.16cm 0.16cm](2.4,-0.5)(5.2,0.395)
\psline[linewidth=0.04cm,linestyle=dashed,dash=0.16cm 0.16cm](2.4,-0.5)(1.8,2.5)
\psbezier[linewidth=0.04,linecolor=red](3.6559284,-2.02)(3.3135412,-2.22)(2.548205,-2.48)(1.8030093,-2.38)(1.0578135,-2.28)(0.68,-1.82)(0.49388167,-1.02)(0.30776334,-0.22)(0.789305,1.2040415)(1.7224476,1.58)(2.6555903,1.9559585)(3.2101226,1.3714887)(3.6156476,0.78)(4.0211725,0.1885113)(4.239899,-0.4818077)(4.21986,-0.92)(4.1998215,-1.3581923)(3.9983156,-1.82)(3.6559284,-2.02)
\psline[linewidth=0.04cm,linecolor=red](4.44,0.56)(1.46,2.46)
\psline[linewidth=0.04cm,linecolor=red](4.44,0.56)(1.46,2.46)
\psline[linewidth=0.04cm,linecolor=red,linestyle=dashed,dash=0.16cm 0.16cm](2.4,-0.5)(3.22,2.7)
\uput{3pt}[45](3.8,2.0){$\xi$}
\rput(2.6,-1.6){$S_0( C^{-1})$}
\put(4,-2.3){$S_0( C)$}
\put(2.7,2.9){\color{red} $\ell_{F}^{-1} \circ \ell_{F_0} \left( \xi \right)$}
\put(1.8,-2.4){\color{red} $S_F(1)$}
\end{pspicture}
\quad
\begin{pspicture}(1,-1.6)(7,2.4)
\psarc[linewidth=0.04,linestyle=dashed,dimen=outer](2.4,-0.5){1.7}{-15}{100}
\psarc[linewidth=0.04,linestyle=dashed,dimen=outer](2.4,-0.5){2.3}{-15}{100}
\psline[linewidth=0.04cm](1.2,2.3)(5.1,-0.2)
\psline[linewidth=0.04cm](2.4,-0.5)(3.8,2.0)
\psline[linewidth=0.04cm](2.4,-0.5)(5.2,0.25)
\psarc[linewidth=0.03cm]{->}(2.4,-0.5){0.6}{17}{59}
\rput(2,0.9){$S_0(C^{-1})$}
\rput(4.7,-1.3){$S_0(C)$}
\end{pspicture}
\end{center}
\caption{Maximum angle between $\ell_{F}^{-1} \circ \ell_{F_0} \left( \xi \right)$ and $\xi$}
\label{fig_the_control_legendre_transform}
\end{figure}
\end{proof}
\section{Behavior at infinity of Regular Hilbert geometries}\label{sec_behavior_at_infinity_regular_hilbert}
In all the following, $(\mathcal{C}, d_{\mathcal{C}})$ will be a regular Hilbert geometry. We will see here that $(\mathcal{C}, d_{\mathcal{C}})$ is asymptotically Riemannian, that is, the space looks more and more like a Riemannian space outside big compact sets:
\begin{defin} \label{def_asympt_riemannian}
A Finsler metric $F$ on a manifold $M$ is called \emph{asymptotically Riemannian} if, for any $C>1$, there exists a compact set $K$ such that, for all $x \in M \smallsetminus K$, there exists a scalar product $g_x$ on $T_xM$ satisfying, for every non-zero vector $v\in T_xM$,
\begin{equation*}
C^{-1} \leqslant \frac{F(x,v)}{\sqrt{g_x(v,v)}} \leqslant C
\end{equation*}
\end{defin}
\begin{rem}
Note that, for this definition to be of any interest, $M$ should be non-compact.
\end{rem}
\subsection{Hessian of a codimension-$1$ submanifold of the projective space}\label{sec_hessian}
Consider a codimension-$1$ $C^2$ submanifold $N$ of the projective space $\mathbb{RP}^n$ (for instance the boundary $\partial \mathcal{C}$ of a convex set $\mathcal{C}$), and pick a point $x\in N$. Choose an affine chart containing $x$ and a Euclidean metric on it. Let $n$ be a unit normal vector to $N$ at $x$ for this metric, that is a unit vector orthogonal to $T_xN$. Now, around $x$, we consider $N$ as the graph of the function, defined on some neighborhood $U$ of $x$ in $T_xN$:
$$G_x \colon u\in U \longmapsto G_x(u)\in\R,$$
such that a neighborhood of $x$ in $N$ is the submanifold $\{u+G(u).n,\ u\in U\}$. The Hessian of $G_x$ at $x$ is a bilinear form on the tangent space $T_xN$. If one chooses an orthonormal basis $(u_1,\cdots,u_{n-1})$ of $T_xN$, then the matrix of this bilinear form is the $(n-1)\times(n-1)$ matrix $\left(\frac{\partial^2 G}{\partial u_i\partial u_j}\right)$ of the second-derivatives of $G$.\\
The definition of the Hessian obviously depends on the choice of the affine chart and of the Euclidean metric. Nevertheless, there are two basic observations which we will use all along this section.
\begin{itemize}
\item The property of the Hessian of $N$ at $x$ to be positive, negative or definite, is independent of the choice of the affine chart and the Euclidean metric. Hence, for example, it is possible to talk about a convex subset of $\mathbb{RP}^n$ whose boundary is $C^2$ with positive definite Hessian.
\item Let $N'$ be another codimension-$1$ $C^2$ submanifold of $\mathbb{RP}^n$, which is tangent to $N$ at $x$. It makes sense to say that $N$ and $N'$ have the same Hessian at $x$. Indeed, choose an affine chart containing $x$, a Euclidean metric on it, a unit vector $n$ normal to $N$ at $x$ and an orthonormal basis $(u_1,\cdots,u_{n-1})$ of $T_xN=T_xN'$. Call $H_x$ and $H_x'$ the Hessians of $N$ and $N'$ at $x$. The fact that they are the same bilinear form on $T_xN$ does not depend on any of the previous choices.
\end{itemize}
\subsection{Busemann functions, horospheres and horoballs}
The \emph{Busemann function} based at $x\in\partial \mathcal{C}$ is defined by
$$b_{x}(z,y) = \lim_{p\to x} d_{\mathcal{C}}(z,p) - d_{\mathcal{C}}(y,p),$$
which, in some sense, measures the (signed) distance from $z$ to $y$ in $\mathcal{C}$ as seen from the point $x\in\partial \mathcal{C}$. A particular expression for $b$ is given by
$$b_{x}(z,y)=\lim_{t\to +\infty} d_{\mathcal{C}}(z,\gamma(t))-t,$$
where $\gamma$ is the geodesic leaving $y$ at $t=0$ to $x$. When $x$ is fixed, then $b_{x}$ is a surjective map from $\mathcal{C}\times\mathcal{C}$ onto $\R$. When $z$ and $y$ are fixed, then $b_.(z,y): \partial \mathcal{C} \rightarrow \R$ is bounded by a constant $C=C(z,y)$.\\
The \emph{horosphere} passing through $z\in \mathcal{C}$ and based at $x\in\partial \mathcal{C}$ is the set
$$\mathcal{H}_{x}(z)=\{y\in\mathcal{C},\ b_{x}(z,y)=0\}.$$
$\mathcal{H}_{x}(z)$ is also the limit when $p$ tends to $x$ of the metric spheres $S(p,d_{\mathcal{C}}(p,z))$ about $p$ passing through $z$. In some sense, the points on $\mathcal{H}_{x}(z)$ are those which are as far from $x$ as $z$ is.\\
The (open) horoball $H_{x}(z)$ defined by $z\in \mathcal{C}$ and based at $x\in\partial \mathcal{C}$ is the ``interior'' of the horosphere $\mathcal{H}_{x}(z)$, that is, the set $$H_{x}(z)=\{y\in\mathcal{C},\ b_{x}(z,y)> 0\}.$$
For example, if ${\mathcal{E}}$ is an ellipsoid, then the horoballs of $({\mathcal{E}}, d_{{\mathcal{E}}})$ are also ellipsoids. We explain this fact in the proof of the following lemma. This proof will introduce the main construction which helps us in understanding the asymptotic behavior of Hilbert geometries.
\begin{lem}\label{busemann}
Let $(\mathcal{C},\d)$ be a regular Hilbert geometry.
\begin{itemize}
\item For any $x \in \partial \mathcal{C}$, the Busemann function $b_{x} \colon \mathcal{C}\times\mathcal{C} \rightarrow \R$ is a $C^2$ function.
\item Let $x\in\partial \mathcal{C}$, $z\in \mathcal{C}$. The set $H_x(z)\cup \{x\}$ is a $C^2$ submanifold of $\mathbb{RP}^n$, whose Hessian at $x$ is the same as the Hessian of $\partial \mathcal{C}$.
\end{itemize}
\end{lem}
\begin{proof}
The first point follows from the following description of the Busemann function $b_{x}(z,y)$, given by Benoist in \cite{Benoist:convexes_div_I}. Let $z'$ and $y'$ be the intersection points of the lines $(xz)$ and $(xy)$ with $\partial \mathcal{C}$, which are distinct from $x$. Let $p$ be the intersection point of $(y'z')$ with $T_x\partial \mathcal{C}$. Then
$$b_{x}(z,y) = \frac{1}{2} \ln [(px):(pz'):(pz):(py)],$$
where $[(px):(pz'):(pz):(py)]$ denotes the cross-ratio of the four lines $(px), (pz'), (pz), (py)$. All these constructions involve only the boundary of $\mathcal{C}$, so the Busemann function has the same regularity as $\partial \mathcal{C}$. This first point implies that horospheres are $C^2$ submanifolds of $\mathcal{C}$.\\
To prove the second point, we first consider the case of an ellipsoid ${\mathcal{E}}$. The Hilbert geometry $({\mathcal{E}},d_{{\mathcal{E}}})$ is a model of the Riemannian hyperbolic space. We will exploit the fact that, for any $x\in\partial{\mathcal{E}}$, $\partial{\mathcal{E}}$ or any horosphere based at $x$ is an orbit of a maximal parabolic group of isometries fixing $x$. We have to prove that the Hessians are the same in all the directions, so we can assume the dimension is $2$.\\
Let then ${\mathcal{E}}$ be an ellipsoid in $\mathbb{RP}^2$ and pick $x\in\partial{\mathcal{E}}$. We can choose a projective basis $(e_1,e_2,e_3)$ such that $e_1=x$, $e_2\in T_x\partial{\mathcal{E}}$ and the maximal parabolic group of isometries fixing $x$ is given by $\mathcal{P} = \{g_t\in \mathrm{SL}(3,\R),\ t\in\R\}$ with
$$\left(\begin{array}{ccc}
1 & t & \frac{t(t-1)}{2}\\
0 & 1 & t \\
0 & 0 & 1
\end{array}\right).$$
The boundary $\partial{\mathcal{E}}$, as well as any horosphere $\mathcal{H}$ based at $x$ is the $\mathcal{P}$-orbit of the point $z=e_1+se_3$ for some $s\in \R\smallsetminus\{0\}$, that is, an ellipse parametrized by
$$t \longmapsto [1+s\frac{t(t-1)}{2}:st:s].$$
In the affine chart given by the intersection with the plane $\{(x_1,x_2,x_3)\in\R^3,\ x_1=1\}$, with origin $x$ and the induced Euclidean metric of $\R^3$, this is the curve
$$t \longmapsto \left(\frac{t}{\frac{1}{s}+\frac{t(t-1)}{2}},\frac{1}{\frac{1}{s}+\frac{t(t-1)}{2}} \right).$$
By making the transformation $t \longmapsto 1/t$, this becomes the curve
$$c \colon t \longmapsto \left(\frac{t}{\frac{t^2}{s}+\frac{1-t}{2}},\frac{t^2}{\frac{t^2}{s}+\frac{1-t}{2}} \right),$$
such that $c(0)=x$. But for $t$ around $0$, we have up to order $2$:
$$c(t) \sim (2t(1+t),2t^2).$$
This implies that the curvature of the curve $c$ at $0$ is independent of $s$, and hence, that, for ellipsoids, the Hessian of the horospheres are all the same at $x$.\\
Now, let $(\mathcal{C},\d)$ be a regular Hilbert geometry. Pick $x\in\partial \mathcal{C}$, and $z\in\mathcal{C}$. Fix an affine chart centered at $x$, containing $\overline\mathcal{C}$, and fix a Euclidean metric $|\cdot|$ on it such that $|zx|=1$ and the Hessian $B_x$ of $\partial \mathcal{C}$ at $x$ is the restriction of the Euclidean scalar product to $T_x\partial \mathcal{C}$.\\
Fix $C>1$. Consider the Euclidean spheres $S_x^+$ and $S_x^-$, whose boundaries are tangent to $\partial \mathcal{C}$ at $x$ and Hessians $B_x^+$ and $B^-_x$ at the point $x$, seen as elements of $\mathrm{GL}(n-1,\R)$, satisfy $B_x^- = C B_x$ and $B_x^+ = C^{-1} B_x$ (this does not depend on the Euclidean metric we use to compute them). For $C$ close enough to $1$, the balls ${\mathcal{E}}_x^-$ and ${\mathcal{E}}_x^+$ they define contain the point $z$.\\
Let $\mathtt{h}_x^-$ and $ \mathtt{h}^+_x$ be the hyperbolic metrics defined by the balls ${\mathcal{E}}_x^-$ and ${\mathcal{E}}_x^+$. There is some neighborhood $U$ of $x$ in $\R^n$, depending on $C$, such that, on $ U\cap {\mathcal{E}}^-_x$, we have
$$\mathtt{h}^+_x\leqslant F_{\mathcal{C}} \leqslant \mathtt{h}^-_x.$$
Denote by $H^-_x(z)$, $H^+_x(z)$ and $H_x(z)$ the horoballs based at $x$ passing through $z$ for the Hilbert geometries defined by ${\mathcal{E}}_x^-$, ${\mathcal{E}}_x^+$ and $\mathcal{C}$ respectively. The previous inequality implies that
$$H^-_x(z)\cap U\subset H_x(z)\cap U\subset H^+_x(z)\cap U.$$
Now, by the result for ellipsoids, the Hessians $B_x^+(z)$ and $B^-_x(z)$ at the point $x$ of the boundaries of $H^+_x(z)$ and $H^-_x(z)$ also satisfy $B_x^-(z) = C B_x$ and $B_x^+(z) = C^{-1} B_x$. This means the horospheres $\mathcal{H}^-_x(z)$ and $\mathcal{H}^+_x(z)$ are ``almost'' osculating for $\mathcal{H}_x(z)$. Since $C>1$ is arbitrary, we see that the horosphere $\mathcal{H}_x(z)$ and $\partial \mathcal{C}$ have the same Hessians at $x$.
\end{proof}
\subsection{Hilbert geometries are asymptotically Riemannian}
\begin{prop}\label{prop_hilbert_are_bilipschitz_at_infinity}
Let $(\mathcal{C},\d)$ be a regular Hilbert geometry, fix a point $o\in\mathcal{C}$ and a constant $C>1$. To each $x\in\partial \mathcal{C}$, we can associate a (non-complete) Riemannian hyperbolic metric $\h_x$ on $\mathcal{C}$ such that
\begin{enumerate}
\item the application $x\longmapsto \h_x$ is continuous;
\item the metric $\mathtt{h}_x$ has the same geodesics as $F_{\mathcal{C}}$ on $\mathcal{C}$;
\item there is $R=R(C) \geqslant 0$ such that, for any $x\in\partial \mathcal{C}$ and $z\in [ox)\smallsetminus B(o,R)$,
$$C^{-1} \leqslant \frac{F_{\mathcal{C}}(z, \cdot)}{\mathtt{h}_x(z, \cdot)} \leqslant C.$$
\end{enumerate}
\end{prop}
\begin{proof}
We more or less repeat the construction used in Lemma \ref{busemann}. By choosing an adapted affine chart, we look at $\mathcal{C}$ as a relatively compact subset of a Euclidean space $\R^n$, with norm $|\cdot|$.\\
Let $x\in \partial \mathcal{C}$. The Hessian of $\partial \mathcal{C}$ at $x$, computed with respect to the metric $|\cdot|$, gives a positive definite bilinear form $B_x$ on $T_x\partial \mathcal{C}$, and the map $x\longmapsto B_x$ is continuous. Define a new Euclidean norm $|\cdot|_x$ on $\R^n$ by setting:
\begin{itemize}
\item the vector $ox$ has norm $1$: $|ox|_x=1$;
\item the restriction of the corresponding scalar product to $T_x\partial \mathcal{C}$ is $B_x$;
\item $T_x\partial \mathcal{C}$ and $ox$ are orthogonal.
\end{itemize}
The map $x\longmapsto |\cdot|_x$ is continuous. The sphere $S_x$ of radius $1$ for the norm $|\cdot|_x$, with center $o$, is tangent to $\mathcal{C}$ at $x$; in fact, it is an osculating sphere.\\
Let $\varepsilon>0$, and consider the spheres $S_x^+$ and $S_x^-$ of respective radius $1+\varepsilon$ and $1-\varepsilon$ for $|\cdot|_x$, whose boundaries are tangent to $\partial \mathcal{C}$ at $x$. Their centers are on the line $(ox)$. Their Hessians $B_x^+$ and $B^-_x$, seen as elements of $\mathrm{GL}(n-1,\R)$, at the point $x$ satisfy $B_x^+ = \frac{1-\varepsilon}{1+\varepsilon} B_x^-$ (and this does not depend on the Euclidean metric we use to compute them).\\
Now, let ${\mathcal{E}}_x^+$ be the smallest ellipsoid which contains $\mathcal{C}$, has $x$ in its boundary, and such that $S_x^+$ is a horosphere at $x$ of the hyperbolic geometry defined by ${\mathcal{E}}_x^+$. In other words, it is the smallest ellipsoid which contains $\mathcal{C}$, is tangent to $\partial \mathcal{C}$ at $x$, has its center on $(ox)$ and the Hessian of its boundary at $x$ is the same as the Hessian of $S_x^+$. Such an ellipsoid exists in the projective space because locally around $x$, $S_x^+$ contains $\mathcal{C}$. However, it might not be an ellipsoid in the affine chart, but could for instance be a paraboloid or a hyperboloid.\\
In the same way, let ${\mathcal{E}}_x^-$ be the largest horosphere at $x$ of the Hilbert geometry defined by $S_x^-$ which is contained in $\mathcal{C}$. We also have that the Hessian of the boundary of ${\mathcal{E}}_x^-$ at $x$ is the same as the Hessian of $S_x^-$.\\
Let $\mathtt{h}_x^-$ and $ \mathtt{h}^+_x$ be the hyperbolic metrics defined by the ellipsoids ${\mathcal{E}}_x^-$ and ${\mathcal{E}}_x^+$. By definition, we have that, on ${\mathcal{E}}^-_x$,
$$\mathtt{h}^+_x\leqslant F_{\mathcal{C}} \leqslant \mathtt{h}^-_x.$$
We will prove that, for $\varepsilon$ small enough, the application $x\longmapsto \h_x^+$ satisfies the desired properties. The property (2) is obvious. To prove (1), we show the following
\begin{lem}\label{lem:continuity}
The maps $x\longmapsto {\mathcal{E}}^{\pm}_x$ are continuous.
\end{lem}
\begin{proof}
We show the continuity of $x\longmapsto{\mathcal{E}}_x^-$ at a given point $x_0\in\partial \mathcal{C}$, the same works for $x\longmapsto{\mathcal{E}}_x^+$. Choose a point $p$ in ${\mathcal{E}}_{x_0}^-$ and let $r\in\R$ such that ${\mathcal{E}}_{x_0}^-$ is the horoball
$${\mathcal{E}}_{x_0}^-=\{z\in\mathcal{C},\ b_{x_0}(o,z)>r\}$$
in the hyperbolic geometry defined by $S^+_{x_0}$. For any $\delta\in\R$, let ${\mathcal{E}}_x^-(\delta)$ be the (open) horoball
$${\mathcal{E}}_x^-(\delta)=\{z\in\mathcal{C},\ b_x(o,z)>r+\delta\}$$
in the hyperbolic geometry defined by $S_x^+$. For any $\delta >0$, the maps $x\longmapsto {\mathcal{E}}_x^-(\delta)$ are continuous, because of the continuity of the Busemann functions. Fix $\delta>0$. The horoball ${\mathcal{E}}_{x_0}^-(\delta)$ is entirely contained in $\mathcal{C}$ while the horoball ${\mathcal{E}}_{x_0}^-(-\delta)$ has a nonempty intersection with $\mathbb{RP}^n\smallsetminus\mathcal{C}$. By continuity of $x\longmapsto {\mathcal{E}}_x^-(\delta)$, the same is true for ${\mathcal{E}}_x^-(\delta)$ and ${\mathcal{E}}_x^-(-\delta)$ for $x$ in some neighborhood of $x_0$. By definition of ${\mathcal{E}}_x^-$, this implies that ${\mathcal{E}}_x^-(\delta) \subset {\mathcal{E}}_x^- \subset {\mathcal{E}}_x^-(-\delta)$ in this neighborhood, hence the continuity of $x\longmapsto{\mathcal{E}}_x^-$ at $x_0$.
\end{proof}
To prove the third point, we consider, for $x\in\partial \mathcal{C}$ and $u\in\R^n\smallsetminus\{0\}$, the function
$$f_{x,u}: r \longmapsto \frac{\h^-_x(o_r,u)}{\h^+_x(o_r,u)},$$
where $o_r$ is the point of $[ox)$ such that $\d(o_r,o)=r$. The function $f_{x,u}$ is defined as soon as $r$ is big enough for $o_r$ to be in ${\mathcal{E}}_x^-$. Remark that $f_{x,u}=f_{x,\l u}$ for all $u\in\R^n\smallsetminus\{0\},\ \l\not= 0$.
\begin{lem}
For $u\in \R.ox\smallsetminus\{0\}$, the function $f_{x,u}$ is decreasing and tends to $1$. For $u\in T_x\partial \mathcal{C}\smallsetminus\{0\}$, the function $f_{x,u}$ is decreasing and tends to $\sqrt{\frac{1+\varepsilon}{1-\varepsilon}}$.
\end{lem}
\begin{proof}
We can choose another affine chart, with coordinates $(t_1, \dots , t_{n-1}, s)$, so that the boundary of ${\mathcal{E}}_x^+$ is the parabola $s=|t|^2$, where $|t|^2 = t_1^2 + \dots +t_{n-1}^2$. In that chart, the boundary of ${\mathcal{E}}_x^-$ has to be an ellipsoid inside of ${\mathcal{E}}_x^+$, and the line $(ox)$, which is an axis of symmetry for both ${\mathcal{E}}_x^+$ and ${\mathcal{E}}_x^-$, is sent to the $y$-axis. The equation of ${\mathcal{E}}_x^-$ is then given by
$$\frac{s^2}{a^2}-\frac{2s}{a}+ \frac{|t|^2}{b^2}=0,$$
for some $a,b>0$.
Let $s=s(r)=|o_rx|$. If $u\in\R.ox\smallsetminus\{0\}$, we have (see Figure \ref{fig_E+_et_E-_dans_carte})
\begin{equation*}
\h^-_x(o_r,u)= \frac{a|u|}{s(2a-s)}, \quad \text{and} \quad \h^+_x(o_r,u) = \frac{|u|}{2s}.
\end{equation*}
Hence
$$f_{x,u}(r) = \frac{2a}{2a-s(r)}$$
which is decreasing and tends to $1$.\\
If $u\in T_x\partial \mathcal{C}$, then (see Figure \ref{fig_E+_et_E-_dans_carte})
\begin{equation*}
\h^-_x(o_r,u)= \frac{a|u|}{b\sqrt{s(2a-s)}}, \quad \text{and} \quad \h^+_x(o_r,u) = \frac{|u|}{\sqrt{s}}.
\end{equation*}
Hence
$$f_{x,u}(r) = \frac{a}{b\sqrt{2a-s(r)}},$$
which is decreasing and tends to $\frac{\sqrt{a}}{b\sqrt{2}}$. Now, direct computations shows that in this chart, $B_x^- = a/b^2$ and $B_x^+ =2$, hence, $f_{x,u}(r)$ converges to $\sqrt{B_x^-(B_x^+)^{-1}} = \sqrt{\frac{1+\varepsilon}{1-\varepsilon}}>1$.
\begin{figure}[h]
\centering
\begin{pspicture}(0,-3.42)(10.52,3.62)
\psline[linewidth=0.04cm,arrowsize=0.25]{->}(0.0,-3.0)(10.0,-3.0)
\psline[linewidth=0.04cm,arrowsize=0.25]{->}(5.0,-3.0)(5.0,3.6)
\psbezier[linewidth=0.04](0.2,2.6)(1.2,-1.6)(2.6,-3.0)(5.0,-3.0)(7.4,-3.0)(8.8,-1.6)(9.8,2.6)
\psellipse[linewidth=0.04,dimen=outer](5.0,-0.1)(2.0,2.9)
\uput[135](5.0,3.6){$s$}
\uput[-45](10.0,-3.0){$t$}
\rput(9.8,2.8){$\mathcal{E}_x^+ : \; s = |t|^2$}
\rput(2.9,2.9){$\mathcal{E}_x^-: \; \frac{s^2}{a^2}-\frac{2s}{a}+ \frac{|t|^2}{b^2}=0$}
\psline[linewidth=0.04cm,linestyle=dashed](1,0.0)(9,0)
\psline[linewidth=0.06cm,linecolor=red]{->}(5,0.0)(5.7,0)
\psline[linewidth=0.06cm,linecolor=red]{->}(5,0.0)(5,-0.7)
\psdots[dotsize=0.14](1,0)
\psdots[dotsize=0.14](3.02,0)
\psdots[dotsize=0.14](5,0)
\uput[135](5,0){$o_r$}
\psdots[dotsize=0.14](6.98,0)
\psdots[dotsize=0.14](9,0)
\psdots[dotsize=0.14](5,2.8)
\psdots[dotsize=0.14](5,-3)
\put(5,-3.3){$x$}
\end{pspicture}
\caption{The ellipsoids ${\mathcal{E}}_x^+$ and ${\mathcal{E}}_x^-$ in a well-chosen chart} \label{fig_E+_et_E-_dans_carte}
\end{figure}
\end{proof}
As a consequence of this lemma, we see that there exists $R\geqslant 0$, depending on $x$ and $\varepsilon$, such that for $r\geqslant R$, we have $f_{x,u}(r)\leqslant \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} + \varepsilon$ for $u\in T_x\partial \mathcal{C}\smallsetminus\{0\}$ or $u\in\R.ox\smallsetminus\{0\}$. Let us define $R(\varepsilon,x)$ as the smallest $R\geqslant 0$ satisfying this property. Now, the continuity of the functions $x \mapsto \h_x^{\pm}$ (Lemma \ref{lem:continuity}) implies that the function $x \mapsto R(\varepsilon,x)$ is also continuous. Hence, if we set
$$R(\eps):= \sup_{x \in \partial \mathcal{C}} R(\varepsilon,x), $$
we have that for any $x\in\partial \mathcal{C}$ and $r\geqslant R(\varepsilon)$, $f_{x,u}(r)\leqslant \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} +\eps$ for $u\in T_x\partial \mathcal{C}\smallsetminus\{0\}$ or $u\in\R.ox\smallsetminus\{0\}$. Now, each $u\in \R^n$ can be decomposed as $u=u_1+u_2$ with $u_1\in T_x\partial \mathcal{C}$ and $u_2\in\R.ox$. Remark that $u_1$ and $u_2$ are orthogonal for $\h_x^+$ as well as for $\h_x^-$, so that
$$\h_x^{\pm}(o_r,u) = \sqrt{\h_x^{\pm}(o_r,u_1)^2 +\h_x^{\pm}(o_r,u_2)^2}.$$
For $r>R(\varepsilon)$, we have
$$
f_{x,u}(r) = \frac{\h^-_x(o_r,u)}{\h^+_x(o_r,u)} = \sqrt{\frac{\h^-_x(o_r,u_1)^2+\h^-_x(o_r,u_2)^2}{\h^+_x(o_r,u_1)^2+\h^+_x(o_r,u_2)^2}} \leqslant \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} + \eps,\ u\in\R^ n\smallsetminus\{0\}.
$$
That means that for any $x\in\partial \mathcal{C}$ and $z\in [ox)$ such that $\d(o,z)\geqslant R(\varepsilon)$, we have
$$1 \leqslant \frac{F_{\mathcal{C}}(z,\cdot)}{\h^+_x(z,\cdot)} \leqslant \frac{\h^-_x(z,\cdot)}{\h^+_x(z,\cdot)} \leqslant \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} + \eps.$$
This proves property (3).
\end{proof}
So we get that Hilbert geometries are asymptotically Riemannian:
\begin{cor}\label{cor_suffit}
Let $(\mathcal{C},\d)$ be a regular Hilbert geometry and $o\in\mathcal{C}$ a base point. For any $C>1$, there exists $R\geqslant 0$ and a continuous Riemannian metric $g$ on $\mathcal{C}\smallsetminus B(o,R)$ such that
$C^{-1} \sqrt{g} \leqslant F_{\mathcal{C}} \leqslant C\sqrt{g}.$
\end{cor}
\begin{proof}
Take the metric $g$ given for $z\in \mathcal{C}\smallsetminus B(o,R)$ by $\sqrt{g_z} = \mathtt{h}_{z^+}$, where $z^+ = [oz)\cap \partial \mathcal{C}$, and $\mathtt{h}_{z^+}$ is given by the last lemma.
\end{proof}
We will need the following version of Proposition \ref{prop_hilbert_are_bilipschitz_at_infinity} in section \ref{sec_bottom_of_essential_spectrum}:
\begin{cor}\label{cor_result_for_essential_spectrum}
Let $(\mathcal{C},\d)$ be a regular Hilbert geometry. To each $x\in\partial \mathcal{C}$, we can associate a (non-complete) Riemannian hyperbolic metric $\h_x$ defined on an open neighborhood $\mathcal{O}_x$ of $[ox)$ which satisfies the following properties.
\begin{enumerate}
\item The application $x\longmapsto \h_x$ is continuous.
\item We have $\displaystyle \bigcup_{x\in\partial \mathcal{C}} \mathcal{O}_x = \mathcal{C}$.
\item The metric $\mathtt{h}_x$ has the same geodesics as $F_{\mathcal{C}}$ on $\mathcal{O}_x$.
\item Let $C>1$ and
$$\mathcal{U}_x(C) = \left\{z\in\mathcal{O}_x,\ C^{-1} \leqslant \frac{F_{\mathcal{C}}(z,\cdot)}{\mathtt{h}_x(z,\cdot)} \leqslant C\right\}.$$
There exists $R=R(C)$ such that, for any $x\in\partial \mathcal{C}$, the intersection of $\mathcal{U}_x(C)$ with $\mathcal{C}\smallsetminus B(o,R)$ is an open neighborhood of $[ox)$ in $\mathcal{C}\smallsetminus B(o,R)$. In particular, we have $\displaystyle \mathcal{C}\smallsetminus B(o,R) \subset \bigcup_{x\in\partial \mathcal{C}} \mathcal{U}_x(C)$.
\end{enumerate}
\end{cor}
\begin{proof}
We use the objects introduced in the proof of Proposition \ref{prop_hilbert_are_bilipschitz_at_infinity}. We let $\h_x$ be the metric defined by the osculating sphere $S_x$ at $x$ which center is $o$. This is a metric on $\mathcal{O}_x = {\mathcal{E}}_x \cap \mathcal{C}$, which is an open neighborhood of $[ox)$. It is immediate that $h_x$ and $\mathcal{O}_x$ satisfy the first three points. For the fourth one, pick $\varepsilon>0$ and consider the ellipsoids ${\mathcal{E}}_x^+$ and ${\mathcal{E}}_x^-$ which depend on $\varepsilon$. Remark that, as ${\mathcal{E}}_x^- \subset S_x \subset {\mathcal{E}}_x^+$ and ${\mathcal{E}}_x^- \subset \mathcal{C} \subset {\mathcal{E}}_x^+$, we always have
$$\frac{\h^+_x(z,\cdot)}{\h^-_x(z,\cdot)} \leqslant \frac{F_{\mathcal{C}}(z,\cdot)}{\mathtt{h}_x(z,\cdot)} \leqslant \frac{\h^-_x(z,\cdot)}{\h^+_x(z,\cdot)},$$
for all $z\in {\mathcal{E}}_x^-$.
Now, we proved above that there is some $R(\varepsilon)\geqslant 0$ such that, for all $z\in [ox)\smallsetminus B(o,R(\varepsilon))$,
$$\frac{\h^-_x(z,\cdot)}{\h^+_x(z,\cdot)}\leqslant \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} + \eps.$$
Hence the intersection of the set $\mathcal{U}_x\left(\sqrt{\frac{1+\varepsilon}{1-\varepsilon}} + 2\eps\right)$ with $\mathcal{C}\smallsetminus B(o,R(\varepsilon))$ is an open neighborhood of $[ox)$ in $\mathcal{C}\smallsetminus B(o,R(\varepsilon))$. Since $\varepsilon>0$ is arbitrary, this proves the fourth point.
\end{proof}
\begin{rem}
Note that the metric $\h_x$ in this Corollary is different from the one in in Proposition \ref{prop_hilbert_are_bilipschitz_at_infinity}. In particular, the metric $\h_x$ of the Corollary is independent of $C$.
\end{rem}
\section{Bottom of the spectrum for asymptotically Riemannian metrics} \label{sec_bottom_of_spectrum}
Let $F$ be a $C^2$ Finsler metric on a manifold $M$. Let $\Omega^F$ be the Holmes--Thompson volume for $F$. The \emph{volume entropy} $h$ of $F$ is defined by
\begin{equation*}
h := \limsup_{R \rightarrow +\infty} \frac{1}{R}\ln \int_{B^F(R)} \Omega^F.
\end{equation*}
In this section, we will show the following
\begin{thm}\label{thm_lambda_1_for_riem_at_infinity}
Let $F$ be an asymptotically Riemannian $C^2$ Finsler metric on a $n$-manifold $M$. Let $h$ be the volume entropy of $F$ and $\lambda_1$ be the bottom of the spectrum of the Finsler Laplacian $-\Delta^F$. Then,
\begin{equation*}
\lambda_1 \leqslant h^2/4 .
\end{equation*}
\end{thm}
The idea of the proof of Theorem \ref{thm_lambda_1_for_riem_at_infinity} follows the Riemannian one: we show that we can choose $s$ such that the function $e^{-s d(O,x)}$ has a Rayleigh quotient as close as we want to $h^2/4$. The difficulty is in the control of the Rayleigh quotient. In Section \ref{subsec_control_of_symbol} we show how we can manage to control the Rayleigh quotient by controlling the symbol of the Finsler Laplacian.
As we proved that regular Hilbert geometries are asymptotically Riemannian (Corollary \ref{cor_suffit}), and we know that the volume entropy is $n-1$ (\cite{ColboisVerovic:hilbert_geometry}), we deduce the upper bound in Theorem \ref{thmintro_bottom}:
\begin{cor}
Let $\left(\mathcal{C}, d_{\mathcal{C}} \right)$ be a regular Hilbert geometry. Let $\lambda_1$ be the bottom of the spectrum of the Finsler Laplacian $-\Delta^{F_{\mathcal{C}}}$. Then
\begin{equation*}
\lambda_1 \leqslant \frac{(n-1)^2}{4} \, .
\end{equation*}
\end{cor}
Note that, for generic asymptotically Riemannian Finsler metrics, we do not always have $\lambda_1 >0$, even when the volume entropy is positive. Indeed, there exists examples of Riemannian metrics on the universal cover of a manifold such that the volume entropy is positive and $\lambda_1=0$ (for instance the solvmanifold described in \cite{BolsinovTaimanov}). However, in the case of regular Hilbert metrics, this is not possible as the next lemma asserts, which gives the lower bound of Theorem \ref{thmintro_bottom}.
\begin{lem}
Let $\left(\mathcal{C}, d_{\mathcal{C}} \right)$ be a regular Hilbert geometry. Let $\lambda_1$ be the bottom of the spectrum of the Finsler Laplacian $-\Delta^{F_{\mathcal{C}}}$. Then $\lambda_1 >0$.
\end{lem}
\begin{proof}
By \cite{ColboisVerovic:hilbert_geometry}, we know that a regular Hilbert metric is bi-Lipschitz equivalent to the hyperbolic space, so, by \cite[Theorem 4]{BarthelmeColbois_eigenvalue_control} (that we recall below) and the Min-Max principle, we deduce that the $\lambda_1$ of $F_{\mathcal{C}}$ is bounded below by $C^{-1} (n-1)$, where $C>1$ is a constant depending on $n$ and the bi-Lipschitz control.
\end{proof}
\begin{thm} [Barthelm\'e--Colbois \cite{BarthelmeColbois_eigenvalue_control}]
Let $F$ and $F_0$ be two Finsler metrics on an $n$-manifold $M$.
Suppose that there exists $C >1$ such that, for any $(x,v) \in TM\smallsetminus \{0\}$,
\begin{equation*}
C^{-1} \leq \frac{F(x,v)}{F_{0}(x,v)} \leq C.
\end{equation*}
Let $C_1$ and $C_2$ be the quasireversibility constants of $F$ and $F_0$ respectively. Then, there exists a constant $K\geq 1$, depending on $C$, $C_1$, $C_2$ and $n$, such that, for any $f \in H^1(M)$,
\begin{equation*}
C^{-K} \leq \frac{E^F(f)}{E^{F_{0}}(f)} \leq C^K.
\end{equation*}
\end{thm}
Note that in \cite{BarthelmeColbois_eigenvalue_control} this Theorem is stated for $M$ compact, but stays true for non-compact manifolds without any change to the proof.
\subsection{Control of the symbol for pointwise bi-Lipschitz metrics} \label{subsec_control_of_symbol}
In this section we prove that, given a bi-Lipschitz control of a Finsler metric by a Riemannian one, we can control the symbol of the Finsler Laplacian by the dual Riemannian metric. Note that this result is not as clear as in Riemannian geometry, as the symbol of the Finsler Laplacian a priori depends on derivatives of the Finsler metric.
\begin{prop} \label{prop_control_of_sigma}
Let $F$ be a Finsler metric on a $n$-manifold $M$, $x \in M$, and $g_x$ a scalar product on $T_xM$.
Suppose that there exists $C \geqslant 1$ such that, for all $v \in T_xM\smallsetminus\{0\}$,
\begin{equation*}
C^{-1} \leqslant \frac{F(x,v)}{\sqrt{g_x(v,v)}} \leqslant C.
\end{equation*}
Then there exists a constant $C'\geqslant 1$, depending only on $C$ and $n$, such that, for all $p \in T^{\ast}_x M$,
\begin{equation*}
C'^{-1} \leqslant \frac{\sigma_F(p,p)}{g_x^{\ast}(p,p)} \leqslant C'.
\end{equation*}
Furthermore,
\begin{equation*}
\lim_{C\rightarrow 1} C'(C,n) = 1.
\end{equation*}
\end{prop}
In \cite{BarthelmeColbois_eigenvalue_control} the first two authors gave a proof of the existence of a $C'$ satisfying the inequality, but not the limit condition. Hence, we here do the proof with a bit more care to ensure this second condition.
Let us fix a $C^2$ Riemannian metric $F_0$ on $M$ such that $F_0(x, \cdot) = \lVert \cdot \rVert_{g_x}$. Let $X_0$ and $X$ be the geodesic vector fields associated with $F_0$ and $F$ respectively. There exists a function $m \colon M \rightarrow (0,+\infty)$ and a vertical vector field $Y \colon HM \rightarrow VHM$ such that $X = m X_0 + Y$. Actually, we have $m=\frac{F_0}{F}$.\\
Before going on to the proof, we start by stating some results that we will need (the proofs are quite elementary and can be found in \cite{BarthelmeColbois_eigenvalue_control}):
\begin{lem} \label{lem_alpha_and_omega}
Let $F$ and $F_0$ be two Finsler metrics on $M$, $X$ and $X_0$ the associated geodesic vector fields. Let $m \colon HM \longrightarrow \R$ be the function $m=\frac{F_0}{F}$ and $\mu \colon M \longrightarrow \R$ be defined by
\begin{equation*}
\mu (x) := \left(\voleucl \left(\mathbb S^{n-1} \right) \right)^{-1} \int_{H_x^{\ast} M} \left(\frac{F_0^{\ast}}{F^{\ast}} \right)^n \left(\ell_{F_0}^{-1}\right)^{\ast} \alpha^{F_0},\ x\in M.
\end{equation*}
Then $X = m X_0 + Y$ for some vertical vector field $Y \in VHM$, $\Omega^F = \mu \Omega^{F_0}$ and
$$\alpha^F_{(x,\xi)} = \frac{1}{\mu(x)} \left(\frac{F_0^{\ast}}{F^{\ast}}(\ell_{F}(\xi)) \right)^n \left(\ell_{F_0}^{-1} \circ \ell_F \right)^{\ast} \alpha^{F_0}_{(x,\xi)}.$$
\end{lem}
\begin{lem} \label{lem_control_of_m_mu}
Let $F$ and $F_0$ be two Finsler metrics on a $n$-manifold $M$. Suppose that for some $x\in M$, there exists $C \geqslant1$ such that, for any $v \in T_xM\smallsetminus\{0\}$,
\begin{equation*}
C^{-1} \leqslant \frac{F(x,v)}{F_{0}(x,v)} \leqslant C.
\end{equation*}
Then for any $v \in T_xM\smallsetminus\{0\}$, $\xi \in H_xM$, we have
\begin{align}
C^{-1} &\leqslant \frac{F^{\ast}(x,v)}{F^{\ast}_{0}(x,v)} \leqslant C, \label{eq_dual_control}\\
C^{-n} &\leqslant \mu(x) \leqslant C^n \label{eq_control_volume} , \\
C^{-1} &\leqslant m(x,\xi) \leqslant C \label{eq_control_m}.
\end{align}
\end{lem}
Note that the result was stated in \cite{BarthelmeColbois_eigenvalue_control} for a uniform bi-Lipschitz control (that is, $C$ was supposed not to depend on $x\in M$), but the proof stays exactly the same in this case of pointwise bi-Lipschitz control.
\begin{proof}[Proof of Proposition \ref{prop_control_of_sigma}]
Let $p \in T_x^{\ast}M\smallsetminus\{0\}$ be fixed. Let $\lVert \cdot \rVert_{g_x^{\ast}}$ be the norm on $T_x^{\ast}M$ dual to the scalar product $g_x$. We suppose that $\lVert p\rVert_{g_{x}^{\ast}} = 1$. Let $\phi \colon M \rightarrow \R$ be a smooth function such that $\phi(x) = 0$ and $d\phi_x = p$. Then the norm of $p$ for the symbol metric $\sigma^F$ is
\begin{equation*}
\lVert p\rVert_{\sigma^F}^2 = \frac{n}{\voleucl\left(\mathbb S^{n-1} \right)} \int_{H_xM} \left(L_X \pi^{\ast} \phi \right)^2 \alpha^F.
\end{equation*}
Let us write $c_n:= n \left(\voleucl\left(\mathbb S^{n-1}\right) \right)^{-1}$.
Let $F_0$ be a $C^2$ Riemannian metric such that $F_0(x, \cdot) = \lVert \cdot \rVert_{g_x}$. Let $X_0$ and $X$ be the geodesic vector fields associated with $F_0$ and $F$ respectively. There exists $m \colon M \rightarrow \R$ and $Y \colon HM \rightarrow VHM$ such that $X = m X_0 + Y$, so, using Lemma \ref{lem_alpha_and_omega} and the change of variable formula, we get
\begin{align*}
\lVert p\rVert_{\sigma^F}^2 &= c_n \int_{H_xM} m^2 \left(L_{X_0} \pi^{\ast} \phi \right)^2 \left(\ell_{F_0}^{-1} \circ \ell_F \right)^{\ast} \left[ \mu^{-1} \left(\frac{F_0^{\ast}}{F^{\ast}} \circ \ell_{F_0} \right)^n \alpha^{F_0} \right] \\
&= c_n \int_{H_xM} \left(m\circ \ell_{F}^{-1} \circ \ell_{F_0} \right)^2 \left(L_{X_0} \pi^{\ast} \phi \circ \ell_{F}^{-1} \circ \ell_{F_0} \right)^2 \mu^{-1} \left(\frac{F_0^{\ast}}{F^{\ast}} \circ \ell_{F_0} \right)^n \alpha^{F_0}.
\end{align*}
Now, using Lemma \ref{lem_control_of_m_mu}, we have that
\begin{align*}
\lVert p\rVert_{\sigma^F}^2 &\leqslant c_n C^{2n +2} \int_{H_xM} \left(L_{X_0} \pi^{\ast} \phi \circ \ell_{F}^{-1} \circ \ell_{F_0} \right)^2 \alpha^{F_0},\\
\lVert p\rVert_{\sigma^F}^2 &\geqslant c_n C^{-2n -2} \int_{H_xM} \left(L_{X_0} \pi^{\ast} \phi \circ \ell_{F}^{-1} \circ \ell_{F_0} \right)^2 \alpha^{F_0}.
\end{align*}
That means we have
\begin{align*}
\frac{\lVert p\rVert_{\sigma^F}^2}{ \lVert p\rVert_{\sigma^{F_0}}^
|
2} &\leqslant
C^{2n +2}\frac{\displaystyle \int_{S^{F_0}_xM} p(\mathcal{L}_{F}^{-1} \circ \mathcal{L}_{F_0} (u))^2 \alpha^{F_0}(u)}{\displaystyle\int_{S^{F_0}_xM} p(u)^2 \alpha^{F_0}(u)}.
\end{align*}
But, by continuity of $\mathcal{L}$ (Lemma \ref{lem_continuity_l}), we have $\mathcal{L}_{F}^{-1} \circ \mathcal{L}_{F_0} (u) = u + \varepsilon(u)$ with $F_0(\varepsilon(u))\leqslant \varepsilon(C)$, where $C\longmapsto \varepsilon(C)$ is a continuous function such that $\varepsilon(0)=0$. This gives $$p(\mathcal{L}_{F}^{-1} \circ \mathcal{L}_{F_0} (u))^2=(p(u) + p(\varepsilon(u)))^2 \leqslant p(u)^2 + p(\varepsilon(u))^2 + |2p(u)p(\varepsilon(u))|$$
and $|p(\varepsilon(u))|\leqslant \varepsilon(C)\|p\|_{\sigma^{F_0}}$. So we get, using Cauchy-Schwarz inequality,
\begin{align*}
\frac{\lVert p\rVert_{\sigma^F}^2}{ \lVert p\rVert_{\sigma^{F_0}}^2} &\leqslant
\frac{C^{2n +2}}{\lVert p\rVert_{\sigma^{F_0}}^2}\left((1+\varepsilon(C)) \lVert p\rVert_{\sigma^{F_0}}^2 + 2\left(\int_{S^{F_0}_xM} p(u)^2 \alpha^{F_0}(u)\right)^{1/2}\left(\int_{S^{F_0}_xM} p(\varepsilon(u))^2 \alpha^{F_0}(u)\right)^{1/2}\right)\\\\
&\leqslant C^{2n +2} (1+2\varepsilon(C)).
\end{align*}
The same computations also gives the lower bound.
\end{proof}
\subsection{$\lambda_1$ and volume entropy} \label{subsec_lambda_and_entropy}
We prove here Theorem \ref{thm_lambda_1_for_riem_at_infinity}. Let $o$ be a base point on $M$. For any $x \in M$, define $\rho(x) := d(o,x)$, with $d$ the Finslerian distance.
\begin{claim}
For any $s\in \R$ such that $2s > h$, we have $e^{-s\rho(\cdot)} \in L^2 \left( M,\Omega^F\right)$.
\end{claim}
\begin{proof}
This fact is straightforward, just using the definition of the volume entropy.
\end{proof}
Choose $C>1$. As $F$ is asymptotically Riemannian, there exists a compact subset $K_C$ of $M$ and, for any $x\in M\smallsetminus K_C$, a scalar product $g_x$ on $T_xM$ such that, for any $v\in T_xM\smallsetminus\{0\}$,
\begin{equation*}
C^{-1} \leqslant \frac{F(x,v)}{\sqrt{g_x(v,v)}} \leqslant C.
\end{equation*}
Now let $R(C)>0$ such that the Finslerian metric ball $B^F(o,R(C)) \subset M$, of center $o$ and radius $R(C)$, contains $K_C$. Set
\begin{equation*}
f_C(x) := \begin{cases}
e^{-s R(C)} \quad \text{if } x \in B^F(o,R(C)) \\
e^{-s \rho(x)} \quad \text{if } x \in M \smallsetminus B^F(o,R(C)).
\end{cases}
\end{equation*}
We will start by giving an upper bound on the energy of $f_C$. Let $\lVert \cdot \rVert_{\sigma^F}$ be the norm given by the symbol of $F$. We have
\begin{equation*}
E^F(f_C) = \frac{n}{\voleucl(\mathbb S^{n-1})} \int_{HM} \left(L_X \pi^{\ast} f \right)^2 \ada = \int_M \lVert df \rVert_{\sigma^F}^2 \Omega^F.
\end{equation*}
Hence, if we set $U_C:= M \smallsetminus B^F(o,R(C))$
\begin{equation*}
E^F(f_C) = \int_{U_C} s^2\lVert d\rho_x \rVert_{\sigma^F_x}^2 e^{-2s \rho(x)} \Omega^F(x).
\end{equation*}
Now, by Proposition \ref{prop_control_of_sigma}, there exists $C'\geqslant 1$ such that, for any $x \in U_C$,
\begin{equation*}
\lVert d\rho_x \rVert_{\sigma^F_x} \leqslant C' \lVert d\rho_x \rVert_{g_x^{\ast}} \leqslant C C' \lVert d\rho_x \rVert_{F^{\ast}_x},
\end{equation*}
where the last inequality holds because a $C$-bi-Lipschitz control of two Finsler metrics implies a $C$-bi-Lipschitz control of their dual metrics (see for instance \cite{BarthelmeColbois_eigenvalue_control}).
By definition,
\begin{equation*}
\lVert d\rho_x \rVert_{F^{\ast},x} = \sup \lbrace d\rho_x(v) \mid v \in T_xM, F(x,v)=1 \rbrace = 1,
\end{equation*}
because $\rho$ is the distance function of $F$.
So we have obtained that
\begin{equation*}
E^F(f_C) \leqslant s^2 C^2 C'^2 \int_{U_C} e^{-2s \rho(x)}\Omega^F(x).
\end{equation*}
We also have that
\begin{equation*}
\int_{M} f_C(x)^2 \Omega^F(x) = \int_{B^F(o,R(C))} e^{-2s R(C)}\Omega^F(x) + \int_{U_C} e^{-2s \rho(x)}\Omega^F(x) \geqslant \int_{U_C} e^{-2s \rho(x)} \Omega^F(x).
\end{equation*}
Therefore,
\begin{equation*}
R^F(f_C) = \frac{E^F(f_C)}{ \int_{M} f_C(x)^2\Omega^F(x)} \leqslant \frac{s^2 C^2 C'^2 \int_{U_C} e^{-2s \rho(x)}\Omega^F(x)}{\int_{U_C} e^{-2s \rho(x)}\Omega^F(x)} = s^2 C^2 C'^2.
\end{equation*}
This is true for any $s > h/2$ and any $C >1$. Since $\lim_{C\rightarrow 1} C' = 1$, we get
\begin{equation*}
\lambda_1 = \inf_{f \in L^2(M, \Omega^F)} R^F(f) \leqslant \frac{h^2}{4}\, .\qedhere
\end{equation*}
This finishes the proof of Theorem \ref{thm_lambda_1_for_riem_at_infinity}.
\subsection{Dirichlet spectrum}
By a slight modification of the above proof, we can show that the same bound holds for the first Dirichlet eigenvalue of an asymptotically Riemannian manifold $M$ from which we removed a compact set $K$, \emph{provided} that we know that the function $e^{-h \rho(x)/2}$ is \emph{not} in $L^2(M)$. For a general (asymptotically) Riemannian manifold, this is probably not true. But it is true for example on the universal cover of a compact negatively curved Riemannian manifold: in this case, Margulis \cite{margulis, margulisbook} proved that, when $R$ goes to infinity, the area of the sphere of radius $R$ is equivalent to $C e^{hR}$, for some constant $C>0$, which allows to conclude. We will see below that this argument also applies to regular Hilbert geometries.
Recall that if $K$ is a compact sub-manifold of $M$ of the same dimension, the Dirichlet spectrum on $M\smallsetminus K$ is the spectrum of the operator $-\Delta^F$ seen on the space obtained by completion of $C^{\infty}_{0}(M\smallsetminus K)$, the space of smooth functions with compact support in $M\smallsetminus K$, under the norm given by the sum of the $L2$-norm and the energy. The first eigenvalue can still be obtained via the infimum of the Rayleigh quotient.
\begin{cor} \label{cor_bound_for_dirichlet_eigenvalue}
Let $(M, F)$ be an asymptotically Riemannian manifold and $K$ a compact sub-manifold of $M$ of the same dimension. Let $\lambda_1(M\smallsetminus K)$ be the bottom of the Dirichlet spectrum of $-\Delta^F$ on $M\smallsetminus K$. Let $o \in M$ and $\rho(x):= d(o,x)$. If the function $x\longmapsto e^{-h \rho(x)/2}$ is not in $L^2(M)$, then
\begin{equation*}
\lambda_1(M\smallsetminus K) \leqslant \frac{h^2}{4} \, .
\end{equation*}
\end{cor}
\begin{proof}
We use the same notations as above: Let $C >1$, and $R(C)$ be such that, outside of $B(o,R(C))$, $F$ is $C$-bi-Lipschitz to a Riemannian metric. By choosing a larger $R(C)$ if necessary, we can assume that $K \subset B(o,R(C))$.
Now, we just need to modify a tiny bit our test function $f_C$ from above so that it is zero on $\partial K$, and show that the Rayleigh quotient is still as close to $h^2/4$ as we want.\\
Let $f_C$ be a function such that
\begin{equation*}
f_C(x) := \begin{cases}
0 \quad \text{if } x \in \partial K\\
e^{-s \rho(x)} \quad \text{if } x \in M \smallsetminus B^F(o,R(C)),
\end{cases}
\end{equation*}
and, furthermore,
\[
\int_{B^F(o,R(C)) \smallsetminus K} \lVert df_C \rVert_{\sigma^F}^2\ \Omega^F \leqslant 1.
\]
Such a function exists if $R(C)$ is large enough.
Hence, if we set again $U_C:= M \smallsetminus B^F(o,R(C))$, we obtain as above that
\begin{equation*}
E^F(f_C) \leqslant 1 + s^2 C^2 C'^2 \int_{U_C} e^{-2s \rho(x)}\ \Omega^F(x).
\end{equation*}
Thus,
\begin{equation*}
R^F(f_C) \leqslant \frac{1}{\int_M e^{-2s\rho(x)}\ \Omega^F(x)} + s^2 C^2 C'^2
\end{equation*}
Now, as $x\longmapsto e^{-h\rho(x)/2}$ is not in $L^2(M)$, $2s$ can be taken close enough to $h$, so that $\int_M e^{-2s\rho(x)} \Omega^F(x)$ arbitrarily large. Finally, as $C$ can be taken arbitrarily close to $1$ and $\lim_{C \rightarrow 1} C'(C) =1$, we obtain that
\[
\inf R^F(f_C) \leq \frac{h^2}{4} \, ,
\]
which ends the proof.
\end{proof}
Using this, we can now prove the corresponding result about regular Hilbert geometries, which will be useful to compute the bottom of the essential spectrum in the next section.
\begin{cor} \label{cor_dirichlet_spectrum_for_hilbert}
Let $\left(\mathcal{C}, d_{\mathcal{C}} \right)$ be a regular Hilbert geometry and $K$ be a compact subset of $\mathcal{C}$ with smooth boundary. Let $\lambda_1(\mathcal{C} \smallsetminus K)$ be the bottom of the Dirichlet spectrum of $-\Delta^{F_{\mathcal{C}}}$ on $\mathcal{C} \smallsetminus K$. Then
\begin{equation*}
\lambda_1(\mathcal{C} \smallsetminus K) \leqslant \frac{(n-1)^2}{4} \, .
\end{equation*}
\end{cor}
\begin{proof}
Let $o \in \mathcal{C}$ and $\rho(x) := \d(o,x)$. Thanks to corollary \ref{cor_bound_for_dirichlet_eigenvalue}, we only have to show that the function $x\longmapsto e^{-(n-1)\rho(x)/2}$ is not in $L^2(\mathcal{C}, \Omega^{F_{\mathcal{C}}})$.\\
In \cite{ColboisVerovic:hilbert_geometry}, the second and fourth authors gave a precise evaluation of the volume form of a regular Hilbert geometry. Their computations imply in particular that there exists some constant $C >0$ such that, for any measurable function $f:[0,+\infty)\longrightarrow \R$,
\[
\int f\circ\rho\ \Omega^{F_C} \geqslant C \int_0^{+\infty} f(r)e^{(n-1)R}\ dr.
\]
(See the proof of Theorem 3.1 in \cite{ColboisVerovic:hilbert_geometry}. The computations are done for the Busemann--Hausdorff volume, but the ratio between Busemann--Hausdorff and Holmes--Thompson volumes is uniformly bounded, with bounds depending only on the dimension (see for instance \cite{BuragoBuragoIvanov}), so their result applies.)\\
The conclusion is immediate:
\[
\int e^{-(n-1)\rho(x)}\Omega^{F_C}(x) \geqslant C \int_0^{+\infty} dr = +\infty \qedhere
\]
\end{proof}
\section{Bottom of the essential spectrum} \label{sec_bottom_of_essential_spectrum}
Coming back to regular Hilbert geometries, we will now study the essential spectrum and prove Theorem \ref{thmintro_essential_bottom}.
\begin{thm}\label{thm_essential_spectrum}
Let $\left(\mathcal{C}, d_{\mathcal{C}} \right)$ be a regular Hilbert geometry. Let $\sigma_{\textrm{ess}}(F_{\mathcal{C}})$ be the essential spectrum of $-\Delta^{F_{\mathcal{C}}}$. Then
\begin{equation*}
\inf \sigma_{\textrm{ess}}(F_{\mathcal{C}}) = \frac{(n-1)^2}{4} \,.
\end{equation*}
\end{thm}
So, if the $\lambda_1$ of a regular Hilbert geometry is strictly less than $(n-1)^2/4$, then it is a true eigenvalue, contrarily to the hyperbolic case where the $\lambda_1$ is just the infimum of the spectrum.
Note that, in the next section, we will construct examples of Hilbert geometries with eigenvalue strictly smaller than $(n-1)^2/4$. Indeed, we will construct examples with arbitrarily many, arbitrarily small eigenvalues.
To prove our result on the essential spectrum, we will use the Cheeger inequality for weighted Laplacians and control the Cheeger constant in regular Hilbert geometries using Corollaries \ref{cor_suffit} and \ref{cor_result_for_essential_spectrum}.
\subsection{Cheeger constant, weighted Laplacians and essential spectrum}
If $F$ is a Finsler metric on a manifold $M$, then (see \cite{moi:natural_finsler_laplace}) $\Delta^F$ is a weighted Laplacian with symbol $\sigma^F$ and symmetric with respect to the volume $\Omega^F$. Hence, we have the following lower bound for the first eigenvalue of $-\Delta^F$:
\begin{prop}[Cheeger Inequality] \label{prop_Cheeger_inequality}
Let $M$ be a non-compact manifold and $F$ a Finsler metric on $M$. Let $d\vol^{\sigma^F}$ be the volume form of the Riemannian metric dual to $\sigma^F$, $d\mathrm{area}^{\sigma^F}$ the associated area element and $\mu \colon M \rightarrow \R$ the function such that $\Omega^F = \mu d\mathrm{vol}^{\sigma^F}$. Set
\begin{equation*}
h_{\mathrm{Cheeger}}^{\sigma,\Omega}(M) := \inf_{D} \left\{ \frac{\int_{\partial D} \mu(x) d\mathrm{area}^{\sigma^F} }{\int_{D} d\mathrm{vol}^{\sigma^F} } \right\},
\end{equation*}
where the infimum is taken over all compact domains $D$ with smooth boundary.
If $\lambda_1$ is the bottom of the spectrum of $-\Delta^F$ on $M$, then
\begin{equation*}
4 \lambda_1 \geqslant h_{\mathrm{Cheeger}}^{\sigma,\Omega}(M)^2.
\end{equation*}
\end{prop}
We do not provide the proof as it is the exact same as for the traditional Cheeger inequality (see for instance \cite{SchoenYau:lectures}). To study the essential spectrum, we also need the decomposition principle of Donnelly and Li, which states that the essential spectrum is independent of the behavior of the operator on any compact subset:
\begin{prop}[Decomposition Principle of Donnelly and Li \cite{DonnellyLi}] \label{prop_decomposition_principle}
Let $M$ be a non-compact manifold and $F$ a Finsler metric on $M$. Let $M'$ be a compact sub-manifold of $M$ of same dimension. Then
\begin{equation*}
\sigma_{\textrm{ess}}(M,F) = \sigma_{\textrm{ess}}(M \smallsetminus M',F).
\end{equation*}
In particular,
\begin{equation*}
h_{\mathrm{Cheeger}}^{\sigma,\Omega}(M \smallsetminus M')^2 \leqslant 4 \inf \sigma_{\textrm{ess}}(F).
\end{equation*}
\end{prop}
We also have the following known result. As we did not find any reference, we provide a proof.
\begin{lem}
Let $\{M'_i\}$ be an increasing family of compact sub-manifolds of $M$ of the same dimension, such that $\cup_i M'_i = M$. Then
$$\inf \sigma_{\textrm{ess}}(M,F) = \lim_{i\to \infty} \l_1(M \smallsetminus M'_i,F),$$
where $\l_1(M \smallsetminus M'_i,F)$ denotes the Dirichlet spectrum of $M \smallsetminus M'_i$.
\end{lem}
\begin{proof}
Let us write $\lambda_1^i := \l_1(M \smallsetminus M'_i,F) $.
By the Decomposition principle, we have that, for all $i$, $\lambda_1^i \leqslant \inf \sigma_{\textrm{ess}}(M,F)$. We suppose that $\lambda_1^i < \inf \sigma_{\textrm{ess}}(M,F)$, otherwise we are done. Let $\lambda = \lim_{i\to \infty} \lambda_1^i $, which exists because, as $\{M'_i\}$ is increasing, the sequence $\{\lambda_1^i\}$ is nondecreasing. To prove that $\lambda$ is in the essential spectrum, we are going to show that, for any $\eps>0$, there exists a family of functions $f_i \in L^2(M)$, with disjoint supports, such that
$$
\lVert -\Delta^F f_i - \lambda f_i \rVert \leqslant \eps \lVert f_i \rVert,
$$
where $\lVert \cdot \rVert$ denotes the $L^2$-norm with respect to $\Omega^F$.
Let $\eps>0$. As $\lambda_1^i$ is an eigenvalue with finite multiplicity of $-\Delta^F$ on $M \smallsetminus M'_i$, we can find a function $f_i \in L^2(M \smallsetminus M'_i)$ with compact support such that
$$
\lVert -\Delta^F f_i - \lambda_1^i f_i \rVert \leqslant \eps \lVert f_i \rVert.
$$
Up to taking a subsequence, we can suppose that $\text{supp} f_i \subset M'_{i+1}$, so that $\text{supp} f_i \subset M'_{i+1} \smallsetminus M'_i$. Hence, for any $i \neq j$, we have $\text{supp} f_i \cap \text{supp} f_j = \emptyset$. So, for $i$ large enough,
\begin{equation*}
\lVert -\Delta^F f_i - \lambda f_i \rVert \leqslant \lVert -\Delta^F f_i - \lambda_1^i f_i \rVert + |\lambda - \lambda_1^i| \lVert f_i \rVert \leqslant 2\eps \lVert f_i \rVert. \qedhere
\end{equation*}
\end{proof}
This gives a part of Theorem \ref{thm_essential_spectrum}.
\begin{cor}
Let $\left(\mathcal{C}, d_{\mathcal{C}} \right)$ be a regular Hilbert geometry. Then
$$\inf \sigma_{\textrm{ess}}(F_{\mathcal{C}}) \leqslant (n-1)^2/4.$$
\end{cor}
\begin{proof}
Pick $o\in\mathcal{C}$. Then Corollary \ref{cor_dirichlet_spectrum_for_hilbert} gives that, for any $i\geqslant 1$, $\l_1(\mathcal{C} \smallsetminus \overline{B(o,i)}) \leqslant (n-1)^2/4$. The previous lemma allows us to conclude.
\end{proof}
\subsection{Essential spectrum of regular Hilbert geometries}
The next few lemmas will allow us to prove the inequality $\inf \sigma_{\textrm{ess}}(F_{\mathcal{C}}) \geqslant (n-1)^2/4$ and thus conclude the proof of Theorem \ref{thm_essential_spectrum}.
Denote by $\sigma$ the symbol of $-\Delta^{F_{\mathcal{C}}}$, by $h_{\mathrm{Cheeger}}^{\sigma,\Omega}$ the weighted Cheeger constant associated with $\sigma$ and $\Omega^{F_{\mathcal{C}}}$ and by $h_{\mathrm{Cheeger}}$ the traditional Cheeger constant for the Riemannian metric dual to $\sigma$.
Let $o \in \mathcal{C}$ be fixed and $K$ a relatively compact open subset of $\mathcal{C}$.
\begin{lem}\label{lem_comparison_between_F_and_sigma}
For any $C>1$, there exists a constant $R = R(C) >0$ and a constant $C_1 = C_1(C) \geqslant1$ such that, on $\mathcal{C} \smallsetminus B(o,R)$, we have:
\begin{equation*}
C_1^{-1} \leqslant \frac{\sigma^{\ast}}{F^2} \leqslant C_1.
\end{equation*}
Furthermore, $C_1$ tends to $1$ as $C$ tends to $1$.
\end{lem}
\begin{proof}
Let $C >1$. According to Corollary \ref{cor_suffit}, there exists $R= R(C)>0$ and a Riemannian metric $g$ on $\mathcal{C}\smallsetminus B(o,R)$ such that $C^{-1} g \leqslant F^2_{\mathcal{C}} \leqslant C g$. By Proposition \ref{prop_control_of_sigma}, there exists a constant $C'=C'(C,n)>1$ such that $(C'C)^{-1} F^2_{\mathcal{C}} \leqslant \sigma^{\ast} \leqslant C'C F^2_{\mathcal{C}}$ on all of $\mathcal{C} \smallsetminus B(o,R) $. Finally, still according to Proposition \ref{prop_control_of_sigma}, $C'C$ tends to $1$ when $C$ tends to $1$, so we can set $C_1=C'C$.
\end{proof}
\begin{lem}\label{lem_cheeger_a_poid_vs_cheeger}
For any $C>1$, there exists a constant $R = R(C) >0$ and a constant $C_2 = C_2(C) >1$ such that
\begin{equation*}
h_{\mathrm{Cheeger}}^{\sigma,\Omega} \left( \mathcal{C} \smallsetminus B(o,R) \right) \geqslant C_2^{-1}h_{\mathrm{Cheeger}} \left( \mathcal{C} \smallsetminus B(o,R) \right).
\end{equation*}
Furthermore, $C_2$ tends to $1$ as $C$ tends to $1$.
\end{lem}
\begin{proof}
Let $C >1$. By Lemma \ref{lem_comparison_between_F_and_sigma}, there exist constants $R= R(C)>0$ and $C_1\geqslant 1$ such that, on $\mathcal{C} \smallsetminus B(o,R) $, we have
$C_1^{-1} F^2_{\mathcal{C}} \leqslant \sigma^{\ast} \leqslant C_1 F^2_{\mathcal{C}}$. Let $\mu \colon M \rightarrow \R$ be the function such that $\Omega^F = \mu d\vol^{\sigma^F}$. By Lemma \ref{lem_control_of_m_mu}, we have $C_1^{-n} \leqslant \mu(x) \leqslant C_1^n$ for any $x \in\mathcal{C} \smallsetminus B(o,R)$. So we get that, for any compact domain $D$ in $\mathcal{C} \smallsetminus B(o,R)$ with smooth boundary,
\begin{align*}
C_1^{-n} \int_{\partial D} d\mathrm{area}^{\sigma} &\leqslant \int_{\partial D} \mu(x) d\mathrm{area}^{\sigma^F} \leqslant C_1^n \int_{\partial D} \mu(x) d\mathrm{area}^{\sigma^F} \\
C_1^{-n} \int_{D} d\mathrm{vol}^{\sigma} &\leqslant \int_{D} \Omega^{F_{\mathcal{C}}} \leqslant C_1^n \int_{D} d\mathrm{vol}^{\sigma}.
\end{align*}
Therefore, setting $C_2 = C_1 ^{2n}$ gives the claim.
\end{proof}
\begin{lem}\label{lem_cheeger_vs_hyperbolic}
For any $C>1$, there exists a constant $R = R(C) >0$ and a constant $C_3 = C_3(C) \geqslant 1$ such that
\begin{equation*}
h_{\mathrm{Cheeger}} \left( \mathcal{C} \smallsetminus B(o,R) \right) \geqslant C_3^{-1} (n-1).
\end{equation*}
Furthermore, $C_3$ tends to $1$ as $C$ tends to $1$.
\end{lem}
\begin{proof}
Let $C>1$. Let $R= R(C)\geqslant0$, $\mathcal{U}_x(C)$ and $\h_x$, $x\in \partial \mathcal{C}$, be given by Corollary \ref{cor_result_for_essential_spectrum}. Let $D$ be a compact domain in $\mathcal{C} \smallsetminus B(o,R)$ with smooth boundary. As the goal is to control the Cheeger constant, we can suppose that $D$ is a convex domain, because convex sets minimize the ratio of area over volume.
For each $x\in \partial \mathcal{C}$, let $K_x$ be a family of open cones with vertex $o$ such that, for any $x \in \partial \mathcal{C}$, $K_x\cap D \subset \mathcal{U}_{x}(C)$ and $ \displaystyle \cup_{x \in \partial \mathcal{C}} K_x$ covers $D$. Such a family exists because $\cup_{x \in \partial \mathcal{C}} \mathcal{U}_x $ openly covers $\mathcal{C} \smallsetminus B(o,R)$. Remark that the boundary of $K_x$ is a union of geodesics of $F_{\mathcal{C}}$, which are also geodesics of $\h_{x}$.
Now, by compactness of $D$, there exist $x_1, \dots, x_k \in \partial \mathcal{C}$ such that $\cup_i K_{x_i}$ openly covers $D$. By choosing the cones $K_{x_i}$ to be smaller if necessary, we can assume that the domain $D$ is partitioned into $\cup_{1 \leqslant i \leqslant k} (K_{x_i} \cap D)$.
\begin{claim} \label{claim_computation_in_hyperbolic}
For any $1 \leqslant i \leqslant k$, we have
\begin{equation*}
\frac{\mathrm{Area}^{\h_{x_{i}}} \left(K_{x_i} \cap \partial D \right) }{ \mathrm{Vol}^{\h_{x_{i}}} \left(K_{x_i} \cap D \right) } \geqslant (n-1).
\end{equation*}
\end{claim}
\begin{proof}[Proof of Claim \ref{claim_computation_in_hyperbolic}]
As $\h_{x_{i}}$ is a hyperbolic metric and the sides of $K_{x_i}$ are geodesics of $\h_{x_{i}}$, we have that
\begin{equation*}
\frac{\mathrm{Area}^{\h_{x_{i}}} \left(K_{x_i} \cap \partial D \right) }{ \mathrm{Vol}^{\h_{x_{i}}} \left(K_{x_i}\right) } \geqslant (n-1).
\end{equation*}
Indeed, as we are in the hyperbolic setting, this can be proved by a direct computation using the divergence formula. As $\mathrm{Vol}^{\h_{x_{i}}} \left(K_{x_i}\right) \geqslant \mathrm{Vol}^{\h_{x_{i}}} \left(K_{x_i} \cap D \right)$, we get the claim.
\end{proof}
For all $1\leqslant i \leqslant k$ holds $ C^{-1} \h_{x_{i}} \leqslant F_{\mathcal{C}}^2 \leqslant C \h_{x_{i}}$ on $\mathcal{U}_{x_{i}}(C)$. As in Lemma \ref{lem_comparison_between_F_and_sigma}, there exists a constant $C_1':=C'_1(C)$ such that, on $\mathcal{U}_{x_{i}}(C)$,
$$C_1'^{-1} \h_{x_{i}} \leqslant \sigma^{\ast} \leqslant C_1' \h_{x_{i}},\ 1\leqslant i \leqslant k,$$ and, furthermore, $\lim_{C\to 1} C'_1(C) =1$.
Hence, for any domain $U$ in $\mathcal{U}_{x_{i}}(C)$, and in particular for $K_{x_i} \cap D $, we have
\begin{align*}
C_1'^{-n-1} \int_{\partial U} d\mathrm{area}^{\h_{x_{i}}} &\leqslant \int_{\partial U} d\mathrm{area}^{\sigma} \leqslant C_1'^{n-1} \int_{\partial U} d\mathrm{area}^{\h_{x_{i}}}, \\
C_1'^{-n} \int_{U} d\mathrm{vol}^{\h_{x_{i}}} &\leqslant \int_{U} d\mathrm{vol}^{\sigma} \leqslant C_1'^n \int_{U} d\mathrm{vol}^{\h_{x_{i}}}.
\end{align*}
So, thanks to Claim \ref{claim_computation_in_hyperbolic}, we get
\begin{equation*}
\frac{\mathrm{Area}^{\sigma} \left(K_{x_i} \cap \partial D \right) }{ \mathrm{Vol}^{\sigma} \left(K_{x_i} \cap D \right) } \geqslant C_1'^{-2n-1} (n-1).
\end{equation*}
Setting $C_3 := C_1'^{2n+1}$, we have
\begin{equation*}
\mathrm{Area}^{\sigma}(\partial D) = \sum_{i=1}^k \mathrm{Area}^{\sigma} (K_{x_i} \cap \partial D)
\
|
. It follows that $f$ is generic.
\end{example}
The next proposition shows that, under mild assumptions, a polyadic set can be recovered from its Stirling kernel (as an appropriate left Kan extension).\footnote{For applications to homomorphism counting in the next section, we will only need the first part of Proposition~\ref{p:reconstruction}. The reader unfamiliar with the concept of Kan extension can safely ignore the second part of the lemma.}
This observation is due to Joyal (for polyadic spaces), see also~\cite{Marques2021}; the proof offered below is a simple adaptation to the discrete case.
For all objects $a\in \A$, we denote by $\Q(a)$ and $\M(a)$, respectively, the poset of quotients of $a$ and the poset of embeddings of $a$; for a definition, see Appendix~\ref{s:fact-systems}. Loosely speaking, the elements of $\Q(a)$ are equivalence classes of quotients from $a$, and the elements of $\M(a)$ are equivalence classes of embeddings into $a$. The category $\A$ is said to be \emph{$\Q$-well-founded} if, for all objects $a\in \A$, the poset $\Q(a)$ is well-founded. That is, any non-empty subset of $\Q(a)$ has a minimal element. Similarly, $\A$ is \emph{$\M$-well-founded} if, for all objects $a\in \A$, the poset $\M(a)$ is well-founded.
\begin{proposition}\label{p:reconstruction}
Let $F\colon \A^\op\to \Set$ be a polyadic set and assume that $\A$ is $\Q$-well-founded. The following statements hold:
\begin{enumerate}[label=(\alph*)]
\item\label{coprod-formula} For all $n\in \A$, \[F(n)\cong \coprod_{n\epi m} \sk{F}(m)\] where the coproduct is indexed by the set $\Q(n)$.
\item\label{Lan-formula} $F\cong \Lan_J \sk{F}$, where $\Lan_J \sk{F}$ is the left Kan extension of $\sk{F}$ along the inclusion functor $J\colon \A_*^\op\into \A^\op$.
\end{enumerate}
\end{proposition}
\begin{proof}
(a) Let $\A$ and $F\colon \A^\op\to\Set$ be as in the statement. For each quotient $f\colon n\epi m$ in $\Q(n)$, the function $F(f)\colon F(m)\to F(n)$ is an injection by Lemma~\ref{l:quot-inj-polyadic}. Let $\xi_f\colon \sk{F}(m)\emb F(n)$ be the obvious restriction of $F(f)$. By the universal property of the coproduct in $\Set$, the family of functions $\{\xi_f \mid f\in \Q(n)\}$ induces a unique map
\[
\xi\colon \coprod_{n\epi m} \sk{F}(m) \to F(n).
\]
We claim that $\xi$ is a bijection. For injectivity, suppose that there are quotients $f_1\colon n\epi m_1$, $f_2\colon n\epi m_2$ and elements $x_1\in \sk{F}(m_1)$ and $x_2\in \sk{F}(m_2)$ such that $\xi(x_1)=\xi(x_2)$. That is, $F(f_1)(x_1)=F(f_2)(x_2)$. By the amalgamation property for $F$ and Lemma~\ref{l:amalg-quotients}, there are quotients $g_1\colon m_1 \epi p$ and $g_2\colon m_2\epi p$ making the square below commute,
\[\begin{tikzcd}
n \arrow[twoheadrightarrow]{r}{f_1} \arrow[twoheadrightarrow]{d}[swap]{f_2} & m_1 \arrow[twoheadrightarrow, dashed]{d}{g_1} \\
m_2 \arrow[twoheadrightarrow, dashed]{r}{g_2} & p
\end{tikzcd}\]
and $y\in F(p)$ such that $F(g_1)(y)=x_1$ and $F(g_2)(y)=x_2$. As $x_1$ and $x_2$ are generic, $g_1$ and $g_2$ must be isomorphisms. In particular, the isomorphism $g_2^{-1}\circ g_1\colon m_1\to m_2$ witnesses the fact that $f_1=f_2$ as elements of $\Q(n)$. Since $\xi$ is injective on each summand, it follows that $x_1=x_2$.
For surjectivity of $\xi$, suppose that $x\in F(n)$ and consider the set
\[
S\coloneqq \{f\in \Q(n)\mid F(f)^{-1}(x)\neq \emptyset\}.
\]
If $S=\emptyset$, then $x\in \sk{F}(n)$ and thus it belongs to the image of $\xi$. Therefore, assume that $S$ is non-empty. Because $\A$ is $\Q$-well-founded, $S$ has a minimal element $g\colon n\epi m$. Let $y\in F(m)$ be such that $F(g)(y)=x$. As $g$ is minimal, $y$ must belong to $\sk{F}(m)$, and so $x$ is in the image of $\xi$.
(b) Fix an arbitrary object $n\in \A$. Consider the composite functor
\[\begin{tikzcd}
G\colon J\down n \arrow{r}{\pi} & \A_*^\op \arrow{r}{\sk{F}} & \Set
\end{tikzcd}\]
where $J\down n$ is the comma category and $\pi$ sends an object $(m, J(m)\to n)$, where $J(m)\to n$ is an arrow in $\A^{\op}$, to its first component $m$. In view of Lemma~\ref{l:amalg-quot}\ref{comma-cat-quot}, we can replace without loss of generality the category $J\down n$ with its full subcategory $\D$ consisting of the pairs $(m, J(m)\to n)$ whose second components correspond to quotients in $\A$. Note that $\D$ is a groupoid, i.e.\ every arrow in $\D$ is an isomorphism. Just observe that a morphism
\[
(m',J(m')\to n)\to (m, J(m)\to n)
\]
in $\D$ corresponds to a commutative triangle
\[\begin{tikzcd}
{} & n \arrow[twoheadrightarrow]{dl} \arrow[twoheadrightarrow]{dr} & {} \\
m \arrow[rightarrowtail]{rr} & {} & m'
\end{tikzcd}\]
in $\A$, and by Lemma~\ref{l:factorisation-properties}\ref{cancellation-e},\ref{isos} the horizontal arrow in necessarily an isomorphism.
Upon choosing representatives, this is equivalent to considering the small diagram
\[\begin{tikzcd}
G'\colon \Q(n)^\op \arrow{r}{\pi'} & \A_*^\op \arrow{r}{\sk{F}} & \Set
\end{tikzcd}\]
where $\Q(n)$ is regarded as a set (i.e., a discrete category) and $\pi'$ sends (the equivalence class of) a quotient $n\epi m$ in $\A$ to its codomain.
The colimit of $G$ thus coincides with the coproduct $\coprod_{n\epi m} \sk{F}(m)$ indexed by the set $\Q(n)$. Moreover, any morphism $h\colon n\to n'$ in $\A$ induces a unique function
\[
\coprod_{n\epi m'} \sk{F}(m') \ \longleftarrow \ \coprod_{n'\epi m} \sk{F}(m)
\]
whose restriction to $\sk{F}(m)$, for all (representatives of equivalence classes of) quotients ${f\colon n'\epi m}$, is the function $\sk{F}(m)\to \sk{F}(m')$ obtained by applying $\sk{F}$ to the bottom horizontal arrow in the following commutative diagram.
\[\begin{tikzcd}
n \arrow{r}{h} \arrow[dashed,twoheadrightarrow]{d} & n' \arrow[twoheadrightarrow]{d}{f} \\
m' \arrow[dashed,rightarrowtail]{r} & m
\end{tikzcd}\]
By item (a), and the colimit formula for pointwise left Kan extensions (cf.\ \cite[Theorem~1 p.~237]{MacLane1998} for the dual statement), we see that $F\cong \Lan_J \sk{F}$.
\end{proof}
We illustrate the previous proposition by means of an example, which also justifies the terminology \emph{Stirling kernel}:
\begin{example}
Let $F\colon \FinSet^\op\to\Set$ be any polyadic set on the category $\FinSet$, and equip the latter with the usual (surjective, injective) factorisation system. We denote by $\uline{n}\in\FinSet$ an $n$-element set. For all non-negative integers $m\leq n$, the number of non-equivalent surjections $\uline{n}\epi \uline{m}$ in $\Q(\uline{n})$ coincides with the number of ways to partition an $n$-element set into $m$ non-empty subsets. This is commonly denoted by $S(n,m)$ and known as the \emph{Stirling number of the second kind} associated with the pair $(n,m)$, see e.g.\ \cite[\S 1.5]{Godsil1993}. Therefore, denoting by $k\, \uline{n}$ the disjoint sum of $k$ copies of $\uline{n}$, Proposition~\ref{p:reconstruction} yields the formula
\[
F(\uline{n})\cong \coprod_{m\leq n} S(n,m) \, \sk{F}(\uline{m})
\]
for all finite sets $\uline{n}$. In particular, if $F$ is a polyadic finite set, we get
\[
|F(\uline{n})|= \sum_{m\leq n} S(n,m) \cdot |\sk{F}(\uline{m})|.
\]
\end{example}
\begin{remark}
The assumption in Proposition~\ref{p:reconstruction} that $\A$ be $\Q$-well-founded is necessary, even when $F$ is a polyadic \emph{finite} set. For instance, let $\A$ be the set $\N$ of natural numbers with the usual total order, regarded as a category. If $\A$ is equipped with the factorisation system $(\Q,\M)$ where $\Q$ consists of all morphisms, and $\M$ consists of the identity morphisms, then $\A_*$ is the category with set of objects $\N$ and no arrows but identities. Note that $\A$ is not $\Q$-well-founded. Just observe that, for all $n\in \A$, the strictly increasing sequence of natural numbers $n < n+1 < n+2 < \cdots$ induces the strictly decreasing sequence
\[
n \epi n+1 \epi n+2 \epi \cdots
\]
in $\Q(n)$. Thus, the latter is not well-founded.
Now, let $a$ be any non-empty finite set, and let $F\colon \A^\op \to \FinSet$ be the constant functor of value $a$. It is not difficult to see that $F$ is a polyadic finite set, and ${\sk{F}\colon \A_*^\op\to\FinSet}$ is the constant functor of value $\emptyset$. Then ${\Lan_J \sk{F}\colon \A^\op\to \FinSet}$ is also the constant functor of value $\emptyset$ and so $F\not\cong \Lan_J \sk{F}$.
\end{remark}
\begin{corollary}\label{cor:kernel-is-polyadic}
Let $\A$ be $\Q$-well-founded. The Stirling kernel of a polyadic (finite) set on $\A$ is a polyadic (finite) set on $\A_*$.
\end{corollary}
\begin{proof}
Let $\A$ be as in the statement and let $F\colon \A^\op\to\Set$ be an arbitrary polyadic set. We claim that $\sk{F}\colon \A_*^\op\to\Set$ has the amalgamation property. Consider embeddings $f_1\colon n\emb m_1$, $f_2\colon n\emb m_2$ and generic elements $s\in \sk{F}(m_1)$ and $t\in\sk{F}(m_2)$ such that $F(f_1)(s)=F(f_2)(t)$. By the amalgamation property for $F$, there exist a commutative square
\[\begin{tikzcd}
n \arrow[rightarrowtail]{r}{f_1} \arrow[rightarrowtail]{d}[swap]{f_2} & m_1 \arrow[dashed]{d}{g_2} \\
m_2 \arrow[dashed]{r}{g_1} & p
\end{tikzcd}\]
in $\A$ and an element $u\in F(p)$ such that $F(g_2)(u)=s$ and $F(g_1)(u)=t$. By Proposition~\ref{p:reconstruction}\ref{coprod-formula}, there exist a quotient $h\colon p\epi q$ and a generic element $v\in \sk{F}(q)$ such that $F(h)(v)=u$. Clearly, we have
\[
F(h\circ g_2)(v)=s \ \text{ and } \ F(h\circ g_1)(v)=t.
\]
It suffices to show that $h\circ g_1$ and $h\circ g_2$ are embeddings, for then the following commutative square in $\A_*$ shows that $\sk{F}$ has the amalgamation property.
\[\begin{tikzcd}
n \arrow[rightarrowtail]{r}{f_1} \arrow[rightarrowtail]{d}[swap]{f_2} & m_1 \arrow[rightarrowtail, dashed]{d}{h\circ g_2} \\
m_2 \arrow[rightarrowtail, dashed]{r}{h\circ g_1} & q
\end{tikzcd}\]
To see that $h\circ g_1$ is an embedding, take its $(\Q,\M)$ factorisation:
\[\begin{tikzcd}[row sep=0.5em]
m_2 \arrow[twoheadrightarrow]{dr}[swap]{e} \arrow{rr}{h\circ g_1} & & q \\
{} & {\cdot} \arrow[rightarrowtail,shorten >=0.5ex]{ur}[swap]{l} &
\end{tikzcd}\]
Let $x\coloneqq F(l)(v)$. As $F(e)(x)=t$ and $t$ is generic, it follows that $e$ is an isomorphism. Therefore, $h\circ g_1$ is an embedding by Lemma~\ref{l:factorisation-properties}\ref{compositions},\ref{isos}. The same proof, mutatis mutandis, shows that $h\circ g_2$ is an embedding.
Finally, note that if $F$ is a polyadic \emph{finite} set, then so is $\sk{F}$.
\end{proof}
\subsection{Pointwise isomorphisms}\label{s:pointwise-isomorphisms}
The main result of this section (Proposition~\ref{pr:pointwise-iso-kernel} below) states that, under appropriate assumptions, whenever two polyadic finite sets are pointwise isomorphic, their Stirling kernels are also pointwise isomorphic. This observation is at the core of the homomorphism counting results presented in Sections~\ref{s:hom-count-finite} and~\ref{s:beyond-loc-finite}.
Recall that two parallel functors $F, G\colon \C\to \D$ are \emph{pointwise isomorphic} if, for all $c\in \C$, there is an isomorphism $\eta_c\colon F(c)\to G(c)$ in $\D$. This contrasts with the concept of \emph{natural isomorphism} between $F$ and $G$, whereby the isomorphisms $\eta_c$ are required to be natural in $c$.
In view of Proposition~\ref{p:reconstruction}, if $F$ and $G$ are polyadic sets whose Stirling kernels $\sk{F}$ and $\sk{G}$ are pointwise isomorphic, then $F$ and $G$ are also pointwise isomorphic. The converse holds for polyadic \emph{finite} sets and follows from an application of the M\"obius inversion formula, due to Gian-Carlo Rota.
Recall that, given a finite\footnote{More generally, throughout this paragraph we could consider a \emph{locally finite} poset, i.e.\ a poset $P$ such that, for all $x,y\in P$, the interval $[x,y]\coloneqq\{z\in P\mid x\leq z\leq y\}$ is finite.} poset $P$, the \emph{incidence algebra} of $P$ has as elements the functions $f\colon P\times P\to \mathbb{R}$ satisfying $f(x,y)=0$ whenever ${x\not\leq y}$. These functions form an associative algebra over the reals with respect to pointwise sum, pointwise multiplication by scalars, and the convolution product $h=fg$ defined as
\[
h(x,y)\coloneqq\sum_{x\leq z\leq y} f(x,z)g(z,y).
\]
The identity element of the incidence algebra is the Kronecker function $\delta(x,y)$ satisfying $\delta(x,y)=1$ if $x=y$ and $\delta(x,y)=0$ otherwise.
The \emph{zeta function} of $P$ is the element $\zeta(x,y)$ of the incidence algebra such that $\zeta(x,y)=1$ if $x\leq y$ and $\zeta(x,y)=0$ otherwise. The function $\zeta$ is invertible in the incidence algebra, see \cite[Proposition~1]{Rota1964}, and its (two-sided) inverse $\mu$ is called the \emph{M\"obius function of $P$}.
\begin{lemma}[M\"obius inversion formula~\cite{Rota1964}]
Let $P$ be a finite poset and let $f_1,f_2\colon P\to \mathbb{R}$ be any two functions. For all $y\in P$,
\[
f_1(y)=\sum_{x\leq y}{f_2(x)} \enspace \Longleftrightarrow \enspace f_2(y)=\sum_{x\leq y}{f_1(x) \mu(x,y)},
\]
where $\mu$ is the M\"obius function of $P$.
\end{lemma}
\begin{proposition}\label{pr:pointwise-iso-kernel}
Suppose that $\A$ is $\Q$-well-founded, and let $F, G$ be polyadic finite sets on $\A$. If $F$ and $G$ are pointwise isomorphic, then so are their Stirling kernels $\sk{F}$ and $\sk{G}$.
\end{proposition}
\begin{proof}
Let $F,G\colon \A^\op\to\FinSet$ be any two polyadic finite sets, and suppose that they are pointwise isomorphic. We must prove that $\sk{F}(n)\cong \sk{G}(n)$ for all $n\in \A$. Let us fix an arbitrary object $n\in \A$. By Proposition~\ref{p:reconstruction}\ref{coprod-formula}, we know that
\[
F(n)\cong \coprod_{n\epi m} \sk{F}(m).
\]
As the left-hand side is a finite set, there exist finitely many quotients
\[
q_1\colon n\epi m_1, \ \ldots, \ q_l\colon n\epi m_l
\]
such that $F(n)\cong \coprod_{i=1}^l\sk{F}(m_i)$. Similarly, there are finitely many quotients
\[
q_{l+1}\colon n\epi m_{l+1}, \ \ldots, \ q_u\colon n\epi m_{u}
\]
such that $G(n)\cong \coprod_{i=l+1}^u\sk{G}(m_i)$. Let $P$ be the sub-poset of $\Q(n)$ defined by the elements $q_1,\ldots, q_u$ and the identity of $n$. For convenience of notation, we let $m_0\coloneqq n$ and let $q_0\colon n \epi m_0$ be the identity. Note that $q_0$ is the top element of $P$. Clearly, $F(n)\cong \coprod_{i=0}^u\sk{F}(m_i)$ and $G(n)\cong \coprod_{i=0}^u\sk{G}(m_i)$.
Consider the functions on $P$ determined, for all $i\in \{0,\ldots, u\}$, by the cardinalities of $F(m_i)$ and $\sk{F}(m_i)$, respectively:
\[
f_1\colon P\to \N, \ \ f_1(q_i)\coloneqq |F(m_i)|
\]
and
\[
f_2\colon P\to \N, \ \ f_2(q_i)\coloneqq |\sk{F}(m_i)|.
\]
Then $f_1(q_0)=\sum_{x\in P} f_2(x)$ and so, by the M\"{o}bius inversion formula,
\[
f_2(q_0)=\sum_{x\in P} f_1(x)\mu(x,q_0)
\]
where $\mu$ is the M\"{o}bius function of the poset $P$. Reasoning in a similar manner for the functions $g_1, g_2\colon P\to \N$ defined by $g_1(q_i)\coloneqq |G(m_i)|$ and $g_2(q_i)\coloneqq |\sk{G}(m_i)|$, we get
\[
g_2(q_0)=\sum_{x\in P} g_1(x)\mu(x,q_0).
\]
By assumption, for all $i\in\{0,\ldots, u\}$, we have $|F(m_i)|=|G(m_i)|$ and so $f_1(q_i)=g_1(q_i)$. We conclude that $f_2(q_0)=g_2(q_0)$, i.e.\ $\sk{F}(n)\cong \sk{G}(n)$.
\end{proof}
If the category $\A$ has pushouts, then Proposition~\ref{pr:pointwise-iso-kernel} can be proved by exploiting the inclusion-exclusion principle (see e.g.\ \cite[\S 2.1]{Stanley1}), rather than the M\"{o}bius inversion formula, in which case we can dispense with the assumption that $\A$ be $\Q$-well-founded. This is an immediate consequence of a result in~\cite{DJR2021}, and will come in handy in Section~\ref{s:beyond-loc-finite}.
\begin{proposition}\label{pr:pointwise-iso-kernel-pushouts}
Suppose that $\A$ has pushouts, and let $F, G$ be polyadic finite sets on $\A$. If $F$ and $G$ are pointwise isomorphic, then so are their Stirling kernels $\sk{F}$ and $\sk{G}$.
\end{proposition}
\begin{proof}
This was proved in \cite[Lemma~12]{DJR2021} whenever $F,G\colon \A^{\op}\to \FinSet$ are functors sending pushout squares in $\A$ consisting of quotients to pullback squares in $\FinSet$. (The proof of the aforementioned result relies on the inclusion-exclusion principle.)
In turn, every polyadic (finite) set has this property. Just observe that, by Lemmas~\ref{l:amalg-quasi-pull} and~\ref{l:quot-inj-polyadic}, a polyadic set $\A^{\op}\to\Set$ sends pushout squares of quotients in $\A$ to quasi-pullbacks of injections. But a quasi-pullback in $\Set$ consisting of injections is a pullback, because the unique mediating map into the pullback is both injective and surjective.
\end{proof}
\section{Homomorphism Counting in Locally Finite Categories}\label{s:hom-count-finite}
We shall now exploit the framework of Section~\ref{s:polyadic-sets}, and in particular Proposition~\ref{pr:pointwise-iso-kernel}, to establish a homomorphism counting result for so-called locally finite categories (Theorem~\ref{thm:right-combinatorial-cats} below).
\begin{definition}
A category $\A$ is \emph{locally finite} if, for all objects $a,b\in\A$, the set $\hom_{\A}(a,b)$ is finite.
\end{definition}
To start with, we record for future reference a well known and easily proved fact about finite monoids. Recall that a monoid $(M,{\cdot},1)$ satisfies the \emph{left-cancellation law} provided that, for all $x,y,z\in M$,
\[
x\cdot y=x\cdot z \enspace \Longrightarrow \quad y=z.
\]
\begin{lemma}\label{l:left-can-group}
A finite monoid satisfying the left-cancellation law is a group.
\end{lemma}
\begin{theorem}\label{thm:right-combinatorial-cats}
Let $\A$ be a locally finite category admitting a proper factorisation system $(\Q,\M)$ such that $\A$ is $\Q$-well-founded. Then $\A$ is right-combinatorial.
\end{theorem}
\begin{proof}
Let $\A$ be as in the statement. For any object $a\in\A$, the representable functor $\mathcal{y}_a\colon \A^\op\to\FinSet$ is a polyadic finite set by Lemma~\ref{l:repr-polyadic-set}, and its Stirling kernel sends an object $c\in \A$ to the finite set $\M(c,a)$ consisting of the embeddings $c\emb a$ (see Example~\ref{ex:kernel-repr}).
Now, if $a,b\in \A$ are any two objects such that $\mathcal{y}_a(c)\cong \mathcal{y}_b(c)$ for all $c\in \A$ then, by Proposition~\ref{pr:pointwise-iso-kernel}, $\M(c,a)\cong \M(c,b)$ for all $c\in \A$. In particular, as $\M(a,a)$ is non-empty (it contains the identity arrow), there exists an embedding $i\in \M(a,b)$. Similarly, there exists an embedding $j\in\M(b,a)$. Note that, by Lemma~\ref{l:factorisation-properties}\ref{compositions}, $j\circ i\in\M(a,a)$. Lemma~\ref{l:left-can-group}, combined with the fact that every embedding is a monomorphism, entails that the set $\M(a,a)$ equipped with the composition operation is a group. So, $j\circ i$ has an inverse. It follows by Lemma~\ref{l:factorisation-properties}\ref{isos},\ref{cancellation-e} that $j$ is an isomorphism.
\end{proof}
\begin{remark}
A variant of Theorem~\ref{thm:right-combinatorial-cats}, where $\Q$ is the collection of extremal epimorphisms and $\M$ is the collection of monomorphisms, and each poset of embeddings $\M(a)$ is finite, was proved by Pultr \cite[Theorem~2.2]{pultr1973isomorphism} exploiting a direct generalisation of Lov\'{a}sz' original counting argument~\cite{Lovasz1967}.
Furthermore, reasoning along the same lines as in the previous proof, we get the following variant of Theorem~\ref{thm:right-combinatorial-cats} by applying Proposition~\ref{pr:pointwise-iso-kernel-pushouts} instead of Proposition~\ref{pr:pointwise-iso-kernel}: \emph{A locally finite category admitting pushouts and a proper factorisation system is right-combinatorial}. This result was first proved in \cite[Theorem~5]{DJR2021}.
\end{remark}
Theorem~\ref{thm:right-combinatorial-cats} admits a dual version (cf.\ Remark~\ref{rem:dual-proper-f-s} in the appendix):
\begin{theorem}\label{thm:left-combinatorial-cats}
Let $\A$ be a locally finite category admitting a proper factorisation system $(\Q,\M)$ such that $\A$ is $\M$-well-founded. Then $\A$ is left-combinatorial.
\end{theorem}
We conclude this section with several examples of applications of Theorems~\ref{thm:right-combinatorial-cats} and~\ref{thm:left-combinatorial-cats}.
\begin{example}
Let $\V$ be any variety of universal algebras (regarded as a category, with morphisms the homomorphisms). By Theorems~\ref{thm:right-combinatorial-cats} and~\ref{thm:left-combinatorial-cats}, the full subcategory $\V_{\fin}$ of $\V$ consisting of the finite members is combinatorial. Just observe that, if $(\Q,\M)$ is the proper factorisation system consisting of surjective and injective homomorphisms, respectively, then $\V_{\fin}$ is both $\Q$-well-founded and $\M$-well-founded. For instance, the following classes of algebras are combinatorial: finite Boolean algebras, finite monoids, finite groups, and finite Abelian groups.
\end{example}
\begin{example}
Generalising the previous example, let $\C$ be any class of universal algebras closed under taking subalgebras. Then the usual factorisation system in the category of all algebras for the given algebraic signature, given by surjective and injective homomorphisms, respectively, restricts to $\C$. Thus, $\C$ is combinatorial. A similar fact holds if $\C$ is closed under taking homomorphic images. For example, the following classes of algebras are closed under homomorphic images and therefore combinatorial: finite regular semigroups, finite inverse semigroups, and finite $p$-groups (cf.\ \cite[Lemma~2.4.4]{Howie1995}, \cite[Lemma~7.35]{CP1967}, and \cite[Exercise~6(c) p.~32]{Serre2002}, respectively).
\end{example}
\begin{example}\label{ex:relational-structures}
Let $\sigma$ be a relational signature, i.e.\ a (possibly infinite) set of relation symbols of finite arity, and let $\R(\sigma)$ be the category of $\sigma$-structures with their homomorphisms. Then the full subcategory $\R_{\fin}(\sigma)$ of $\R(\sigma)$ defined by the \emph{finite} $\sigma$-structures is combinatorial; for a finite signature~$\sigma$, this is precisely Lov\'{a}sz' homomorphism counting theorem~\cite{Lovasz1967}.
First, recall that epimorphisms and monomorphisms in $\R(\sigma)$ coincide, respectively, with the surjective and injective homomorphisms. Further, strong epimorphisms (respectively, strong monomorphisms) coincide with the epimorphisms (respectively, monomorphisms) that reflect the relation symbols. The same holds in $\R_{\fin}(\sigma)$.
Now, note that the factorisation system $(\Q,\M)$ in $\R_{\fin}(\sigma)$, where $\Q$ consists of the strong epimorphisms and $\M$ of the monomorphisms, is proper and $\R_{\fin}(\sigma)$ is $\Q$-well-founded. Thus, $\R_{\fin}(\sigma)$ is right-combinatorial by Theorem~\ref{thm:right-combinatorial-cats}. On the other hand, the factorisation system $(\Q',\M')$ in $\R_{\fin}(\sigma)$, where $\Q'$ consists of the epimorphisms and $\M'$ of the strong monomorphisms, is also proper and $\R_{\fin}(\sigma)$ is $\M'$-well-founded. So, $\R_{\fin}(\sigma)$ is left-combinatorial in view of Theorem~\ref{thm:left-combinatorial-cats}.
Therefore, $\R_{\fin}(\sigma)$ is combinatorial. Observe that, when $\sigma$ is infinite, the category $\R_{\fin}(\sigma)$ need not be $\Q'$-well-founded nor $\M$-well-founded.
\end{example}
\section{Beyond Locally Finite Categories}\label{s:beyond-loc-finite}
In Theorem~\ref{thm:right-combinatorial-cats} we saw that a large class of locally finite categories is right-combinatorial. The main result of this section (Theorem~\ref{th:Lovasz-finite-type} below) is a `local' extension of this fact to categories that need not be locally finite. This result is then specialised to the case of locally finitely presentable categories.
\subsection{Nerves and hom-spaces}\label{s:nerves-hom-spaces}
Throughout this section we fix a category $\D$ admitting a proper factorisation system $(\Q,\M)$, and a full subcategory $\C$ of $\D$ such that:
\begin{enumerate}[label=(\roman*)]
\item $\C$ is a \emph{dense} subcategory of $\D$, i.e.\ every $a\in \D$ is the colimit of the canonical diagram given by the forgetful functor $\C\down a \to \D$.
\item $\C$ has all finite colimits and they are preserved by the inclusion functor $\C\hookrightarrow \D$.
\item For every composite $a \epi b \emb c$ in $\D$, if $a,c\in \C$ then also $b\in \C$.
\end{enumerate}
\begin{remark}\label{rem:assumptions-dense-subcat}
Because $\C$ is closed in $\D$ under finite colimits by item~(ii), the diagrams of the form $\C\down a \to \D$ in item~(i) are automatically directed.
Further, item~(iii) amounts to saying that the proper factorisation system $(\Q,\M)$ in $\D$ restricts to a (proper) factorisation system in $\C$. This condition is satisfied, e.g., if $\C$ is closed in $\D$ under $\Q$-images (i.e., given a quotient $a\epi b$ in $\D$, if $a\in \C$ then also $b\in\C$) or under $\M$-subobjects.
Also, note that item~(iii) implies that $\C$ is closed in $\D$ under isomorphisms.
\end{remark}
Given an object $a\in\D$, let $N_a\colon \C^\op\to\Set$ be the restriction of the presheaf $\mathcal{y}_a\colon \D^\op\to\Set$ to $\C^\op$.
We consider the \emph{nerve functor}
\[
N\colon \D\to \widehat{\C}, \quad a\mapsto N_a.
\]
In Theorem~\ref{th:Lovasz-finite-type}, we will identify a class of objects of $\D$ such that any two objects in this class are isomorphic precisely when their images under the nerve functor are pointwise isomorphic.
It is useful to observe that the nerve functor is full and faithful because $\C$ is a dense subcategory of $\D$ (see e.g.\ \cite[p.~218]{AHS1990}). Hence, $N$ is conservative (that is, it reflects isomorphisms). Explicitly, this means that a morphism $f\colon a\to b$ in $\D$ is an isomorphism if, and only if, the function
\[
f\circ -\colon N_a(c)\to N_b(c)
\]
is a bijection for every $c\in \C$.
Since we want to be able to count morphisms from objects of $\C$, it makes sense to restrict our attention to those objects of $\D$ that look `finite' from the viewpoint of every object of $\C$:
\begin{definition}
An object $a\in\D$ is of \emph{finite $\C$-type} provided that the set $\hom_{\D}(c,a)$ is finite for each $c\in\C$.
\end{definition}
\begin{example}
The previous definition makes sense for all categories $\D$ equipped with a (full) subcategory $\C$. For instance, let $\D$ be the opposite of the category of groups and group homomorphisms, and let $\C$ be the full subcategory of $\D$ defined by the finite groups. Then any finitely generated group is of finite $\C$-type, because there are finitely many homomorphisms from a finitely generated group to a finite one.
\end{example}
\begin{lemma}\label{lem:nerve-discrete-polyadic-space}
Let $a\in\D$ be an object of finite $\C$-type. The following statements hold:
\begin{enumerate}[label=(\alph*)]
\item\label{nerv-pol-set} The nerve $N_a\colon \C^\op\to\FinSet$ is a polyadic finite set.
\item\label{Stirling-nerve} The Stirling kernel of $N_a$ sends an object $c\in\C$ to the set $\M(c,a)$ of embeddings $c\emb a$.
\end{enumerate}
\end{lemma}
\begin{proof}
First we show that, for any polyadic set $F\colon \D^\op\to\Set$, its restriction to $\C^\op$ is also a polyadic set. Item (a) then follows at once from Lemma~\ref{l:repr-polyadic-set}.
Let $F\colon \D^\op\to\Set$ be an arbitrary polyadic set. Since $\C$ has pushouts, by Lemma~\ref{l:amalg-quasi-pull} it suffices to show that $F$ sends pushout squares in $\C$ to quasi-pullbacks in $\Set$. Consider a pushout square in $\C$ as on the left-hand side below and the corresponding square in $\Set$ as on the right-hand side.
\begin{equation*}
\begin{tikzcd}
{\cdot} \arrow{d}[swap]{g} \arrow{r}{f} & a \arrow{d} \\
b \arrow{r} & c
\end{tikzcd}
\ \ \ \ \ \ \ \ \ \ \ \ \
\begin{tikzcd}
F(\cdot) & F(a) \arrow{l}[swap]{F(f)} \\
F(b) \arrow{u}{F(g)} & F(c) \arrow{u} \arrow{l}
\end{tikzcd}
\end{equation*}
If $x\in F(a)$ and $y\in F(b)$ are such that $F(f)(x)=F(g)(y)$, by the amalgamation property of $F$ there exist $d\in \D$, morphisms $f'\colon b\to d$ and $g'\colon a\to d$ such that $f'\circ g=g'\circ f$, and an element $z\in F(d)$ such that $F(g')(z)=x$ and $F(f')(z)=y$. Because the inclusion $\C\hookrightarrow \D$ preserves pushouts, there exists a unique morphism $h\colon c\to d$ making the ensuing diagram commute. The element $F(h)(z)\in F(c)$ then witnesses the fact that the rightmost square above is a quasi-pullback.
Item (b) follows by reasoning as in Example~\ref{ex:kernel-repr}, using the fact that the factorisation system in $\D$ restricts to a proper factorisation system in $\C$.
\end{proof}
Consider any two objects $a,b\in\D$ and the canonical colimit cocone
\[
\{b_i\to b\mid i\in I\}
\]
in $\D$ where each $b_i$ belongs to $\C$. We have an isomorphism
\[
\mathcal{y}_a(b)\cong \lim_{i\in I}\mathcal{y}_a(b_i)
\]
in the category of sets. Let us assume that $a$ is of finite $\C$-type, so that each $\mathcal{y}_a(b_i)=N_a(b_i)$ is finite. Note that the diagram consisting of the $b_i$'s is directed (see Remark~\ref{rem:assumptions-dense-subcat}). So, if we equip the sets $\mathcal{y}_a(b_i)$ with the discrete topologies, the hom-set $\mathcal{y}_a(b)$ carries a natural Stone (i.e.\ compact, Hausdorff and zero-dimensional) topology, namely the inverse limit topology. Explicitly, this is the topology generated by the sets of the form
\begin{equation*}
\begin{tikzcd}
\mathscr{U}_{\langle u,v\rangle}\coloneqq \{h\in \mathcal{y}_a(b)\mid h\circ u=v\}
\end{tikzcd}
\quad \quad \quad \quad
\begin{tikzcd}[column sep=1em]
{} & c \arrow{dl}[swap]{u} \arrow{dr}{v} & {} \\
b \arrow{rr}{h} & {} & a
\end{tikzcd}
\end{equation*}
for $c\in \C$, $u\in N_b(c)$ and $v\in N_a(c)$.
Let us denote by $E_a(b)$ the `hom-space' obtained by endowing the set $\mathcal{y}_a(b)$ with the Stone topology just described.\footnote{Although we shall not need this fact in the following, let us point out that the assignment $b\mapsto E_a(b)$ yields a functor $E_a\colon \D^\op\to\Stone$ which extends $\mathcal{y}_a\colon \D^\op\to\Set$.}
Next, we prove some useful properties of the space of endomorphisms $E_a(a)$:
\begin{lemma}\label{lem:finite-type-topology}
The following statements hold for every $a\in\D$ of finite $\C$-type:
\begin{enumerate}[label=(\alph*)]
\item $E_a(a)$ is a topological monoid with respect to composition.
\item $\M(a,a)$ is a closed submonoid of $E_a(a)$.
\end{enumerate}
\end{lemma}
\begin{proof}
For item (a), we must prove that the composition operation
\[
\circ \colon E_a(a)\times E_a(a)\to E_a(a), \enspace (f,g)\mapsto f\circ g
\]
is continuous. Consider $(f,g)\in E_a(a)\times E_a(a)$ and an open neighbourhood $\mathscr{U}_{\langle u,v\rangle}$ of $f\circ g$. Then the set $\mathscr{U}_{\langle g\circ u,v\rangle}\times \mathscr{U}_{\langle u,g\circ u \rangle}$ is an open neighbourhood of $(f,g)$ whose image is contained in $\mathscr{U}_{\langle u,v\rangle}$. Just observe that, for all $(f',g')\in \mathscr{U}_{\langle g\circ u,v\rangle}\times \mathscr{U}_{\langle u,g\circ u \rangle}$, we have $f'\circ g'\circ u=f'\circ g\circ u=v$. Hence, the composition operation is continuous.
For item (b), suppose that $f\in E_a(a)$ is not an embedding. We must find an open neighbourhood $V$ of $f$ disjoint from $\M(a,a)$. Because $f$ is not an isomorphism and the nerve functor $N\colon \D\to\widehat{\C}$ is conservative, there exists $c\in\C$ such that the map
\[
f\circ -\colon N_a(c)\to N_a(c)
\]
is not a bijection. Since $N_a(c)$ is a finite set, $f\circ -$ cannot be injective. Hence, there exist distinct morphisms $u,v\in N_a(c)$ such that $f\circ u=f\circ v$. The set $V\coloneqq \mathscr{U}_{\langle u,f\circ u\rangle}\cap \mathscr{U}_{\langle v,f\circ v\rangle}$ is clearly an open neighbourhood of $f$. We claim that $V$ is disjoint from $\M(a,a)$. Assume, by way of contradiction, that $g\colon a\emb a$ is an embedding that belongs to $V$. Then $g\circ u=f\circ u=f\circ v=g\circ v$. As $g$ is a monomorphism we get $u=v$, a contradiction.
\end{proof}
\begin{remark}
Recall that a topological monoid is profinite if, and only if, its underlying space is Stone~\cite{Numakura1957}. Thus, it follows by Lemma~\ref{lem:finite-type-topology} that $E_a(a)$ and $\M(a,a)$ are profinite monoids.
\end{remark}
Lemma~\ref{l:left-can-group} for finite monoids admits a non-trivial generalisation to topological monoids, which we now recall. This result will be applied in (the proof of) Theorem~\ref{th:Lovasz-finite-type} to the topological monoids $\M(a,a)$ as in Lemma~\ref{lem:finite-type-topology}.
\begin{lemma}[{Numakura's Lemma~\cite[Lemma 2L]{Numakura1952}}]\label{lem:numakura-lemma}
Let $S$ be a topological monoid whose topology is compact and Hausdorff. If $S$ satisfies the left-cancellation law then it is a group.
\end{lemma}
We are interested in determining when two non-isomorphic objects $a,b\in\D$ of finite $\C$-type can be distinguished by counting the number of morphisms from objects of $\C$. To this end, we will assume that $a$ and $b$ satisfy an additional `separability' property. (The latter is satisfied by any object of a locally finitely presentable category $\D$, for an appropriate choice of the subcategory $\C$, cf.~the proof of Theorem~\ref{th:Lovasz-lfp-categories}.)
\begin{definition}\label{def:C-separable}
An object of $\D$ is \emph{$\C$-separable} if it is the colimit of a directed diagram $J$ in $\C$ such that:
\begin{enumerate}[label=(\roman*)]
\item The colimit cocone consists of embeddings.
\item For every compatible cocone of embeddings over $J$, the unique connecting morphism is also an embedding.
\end{enumerate}
\end{definition}
Intuitively, an object $a$ is $\C$-separable if there exists a (directed) family $S$ of $\M$-subobjects of $a$ such that, for all objects $d\in\D$, $a$ can be embedded into $d$ precisely when all objects in $S$ can be embedded coherently into $d$ (and, moreover, $a$ is the union of its subobjects in $S$).
\begin{remark}\label{rem:small-diag-embeddings}
Note that, by Lemma~\ref{l:factorisation-properties}\ref{cancellation-m}, item~(i) in the previous definition implies that the diagram $J$ consists entirely of embeddings.
\end{remark}
We are now in a position to prove the main result of this section:
\begin{theorem}\label{th:Lovasz-finite-type}
For any two $\C$-separable objects $a,b\in\D$ of finite $\C$-type,
\[
a\cong b \enspace \Longleftrightarrow \enspace \hom_{\D}(c,a)\cong \hom_{\D}(c,b) \ \text{ for all } c\in \C.
\]
\end{theorem}
\begin{proof}
Let $a,b\in \D$ be as in the statement. For the non-trivial direction, suppose that $\hom_{\D}(c,a)\cong \hom_{\D}(c,b)$ for all $c\in\C$. By assumption, $b$ is the colimit of a directed diagram $\{b_i\mid i\in I\}$ in $\C$ satisfying the conditions in Definition~\ref{def:C-separable}.
%
Therefore, $b\cong \colim_{D} b_i$ entails
\begin{equation*}
\mathcal{y}_a(b)\cong \lim_{i\in I} N_a(b_i),
\end{equation*}
i.e.\ the map associating with a compatible cocone $\{b_i\to a\mid i\in I\}$ the unique connecting morphism $b\to a$ is a bijection. If each finite set $N_a(b_i)$ is equipped with the discrete topology, then the induced inverse limit topology $\tau$ on $\mathcal{y}_a(b)$ coincides with the topology of $E_a(b)$ and so we have a homeomorphism of Stone spaces
\begin{equation}\label{eq:homeo-E_a(b)-inv-lim}
E_a(b)\cong \lim_{i\in I} N_a(b_i).
\end{equation}
Just observe that the diagram of the $b_i$'s is a subdiagram of the canonical diagram given by (the image of) the forgetful functor $\C\down b \to \D$, and so the identity map $E_a(b)\to (\mathcal{y}_a(b),\tau)$ is continuous. In turn, since any two comparable compact Hausdorff topologies on a set must coincide, the two topologies are one and the same.
The space $\lim_{i\in I} \M(b_i,a)$, endowed with the inverse limit topology, can be identified with a subspace of $\lim_{i\in I} N_a(b_i)$. We claim that the homeomorphism in~\eqref{eq:homeo-E_a(b)-inv-lim} restricts to a homeomorphism between $\M(a,b)$ and $\lim_{i\in I} \M(b_i,a)$.
If a compatible cocone $\{b_i\to a\mid i\in I\}$ induces a connecting morphism $b\to a$ that is an embedding then Lemma~\ref{l:factorisation-properties}\ref{compositions}, combined with item~(i) in the definition of a $\C$-separable object and Remark~\ref{rem:small-diag-embeddings}, entails that each $b_i\to a$ is an embedding. Conversely, if $\{b_i\to a\mid i\in I\}$ is a compatible cocone consisting of embeddings then the unique connecting morphism is an embedding by item~(ii) in the definition of a $\C$-separable object. Hence, $\M(b,a)$ is homeomorphic to $\lim_{i\in I} \M(b_i,a)$.
By assumption, the nerves $N_a, N_b\colon \C^\op\to\FinSet$ are pointwise isomorphic. It follows by Proposition~\ref{pr:pointwise-iso-kernel-pushouts} and Lemma~\ref{lem:nerve-discrete-polyadic-space}\ref{nerv-pol-set} that their Stirling kernels are also pointwise isomorphic. So, by Lemma~\ref{lem:nerve-discrete-polyadic-space}\ref{Stirling-nerve},
\[
\M(b_i,a)\cong \M(b_i,b)
\]
for every $i\in I$. As the sets $\M(b_i,b)$ are non-empty (they contain the colimit maps), we conclude that the space $\M(b,a)$ is the inverse limit of non-empty finite discrete spaces and thus $\M(b,a)\neq\emptyset$ (see e.g.\ \cite[Theorem~2-85]{HY1988}). That is, there exists an embedding $\alpha\colon b\emb a$. By symmetry, there exists also an embedding $\beta\colon a\emb b$.
The composite $\alpha\circ \beta$ belongs to the monoid $\M(a,a)$, which is a Stone topological monoid by Lemma~\ref{lem:finite-type-topology}. It follows by Numakura's Lemma that $\alpha\circ\beta$ has an inverse and so $\alpha$ is an isomorphism by Lemma~\ref{l:factorisation-properties}\ref{isos},\ref{cancellation-e}.
\end{proof}
\subsection{Locally finitely presentable categories}\label{s:lfp}
In this section we specialise Theorem~\ref{th:Lovasz-finite-type} to the case of locally finitely presentable categories. To start with, we recall some basic definitions; for a more thorough treatment, the reader can consult e.g.~\cite{AR1994}.
An object $a$ of a category $\A$ is \emph{finitely presentable} (respectively, \emph{finitely generated}) if the associated covariant hom-functor
\[
\hom_{\A}(a,-)\colon \A\to\Set
\]
preserves directed colimits (respectively, directed colimits of monomorphisms). A category $\A$ is said to be \emph{locally finitely presentable} if it is cocomplete, every object is a directed colimit of finitely presentable objects, and there exists, up to isomorphism, only a set of finitely presentable objects.
Let $\A$ be a locally finitely presentable category. Then $\A$ admits a proper factorisation system $(\Q,\M)$ where $\Q$ consists of the strong epimorphisms and $\M$ of the monomorphisms. Further, an object $a\in \A$ is finitely generated if, and only if, there exist a finitely presentable object $b\in \A$ and a quotient (i.e., a strong epimorphism) $b\epi a$. For a proof of these facts see, e.g., \cite[Propositions~1.61 and~1.69(ii)]{AR1994}. Throughout, we denote by $\A_{\fp}$ and $\A_{\fg}$ the full subcategories of $\A$ consisting, respectively, of the finitely presentable and finitely generated objects.
In this context, the role of the dense subcategory $\C$ in Theorem~\ref{th:Lovasz-finite-type} is played by $\A_{\fg}$. However, it is an easy observation that any object of finite $\A_{\fp}$-type is also of finite $\A_{\fg}$-type:
\begin{lemma}\label{l:finite-fp-fg-type}
Let $\A$ be a locally finitely presentable category. An object of $\A$ is of finite $\A_{\fp}$-type if, and only if, it is of finite $\A_{\fg}$-type.
\end{lemma}
\begin{proof}
For the non-trivial direction, suppose that $a$ is of finite $\A_{\fp}$-type and consider an arbitrary $b\in \A_{\fg}$. If $f\colon c\epi b$ is a quotient with $c$ finitely presentable, the map $-\circ f \colon \hom_{\A}(b,a)\to \hom_{\A}(c,a)$ is injective. Since the latter set is finite, so is the former.
\end{proof}
\begin{theorem}\label{th:Lovasz-lfp-categories}
Let $\A$ be a locally finitely presentable category. For any two objects $a,b\in\A$ of finite $\A_{\fp}$-type,
\[
a\cong b \enspace \Longleftrightarrow \enspace \hom_{\A}(c,a)\cong \hom_{\A}(c,b) \ \text{ for all } c\in \A_{\fg}.
\
|
]
\end{theorem}
\begin{proof}
Recall that $\A_{\fg}$ is a dense subcategory of $\A$ because so is $\A_{\fp}$ (see e.g.\ \cite[Proposition~1.22]{AR1994}), and it is closed in $\A$ under $\Q$-images (see e.g.\ \cite[Proposition~1.69(i)]{AR1994}). Further, it is a folklore result that $\A_{\fg}$ has all finite colimits and these are preserved by the inclusion functor $\A_{\fg}\hookrightarrow \A$.
Finally, every object of $\A$ is $\A_{\fg}$-separable (cf.\ \cite[Proposition~1.62 and Theorem~1.70]{AR1994}). Therefore, an application of Theorem~\ref{th:Lovasz-finite-type} with $\D\coloneqq \A$ and $\C\coloneqq \A_{\fg}$, combined with Lemma~\ref{l:finite-fp-fg-type}, yields the statement.
\end{proof}
We now specialise Theorem~\ref{th:Lovasz-lfp-categories} to ind- and pro-categories. The ensuing results are then applied in concrete cases in Section~\ref{s:examples}. A further application of Theorem~\ref{th:Lovasz-lfp-categories}, this time to categories of coalgebras for certain comonads, is presented in Section~\ref{s:coalg-finite-rank}.
Given a (essentially small) category $\C$, denote its \emph{ind-completion} and \emph{pro-completion} by $\ind(\C)$ and $\pro(\C)$, respectively (see e.g.~\cite[\S VI.1]{Johnstone1986}). These are, respectively, the free cocompletion of $\C$ under filtered colimits, and the free completion of $\C$ under cofiltered limits. Up to an equivalence of categories, we can and will identify $\C$ with a full subcategory of $\ind(\C)$ and $\pro(\C)$, respectively.
\begin{corollary}\label{cor:counting-indC}
Let $\C$ be an essentially small category with finite colimits that is closed under strong epimorphic images in $\ind(\C)$. For all objects $a,b\in\ind(\C)$ of finite $\C$-type,
\[
a\cong b \enspace \Longleftrightarrow \enspace \hom_{\ind(\C)}(c,a)\cong \hom_{\ind(\C)}(c,b) \ \text{ for all } c\in \C.
\]
\end{corollary}
\begin{proof}
If $\C$ is essentially small and has finite colimits then $\ind(\C)$ is a locally finitely presentable category; see e.g.\ \cite[Corollary~VI.1.3]{Johnstone1986} and \cite[Theorem~1.46]{AR1994}.
Furthermore, every finitely presentable object of $\ind(\C)$ is isomorphic to an object of $\C$ (this follows from the fact that $\C$ has finite colimits, hence it is idempotent-complete, combined with \cite[Exercise~6.1(iii)]{KS2006}). The same is true of finitely generated objects of $\ind(\C)$, as $\C$ is closed under strong epimorphic images in $\ind(\C)$. Therefore, the statement follows directly from Theorem~\ref{th:Lovasz-lfp-categories}.
\end{proof}
We record for future reference the dual version of Corollary~\ref{cor:counting-indC}.
\begin{corollary}\label{cor:counting-proC}
Let $\C$ be an essentially small category with finite limits that is closed under strong subobjects in $\pro(\C)$. Let $a,b\in\pro(\C)$ be such that the sets $\hom_{\pro(\C)}(a,c)$ and $\hom_{\pro(\C)}(b,c)$ are finite for all $c\in\C$. Then
\[
a\cong b \enspace \Longleftrightarrow \enspace \hom_{\pro(\C)}(a,c)\cong \hom_{\pro(\C)}(b,c) \ \text{ for all } c\in \C.
\]
\end{corollary}
\subsection{Coalgebras for comonads of finite rank}\label{s:coalg-finite-rank}
In this section we specialise Theorem~\ref{th:Lovasz-lfp-categories} to categories of coalgebras for comonads on locally finitely presentable categories.
Whenever $T$ is a comonad on a category $\A$, we write $\EM(T)$ for the category of Eilenberg-Moore coalgebras for $T$, and
\[\begin{tikzcd}[column sep=1.5em]
\A \arrow[yshift=-6pt]{rr}[swap]{F} & {\text{\scriptsize{$\bot$}}} & \EM(T) \arrow[yshift=6pt]{ll}[swap]{U}
\end{tikzcd}\]
for the associated adjunction. For the latter notions, see Appendix~\ref{s:finite-rank}.
In the next result, we will assume that $\A$ is locally finitely presentable and $T$ is of \emph{finite rank}, and so the category $\EM(T)$ is also locally finitely presentable. Comonads of finite rank were defined by Diers in~\cite{Diers1986}; for a definition and some basic facts, we refer the reader to Appendix~\ref{s:finite-rank}.
\begin{corollary}\label{cor:hom-counting-comonads-finite-rank}
Let $\A$ be a locally finitely presentable category and let $T$ be a comonad of finite rank on $\A$ with associated adjunction $U\dashv F$. The following statements are equivalent for all objects $a,b\in\A$ of finite $\A_{\fp}$-type:
\begin{enumerate}
\item $F(a)\cong F(b)$.
\item For all $x\in \EM(T)_{\fg}$,
\[
\hom_{\A}(U(x),a)\cong \hom_{\A}(U(x),b).
\]
\end{enumerate}
\end{corollary}
\begin{proof}
Since $T$ is of finite rank, Theorem~\ref{t:EM-lfp} in the appendix entails that the category $\EM(T)$ is locally finitely presentable and the forgetful functor
\[
U\colon \EM(T)\to \A
\]
preserves (and reflects) finitely presentable objects.
Note that the object $F(a)$ is of finite $\EM(T)_{\fp}$-type whenever $a$ is of finite $\A_{\fp}$-type. Just observe that, for all $x\in \EM(T)_{\fp}$,
\[
\hom_{\EM(T)}(x,F(a))\cong \hom_{\A}(U(x),a)
\]
which is a finite set because $U(x)$ is finitely presentable.
Therefore, the statement follows by an application of Theorem~\ref{th:Lovasz-lfp-categories}.
\end{proof}
\begin{remark}\label{rem:replace-fg-with-fp}
Suppose we are in the situation of the previous corollary. In view of Corollary~\ref{cor:fp-equal-fg} in the appendix, if the finitely presentable objects in $\A$ coincide with the finitely generated ones, then the same holds in $\EM(T)$.
In this case, Corollary~\ref{cor:hom-counting-comonads-finite-rank} states that, for all objects $a,b\in\A$ of finite $\A_{\fp}$-type, $F(a)\cong F(b)$ if, and only if,
\[
\hom_{\A}(U(x),a)\cong \hom_{\A}(U(x),b)
\]
for all $x\in\EM(T)$ such that $U(x)$ is finitely presentable.
This occurs, for instance, when $\A=\Set$ or $\A=\R(\sigma)$ is the category of $\sigma$-structures for a finite relational signature $\sigma$. A similar remark applies to Corollary~\ref{cor:hom-counting-comonads-finite-rank-relative} below.
\end{remark}
We include a `relative' version of Corollary~\ref{cor:hom-counting-comonads-finite-rank} which will be needed in Section~\ref{s:finite-variable-logic} for applications to finite-variable logics. To this end, given a functor $G\colon \C\to \D$ and a full subcategory $\tilde{\C}$ of~$\C$, we write $G[\tilde{\C}]$ for the full subcategory of $\D$ defined by the objects of the form $G(c)$ with $c\in\tilde{\C}$.
\begin{corollary}\label{cor:hom-counting-comonads-finite-rank-relative}
Let $\A'$ be a locally finitely presentable category and assume that there is a full and faithful functor $J\colon \A\hookrightarrow \A'$ with a left adjoint $H$. Let $T,T'$ be comonads on $\A$ and $\A'$, respectively, with associated adjunctions ${U\dashv F}$ and ${U'\dashv F'}$.
Suppose $T'$ is of finite rank and the adjunction $H\dashv J$ restricts to functors $U[\EM_\fg(T)]\leftrightarrows U'[\EM_\fg(T')]$:
\[\begin{tikzcd}[column sep=1em]
\A \arrow[yshift=-6pt]{rr}[swap]{J} & {\text{\scriptsize{$\bot$}}} & \A' \arrow[yshift=6pt]{ll}[swap]{H} \\
U[\EM_\fg(T)] \arrow[hookrightarrow]{u} \arrow[dashed,yshift=-4pt]{rr} & {} & U'[\EM_\fg(T')] \arrow[hookrightarrow]{u} \arrow[dashed,yshift=4pt]{ll}
\end{tikzcd}\]
The following are equivalent for all objects $a,b\in\A$ of finite $\A_{\fp}$-type:
\begin{enumerate}
\item $F'J(a)\cong F'J(b)$.
\item For all $x\in \EM(T)_{\fg}$,
\[
\hom_{\A}(U(x),a)\cong \hom_{\A}(U(x),b).
\]
\end{enumerate}
\end{corollary}
\begin{proof}
In view of Corollary~\ref{cor:hom-counting-comonads-finite-rank}, item~1 in the statement is equivalent to saying that, for all $x'\in\EM(T')_{\fg}$,
\begin{equation}\label{eq:bij-T'-coalg}
\hom_{\A'}(U'(x),J(a))\cong \hom_{\A'}(U'(x),J(b)).
\end{equation}
In turn, it is an easy observation that the latter condition is equivalent to item 2 in the statement. This is essentially the content of \cite[Lemma~27]{DJR2021}; for the sake of completeness, we provide a proof. Suppose that equation~\eqref{eq:bij-T'-coalg} holds for all $x'\in\EM(T')_{\fg}$, and fix an arbitrary $x\in \EM(T)_{\fg}$. We have
\begin{align*}
\hom_{\A}(U(x),a) &\cong \hom_{\A'}(JU(x),J(a)) \tag{$J$ full and faithful} \\
& \cong \hom_{\A'}(JU(x),J(b)) \tag{$JU(x)\in U'[\EM_\fg(T')]$} \\
&\cong \hom_{\A}(U(x),b). \tag{$J$ full and faithful}
\end{align*}
Conversely, suppose item 2 in the statement holds. For all $x'\in\EM(T')_{\fg}$,
\begin{align*}
\hom_{\A'}(U'(x),J(a)) &\cong \hom_{\A}(HU'(x),a) \tag{$H\dashv J$} \\
& \cong \hom_{\A}(HU'(x),b) \tag{$HU'(x)\in U[\EM_\fg(T)]$} \\
&\cong \hom_{\A'}(U'(x),J(b)) \tag{$H\dashv J$}
\end{align*}
and so equation~\eqref{eq:bij-T'-coalg} holds. This concludes the proof.
\end{proof}
\section{Examples}\label{s:examples}
\subsection{Trees}
If $(P, {\leq})$ is a poset, then $C \subseteq P$ is a \emph{chain} if it is linearly ordered. A \emph{forest} is a poset $(P,\leq)$ such that, for all $u\in P$, the set
\[
\down u\coloneqq \{v\in P\mid v\leq u\}
\]
is a finite chain.
The \emph{covering relation} $\cvr$ associated with a partial order $\leq$ is defined by $u\cvr v$ if and only if $u<v$ and there is no $w$ such that $u<w< v$.
The \emph{roots} of a forest are the minimal elements. A \emph{tree} is a forest with at most one root (note that a tree is either empty, or has a unique root, the least element in the order).
Morphisms of trees are maps which preserve the root and the covering relation.
The category of trees is denoted by $\T$. Monomorphisms and strong epimorphisms in $\T$ coincide, respectively, with the injective and surjective tree morphisms.
It is well known that $\T$ is a locally finitely presentable category in which the finitely presentable objects, which coincide with the finitely generated ones, are precisely the finite trees. Moreover, it is not difficult to see that a tree $(P,\leq)$ has finite $\T_{\fp}$-type if, and only if, it is \emph{finitely branching}. That is, for every $u\in P$, the set $\{v\in P\mid u\cvr v\}$ is finite.
Thus, Theorem~\ref{th:Lovasz-lfp-categories} entails at once the following result:
\begin{theorem}
Let $P,Q$ be any two finitely branching trees. Then $P\cong Q$ if and only if, for all finite trees $R$, the number of tree morphisms $R\to P$ is the same as the number of tree morphisms $R\to Q$.
\end{theorem}
\begin{remark}
In the last part of the proof of Theorem~\ref{th:Lovasz-finite-type}, we used the fact that
\[
\forall i\in I. \ \M(b_i, a)\neq \emptyset \enspace \Longrightarrow \enspace \M(\colim_{i\in I} b_i, a)\neq\emptyset
\]
where, using the notation of the aforementioned theorem, the objects $b_i$ sit in the category $\C$ and $a$ is a $\C$-separable object of finite $\C$-type.
This can be regarded as a generalisation of K\"{o}nig's Lemma for trees, stating that every finitely branching infinite tree contains an infinite simple path. Just observe that a countably infinite simple path $P_\omega$ is a colimit in $\T$ of finite simple paths $P_n$ of (increasing) length $n$. If $Q$ is an infinite tree, then for all $n$ there exists an embedding $P_n \emb Q$. If, in addition, $Q$ is finitely branching, then it has finite $\T_{\fp}$-type (and is $\T_{\fp}$-separable) and so there is an embedding $P_\omega\emb Q$.\footnote{In the specific case of trees, embeddings could be replaced by arbitrary arrows. Just note that every morphism in $\T$ whose domain is linearly ordered is automatically injective.}
In this sense, a version of K\"{o}nig's Lemma holds, in particular, in every locally finitely presentable category.
\end{remark}
\subsection{Profinite algebras}
In this section we focus on profinite universal algebras; a nice expository paper on the subject is~\cite{Banaschewski1972}.
Let $\V$ be an arbitrary variety of universal algebras, regarded as a category with morphisms the homomorphisms. The full subcategory $\V_{\fin}$ of $\V$ defined by the finite algebras is essentially small and has finite limits, which are computed in the category of sets. The same holds for any full subcategory $\C$ of $\V_{\fin}$ that is closed under subalgebras and finite products.
Let us fix a full subcategory $\C$ of $\V_{\fin}$ closed under subalgebras and finite products. The category $\pro(\C)$ can be identified with a full subcategory of the category $\K_{\V}$, whose objects are the topological $\V$-algebras carrying a compact Hausdorff topology and whose morphisms are the continuous homomorphisms.\footnote{Under this identification, an algebra in $\C$ is regarded as a topological algebra with respect to the discrete topology.} Explicitly, a topological algebra $A\in \K_{\V}$ belongs to $\pro(\C)$ if and only if, whenever $f, g\colon B\to A$ are distinct morphisms in $\K_{\V}$, there exists a morphism $h\colon A\to C$ with $C\in \C$ such that $h\circ f\neq h\circ g$. We shall refer to the objects of $\pro(\C)$ as \emph{pro-$\C$} algebras. If $\C=\V_{\fin}$, these coincide with the usual profinite $\V$-algebras.
The monomorphisms in $\K_{\V}$ are precisely the injective maps, i.e.\ the maps that provide (topological and algebraic) isomorphisms with the image. This holds in $\pro(\C)$ as well, because the latter is a reflective subcategory of $\K_{\V}$. In particular, the subobjects in $\pro(\C)$ can be identified with the closed subalgebras. For the previous assertions, cf.~\cite{Banaschewski1972} and the references therein. It follows that $\C$ is closed under (strong) subobjects in $\pro(\C)$.
Finally, recall that a universal algebra $A$ is said to be \emph{finitely generated} if there exists a finite subset $S\subseteq A$ such that the inclusion-smallest subalgebra $\langle S\rangle$ of $A$ containing $S$ is $A$ itself. In the setting of topological algebras it is customary to relax the previous condition and say that a topological algebra $A$ is \emph{topologically finitely generated} if there exists a finite subset $S\subseteq A$ such that $\langle S\rangle$ is dense in the topology of $A$.
Now, if $A\in \pro(\C)$ is topologically finitely generated and $C\in \C$, the set $\hom_{\pro(\C)}(A, C)$ is finite. Just observe that, if $S\subseteq A$ is a finite subset such that $\langle S\rangle$ is dense in $A$, the obvious restriction function
\[
\hom_{\pro(\C)}(A, C)\to C^S
\]
is injective because a continuous map into a Hausdorff space is completely determined by its behaviour on any dense subset of its domain.
Hence, the following is an immediate consequence of Corollary~\ref{cor:counting-proC}:
\begin{theorem}\label{t:counting-pro-C}
Let $\V$ be a variety of universal algebras, $\C\subseteq \V_{\fin}$ a class of finite algebras closed under subalgebras and finite products, and $A,B$ two topologically finitely generated pro-$\C$ algebras. Then $A\cong B$ if and only if, for all (discrete) algebras $C\in \C$, the number of continuous homomorphisms $A\to C$ is the same as the number of continuous homomorphisms $B\to C$.
\end{theorem}
The previous result applies, e.g., when $\C$ is the class of finite lattices, or finite Heyting algebras, or finite semigroups, and provides a characterisation of the isomorphism relation for (topologically finitely generated) profinite lattices, profinite Heyting algebras, and profinite semigroups, respectively.
\vspace{0.5em}
Next, we restrict our attention to topologically finitely generated profinite groups, whose topological structure is determined by the algebraic one.
Instantiating Theorem~\ref{t:counting-pro-C} with $\C$ the class of finite groups, we obtain the following result: \emph{Two topologically finitely generated profinite groups $G,H$ are isomorphic (as topological groups) if and only if, for all finite discrete groups $K$, the number of continuous group homomorphisms $G\to K$ is the same as the number of continuous group homomorphisms $H\to K$.}\footnote{This fact was first conjectured by Mima Stanojkovski (private e-mail communication).}
In turn, a remarkable result of Nikolov and Segal~\cite{NS2007a,NS2007b} states that the topological structure of a topologically finitely generated profinite group is completely determined by its algebraic structure. More precisely, the subgroups of finite index of a topologically finitely generated profinite group coincide with the open subgroups. (In the special case of pro-$p$-groups, this was proved by Serre in the 1970s, cf.\ \cite[Exercise~6 p.~32]{Serre2002}.) This implies that (i) any group homomorphism from a topologically finitely generated profinite group to a finite discrete group is continuous, and (ii) any two topologically finitely generated profinite groups are isomorphic as topological groups if, and only if, they are isomorphic as abstract groups. The homomorphism counting result in the previous paragraph can then be restated as follows: \emph{Two topologically finitely generated profinite groups $G,H$ are isomorphic if and only if, for all finite groups $K$, the number of group homomorphisms $G\to K$ is the same as the number of group homomorphisms $H\to K$.}
A similar result, whose statement we shall omit, holds for (topologically finitely generated) Abelian profinite groups, also known as \emph{proabelian groups}, by taking as $\C$ the class of finite Abelian groups.
For an example where $\C$ is a proper subclass of $\V_{\fin}$, let $\C$ consist of the finite $p$-groups. We deduce that: \emph{Two topologically finitely generated pro-$p$-groups $G,H$ are isomorphic if and only if, for all finite $p$-groups $K$, the number of group homomorphisms $G\to K$ is the same as the number of group homomorphisms $H\to K$.}
\vspace{0.5em}
A typical homomorphism counting result will count the number of morphisms (into, or from a given object) in the monoid $\N$ of natural numbers. Now, recall that the proof of Theorem~\ref{th:Lovasz-finite-type} (and similarly, the proof of Theorem~\ref{thm:right-combinatorial-cats} in the locally finite case) consists essentially of two steps. First, using the Stirling kernel construction, we show that the functors $\M(-,a)$ and $\M(-,b)$ are pointwise isomorphic whenever the functors $\hom(-,a)$ and $\hom(-,b)$ are pointwise isomorphic. Second, Numakura's Lemma is invoked to show that $a\cong b$ if, for all $c$,
\[
\M(c,a)\neq\emptyset \enspace \Longleftrightarrow \enspace \M(c,b)\neq\emptyset.
\]
Dispensing with the first step, we obtain a (weaker) result whereby the isomorphism type of an object is determined by counting the number of embeddings (or, dually, quotients) in the two-element monoid $(\{0,1\},\vee,0)$. We state a special case of this result, which generalises a known fact in profinite group theory (cf.\ \cite[Theorems~3.2.7 and~3.2.9]{RZ2010}).
\begin{proposition}
Let $\V$ be a variety of universal algebras, $\C\subseteq \V_{\fin}$ a class of finite algebras closed under subalgebras and finite products, and $A,B$ two topologically finitely generated pro-$\C$ algebras. Suppose that, for all (discrete) algebras $C\in \C$, there exists a quotient\footnote{That is, a continuous surjective homomorphism.} $A\epi C$ if and only if there exists a quotient $B\epi C$. Then $A\cong B$.
\end{proposition}
\subsection{Finite-variable logics}\label{s:finite-variable-logic}
In this section we assume familiarity with the basic notions of first-order logic and finite model theory. We shall present an application of the results in Section~\ref{s:beyond-loc-finite} to homomorphism counting in finite model theory; this is the topic of the recent work~\cite{DJR2021}. Following~\cite{DJR2021}, we apply the framework of \emph{game comonads} in (finite) model theory introduced by Abramsky, Dawar \emph{et al.} in~\cite{Abramsky2017b,AbramskyShah2018}.
Let us fix a finite relational signature $\sigma$ and a positive integer $k$. We recall from~\cite{Abramsky2017b} the \emph{pebbling comonad} $\Pk$ on the category $\R(\sigma)$ of $\sigma$-structures, which models $k$-pebble games.
Set $\k\coloneqq \{1,\dots,k\}$. Given a $\sigma$-structure $A$, we consider the set ${(\k\times A)^+}$ of \emph{plays} in $A$, i.e.\ all non-empty finite sequences of elements of $\k\times A$. A pair $(p,a) \in {\k\times A}$ is called a \emph{move}. Intuitively, the move $(p,a)$ corresponds to placing the pebble $p$ on the element $a$. Whenever $[(p_1,a_1),\dots,(p_l,a_l)]$ is a play, $p_i$ is called the \emph{pebble index} of the move $(p_i,a_i)$. Define the map
\[
\epsilon_A\colon (\k\times A)^+\to A, \ \ [(p_1,a_1),\dots,(p_l,a_l)] \mapsto a_l
\]
sending a play to the element of $A$ in its last move. Let $\Pk(A)$ be the $\sigma$-structure with universe $(\k\times A)^+$ and such that, for every $R\in\sigma$ of arity~$j$, its interpretation $R^{\Pk(A)}$ consists of the tuples of plays $(s_1,\ldots,s_j)$ such that:
\begin{enumerate}[label=(\roman*)]
\item The $s_i$'s are pairwise comparable in the prefix order.
\item Whenever $s_i$ is a prefix of~$s_{i'}$, the pebble index of the last move in $s_i$ does not appear in the suffix of $s_i$ in $s_{i'}$.
\item ${(\epsilon_A(s_1),\dots,\epsilon_A(s_j))\in R^A}$.
\end{enumerate}
This assignment extends to a functor $\R(\sigma)\to\R(\sigma)$ by setting, for all homomorphisms of $\sigma$-structures $h\colon A\to B$,
\[
\Pk(h)\colon \Pk(A)\to \Pk(B), \ \ [(p_1,a_1),\dots,(p_l,a_l)] \mapsto [(p_1,h(a_1)),\dots,(p_l,h(a_l))].
\]
In fact, $\Pk$ is a comonad on $\R(\sigma)$ when equipped with the counit $\epsilon$ described above and the comultiplication $\delta_A \colon \Pk(A)\to \Pk\Pk(A)$ given by
\[
[(p_1,a_1),\dots,(p_l,a_l)]\mapsto [(p_1,s_1),\ldots,(p_l,s_l)],
\]
where $s_i\coloneqq [(p_1,a_1),\ldots,(p_i,a_i)]$ for $i\in\{1,\ldots,l\}$.
Moreover, it follows directly from \cite[Proposition~22]{Abramsky2017b} that a $\sigma$-structure $A$ admits a coalgebra structure for $\Pk$ if, and only if, it has \emph{tree-width} less than $k$ (the notion of tree-width for relational structures was introduced in~\cite{FV1999} and generalises the homonymous concept for graphs).
Denote by $\CL$ (respectively, $\CL(\text{\small{$\overset{w.o.}=$}})$) the extension of first-order logic (respectively, first-order logic \emph{without equality}) obtained by adding \emph{counting quantifiers} $\exists_{{\geq}i}$ for all natural numbers $i$. The $k$-variable fragments of $\CL$ and $\CL(\text{\small{$\overset{w.o.}=$}})$ are denoted by $\CL^k$ and $\CL^k(\text{\small{$\overset{w.o.}=$}})$, respectively.
Further, consider the adjunction $U\dashv F\colon \R(\sigma)\to\EM(\Pk)$ associated with the comonad $\Pk$. For any two finite $\sigma$-structures $A$ and $B$ we have
\[
F(A)\cong F(B) \enspace \Longleftrightarrow \enspace A\equiv_{\CL^k(\text{\scriptsize{$\overset{w.o.}=$}})} B,
\]
i.e.\ $F(A)\cong F(B)$ precisely when $A$ and $B$ satisfy the same sentences of $\CL^k(\text{\small{$\overset{w.o.}=$}})$. For the latter assertion, cf.\ \cite[Theorem~18]{Abramsky2017b} and \cite[\S VI]{DJR2021}.
It is well known that $\R(\sigma)$ is a locally finitely presentable category, and every finitely presentable $\sigma$-structure is finite. Since we assumed that the signature $\sigma$ is finite, the converse holds as well, and so the finitely generated objects in $\R(\sigma)$ coincide with the finitely presentable ones and are precisely the finite $\sigma$-structures. See e.g.\ \cite[pp.~200--201]{AR1994}.
It follows easily from the criterion in Lemma~\ref{l:sufficient-cond-finite-rank} in the appendix that $\Pk$ is finitary, and it can be verified that it satisfies the conditions in the definition of comonad of finite rank (cf.\ also Remark~\ref{rem:item-iii-monic}). As the forgetful functor $U\colon\EM(\Pk)\to \R(\sigma)$ preserves and reflects finitely presentable objects by Lemma~\ref{l:preserves-reflects-fp-objects}, a coalgebra $x\in\EM(\Pk)$ is finitely presentable if and only if $U(x)$ is a finite $\sigma$-structure.
Therefore, in view of Corollary~\ref{cor:hom-counting-comonads-finite-rank} and Remark~\ref{rem:replace-fg-with-fp}, the following statements are equivalent for all finite $\sigma$-structures $A,B$:
\begin{enumerate}
\item $A\equiv_{\CL^k(\text{\scriptsize{$\overset{w.o.}=$}})} B$.
\item For all finite $\sigma$-structures $C$ with tree-width less than $k$, the number of homomorphisms $C\to A$ is the same as the number of homomorphisms $C\to B$.
\end{enumerate}
To characterise equivalence in the logic $\CL^k$ \emph{with equality}, we proceed as follows. Let $\sigma'$ be the relational signature obtained by adding a binary relation symbol $I$ to $\sigma$. There is an adjunction ${H\dashv J\colon \R(\sigma)\to \R(\sigma')}$, where:
\begin{itemize}
\item $J$ sends a $\sigma$-structure $A$ to the $\sigma'$-structure obtained by interpreting $I$ as the identity relation on $A$;
\item $H$ sends a $\sigma'$-structure $B$ to the quotient structure $B^-/{\sim}$, where $B^-$ is the $\sigma$-reduct of $B$ and $\sim$ is the equivalence relation generated by the interpretation of $I$ in $B$.
\end{itemize}
For a proof of this fact, see \cite[Lemma~25]{DJR2021}.
Since the comonad $\Pk$ was defined for an arbitrary relational signature $\sigma$, we have a corresponding comonad $\Pk'$ on $\R(\sigma')$ with associated adjunction $U'\dashv F'$. It follows from \cite[Theorem~18]{Abramsky2017b} that, for all finite $\sigma$-structures~$A,B$,
\[
F'J(A)\cong F'J(B) \enspace \Longleftrightarrow \enspace A\equiv_{\CL^k} B.
\]
Moreover, the adjunction $H\dashv J$ restricts to the full subcategories of $\R(\sigma)$ and $\R(\sigma')$ defined by the structures with tree-width less than $k$ (this assertion is trivial for $J$; for $H$, this follows directly from \cite[Proposition~23]{DJR2021}).
Thus, Corollary~\ref{cor:hom-counting-comonads-finite-rank-relative} and Remark~\ref{rem:replace-fg-with-fp} entail the following result, which was first proved in \cite[Theorem~21]{DJR2021} for an arbitrary relational signature $\sigma$ (by means of an indirect argument) and generalises a result of Dvo\v{r}\'{a}k for graphs~\cite{dvovrak2010recognizing}:
\begin{theorem}
Let $\sigma$ be a finite relational signature and let $A,B$ be any two finite $\sigma$-structures. Then $A\equiv_{\CL^k} B$ if and only if, for all finite $\sigma$-structures $C$ with tree-width less than $k$, the number of homomorphisms $C\to A$ is the same as the number of homomorphisms $C\to B$.
\end{theorem}
|
\section{Introduction}
Grand unified theories (GUTs)~\cite{GUT} predict a very attractive unification of the
strong and electroweak interactions and, when embedded in a supersymmetric framework,
they also lead to a very successful gauge coupling unification~\cite{gauge_coupl_unif}.
Since a direct access to the GUT scale ($M_G$) is not possible, experimental signals
for GUTs may be looked for either by confronting their predictions for quantities that
are not predicted in the SM, such as the weak mixing angle and the $m_b/m_\tau$ ratio
or by the observation of processes which are forbidden or highly suppressed in the
SM, such as the proton decay.
In particular, the baryon number $B$, the family lepton numbers $L_e, L_\mu, L_\tau$ and
the total lepton number $L = L_e+L_{\mu}+L_{\tau}$, which are accidental symmetries of
the SM, are broken in a GUT context.
As a result, processes violating $B$, $L$ or $L_i$ are generated and they turn out to be
suppressed by powers of $1/M_{G}$. Still, the experimental sensitivity to proton
decay and to neutrino masses (assuming an underlying see-saw mechanism~\cite{seesaw})
allow to probe $M_{G}$. Indeed, the observed neutrino masses and mixings give solid
indications for the existence of some New Physics (NP) at $M_{G}$. In contrast,
$L_i$ violating processes, such as $\mu \to e\gamma$, are predicted to be far below
any realistic experimental sensitivity.
If the theory above $M_{G}$ is supersymmetric, the situation drastically changes.
The dynamics at the scale $M_{G}$ leave indelible traces on the soft terms of the
light sparticles by means of interactions not suppressed by inverse powers of
$M_{G}$~\cite{hkr,borzumatimasiero}.
Processes which violate $L_i$ are now suppressed only by powers of $1/{\tilde m}$, where
${\tilde m}$ is the scale of supersymmetry breaking, and therefore they might be measurable.
Moreover, since within GUTs quarks and leptons sit in the same multiplets, the quark-lepton
unification feeds into the SUSY breaking soft sector~\cite{hkr,barbieri,correlations}.
This implies relations between lepton and quark flavor changing transitions at the
weak scale. For example, one can expect correlations between $\mu\to e\,\gamma$ and $K^0-\bar{K}^0$ mixing, $\tau\to\mu\,\gamma$ and $b \to s$ transitions such as
$B_d\to\phi K_s$ and $B^{0}_s-\bar{B}^{0}_s$ mixing and so on. Therefore, hadronic
and leptonic FCNC processes provide a splendid opportunity to link the weak scale
to the GUT scale, where the fundamental SUSY Lagrangian is defined.
Despite of the remarkable agreement of flavour data with the SM predictions in the $K$
and $B_d$ systems, a closer look at the data might indicate some tensions especially in
CP violating observables. In particular:
i) the most recent UT analyses~\cite{SL,Buras:2008nn,Lenz:2010gu,Bevan:2010gi} show that
the size of CP violation determined through the $B^{0}_d-\overline{B}^{0}_d$ system,
appears insufficient to describe the experimental value of $\epsilon_K$ within the SM
if the $\Delta M_d/\Delta M_s$ constraint is taken into account~\cite{Buras:2008nn}.
Vice versa, the simultaneous SM description of $\epsilon_K$ and $\Delta M_d/\Delta M_s$
requires a $\sin 2\beta$ significantly larger than the measured value of
$S_{\psi K_S}$~\cite{SL}.
ii) The recent messages from the Tevatron seem to hint the presence of new sources of
CPV entering the $B^{0}_s$ system~\cite{Aaltonen:2007he,Abazov:2010hv,Abazov:2008fj}.
iii) The value of $\sin 2\beta$ extracted from some penguin dominated modes, such as
$B_d\to\phi K_S$, is significantly lower than the value from $B_d\to\psi K_S$.
iv) Last but not least, we remind the $(g-2)_{\mu}$ anomaly~\cite{g_2_th}. Interestingly,
the possibility that the present $3\sigma$ discrepancy may arise from errors in the
determination of the hadronic leading-order contribution to $\Delta a_{\mu}$ seems
to be unlikely~\cite{passera_mh}.
In the light of the above considerations, in this work we focus on the SUSY $SU(5)$
GUT model plus right-handed neutrinos~\cite{Hisano:1997tc} ($SSU(5)_{RN}$),
accounting for the neutrino masses and mixings via a type-I sees-saw model~\cite{seesaw},
with the main goals:
\begin{itemize}
\item[\bf i)] to analyze how the $SSU(5)_{RN}$ model faces the above tensions,
monitoring the low energy consequences implied by their possible solutions;
\item[\bf ii)] to quantify the NP room left for $b\to s$ transitions compatible
with all the available experimental data on $\Delta F=2$ and $\Delta F=1$ processes,
\item[\bf iii)] to outline strategies aimed to probe or to falsify the $SSU(5)_{RN}$
model by means of a correlated analysis of low energy observables, including also
$K$ and $D$ systems as well as lepton flavour violation, electric dipole moments
(EDMs) and the $(g-2)_{\mu}$.
\end{itemize}
Since many analyses along this subject appeared in the literature~\cite{Parry:2007fe},
we want to emphasize here that the current study goes well beyond previous works as for
i) the inclusion of all relevant SUSY contributions, ii) the number of processes considered,
and iii) the special attention given to the UT analyses. Moreover, we point out many new
correlations among observables that should enable us to probe or falsify the scenario in
question once improved data will be available.
In Section 2 we update our UT analysis of~\cite{Altmannshofer:2009ne} using the $(R_b,\gamma)$
plane and stressing that a large value of $\gamma$ in the UT would provide a natural solution
to the observed tensions within the $SSU(5)_{RN}$ model. In Section 3 we summarize the flavour
structure of the SUSY GUT considered in this paper. In Sections 4 and 5 the relevant formulae
for the hadronic and the leptonic sectors are given, respectively.
Section 6 is devoted to approximate analytical expressions for various correlations between
hadronic and leptonic observables that are then analyzed numerically in Section 7.
In Section 8 we present a DNA-table for this GUT scenario and compare it with the ones of supersymmetric flavour models analyzed by us in~\cite{Altmannshofer:2009ne}. Finally we end
our paper with a list of the most important findings.
\begin{table*}
\begin{center}
\begin{tabular}{|l|l||l|l|}
\hline
parameter & value & parameter & value \\
\hline\hline
$F_K$ & $(155.8\pm 1.7) \text{MeV}$ \cite{Laiho:2009eu,Antonio:2007pb} & $m_s(2\,\text{GeV})$&$0.105\,\text{GeV}$~\cite{Amsler:2008zzb}\\
$F_{B_d}$ & $(192.8 \pm 9.9) \text{MeV}$ \cite{Laiho:2009eu,Antonio:2007pb} & $m_d(2\,\text{GeV})$&$0.006\,\text{GeV}$~\cite{Amsler:2008zzb}\\
$F_{B_s}$ & $(238.8 \pm 9.5) \text{MeV}$\cite{Laiho:2009eu,Antonio:2007pb} & $|V_{ts}|$&$0.040\pm 0.003$~\cite{Bona:2007vi}\\
$\hat B_K$ & $0.725 \pm 0.026$ \cite{Laiho:2009eu,Antonio:2007pb} & $|V_{tb}|$&$ 1\pm 0.06$~\cite{Bona:2007vi}\\
$\hat B_{B_d}$ & $1.26\pm 0.11$ \cite{Laiho:2009eu,Antonio:2007pb} & $|V_{td}|_{\rm tree}$&$(8.3\pm 0.5)\cdot 10^{-3}$~\cite{Bona:2007vi}\\
$\hat B_{B_s}$ & $1.33\pm 0.06$ \cite{Laiho:2009eu,Antonio:2007pb} & $|V_{us}|$&$ 0.2255 \pm 0.0019 $\cite{Amsler:2008zzb}\\
$M_{B_s}$&$5.3664$ GeV~\cite{Amsler:2008zzb} & $|V_{cb}|$&$ (40.6 \pm 1.1)\times 10^{-3}$\cite{Nakamura:2010zzi}\\
$M_{B_d}$&$5.2795$ GeV~\cite{Amsler:2008zzb} & $\sin(2\beta)_{\rm tree}$ & $0.734\pm 0.038$~\cite{Bona:2007vi}\\
$M_K$&$0.497614\,\text{GeV}$~\cite{Amsler:2008zzb} & $\sin(2\beta_s)$&$0.038\pm 0.003$~\cite{Bona:2007vi}\\
$\eta_{cc}$ & $1.43\pm 0.23$\cite{Herrlich:1996vf} & $\alpha_s(m_Z)$&$ 0.1184$~\cite{Bethke:2009jm}\\
$\eta_{tt}$ & $0.5765\pm 0.0065$\cite{Buras:1990fn} & $\Delta M_s$ & $(17.77\pm 0.12)~{\rm ps}^{-1}$ ~\cite{Barberio:2008fa}\\
$\eta_{ct}$ & $0.496\pm 0.047$\cite{Brod:2010mj} & $\Delta M_d$ & $(0.507\pm 0.005)~ {\rm ps}^{-1}$ ~\cite{Barberio:2008fa}\\
$\eta_{B}$ & $0.551\pm 0.007$\cite{Buras:1990fn} & $\Delta M_K$&$(5.292\pm 0.009)\cdot 10^{-3} ps^{-1}$~\cite{Amsler:2008zzb}\\
$\xi$ & $1.243 \pm 0.028$\cite{Laiho:2009eu,Antonio:2007pb} & $\kappa_\varepsilon$& $0.94\pm 0.02$~\cite{Buras:2010pz} \\
$m_c(m_c)$ & $(1.268\pm 0.009) {\rm GeV}$\cite{Allison:2008xk} & $\varepsilon_K^{\text{exp}}$&$(2.229\pm 0.01)\cdot 10^{-3}$~\cite{Amsler:2008zzb}\\
$m_t(m_t)$ & $(163.7\pm 1.1) {\rm GeV}$\cite{CDF_D0_mt} & $S_{\psi K_S}^{\text{exp}}$& $0.672\pm0.023$~\cite{Barberio:2008fa}\\
$m_b(m_b)$& $ (4.2+0.17-0.07) {\rm GeV}$~\cite{Amsler:2008zzb} & & \\
\hline
\end{tabular}
\caption{Values of the input parameters used in our analysis. The subscript ``tree''
in $|V_{td}|$ and $\sin(2\beta)$ stands for the inputs extracted from data using only
tree-level observables~\cite{Bona:2007vi}.}
\label{tab:eps}
\end{center}
\end{table*}
\begin{figure*}[th]
\includegraphics[width=0.4\textwidth]{Rb_gamma.pdf}~~
\includegraphics[width=0.4\textwidth]{Rb_gamma2.pdf}\\
\includegraphics[width=0.4\textwidth]{Rb_gamma4.pdf}~~
\includegraphics[width=0.4\textwidth]{Rb_gamma3.pdf}
\caption{
The $R_b-\gamma$ plane assuming: {\bf i)} the SM situation with the input parameters of
Table~\ref{tab:eps} (upper left), {\bf ii)} $\sin 2\beta$ and $R_t$ NP free while $\epsilon_K$
affected by a $+18\%$ NP effect compared to the SM contribution (upper right), {\bf iv)}
$\epsilon_K$ and $R_t$ NP free while $\sin 2\beta$ affected by a NP phase in $B_d$ mixing
with $\varphi_{B_d}\approx -4.5^\circ$ (lower left), {\bf iii)} $\epsilon_K$ and $\sin 2\beta$
NP free while $\Delta M_d/\Delta M_s$ affected by a $-20\%$ NP contribution compared to the
SM contribution (lower right). The black star stands for the values obtained from the NP UT fit~\cite{Bona:2005eu}.}
\label{fig1}
\end{figure*}
\section{UT analysis} \label{sec:UT_analysis}
In this section, we perform a unitarity triangle (UT) analysis in the framework of the
$SSU(5)_{RN}$ model. We remind that there exist two different UTs: 1) the so-called
reference unitarity triangle (RUT)~\cite{Goto:1995hj}, determined entirely from tree level
decays hence, likely unaffected by any significant NP pollution, and 2) the universal
unitarity triangle (UUT)~\cite{Buras:2000dm} of models with constrained MFV, determined
by means of loop-induced FCNC processes and hence potentially sensitive to NP effects.
Therefore, a comparative UT analysis performed by means of the RUT and UUT may unveil
NP effects.
In particular, the above UTs are characterized by the following parameters
\begin{eqnarray}
\label{eq:UT_parameter_1}
V_{us} &\equiv&
\lambda ~,~ V_{cb} ~,~ R_b ~,~ \gamma \qquad {\rm RUT}\,,
\nonumber\\
\label{eq:UT_parameter_2}
V_{us} &\equiv&
\lambda ~,~ V_{cb} ~,~ R_t ~,~ \beta \qquad {\rm UUT}\,,
\end{eqnarray}
where $R_b\equiv|V_{ud}V^*_{ub}|/|V_{cd}V^*_{cb}|$, $R_t\equiv|V_{td}V^*_{tb}|/|V_{cd}V^*_{cb}|$
and the angles $\beta$ and $\gamma$ are such that $V_{td}=|V_{td}|e^{-i\beta}$ and $V_{ub}=|V_{ub}|e^{-i\gamma}$. Moreover, the dictionary between $(R_t,\beta)$ and
$(R_b,\gamma)$ reads
\begin{equation} \label{eq:Rt_beta}
R_t\!=\!\sqrt{1+R_b^2-2 R_b\cos\gamma} ~,~~
\cot\beta\!=\!\frac{1-R_b\cos\gamma}{R_b\sin\gamma}~,
\end{equation}
\begin{equation} \label{eq:Rb_gamma}
R_b\!=\!\sqrt{1+R_t^2-2 R_t\cos\beta} ~,~~
\cot\gamma\!=\!\frac{1-R_t\cos\beta}{R_t\sin\beta}~.
\end{equation}
In terms of physical observables we can write
\begin{equation}
\label{eq:Rt_sin2beta_NP}
R_t = \frac{\xi}{\lambda} \sqrt{\frac{m_{B_s}}{m_{B_d}}} \sqrt{\frac{\Delta M_d}{\Delta M_s}} \sqrt{\frac{C_{B_s}}{C_{B_d}}} ~,~~~ \sin(2 \beta + 2 \varphi_{B_d}) = S_{\psi K_S} ~,
\end{equation}
with the SM limit recovered for $C_{B_q}=1$ and $\varphi_{B_d}=0$.
The last observable that is relevant for our UT analysis is $\epsilon_K$.
In the SM, $\epsilon_K$ can be written as~\cite{Buras:2008nn}
\begin{eqnarray}
\label{eq:epsK_SM}
|\epsilon_K|\!&=&\!
\kappa_\epsilon C_\epsilon\hat B_K |V_{cb}|^2 |V_{us}|^2
\bigg(\frac{|V_{cb}|^2}{2} R_t^2 \sin 2\beta \eta_{tt}S_0(x_t)+
\nonumber\\
&+&R_t\sin\beta(\eta_{ct}S_0(x_c,x_t)-\eta_{cc}x_c)\bigg),
\end{eqnarray}
where $C_\epsilon \simeq3.658\times 10^4$ and all the parameters entering the above
expression are reported in Table~\ref{tab:eps}.
As stressed in~\cite{Buras:2008nn}, the SM prediction for $\epsilon_K$ implied by the
measured value of $S_{\psi K_S}=\sin2\beta$ may be too small to agree with experiment.
The main reasons are the decreased value of $\hat B_K$ and the decreased value of
$\epsilon_K$ in the SM arising from a multiplicative factor, estimated as
$\kappa_\epsilon=0.94\pm 0.02$~\cite{Buras:2010pz}.
Taking into account also the recent calculation of the QCD factor $\eta_{ct}$ at the
NNLO~\cite{Brod:2010mj}, that enhances the value of $\varepsilon_K$ by $3\%$~\cite{Brod:2010mj},
the total suppression of $\epsilon_K \propto\hat B_K \kappa_\epsilon$ compared to the
commonly used formulae is typically of order 15\%. Using the inputs of Table~\ref{tab:eps}, eq.~(\ref{eq:Rt_sin2beta_NP}) for the SM case (where $C_{B_s}=C_{B_d}=1$), and
eq.~(\ref{eq:epsK_SM}), one finds~\cite{Brod:2010mj}
\begin{equation} \label{eq:epsK_SMnumber}
|\epsilon_K|^{\rm SM} = (1.90 \pm 0.26) \times 10^{-3}~,
\end{equation}
to be compared with the experimental measurement \cite{Amsler:2008zzb}
\begin{equation} \label{eq:epsK_exp}
|\epsilon_K|^{\rm exp} = (2.229 \pm 0.010) \times 10^{-3}~.
\end{equation}
In fig.~\ref{fig1}, we show the above tensions in the $R_b-\gamma$ plane updating
the analysis of~\cite{Altmannshofer:2009ne} by the inclusion of the new values for $\kappa_\epsilon$~\cite{Buras:2010pz} and $\eta_{ct}$~\cite{Brod:2010mj}.
In the upper left plot of fig.~\ref{fig1}, we show the regions corresponding to the 1$\sigma$
allowed ranges for $\sin2\beta$, $R_t$ and $|\epsilon_K|^{\rm SM}$ as calculated by means of~(\ref{eq:Rt_sin2beta_NP}) and (\ref{eq:epsK_SM}), respectively, using the numerical
input parameters of tab.~\ref{tab:eps}. As shown, there are three different values of
$(R_b,\gamma)$, dependently which two constraints are simultaneously applied.
Possible solutions to this tension can be obtained assuming:
\begin{itemize}
\item[\bf 1)] a positive NP contribution to $\epsilon_K$, at the level of $\approx +20\%$,
leaving $\sin 2\beta$ and $\Delta M_d/\Delta M_s$ SM-like~\cite{Buras:2008nn}.
\item[\bf 2)] $\epsilon_K$ and $\Delta M_d/\Delta M_s$ NP free while $S_{\psi K_S}$
affected by a NP phase in $B_d$ mixing with $\varphi_{B_d}\approx -5^\circ$~\cite{SL}.
\item[\bf 3)] $\epsilon_K$ and $S_{\psi K_S}$ NP free while $\Delta M_d/\Delta M_s$
affected by NP at the level of $\approx -20\%$~\cite{Altmannshofer:2009ne},
requiring in turn an increased value of $R_t$ to fit the data.
\end{itemize}
Of course all these effects could be simultaneously at work.
As stressed in~\cite{Altmannshofer:2009ne}, the possibility 3) implies a large value of
$\gamma$, as shown in fig.~\ref{fig1} (see the lower plot on the right), and $\alpha$
significantly below $90^\circ$. Therefore, this scenario can be easily probed or falsified
by means of improved determinations of $\gamma$, as expected at the LHCb, and $\alpha$.
Moreover, the possibility 3) is particularly relevant within the $SSU(5)_{RN}$ model,
as we will discuss in detail later. In such a case, if the mixing angle regulating the
$b\to s$ transition contains a natural $\mathcal{O}(1)$ CPV phase, then solution 3)
also implies a non-standard value for $S_{\psi\phi}$ in the $B_s^0$ system.
\section{SUSY GUTs and flavour phenomenology}
The quark-lepton unification predicted by GUTs implies, in a SUSY framework,
a unification of the squark and slepton mass matrices at the GUT scale.
For instance, within SU(5), the multiplet ${\bar {\bf 5}}$ contains the right-handed
down-type quarks ($D^c$) and the left-handed lepton doublets ($L$) while the $\bf 10$
multiplet contains the left-handed quark doublets ($Q$), the right-handed up-type
quarks ($U^c$), and the right-handed charged-leptons ($E^c$).
Even if we assume a complete flavour blindness for the sfermion masses at the high
scale (either $M_P$ or $M_G$), we can still expect sizable sources of FCNC if the
theory contains neutrino Yukawa interactions accounting for the neutrino masses and
mixings by means of a see-saw mechanism.
Assuming a type-I see-saw with three heavy right-handed neutrinos~\cite{seesaw}, the
effective light-neutrino mass matrix resulting after integrating out the heavy fields
is $m_\nu \!=\! y_\nu^T\hat{M}_\nu^{-1} y_\nu \langle H_u \rangle^2$ where $\hat{M}_\nu$
is the right-handed neutrino mass matrix, $y_{\nu}$ is the unknown neutrino Yukawa
matrix, and $\langle H_u \rangle$ is the up-type Higgs VEV. Hereafter, we take a basis
where $\hat{M}_\nu$ is diagonal and symbols with hat mean they are diagonal matrices.
In the mSUGRA scenario and specializing to the case of $SSU(5)_{RN}$ model, low-energy flavor-violating SUSY-breaking terms are radiatively induced, and they are qualitatively
given as
\begin{eqnarray}
(m_{{\tilde d}_R}^2)_{ij}\!\!&=&\!\!
- \frac{(3m_0^2\!+\!A_0^2)}{8\pi^2}
(e^{i\hat{\phi}_{d}} y^T_\nu y^*_\nu
e^{-i\hat{\phi}_{d}})_{ij}\ln \frac{ M_{\rm P} }{ M_{\rm G} },
\nonumber\\
(m_{{\tilde d}_L}^2)_{ij}\!\!&=&\!\!
- \frac{(3m_0^2\!+\!A_0^2)}{8\pi^2} (V^\dagger \hat{y}_u^2 V)_{ij}
\left ( 3\ln \frac{ M_{\rm P} }{ M_{\rm G} } \!+\! \ln \frac{ M_{\rm G} }{\tilde m} \right ) ,
\nonumber\\
(m_{{\tilde e}_R}^2)_{ij}\!\!&=&\!\!
-3\frac{(3m_0^2\!+\!A_0^2)}{8\pi^2}
(e^{i\hat{\phi}_d} V^T \hat{y}_u^2 V^* e^{-i\hat{\phi}_d})_{ij}
\ln\frac{M_{\rm P}}{M_{\rm G}}\,,
\nonumber\\
(m_{{\tilde l}_L})_{ij}\!\!&=&\!\!
- \frac{(3m_0^2\!+\!A_0^2)}{8\pi^2}
(y^{\dagger}_{\nu})_{ik} (y_{\nu})_{kj}\ln\frac{ M_{\rm P} }{M_{\nu_k}},
\label{Eq:SU5RN_FV}
\end{eqnarray}
where $V$ is the CKM matrix, $\hat{\phi}_d$ is a GUT phase and $m_0$ ($A_0$) is the
universal scalar mass (trilinear coupling).
Within $SU(5)$, as both $Q$ and $E^c$ are hosted in the ${\bf 10}$ representation, the CKM
matrix mixing of the left-handed quarks will give rise to off-diagonal entries in the running
of the right-handed slepton soft masses $(m_{{\tilde e}_R}^2)_{ij}$ due to the interaction
of the colored Higgs~\cite{hkr,barbieri}. Vice versa, $Y_{\nu}$ enters the mass matrices
of both right-handed down squark and left-handed sleptons, as $D^c$ and $L$ lie in the
${\bar {\bf 5}}$ multiplet of $SU(5)$.
We remind that, within a type-I see-saw scenario, $y_\nu$ can be written in the general form~\cite{Casas:2001sr} $y_\nu=\sqrt{\hat{M}_\nu}R\sqrt{\hat{m}_\nu}U^{\dagger}/\langle H_u\rangle$
where $R$ is an arbitrary complex orthogonal matrix while $U$ is the PMNS matrix.
The determination of $(m_{{\tilde{l}_L}}^2)_{i\neq j}$ would require a complete knowledge
of the neutrino Yukawa matrix $y_\nu$, which is not possible using only low-energy
observables from the neutrino sector.
In particular, the ratio $(m_{{\tilde{l}_L}}^2)_{12}/(m_{{\tilde{l}_L}}^2)_{23}$ is highly
model dependent while $(m_{{\tilde{l}_L}}^2)_{12}\simeq(m_{{\tilde{l}_L}}^2)_{13}$ to a good approximation~\cite{Hisano:2009ae}. The situation is completely different in a
type-II see-saw scenario where the mixing angles of $(m_{{\tilde{l}_L}}^2)_{i\neq j}$ are
entirely governed by the PMNS matrix and low energy LFV processes relative to different
family transitions are strikingly correlated~\cite{Rossi:2002zb}.
Furthermore, we remind that a successful Yukawa coupling unification requires either the
introduction of some GUT breaking effects or some new non-renormalizable contributions.
In both cases, we are forced to introduce a relative rotation matrix $V^{(ql)}$ between
the quark and leptonic fields (defined in the super-CKM basis) such that the quark-lepton
correlations might be modified~\cite{Borzumati:2009hu}. However, since the matrix $V^{(ql)}$
is unknown, in the following, we assume the case where the naive quark-lepton correlations
are still valid.
\section{Hadronic sector}
In what follows, we discuss the relevant observables for our study, including
$\Delta F =2,1,0$ transitions.
{\bf 1.} The complete set of operators for $\Delta B=2$ transitions is~\cite{Buras:2001ra}
\begin{eqnarray}
Q_1^{VLL}&=&(\bar b_L \gamma_\mu q_L)(\bar b_L \gamma^\mu q_L)\,, \nonumber\\
Q_1^{SLL}&=&(\bar b_R q_L)(\bar b_R q_L)\,, \nonumber\\
Q_2^{SLL}&=&(\bar b_R \sigma_{\mu\nu} q_L)(\bar b_R \sigma^{\mu\nu} q_L)\,, \nonumber\\
Q_1^{LR}&=&(\bar b_L \gamma_\mu q_L)(\bar b_R \gamma^\mu q_R)\,, \nonumber\\
Q_2^{LR}&=&(\bar b_R q_L)(\bar b_L q_R)\,,
\label{eq:operatorsDF2}
\end{eqnarray}
where $q=s,d$, $\sigma_{\mu\nu}=\frac{1}{2}\left[\gamma_\mu,\gamma_\nu\right]$.
In Eq.~(\ref{eq:operatorsDF2}) we did not show the operators $Q_1^{\rm VRR}$ and
$Q_{1,2}^{SRR}$ that are obtainable from $Q_1^{\rm VLL}$ and $Q_{1,2}^{SLL}$,
respectively, with the exchange $q_L\to q_R$.
The low energy effective Hamiltonian reads~\cite{Buras:2001ra}
\begin{equation}
{\cal H}_{\rm eff}=\sum_{i,a}C_i^a(\mu_B,B)Q_i^a~,
\end{equation}
where the summation is performed over contributing operators. The off-diagonal
element in the $B^{0}_q$-meson mixing is given by
\begin{equation}
\label{eq:M12q}
M_{12}^q = \frac{1}{3}M_{B_q} F_{B_q}^2\sum_{i,a} C_i^{a*}(\mu_H,B_q) P_i^a(B_q)\,,
\end{equation}
where $P_i^a(B^{0}_q)$ collect all RG effects from the high scale, where heavy degrees of freedom
are integrated out, down to the $B$ meson scale as well as hadronic matrix elements obtained
by lattice QCD techniques. Updating the results of Ref.~\cite{Buras:2001ra}, it turns out that
$P_2^{LR} (B^{0}_q)\approx 3.4$ and $P_1^{SLL} (B^{0}_q)\approx -1.4$ for $\mu_H=246$~GeV.
Introducing the notation
\begin{equation}
M_{12}^q=\left(M_{12}^q\right)_{\text{SM}} C_{B_q}e^{2 i\varphi_{B_q}}~,
\qquad (q=d,s)~,
\label{eq:M12}
\end{equation}
the $B^{0}_{s,d}$ mass differences and the CP asymmetries $S_{\psi K_S}$ and $S_{\psi\phi}$ are
\begin{eqnarray}
\Delta M_q &=& 2\left|M_{12}^q\right| = (\Delta M_q)_{\text{SM}}C_{B_q}~,
\label{eq:Delta_Mq} \\
S_{\psi K_S} &=& \sin( 2\beta + 2\varphi_{B_d} )~,\\
S_{\psi\phi} &=& \sin( 2|\beta_s| - 2\varphi_{B_s} )~,
\label{eq:CPV_Bq}
\end{eqnarray}
where $\sin(2\beta)_{\rm tree}=0.734\pm 0.038$~\cite{Bona:2007vi} and
$\sin(2\beta_s)=0.038\pm 0.003$~\cite{Bona:2007vi}.
The $\Delta S=2$ operators are obtained from Eq.~(\ref{eq:operatorsDF2}) by means of $b\to s$
and $q=d$. The off-diagonal element in $K^0-\bar K^0$ mixing $M_{12}^K$ is then given by
\begin{equation}
2 M_K M_{12}^K=\langle \bar K^0|{\cal H}_{\rm eff}|K^0\rangle^*
\end{equation}
and the observables $\Delta M_K$ and $\epsilon_K$ can be evaluated through
\begin{eqnarray}
\Delta M_K&=&2 \text{Re}\left( M_{12}^K\right)\,,\\
\epsilon_K&=&e^{i\varphi_\epsilon}\frac{\kappa_{\epsilon}}{\sqrt 2\,
\Delta M_K}{\text{Im}}\left(M_{12}^K\right)\,,
\end{eqnarray}
where $\kappa_\varepsilon=0.94\pm0.02$~\cite{Buras:2008nn,Buras:2010pz} accounts
for $\varphi_\varepsilon=(43.51\pm0.05)^\circ\neq\pi/4$ and includes long distance
contributions.
Let us discuss now the semileptonic asymmetry in $B^{0}$ decays $A^{b}_{\text{SL}}$, described
by the quantity~\cite{Abazov:2010hv}
\begin{equation}
A^{b}_{\text{SL}} = (0.506 \pm 0.043) A^{d}_{\text{SL}} + (0.494 \pm 0.043) A^{s}_{\text{SL}}~,
\end{equation}
where $A^{d}_{\text{SL}}$ and $A^{s}_{\text{SL}}$ are the asymmetries in $B^0_d$ and $B^0_s$
decays, respectively. Within the SM, it is predicted that
$A^{d}_{\text{SL}}({\rm SM})=(-0.48 ^{+0.1}_{-0.12}) \times 10^{-3}$~\cite{Nierste},
$A^{s}_{\text{SL}}({\rm SM})=(2.1 \pm 0.6) \times 10^{-5}$~\cite{Nierste} and
therefore~\cite{Nierste}
\begin{equation}
A^{b}_{\text{SL}}({\rm SM}) = (-0.23^{+0.05}_{-0.06}) \times 10^{-3}~.
\label{aslbsm}
\end{equation}
This has to be compared with the recently measured value by the D0 collaboration~\cite{Abazov:2010hv}
\begin{equation}
A^{b}_{\text{SL}}({\rm D0}) = (-9.57 \pm 2.51 \pm 1.46)\times 10^{-3}~,
\label{ASL_exp}
\end{equation}
that differs by 3.2 standard deviations from the SM prediction, providing the first
evidence for anomalous CP-violation in the mixing of neutral $B^{0}$ mesons.
In the presence of NP, the asymmetries $A^{d}_{\text{SL}}$ and $A^{s}_{\text{SL}}$
can be evaluated by means of the following expression~\cite{Nierste}
\begin{equation}
\label{eq:A_SL_corr}
A^{q}_{\text{SL}} =
{\rm Im}\left(\!\frac{\Gamma^{q}_{12}}{M^{q}_{12}}\!\right)^{\rm \!SM}
\!\frac{\cos 2\varphi_{B_q}}{C_{B_q}}-
{\rm Re}\left(\!\frac{\Gamma^{q}_{12}}{M^{q}_{12}}\!\right)^{\rm \!SM}
\!\frac{\sin 2\varphi_{B_q}}{C_{B_q}}~,
\end{equation}
where the updated values for ${\rm Re}(\Gamma^{q}_{12}/M^{q}_{12})^{\rm SM}$ and
${\rm Im}(\Gamma^{q}_{12}/M^{q}_{12})^{\rm SM}$ can be found in~\cite{Nierste}.
Moreover, we recall that $A^{s}_{\text{SL}}$, in the presence of NP and neglecting
$\beta_s$, is correlated model-independently with $S_{\psi\phi}$ as
$A^{s}_{\text{SL}}\simeq S_{\psi\phi}\!\times\!({\rm R}^{s}_{12}/C_{B_s})$~\cite{Ligeti:2006pm}
(for an alternative model-independent formula, see~\cite{Grossman:2009mn}).
On general grounds, we observe that the $A^{d}_{\text{SL}}$ contribution to
$A^{b}_{\text{SL}}$ is constrained model-independently by the limited NP room
left to $S_{\psi K_S}$. Furthermore, since in the $SSU(5)_{RN}$ model
$(m_{\tilde{l}_L}^{2})_{12}\simeq (m_{\tilde{l}_L}^{2})_{13}$, implying that $(m_{\tilde{d}_R}^{2})_{12}\simeq (m_{\tilde{d}_R}^{2})_{13}$, it turns out
that $S_{\psi K_S}$ is SM-like to a very good extent after imposing the experimental
bound on $\epsilon_K$. As a result we conclude that, within the $SSU(5)_{RN}$
model, the NP contributions to $A^{d}_{\text{SL}}$ are completely negligible.
By contrast, the NP contributions to $A^{s}_{\text{SL}}$ might be large since in
this case the limits from $S_{\psi K_S}$ and $\epsilon_K$ do not generally apply.
Therefore, within the $SSU(5)_{RN}$ model, it turns out that
\begin{equation}
\label{eq:Ab_SL_Y}
A^{b}_{\text{SL}} \approx 0.5~A^{s}_{\text{SL}}
\approx - 10^{-3}~\frac{S_{\psi\phi}}{C_{B_s}}\,,
\end{equation}
where $S_{\psi\phi}\lesssim 1$ and therefore $A^{b}_{\text{SL}}\gtrsim -10^{-3}$.
We briefly recall now the leading SUSY contributions to the $K^0$, $B^{0}_{d}$
and $B^{0}_{s}$ mixing amplitudes coming from gluino/squark boxes and, in the
large $\tan\beta$ regime, from double Higgs penguin contributions (we refer to ref.~\cite{Altmannshofer:2009ne} and therein references for the full expressions).
The dominant gluino/squark boxes read
\begin{eqnarray}
\label{eq:MIA2_C4}
\left(C_{2}^{LR}\right)_{\tilde g}
\simeq
-\frac{\alpha_s^2}{m_{\tilde q}^{2}}
\left[(\delta_d^{LL})_{ij}(\delta_d^{RR})_{ij}\right]
g_4^{(1)}(x_g)~,
\end{eqnarray}
where $(\delta_d^{LL})_{ij},(\delta_d^{RR})_{ij}$ are the mass insertion (MI) parameters
(as defined in ref.~\cite{Altmannshofer:2009ne}), $x_g=M_{\tilde g}^2/m_{\tilde q}^{2}$,
the loop function is such that $g_4^{(1)}(1)=23/180$ and $ij=23,13,12$ for the $B^{0}_s$,
$B^{0}_d$ and $K^0$ systems, respectively.
The dominant Higgs mediated contributions for the $B^{0}_s$ system read
\begin{eqnarray}
\left(C_{2}^{LR}\right)_{H}
\!\!&\simeq&\!\!
\frac{\alpha_2^2 \alpha_s}{4\pi}\frac{m_b^2 m_t^2}{2 M_W^4}
\frac{t^{4}_{\beta}}{(1+\epsilon t_{\beta})^{4}}
\frac{|\mu|^2}{M_A^2 m_{\tilde q}^{2}}
\nonumber\\
&\times&\!\!\!
\left[\frac{A_t^* M_{\tilde g}}{m_{\tilde q}^{2}}
h_1(x_g) h_3(x_\mu)\right](\delta^{RR}_d)_{23}V_{ts}~,
\label{eq:C4HB}
\end{eqnarray}
where $t_{\beta}=\tan\beta$, $x_\mu=|\mu|^2/m_{\tilde q}^{2}$, $h_1(1)=4/9$, $h_3(1)=-1/4$, $|\epsilon|\approx 10^{-2}$ is the well known resummation factor stemming from ($t_{\beta}$
enhanced) non-holomorphic threshold corrections and $m_b$ has to be evaluated at the scale
$M_A$. Notice that, the presence of the RR MIs is crucial to avoid the typical
$m_s/m_b$ suppression of $\left(C_{2}^{LR}\right)_{H}$ arising within a MFV scenario~\cite{Buras:2001mb}.
The Wilson coefficient for $B^{0}_d$ mixing can be obtained by replacing 23 with 13 and
$V_{ts}$ by $V_{td}$.
In the case of $K^0$ mixing, the most relevant effect from the neutral Higgses arises
only at the fourth order in the MI expansion. Since in the $SSU(5)_{RN}$
model the MI are radiatively induced, this effect can be safely neglected.
Moreover, we remind that in contrast to the case with gluino box contributions, there
are no analogous Higgs mediated contributions for $D^0-\bar D^0$ mixing as there are
no $\tan\beta$ enhanced non-holomorphic threshold corrections in the up-quark sector.
{\bf 2.} The CP asymmetries $S_f$ in the decays of neutral $B^{0}_d$ mesons into final
CP eigenstates $f$ can be affected by NP both in the $B^{0}_d$ mixing amplitude and
the decay amplitudes $\bar b\to\bar qq\bar s$ ($q=s,d,u$).
In the $SSU(5)_{RN}$ model, only the latter effect can be sizable.
The asymmetries are defined as
\begin{equation}
S_f=\frac{2{\rm Im}(\lambda_f)}{1+|\lambda_f|^2}~.
\label{eq:Sf}
\end{equation}
where $\lambda_f=e^{-2i(\beta + \varphi_{B_d})}(\overline{A}_f/A_f)$ with $\varphi_{B_d}$
being the NP phase of the $B^{0}_d$ mixing amplitude, $M_{12}^d$, and $A_f$ ($\overline{A}_f$)
is the decay amplitude for $B^{0}_d(\overline{B^{0}_d})\to f$. Within the SM it turns out that
$A_f^{\rm SM}=A_f^c(1+a_f^u e^{i\gamma})$ where the $a_f^u$ parameters have been evaluated
in the QCD factorization approach at the leading order and to zeroth order in $\Lambda/m_b$ in~\cite{buchalla}.
In the presence of NP one can generally define the amplitude $A_f$ as
\begin{equation}
A_f = A_{f}^{\rm SM} + A_f^c\sum_i\left(b_{fi}^c+b_{fi}^ue^{i\gamma}\right)
\left(C_i^{*}+\zeta \tilde C_i^{*}\right)~,
\label{eq:def_b_fu}
\end{equation}
where $C_i$ and $\tilde{C}_i$ are the NP contributions to the Wilson coefficients evaluated
at the scale $M_W$ (see Ref.~\cite{Altmannshofer:2009ne} for the notation), the parameters
$b_{fi}^u$ and $b_{fi}^c$ calculated in~\cite{buchalla}
and $\zeta=\pm 1$ depending on the parity of the final state; for instance $\zeta=1$ for
$\phi K_S$ and $\zeta=-1$ for $\eta^\prime K_S$. Within the $SSU(5)_{RN}$ model the by far
dominant contribution to $A_f$ is provided by $\tilde C_8$ which reads~\cite{Altmannshofer:2009ne}
\begin{eqnarray}
\frac{4 G_F}{\sqrt{2}}~\tilde C_{8}^{\tilde g}
\simeq
\frac{g_s^2}{m_{\tilde q}^{2}}
\frac{M_{\tilde g}\mu^*}{m_{\tilde q}^{2}} \frac{t_{\beta}}{(1+\epsilon t_{\beta})}
\frac{(\delta_d^{RR})_{32}}{V_{ts}^*}g_{8}(x_g)~.
\label{eq:tildeC7_g}
\end{eqnarray}
where the loop function is such that $g_{8}(1)=-7/120$.
The branching ratio for $B_q\to\mu^+\mu^-$ (with $q=s,d$) in the presence of NP scalar
currents can be expressed as~\cite{Altmannshofer:2009ne}
\begin{equation}
\frac{{\rm BR}(B_q \!\to\! \mu^+ \mu^-)}{{\rm BR}(B_q \!\to\! \mu^+ \mu^-)_{\rm SM}}
\simeq \left|1 - C^q_S -\tilde C^q_S\right|^2 + \left|C^q_S - \tilde C^q_S\right|^2~,
\end{equation}
where the relevant contributions to the Wilson coefficients $C^q_S$ and $\tilde C^q_S$
arising in the $SSU(5)_{RN}$ model read
\begin{eqnarray}
\tilde C^q_S \!\!&\simeq&\!\! \frac{\alpha_2 \alpha_s}{8M^2_A}
\frac{m^{2}_{B_q}}{M_W^2 C_{10}^{\rm SM}}
\frac{t^{3}_{\beta}}{(1\!+\!\epsilon t_{\beta})^{2}}
\frac{M_{\tilde g}\mu^*}{m_{\tilde q}^{2}} (\delta_d^{RR})_{3q} h_1(x_g)~,
\label{eq:CStilde}
\nonumber\\
C^q_S \!\!&\simeq&\!\! -\frac{\alpha_2^2}{8M^2_A}
\frac{m^{2}_{B_q}}{M_W^4 C_{10}^{\rm SM}}
\frac{m_t^2 t^{3}_{\beta}}{(1\!+\!\epsilon t_{\beta})^{2}}
\frac{A_t\mu}{m_{\tilde q}^{2}}V_{tq}^* h_3(x_\mu)~,
\label{eq:CS}
\end{eqnarray}
with $C^{\rm SM}_{10}$ given for instance in~\cite{Altmannshofer:2009ne}.
{\bf 3.} Also the hadronic and leptonic EDMs might be generated by {\it flavor dependent} phases
(flavored EDMs)~\cite{Hisano:2008hn}. In particular, the neutron EDM $d_n$ can be estimated
from the naive quark model as $d_n\approx\frac{4}{3} d_d-\frac{1}{3} d_u$ (with $d^{(c)}_{f}$
evaluated at $1$~GeV) or, alternatively, by means of QCD sum rule techniques~\cite{qcdsumrules,Demir:2002gg,Demir:2003js,Olive:2005ru}.
In the latter case, it turns out that
\begin{equation}
d_n \!= (1\pm 0.5)\Big[ 1.4 (d_d-0.25 d_u) + 1.1 e\, (d^c_d+0.5 d^c_u)\Big]\,.
\label{Eq:dn_odd}
\end{equation}
Similarly, the prediction for the Mercury EDM in the QCD sum rule approach reads~\cite{Demir:2003js,Olive:2005ru}
\begin{eqnarray}
\label{eq:mercuryedm}
d_{\rm Hg}
&\simeq&
7\times 10^{-3}\,e\,(d_u^c-d_d^c)\,,
\end{eqnarray}
where, in eq.~(\ref{eq:mercuryedm}), we have retained only the contributions relevant to
our analysis. Notice that, the values of $d^{(c)}_{f}$ entering the EDM's predictions of
eqs.~(\ref{Eq:dn_odd},\ref{eq:mercuryedm}) are assumed to be evaluated at $1$~GeV
by means of QCD renormalization group evolution~\cite{Degrassi:2005zd} from the corresponding
values at the electroweak scale.
The dominant gluino/squark contribution to the down-quark (C)EDMs at the SUSY scale reads
\begin{equation}
\left\{\frac{d_{d_i}}{e},~d^c_{d_i}\right\}\!\approx\!
\frac{\alpha_{s}}{4\pi}\frac{m_{b}}{m_{\tilde q}^{2}}
\frac{M_{\tilde g}\mu}{m_{\tilde q}^{2}} t_{\beta}
{\rm Im}\left[(\delta^{LL}_{d})_{i3}(\delta^{RR}_{d})_{3i}\right]
f(x_g)\,,
\label{Eq:edm_d_gluino}
\end{equation}
where the loop functions satisfy $f(1)=$ $\{4/135,~11/180\}$.
\begin{figure*}[th]
\includegraphics[width=0.4\textwidth]{epsK_meg.pdf}~
\includegraphics[width=0.4\textwidth]{contour.pdf}\\
\includegraphics[width=0.4\textwidth]{epsK_de.pdf}~
\includegraphics[width=0.4\textwidth]{epsK_dn.pdf}
\caption{
Upper left: ${\rm BR}(\mu\to e\gamma)$ vs. $\epsilon^{SUSY}_{K}/(\epsilon^{SM}_{K})$.
The blue (red) points can explain the $(g-2)_{\mu}$ anomaly at the level of
$\Delta a^{\rm SUSY}_{\mu}\gtrsim 1(2)\times 10^{-9}$.
Upper right: ${\rm BR}(\mu\to e\gamma)$ in the $(m_0,M_{1/2})$ plane imposing
a $+20\%$ NP effect in $\epsilon^{SUSY}_{K}$.
Lower left (right): electron (neutron) EDM vs. $\epsilon^{SUSY}_{K}/(\epsilon^{SM}_{K})$
for different values of ${\rm BR}(\mu\to e\gamma)$. In all the plots we vary the SUSY
parameters in the ranges $m_0<1\,{\rm TeV}$, $M_{1/2}<1\,{\rm TeV}$, $|A_0|<3m_0$
(in the only upper right plot we set $|A_0|=0$), $\tan\beta=10$ and $\mu>0$.
We assume a hierarchical spectrum for both light and heavy neutrinos setting
$m_{\nu_3}=0.05{\rm eV}$ and varying the neutrino parameters in the ranges
$10^{11}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$, $10^{-5}\leq U_{e3}\leq 0.1$.}
\label{fig2}
\end{figure*}
\begin{figure*}[th]
\includegraphics[width=0.4\textwidth]{Spsiphi_BrTMG.pdf}~
\includegraphics[width=0.4\textwidth]{DMdDMs_BrTMG.pdf}\\
\includegraphics[width=0.4\textwidth]{gamma_BrTMG.pdf}~
\includegraphics[width=0.4\textwidth]{DMdDMs_gamma.pdf}
\caption{
Upper left: ${\rm BR}(\tau\to\mu\gamma)$ vs. $S_{\psi\phi}$.
Upper right: ${\rm BR}(\tau\to\mu\gamma)$ vs. $R_{sd}$.
Lower left: ${\rm BR}(\tau\to\mu\gamma)$ vs. $\gamma$.
Lower right: $\gamma$ vs. $R_{sd}$.
Blue points satisfy the condition $\Delta a^{\rm SUSY}_{\mu}\gtrsim 1\times 10^{-9}$
while the grey bands indicate the experimentally excluded regions.
In all the plots we vary the SUSY parameters in the ranges $m_0<1\,{\rm TeV}$,
$M_{1/2}<1\,{\rm TeV}$, $|A_0|<3m_0$, $\tan\beta=10$ and $\mu>0$.
We assume a hierarchical spectrum for both light and heavy neutrinos setting
$m_{\nu_3}=0.05{\rm eV}$, $U_{e3}= 0$ and varying the heaviest heavy neutrino
mass in the ranges $10^{13}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$.}
\label{fig3}
\end{figure*}
\begin{figure*}[th]
\includegraphics[width=0.4\textwidth]{Spsiphi_bsmm.pdf}~
\includegraphics[width=0.4\textwidth]{Spsiphi_AsSL.pdf}\\
\includegraphics[width=0.4\textwidth]{bsmm_bdmm.pdf}~
\includegraphics[width=0.4\textwidth]{Spsiphi_SphiKs.pdf}\\
\includegraphics[width=0.4\textwidth]{SphiKs_SetapKs.pdf}~
\includegraphics[width=0.4\textwidth]{SphiKs_BrTMG.pdf}
\caption{
Upper left: ${\rm BR}(B_s\to \mu^+\mu^-)$ vs. $S_{\psi\phi}$.
Upper right: $S_{\psi\phi}$ vs. $A^{s}_{SL}$.
Central left: ${\rm BR}(B_d\to \mu^+\mu^-)$ vs. ${\rm BR}(B_s\to \mu^+\mu^-)$.
Central right: $S_{\psi\phi}$ vs. $S_{\phi K_S}$.
Lower left: $S_{\eta^{\prime} K_S}$ vs. $S_{\phi K_S}$.
Lower right: ${\rm BR}(\tau\to\mu\gamma)$ vs. $S_{\phi K_S}$.
The grey bands indicate the experimentally excluded regions.
In all the plots we vary the SUSY parameters in the ranges $m_0<2\,{\rm TeV}$,
$M_{1/2}<1\,{\rm TeV}$, $|A_0|<3m_0$, $\tan\beta\leq 60$ and $\mu>0$.
We assume a hierarchical spectrum for both light and heavy neutrinos setting
$m_{\nu_3}=0.05{\rm eV}$, $U_{e3}= 0$ and varying the heaviest heavy neutrino
mass in the ranges $10^{13}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$.}
\label{fig4}
\end{figure*}
\section{Leptonic sector}
The branching ratio for $\ell_{i}\rightarrow \ell_{j}\gamma$ can be written as
\begin{eqnarray}
\frac{{\rm BR}(\ell_{i}\rightarrow \ell_{j}\gamma)}{{\rm BR}(\ell_{i}\rightarrow
\ell_{j}\nu_i\bar{\nu_j})} =\frac{48\pi^{3}\alpha}{G_{F}^{2}}(|A_L^{ij}|^2+|A_R^{ij}|^2)\,.
\nonumber
\end{eqnarray}
Starting from the full expressions of ref.~\cite{Hisano:1995cp} (which we use in our numerical analysis), and specializing to the illustrative case of a degenerate SUSY spectrum with a common
mass $m_{\tilde\ell}$, one can find
\begin{equation}
A^{ij}_L \simeq
\frac{\alpha_2}{60\pi}\frac{t_{\beta}}{m^{2}_{\tilde\ell}}
(\delta^{LL}_{\ell})_{ji}~.
\label{MI_degenerate}
\end{equation}
The main SUSY contribution to $a^{\rm MSSM}_\mu$ is usually provided by the loop
exchange of charginos and sneutrinos. In the limit of degenerate SUSY masses one
finds
\begin{equation}
\frac{a^{\rm MSSM}_\mu}{ 1 \times 10^{-9}}
\approx 1.5\left(\frac{t_{\beta}}{10} \right)
\left( \frac{300~\rm GeV}{m_{\tilde \ell}}\right)^2 \text{sgn}\,\mu\,.
\label{eq:g_2}
\end{equation}
Assuming a degenerate SUSY spectrum, it is straightforward to find the
correlation between $\Delta a^{\rm SUSY}_{\mu}$ and the branching ratios
for $\ell_i\to\ell_j\gamma$~\cite{Hisano:2001qz,Hisano:2009ae}
\begin{eqnarray}
{\rm BR}(\mu\to e\gamma)&\approx&
2\times 10^{-12}
\left[\frac{\Delta a^{\rm SUSY}_{\mu}}{ 3 \times 10^{-9}}\right]^{2}
\bigg|\frac{(\delta^{LL}_{\ell})_{21}}{10^{-4}}\bigg|^2\,,\nonumber\\
{\rm BR}(\tau\to\mu\gamma)&\approx&
8\times 10^{-8}
\left[\frac{\Delta a^{\rm SUSY}_{\mu}}{ 3 \times 10^{-9}}\right]^{2}
\bigg|\frac{(\delta^{LL}_{\ell})_{32}}{10^{-2}}\bigg|^2\,,
\label{lfvgm2}
\end{eqnarray}
where we have assumed that the MIs $(\delta^{LL}_{\ell})_{ij}$ provide the
dominant contributions to ${\rm BR}(\ell_i\to \ell_j\gamma)$, as it happens in the GUT
framework analyzed here.
Within the $SSU(5)_{RN}$ model, leptonic EDMs are generated via {\it flavour dependent}
phases (flavoured EDMs). It turns out that
\begin{equation}
\frac{d_{\ell_i}}{e}\simeq
-\frac{\alpha_Y}{4\pi}\bigg(\frac{m_\tau}{m_{\tilde \ell}^{2
|
}}\bigg)\,t_{\beta}~
\frac{{\rm Im}[\left(\delta^{RR}_{\ell}\right)_{i3}\left(\delta^{LL}_{\ell}\right)_{3i}]}{30}\,,
\label{edm_flavor}
\end{equation}
where a common SUSY mass $m_{\tilde \ell}$ has been assumed. If $t_{\beta}=10$ and
$m_{\tilde \ell}=300\,{\rm GeV}$, it turns out that $d_{\ell_i}\sim 10^{-22}\times
{\rm Im}[(\delta^{RR}_{\ell})_{i3}(\delta^{LL}_{\ell})_{3i}]\,e\,$cm.
\section{Hadron-lepton correlations}
\label{correlations}
As already anticipated in the Introduction, SUSY GUT models link flavor-violating observables
of the leptonic and hadronic sectors. In the following, we provide approximate analytical
expressions for these hadron-lepton correlations in order to get an idea of where we stand.
In particular, the GUT relation $(\delta_e^{LL})_{ij}=(\delta_d^{RR})_{ji}$, which is modified
at the electroweak scale as $(\delta_e^{LL})_{ij}=(\delta_d^{RR})_{ji} \times (m_{\tilde{q}}/m_{\tilde{\ell}})^2$, implies
\begin{eqnarray}
{\rm BR}(\mu\to e\gamma)
&\approx&
2\times 10^{-12}
\left(\frac{\epsilon^{SUSY}_{K}}{10^{-4}}\right)^{2}
\left(\frac{t_\beta}{10}\right)^2
\nonumber\\
&\times&
\bigg|\frac{(\delta_d^{LL})_{12}}{10^{-4}}\bigg|^{-2}
\left(\frac{m_{\tilde{q}}}{m_{\tilde{\ell}}}\right)^8\,,
\label{mueg_epsk}
\end{eqnarray}
and similarly for the 23 sector
\begin{eqnarray}
{\rm BR}(\tau\to\mu\gamma)&\approx&
7\times 10^{-8}
\bigg[ (C_{B_s}-1)^2 + C_{B_s} S_{\psi\phi}^2 \bigg]
\nonumber\\
&\times&
\left(\frac{t_\beta}{10}\right)^2
\bigg|\frac{(\delta_d^{LL})_{23}}{10^{-2}}\bigg|^{-2}
\left(\frac{m_{\tilde{q}}}{m_{\tilde{\ell}}}\right)^8\,,
\label{tmug_spsiphi}
\end{eqnarray}
where we have assumed a common SUSY mass $m_{\tilde{q}}$ ($m_{\tilde{\ell}}$) for the hadron
(lepton) sector and $(\delta_d^{LL})_{12,23}$ have been normalized to the typical values they
attain when they are radiatively generated by the large top Yukawa coupling and the CKM matrix
(see Eq.\,(\ref{Eq:SU5RN_FV})).
Finally, there is an even more direct correlation between ${\rm BR}(\tau\to\mu\gamma)$
and the NP effects entering $S_{\phi K_S}$.
Defining $S_{\phi K_S}=S_{\psi K_S}+\Delta S_{\phi K_S}$, we find
\begin{eqnarray}
{\rm BR}(\tau\to\mu\gamma)&\approx&
3\times 10^{-8}\left(\frac{m_{\tilde{q}}}{m_{\tilde{\ell}}}\right)^8
\left|\Delta S_{\phi K_S}\right|^2\,,
\label{tmug_sphiks}
\end{eqnarray}
where, starting from eqs.~(\ref{eq:Sf}),(\ref{eq:def_b_fu}),(\ref{eq:tildeC7_g})
and keeping only linear terms in the NP contributions, it turns out that
$\Delta S_{\phi K_S}\approx -2 b_{\phi K_S}^c \cos 2\beta ~{\rm Im}\tilde{C_8}$
with $b_{\phi K_S}^c\approx 1.4$~\cite{buchalla}.
\section{Numerical analysis}
\label{sec:num_analysis}
In this section, we present the numerical results for the observables discussed in the
previous sections in the context of the $SSU(5)_{RN}$ model, assuming a gravity
mediated mechanism for the SUSY breaking terms with $M_P = 2.4\times 10^{18}$~GeV.
In the upper (left) plot of Fig.~\ref{fig2}, we show the predictions for ${\rm BR}(\mu\to e\gamma)$ vs. $\epsilon^{SUSY}_{K}/(\epsilon^{SM}_{K})$ varying the SUSY parameters in the
ranges $(m_0,M_{1/2})<1\,{\rm TeV}$, $|A_0|<3m_0$, $\tan\beta=10$ and $\mu>0$.
Concerning the neutrino sector, hereafter, we assume a hierarchical spectrum for both light
and heavy neutrinos such that $m_{\nu_3}=0.05{\rm eV}$ and we vary the neutrino parameters
in the ranges $10^{11}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$, $10^{-5}\leq U_{e3}\leq 0.1$.
The blue (red) points can explain the $(g-2)_{\mu}$ anomaly at the level of $\Delta a^{\rm SUSY}_{\mu}\gtrsim 1(2)\times 10^{-9}$ while satisfying the constraints from ${\rm BR}(B\to X_s\gamma)$~\cite{Amsler:2008zzb} at the $99\%$ C.L.. As we can see, sizable SUSY
effects in $\epsilon_{K}$, that might be desirable to solve the UT anomaly, generally
imply a lower bound for ${\rm BR}(\mu\to e\gamma)$ in the reach of the MEG experiment.
The above statement is even more strengthened if we further require to explain the
$(g-2)_{\mu}$ anomaly.
In the upper (right) plot of Fig.~\ref{fig2}, we show the values reached by
${\rm BR}(\mu\to e\gamma)$ in the $(m_0,M_{1/2})$ plane setting $\mu>0$, $A_0=0$,
$\tan\beta=10$ and imposing a NP effect in $\epsilon^{SUSY}_{K}$ at the level
of $+20\%$ compared to the SM contribution to solve the UT tension.
The grey region is excluded by the constraint from the lower bound on the lightest Higgs
boson mass $m_{h^0}$ (we impose $m_{h^0}>111.4$~GeV to take into account the theoretical
uncertainties in the evaluation of $m_{h^0}$), the orange region is excluded by the
constraints on ${\rm BR}(B\to X_s\gamma)$ at the $99\%$ C.L. (we have evaluated
${\rm BR}(B\to X_s\gamma)$ including the SM effects at the NNLO~\cite{misiak} and the
NP contributions at the LO), the light blue (blue) region satisfies
$\Delta a^{\rm SUSY}_{\mu}\gtrsim 1(2)\times 10^{-9}$.
In the lower plots of Fig.~\ref{fig2}, on the left (right), we show the electron (neutron)
EDM vs. $\epsilon^{SUSY}_{K}/(\epsilon^{SM}_{K})$ for different values of ${\rm BR}(\mu\to e\gamma)$.
The requirement of sizable non-standard effects in $\epsilon^{SUSY}_{K}$ always implies large
values for $d_{e,n}$, in the reach of the planned experimental resolutions, as well as values
for ${\rm BR}(\mu\to e\gamma)$ that are most likely within the MEG reach.
The correlations between leptonic and hadronic observables in the plots on the left
in Fig.~\ref{fig2} demonstrate very clearly that we deal here with a GUT scenario.
In figs.~\ref{fig3},~\ref{fig4}, we present the predictions for B-physics observables.
As discussed in the previous sections, within a $SSU(5)_{RN}$ model, $b\to s$
and $\tau\to\mu$ transitions are linked, therefore, processes like $B^{0}_s$ mixing
and $\tau\to\mu\gamma$ turn out to be related.
In the plots of fig.~\ref{fig3} we vary the SUSY parameters in the ranges
$(m_0,M_{1/2})<1\,{\rm TeV}$, $|A_0|<3m_0$, $\tan\beta=10$ and $\mu>0$.
We assume a hierarchical spectrum for both light and heavy neutrinos setting
$m_{\nu_3}=0.05{\rm eV}$, $U_{e3}= 0$ and varying the heaviest heavy neutrino
mass in the range $10^{13}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$.
In the upper plot of fig.~\ref{fig3} on the left, we show the correlation between
${\rm BR}(\tau\to\mu\gamma)$ and $S_{\psi\phi}$. We see that, non-standard values for
$S_{\psi\phi}$ imply a lower bound for ${\rm BR}(\tau\to\mu\gamma)$ within the SuperB
reach. However, it seems unlikely to simultaneously explain the $(g-2)_{\mu}$ anomaly
(blue points correspond to $\Delta a^{\rm SUSY}_{\mu}\gtrsim 1\times 10^{-9}$) while
generating a large $S_{\psi\phi}$.
The situation can slightly change for large values of $\tan\beta$. In this case,
the constraints from ${\rm BR}(\tau\to\mu\gamma)$ might be still compatible with
$|S_{\psi\phi}|\leq 0.2$ and $\Delta a^{\rm SUSY}_{\mu}\gtrsim 1\times 10^{-9}$
(see fig.~\ref{fig4}).
In the upper plot of fig.~\ref{fig3} on the right, we show the correlation between
${\rm BR}(\tau\to\mu\gamma)$ vs. $R_{sd}$ defined as
\begin{equation}
R_{sd}=\frac{\Delta M_d/\Delta M_s}{(\Delta M_d/\Delta M_s)_{\rm SM}}\,.
\end{equation}
We see that non-standard effects in $R_{sd}$ may be easily generated, providing a
possible solution to the UT anomaly. This will imply in turn a lower bound for
${\rm BR}(\tau\to\mu\gamma)$ within the expected experimental reach of a SuperB.
In the lower plot of fig.~\ref{fig3} on the left, we show the correlation between
${\rm BR}(\tau\to\mu\gamma)$ and the CKM angle $\gamma$. This correlation can
be understood looking at the explicit expressions for $\gamma$ and $R_t$ in the presence
of NP, see eqs.(\ref{eq:Rb_gamma},\ref{eq:Rt_sin2beta_NP}). In particular,
NP effects in $R_{sd}$ would affect $R_t$ and therefore the determination of $\gamma$.
Moreover, since $b\to s$ and $\tau\to\mu$ transitions are linked in our framework,
it turns out that non-standard values for $\gamma$ imply a lower bound for
${\rm BR}(\tau\to\mu\gamma)$.
It will be exciting to monitor such a correlated NP effect at the LHCb.
In the lower plot of fig.~\ref{fig3} on the right, we report the correlation between
$\gamma$ vs. $R_{sd}$ clearly showing that negative NP effects in $R_{sd}$, accounting
for the UT anomaly, would imply large non-standard values for the angle $\gamma$.
Such large non-standard effects would also imply large (visible) values for
${\rm BR}(\tau\to\mu\gamma)$ as seen in the plot on the left.
In fig.~\ref{fig4}, we show the predictions for B-physics and lepton observables
including the large $\tan\beta$ regime, in order to make Higgs mediated effects for
$\Delta F = 1,2$ processes like ${\rm BR}(B_{s,d}\to\mu^+\mu^-)$ and $S_{\psi\phi}$
visible.
In particular, the plots of fig.~\ref{fig4} have been obtained employing the following
scan over the SUSY parameters: $m_0 (M_{1/2})<2(1)\,{\rm TeV}$, $|A_0|<3m_0$,
$\tan\beta<60$ and $\mu>0$. We assume a hierarchical spectrum for both light and heavy
neutrinos setting $m_{\nu_3}=0.05{\rm eV}$, $U_{e3}= 0$ and varying the heaviest heavy
neutrino mass in the range $10^{13}\leq M_{\nu_3}({\rm GeV})\leq 10^{15}$.
In the upper plot of fig.~\ref{fig4} on the left, we show ${\rm BR}(B_s\to \mu^+\mu^-)$
vs. $S_{\psi\phi}$ visualizing also the values attained by ${\rm BR}(\tau\to\mu\gamma)$
with different colours.
As we can see, $S_{\psi\phi}$ can depart from the SM expectations irrespective of whether
${\rm BR}(B_s\to \mu^+\mu^-)$ is SM-like or not. The reason is that $S_{\psi\phi}$ receives
large effects from both gluino/squark box contributions and from $\tan\beta$ enhanced double penguin Higgs contributions. In contrast, in the case of ${\rm BR}(B_s\to \mu^+\mu^-)$ only
the latter contribution can be effective.
In the upper plot of fig.~\ref{fig4} on the right, we show the (almost) model-independent
correlation between $A^{s}_{SL}$ and $S_{\psi\phi}$ (see eq.~(\ref{eq:Ab_SL_Y})).
Green points fulfill all the current available constraints, while blue points further
explain the $(g-2)_{\mu}$ anomaly at the level of $\Delta a^{\rm SUSY}_{\mu}\gtrsim
1\times 10^{-9}$.
While large departures from the SM expectations for $A^{s}_{SL}$ are still allowed,
the large value reported by the Tevatron~\cite{Abazov:2010hv} (see eq.~(\ref{ASL_exp}))
cannot be accounted for within the $SSU(5)_{RN}$ model.
In the central plot of fig.~\ref{fig4} on the left, we show the correlation between
${\rm BR}(B_s\to\mu^+\mu^-)$ and ${\rm BR}(B_d\to\mu^+\mu^-)$. Interestingly enough,
we notice that sizable departures from the MFV predictions $|V_{ts}/V_{td}|^2$ imply
large values for ${\rm BR}(\tau\to\mu\gamma)$, well within the SuperB reach.
In the central plot of fig.~\ref{fig4} on the right, we show the predictions for
$S_{\psi\phi}$ and $S_{\phi K_S}$. Noteworthy enough, these observables can sizably
depart from the SM expectations in a correlated manner.
The pattern of correlation is twofold: for moderate/low $\tan\beta$ values
(corresponding to the almost vertical band), $S_{\psi\phi}$ receives the dominant
contributions from gluino/squark boxes and the correlation might be in agreement
with the current non-standard experimental data while for large $\tan\beta$ values
(corresponding to the almost horizontal band), $S_{\psi\phi}$ receives the dominant
contributions from double penguin Higgs exchanges and the correlation is opposite.
In any case, large non-standard effects for $S_{\psi\phi}$ and/or $S_{\phi K_S}$
always imply experimentally visible values for ${\rm BR}(\tau\to\mu\gamma)$.
In the lower plot of fig.~\ref{fig4} on the left, we report the correlation between
$S_{\eta^{\prime} K_S}$ and $S_{\phi K_S}$ clearly showing that these observables
exhibit opposite deviations with respect to the SM expectations.
This is understood remembering that the NP amplitudes for these processes can be written
as $A_{\rm NP}\sim C_i+\zeta \tilde C_i$ where $C_i$ and $\tilde C_i$ are the NP Wilson
coefficients and $\zeta=\pm 1$ depending on the parity of the final state which is $\zeta=1$
for $\phi K_S$ and $\zeta=-1$ for $\eta^\prime K_S$ (see Section~4).
Since in the $SSU(5)_{RN}$ model $\tilde C_i$ provide the largely dominant
effects, $S_{\eta^{\prime} K_S}$ and $S_{\phi K_S}$ are expected to show opposite departures
from the SM predictions as confirmed numerically. This is in contrast to scenarios like
the flavour-blind MSSM~\cite{Altmannshofer:2008hc}, the MSSM with MFV or models with purely left-handed currents where $C_i$ are dominant~\cite{Altmannshofer:2009ne}.
In the lower plot of fig.~\ref{fig4} on the right, we also show the correlation
between ${\rm BR}(\tau\to\mu\gamma)$ and $S_{\phi K_S}$ confirming that sizable NP
effects for $S_{\phi K_S}$ imply a lower bound for ${\rm BR}(\tau\to\mu\gamma)$ within
the SuperB reach. However, we notice that an explanation of the $(g-2)_{\mu}$ anomaly
would prevent large non-standard effects for $S_{\phi K_S}$.
\section{DNA-Flavour Test of $SSU(5)_{\rm RN}$}\label{sec:dna}
The pattern of flavour violation predicted by specific NP model represents one of the
most powerful tools in the attempt to probe or to falsify the model in question. Motivated
by this consideration, in Ref.~\cite{Altmannshofer:2009ne} a ``DNA-Flavour Test'' has
been introduced with the aim of summarizing the potential size of deviations from the SM
results for the most interesting observables in a selection of SUSY and non-SUSY models.
In tab.~\ref{tab:DNA}, we extend such a ``DNA-Flavour Test'' to the $SSU(5)_{RN}$
model. We remind that we distinguish among large, moderate (but still visible) and
vanishingly small effects by three {\it red} stars, two {\it blue} stars and one
{\it black} star, respectively.
While we refer to Ref.~\cite{Altmannshofer:2009ne} for a detailed description of the
pattern of NP effects in various SUSY models, we want to comment here about one of
the most remarkable difference we found between the $SSU(5)_{RN}$ model and the
SUSY flavour models discussed in Ref.~\cite{Altmannshofer:2009ne}.
In fact, none of the models discussed in Ref.~\cite{Altmannshofer:2009ne} was able to
simultaneously account for the current data for $S_{\psi\phi}$ and $S_{\phi K_S}$, in
contrast to the $SSU(5)_{RN}$ model discussed here.
The reason for this can be traced back recalling that $S_{\phi K_S}$ receives the dominant
effects from gluino/squark penguins while $S_{\psi\phi}$ either from gluino/squark boxes
(at moderate/low $\tan\beta$ values) or from double Higgs penguins (at large $\tan\beta$).
However, only the moderate/low $\tan\beta$ solution can simultaneously account for an
enhancement of $S_{\psi\phi}$ and a suppression of $S_{\phi K_S}$ (relative to $S_{\psi K_S}$)
as required by the data.
Yet such effects are strongly constrained either by $D^0-\bar D^0$ mixing (in case of
Abelian flavour models) or by $K^0-\bar K^0$ mixing (in case of non-Abelian flavour models). Consequently, in this region of parameter space $S_{\psi\phi}$ cannot be large in these
models. In fact $S_{\psi\phi}$ receives in these models large values only at large
$\tan\beta$ where the sign of the correlation between $S_{\psi\phi}$ and $S_{\phi K_S}$
is found to be opposite to data ~\cite{Altmannshofer:2009ne}, that is $S_{\phi K_S}$ is
enhanced rather than suppressed when $S_{\psi\phi}$ is enhanced.
In contrast, the $SSU(5)_{RN}$ model predicts unobservable effects for $D^0-\bar D^0$
mixing while the NP effects in $K^0-\bar K^0$ are generally unrelated to those entering
$B^0_s-\bar B^0_s$ mixing and therefore the tight bounds from $\epsilon_K$ can be always
avoided. Therefore at moderate/low $\tan\beta$ the suppression of $S_{\phi K_S}$ and
simultaneous sizable enhancement of $S_{\psi\phi}$ can be obtained.
In this context let us recall that within the SM4, the SM with fourth sequential generation,
the correlation between $S_{\psi\phi}$ and in $S_{\phi K_S}$ is qualitatively similar to the
one found in the $SSU(5)_{RN}$ model, that is with increasing $S_{\psi\phi}$ the asymmetry
$S_{\phi K_S}$ decreases in accordance with the data~\cite{Hou:2005yb,Soni:2008bc,Buras:2010pi}.
However, in this model the absence of right-handed currents implies, in contrast to
$SSU(5)_{RN}$, that also $S_{\eta' K_S}$ decreases with increasing $S_{\psi\phi}$.
Finally, it has to be stressed that the ``DNA-Flavour Test'' table doesn't account for
possible correlations among observables. Therefore, since simultaneous large effects are
not always possible for certain sets of observables, it will be interesting to monitor
the changes in this table with improved experimental results.
\definecolor{green1}{rgb}{0.06,0.66,0.06}
\definecolor{orange1}{rgb}{0.98,0.60,0.07}
\newcommand{{\color{green1}$\bigstar$}}{{\color{green1}$\bigstar$}}
\newcommand{{\color{orange1}\LARGE \protect\raisebox{-0.1em}{$\bullet$}}}{{\color{orange1}\LARGE \protect\raisebox{-0.1em}{$\bullet$}}}
\newcommand{{\color{red}\small \protect\raisebox{-0.05em}{$\blacksquare$}}}{{\color{red}\small \protect\raisebox{-0.05em}{$\blacksquare$}}}
\newcommand{{\color{red}$\bigstar\bigstar\bigstar$}}{{\color{red}$\bigstar\bigstar\bigstar$}}
\newcommand{{\color{blue}$\bigstar\bigstar$}}{{\color{blue}$\bigstar\bigstar$}}
\newcommand{{\color{black}$\bigstar$}}{{\color{black}$\bigstar$}}
\begin{table}[t]
\addtolength{\arraycolsep}{4pt}
\renewcommand{\arraystretch}{1.5}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Observable & $SSU(5)_{\rm RN}$ model
\\
\hline\hline
$D^0-\bar D^0$ & {\color{black}$\bigstar$}
\\
\hline
$\epsilon_K$& {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$ S_{\psi K_S}$ & {\color{black}$\bigstar$}
\\
\hline
$ \gamma $ & {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$ R_t $ & {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$ S_{\psi\phi}$ & {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline\hline
$S_{\phi K_S}$ & {\color{red}$\bigstar\bigstar\bigstar$} \\
\hline
$S_{\eta^{\prime} K_S}$ & {\color{blue}$\bigstar\bigstar$} \\
\hline
$A_{\rm CP}\left(B\rightarrow X_s\gamma\right)$ & {\color{black}$\bigstar$}
\\
\hline
$A_{7,8}(B\to K^*\mu^+\mu^-)$ & {\color{black}$\bigstar$}
\\
\hline
$A_{9}(B\to K^*\mu^+\mu^-)$ & {\color{black}$\bigstar$}
\\
\hline
$B\to K^{(*)}\nu\bar\nu$ & {\color{black}$\bigstar$}
\\
\hline
$B_s\rightarrow\mu^+\mu^-$ & {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$K^+\rightarrow\pi^+\nu\bar\nu$ & {\color{black}$\bigstar$}
\\
\hline
$K_L\rightarrow\pi^0\nu\bar\nu$ & {\color{black}$\bigstar$}
\\
\hline
$\mu\rightarrow e\gamma$& {\color{red}$\bigstar\bigstar\bigstar$} \\
\hline
$\tau\rightarrow \mu\gamma$ & {\color{red}$\bigstar\bigstar\bigstar$} \\
\hline
$\mu + N\rightarrow e + N$& {\color{red}$\bigstar\bigstar\bigstar$} \\
\hline\hline
$d_n$& {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$d_{Hg}$& {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$d_e$& {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
$\left(g-2\right)_\mu$& {\color{red}$\bigstar\bigstar\bigstar$}
\\
\hline
\end{tabular}
\renewcommand{\arraystretch}{1}
\caption{\small
``DNA'' of flavour physics effects for the $SSU(5)_{RN}$ model. {\color{red}$\bigstar\bigstar\bigstar$}\ signals large
effects, {\color{blue}$\bigstar\bigstar$}\ visible but small effects and {\color{black}$\bigstar$}\ implies vanishingly small effects.}
\label{tab:DNA}
\end{table}
\section{Conclusions}
Despite of the remarkable agreement of flavour data with the SM predictions in the $K$
and $B_d$ systems, a closer look at the data might indicate some tensions especially in
CP violating observables. In particular, the most recent UT analyses show some tensions
at the level of $(2-3)\sigma$~\cite{SL,Buras:2008nn,Lenz:2010gu,Bevan:2010gi} and recent
messages from the Tevatron seem to hint the presence of new sources of CPV entering the
$B^{0}_s$ systems~\cite{Aaltonen:2007he,Abazov:2010hv,Abazov:2008fj}.
Motivated by the above facts, in the present work, we have analyzed the low energy implications of a supersymmetric $SU(5)$ GUT scenario with right-handed neutrinos~\cite{Hisano:1997tc} ($SSU(5)_{RN}$) accounting for the neutrino
masses and mixing angles by means of a type-I see-saw mechanism~\cite{seesaw}.
Since supersymmetric Grand Unified theories generally predict FCNC and CP violating
processes to occur both in the leptonic and hadronic sectors, we have performed
an extensive study of FCNC and CP Violation in both sectors, analyzing possible
hadron/lepton correlations among observables. In particular, we have monitored
the low energy consequences implied by the solutions to the above tensions.
However, within the $SSU(5)_{RN}$ model, it is not possible to link model
independently different flavour transitions like $s\to d$ and $b\to s$. In fact,
the neutrino Yukawa couplings, which regulate the size of the flavour violation
both in the hadronic and leptonic sectors, are unknown. Therefore, we have analyzed
the phenomenology related to $s\to d$ and $b\to s$ transitions separately.
The main results of our study of the $s\to d$ transitions and their correlation
with $\mu\to e$ transitions are
\begin{itemize}
\item Sizable SUSY effects in $\epsilon_{K}$, that might be desirable to solve
the UT anomaly, generally imply a lower bound for ${\rm BR}(\mu\to e\gamma)$ in
the reach of the MEG experiment. Furthermore, the simultaneous requirement of an
explanation for both the $(g-2)_\mu$ and the UT anomalies would typically imply
${\rm BR}(\mu\to e\gamma)\geq 10^{-12}$.
\item The requirement of sizable non-standard effects in $\epsilon^{SUSY}_{K}$ always
implies large values for the electron and neutron EDMs, in the reach of the planned
experimental resolutions.
\end{itemize}
The main results of our study of the $b\to s$ transitions and of their correlations with
$\tau\to\mu$ transitions are
\begin{itemize}
\item Non-standard values for $S_{\psi\phi}$ imply a lower bound for ${\rm BR}(\tau\to\mu\gamma)$
within the SuperB reach. However, the $(g-2)_\mu$ anomaly can be solved only for large $\tan\beta$
values where we find $|S_{\psi\phi}|\leq 0.2$ for $\Delta a^{\rm SUSY}_{\mu}\gtrsim 1\times 10^{-9}$ while being still compatible with the constraints from ${\rm BR}(\tau\to\mu\gamma)$.
\item The UT anomaly can be solved by means of negative NP effects in $\Delta M_d/\Delta M_s$
which, in turn, also indirectly enhance $\epsilon_K$ via the increased value of $R_t$.
This scenario implies a lower bound for ${\rm BR}(\tau\to\mu\gamma)$ within the SuperB
reach and large values for the angle $\gamma$ and it will be probed or falsified quite
soon at the LHCb.
\item Both ${\rm BR}(B_s\to\mu^+\mu^-)$ and ${\rm BR}(B_d\to\mu^+\mu^-)$ can reach
large non-standard values. However, sizable departures from the MFV prediction
${\rm BR}(B_s\to\mu^+\mu^-)/{\rm BR}(B_d\to\mu^+\mu^-)\approx|V_{ts}/V_{td}|^2$ would
imply large values for ${\rm BR}(\tau\to\mu\gamma)$, well within the SuperB reach.
\item The dileptonic asymmetry $A^{b}_{\text{SL}}$ can sizably depart from the SM
expectations but the large value reported by the Tevatron~\cite{Abazov:2010hv} cannot
be accounted for within the $SSU(5)_{RN}$ model. In particular, we find that
$A^{b}_{\text{SL}} \approx 0.5~A^{s}_{\text{SL}}$ since $A^{d}_{\text{SL}}$ remains
SM-like.
\item The asymmetry $S_{\phi K_S}$ can sizably depart from the SM expectations
and it turns out to be correlated with $S_{\psi\phi}$. In particular, it is possible
to simultaneously account for an enhancement of $S_{\psi\phi}$ and a suppression of
$S_{\phi K_S}$ (relative to $S_{\psi K_S}$) as required by the data. This is in
contrast to the SUSY flavour models discussed in Ref.~\cite{Altmannshofer:2009ne}.
Moreover, the asymmetries $S_{\eta^{\prime} K_S}$ and $S_{\phi K_S}$ exhibit opposite
deviations with respect to the SM expectations.
\end{itemize}
Finally, we provided a ``DNA-Flavour Test'' (proposed in Ref.~\cite{Altmannshofer:2009ne})
for the $SSU(5)_{RN}$ model, with the aim of showing a tool to distinguish between
NP scenarios, once additional data on flavour changing processes become available.
As shown in tab.~\ref{tab:DNA}, further important predictions of the $SSU(5)_{RN}$ model
are that i) $S_{\psi K_S}$ remains SM-like to a very good extent (therefore, the solution of
the UT anomaly by means of CPV effects in $b\to d$ mixing is not possible), ii) CPV effects
in $D^{0}-\bar{D}^{0}$ are negligibly small, and iii) $BR(K^{0}_{L}\to\pi^{0}\nu\bar{\nu})$
and $BR(K^{+}\to \pi^{+}\nu\bar{\nu})$ also remain SM-like.
In conclusion, the above results show the richness which is present in flavour physics
once we embed a GUT group within a gravity mediated SUSY breaking scenario. It will be
exciting to monitor upcoming results from the Tevatron, LHC(b), the MEG experiment at
PSI and SuperB machines to establish whether some patterns of deviations from the SM
expectations we have pointed out in this work are at work or not.
The interplay of all these efforts with the direct searches for NP will be most exciting.
\vspace*{20pt}
\noindent
\textit{Acknowledgments:}
AJB would like to thank the Particle Theory Institute of Vienna University for its
hospitality during the final steps of this work.
This work has been supported in part by the Cluster of Excellence ``Origin and Structure
of the Universe'' and by the German Bundesministerium f{\"u}r Bildung und Forschung under
contract 05H09WOE.
|
\section{Introduction}\label{intro}
In the recent paper \cite{Roberts}, Roberts introduced a ``totally
twisted'' version of $\delta$-graded characteristic-2 Khovanov
homology for links. Jaeger~\cite{Jaeger} then showed that for knots,
the reduced totally twisted Khovanov homology actually coincides with
the ordinary reduced Khovanov homology (tensored with a suitable
coefficient field). We show how to extend the totally twisted
construction over $\Z$, in the context of the odd Khovanov homology of
Ozsv{\'a}th, Rasmussen and Szab{\'o}~\cite{Odd}. The result is a chain
complex whose homology computes the reduced odd Khovanov homology of
knots (again, tensored with a suitable ring). Cancelling some
differentials in the complex leads to an equivalent complex whose
generators are in bijection with spanning trees of the Tait graph. The
coefficient of the differential between two spanning trees is
determined up to a sign; the sign ambiguity comes from an analogous
ambiguity in odd Khovanov homology, where one must choose a sign
assignment on the edges of a cube of resolutions.
\subsection{Acknowledgements.}
The author would like to thank John Baldwin, Zolt{\'a}n Szab{\'o},
Cotton Seed, and Kevin Wilson for several very helpful discussions.
\section{The construction}\label{construction}
We assume the reader is familiar with odd Khovanov homology, as
described in \cite{Odd}. Here we briefly fix notation. Let $D$ be an
$n$-crossing diagram for a link $L$, with marks $m_i$ assigned to
edges. For the general construction, we allow any assignment of marks;
to obtain the relationship with spanning trees in
Section~\ref{spanningtrees}, it will be imporant that each edge has at
least one mark. Let $R$ be the polynomial ring $\Z [x_i]$, with one
variable for each mark. Since we will be working with odd Khovanov
homology, we also want to choose an orientation for each
crossing. Figure~\ref{crossing} below shows one possible choice; the
other has the arrow reversed.
Each crossing in $D$ has a $0$-resolution and a $1$-resolution. A
complete resolution of $D$ gives rise to a diagram with no crossings,
which consists of $k$ unlinked circles. To a complete resolution
$\rho$, associate the group $V_{\rho} = H^*(S^1 \times \ldots \times
S^1)$, where there are $k$ $S^1$ factors. This group is actually a
ring, and if we label the circles of the resolution $a_1, \ldots,
a_k$, a convenient set of multiplicative generators for $V_{\rho}$ may
be labelled $\{a_1, \ldots, a_k\}$ as well. The chain complex
computing odd Khovanov homology may be written
$(C_*^{odd}(D),d_{odd})$, where $C_*^{odd}(D) = \oplus_{\rho \in
\{0,1\}^n} V_{\rho}$. We refer to \cite{Odd} for the definition of
$d_{odd}$. Here we only remind the reader that while the orientations
on the crossings determine split and merge maps with well-defined
signs, these naive maps do not automatically fit into a differential
satisfying $d_{odd}^2 = 0$. Rather, one must correct by putting signs
on the edges of the cube of resolutions. We will denote the naive maps
collectively as $d'_{odd}$ and reserve the name $d_{odd}$ for the
sign-corrected differential.
\begin{figure}
\labellist
\small \hair 2pt \pinlabel $X$ at 173 43 \pinlabel $Y$ at 309 43
\endlabellist
\centering
\includegraphics[scale=0.3]{XandY}
\caption{The configurations $X$ and $Y$.}
\label{XandY}
\end{figure}
The method of correcting the signs makes use of the concept of
2-dimensional (oriented) configurations, which are pictures of a
complete resolution along with two (oriented) arcs such that surgery
along an arc corresponds to switching a crossing. Two examples of
these configurations are depicted in Figure~\ref{XandY}, and many more
are shown below in Figure~\ref{bubbles}. In fact, the configurations
$X$ and $Y$ in Figure~\ref{XandY} are special; they correspond to
faces of the cube of resolutions which both commute and
anticommute. The convention we will use is that that the configuration
labelled $X$ anticommutes and the one labelled $Y$ commutes. We only
mention this here because we will need to deal with 2-dimensional
configurations in Section~\ref{movemarksec}, and there it will be
important that we use this specific convention and not its opposite.
Note that tensoring the odd Khovanov complex with $R$ amounts simply
to taking cohomology with $R$ coefficients when defining $V_{\rho}$,
and we will use these coefficients from here on.
The construction we discuss amounts to defining a differential
$d_{v,\rho}$ on each $V_{\rho}$ (the small $v$ is meant to suggest
``vertical,'' in contrast with the ``horizontal'' maps of
$d_{odd}$). Setting $d_v = \sum_{\rho} d_{v,\rho}$, we consider the
complex $(C_*^{odd},d_v + d_{odd})$. In the remainder of this section,
we will define $d_{v,\rho}$ and show that $(d_v + d_{odd})^2 =
0$. When coefficients are taken modulo $2$, we will get the complex
from \cite{Jaeger}.
Fix a resolution $\rho$ with circles $a_1, \ldots, a_k$. The marks
$m_i$ on $D$ pass to marks on these circles. For each $i$, define $w_i
\in R$ to be the sum of the variables $x_j$ corresponding to those
marks $m_j$ lying on $a_i$. As a preliminary definition, define
\[
d'_{v,\rho} = \sum_i m(w_i a_i, \cdot),
\]
where $m(\cdot, \cdot)$ denotes multiplication in the ring
$V_{\rho}$. (Note that the multiplication is anti-commutative, so
order matters, and we are multiplying by $w_i a_i$ on the left.) It is
clear that $(d'_{v,\rho})^2 = 0$, and the same holds for $d'_v =
\sum_{\rho} d'_{v,\rho}$. Taking coefficients modulo $2$, we get the
twisted complex in Jaeger's form (\cite{Jaeger}): his
dot-multiplication maps correspond to our left multiplication.
The definition of $d'_{v,\rho}$ is only preliminary since we will need
to modify $d'_{v,\rho}$ by an overall sign, depending on $\rho$. We
now describe this modification. It will be based on the following
lemma:
\begin{lemma}\label{naivedv} Write $d'_{odd} = d'_{odd,split} +
d'_{odd,join}$. Then $d'_v$ anticommutes with $d'_{odd,split}$ and
commutes with $d'_{odd,join}$.
\end{lemma}
\begin{proof} We will do the split case and leave the (very similar)
join case to the reader. Consider two resolutions $\rho$ and $\rho'$
with a nonzero component of $d'_{odd,split}$ between them. Then
$\rho'$ is obtained from $\rho$ by splitting one circle $a$ into two
circles $b$ and $c$. Choose the labels $b$ and $c$ such that
$d'_{odd}(a) = bc$. Denote the passive circles in $\rho$ by
$\{p_i\}$; then the remaining circles in $\rho'$ may also be
labelled $\{p_i\}$.
Homogeneous generators of $V_{\rho}$ take the form $\pi$ or $a\pi$,
where here $\pi$ denotes any product of the $p_i$. In the following
computation, all sums over $q$ indicate sums over those passive
circles $q$ which are not contained in $\pi$.
\begin{align*}
d'_v d'_{odd}(\pi) &= d'_v(b-c)\pi = \sum_q w_q q(b-c)\pi,
\end{align*}
while
\begin{align*}
d'_{odd} d'_v(\pi) &= d'_{odd} (\sum_q w_q q \pi) = \sum_q w_q (b-c)q \pi \\
&= - d'_v d'_{odd}(\pi).
\end{align*}
Similarly,
\begin{align*}
d'_v d'_{odd}(a\pi) &= d'_v(bc\pi) = \sum_q w_q qbc\pi
\end{align*}
while
\begin{align*}
d'_{odd} d'_v(a\pi) &= d'_{odd} (\sum_q w_q qa\pi) = -d'_{odd}(\sum_q w_q aq\pi) = -\sum_q w_q bcq\pi = -\sum_q w_q qbc\pi \\
&= -d'_v d'_{odd}(a\pi).
\end{align*}
\end{proof}
The preceding lemma tells us that, to properly define $d_v$, we should
flip the signs on $d'_{v,\rho}$ for some vertices $\rho$ of the cube
of resolutions. For each split edge of the cube, we want the endpoints
to receive the same sign-change, while for each join edge, we want the
endpoints to receive the opposite sign-change. As with odd Khovanov
homology, a cohomological argument allows us to make these choices:
\begin{proposition} It is possible to flip the signs on some of the
$d'_{v,\rho}$ such that the above conditions hold, and the flips are
unique up to an overall sign.
\end{proposition}
\begin{figure}
\begin{center}
\includegraphics[scale=0.3]{unoriented}
\end{center}
\caption{The five types of unoriented 2-dimensional
configurations. Dotted lines indicate the arcs of the each
configuration.}
\label{unoriented}
\end{figure}
\begin{proof} Consider the usual CW structure of the cube $Q =
[0,1]^n$ of resolutions, with the $k$-skeleton consisting of the
$k$-dimensional faces. Define a cochain $\tau \in C^1(Q; \Z/2\Z)$ by
labelling split edges $0$ and join edges $1$. We want to show that
$\tau$ is a coboundary, but since $Q$ is contractible, it suffices
to show $\tau$ is a cocycle. To compute $\delta \tau$, we look at
the $2$-dimensional faces of $Q$. There are five types of these,
corresponding to the five possible unoriented 2-dimensional
configurations. These are shown in Figure~\ref{unoriented}. For each
of these five configurations, the sum of $\tau$ along the boundary
edges of the corresponding $2$-dimensional face is $0$ (mod $2$);
this can easily be checked. Hence $\delta \tau = 0$, so $\tau =
\delta \sigma$ for some $\sigma \in C^0(Q; \Z/2\Z)$. Flip the sign
on a vertex $\rho$ if $\sigma(\rho) = 1$, and leave it alone if
$\sigma(\rho) = 0$. Note that $\sigma$ is unique up to an overall
sign since $H^0(Q; \Z/2\Z) = \Z/2\Z$.
\end{proof}
If we define $d_v$ by making the appropriate sign flips, then $d_v$
anticommutes with $d'_{odd}$, and of course $d_v^2 = 0$. Finally,
correct the signs in $d'_{odd}$ as in standard odd Khovanov homology
(using another cohomological argument; see \cite{Odd}). It is still
true that $d_v$ anticommutes with $d_{odd}$, so we have $(d_v +
d_{odd})^2 = 0$. This completes the construction of a complex which
reduces to Jaeger's twisted complex modulo 2.
\begin{remark} If we have a basepoint on our link, we can define
reduced versions of everything above in the standard way.
\end{remark}
\subsection{Invariance.}
The homotopy invariance of $(C_*^{odd},d_v + d_{odd})$ under the
Reidemeister moves can be proved using an argument of Baldwin, similar
to that used in \cite{Baldwin}. For R1 and R2, one writes down the
complexes before and after the move, and then cancels differentials in
the ``before'' complex to obtain the ``after'' complex. Once
invariance under R2 is proven, invariance under R3 amounts to
considering the braid word $xyxy^{-1}x^{-1}y^{-1}$, where $x$ and $y$
are elementary $3$-strand braid group generators, and showing its
appearance in a Khovanov complex is equivalent to the identity. The
relevant ``before'' complex comes from a $64$-vertex cube of
resolutions which was dealt with in \cite{Baldwin}. In our case, the
presence of $d_v$ does not make anything harder for R2 or R3, and R1
does not change much either. In this section we will briefly sketch
the proof for invariance under R1.
\begin{figure}
\centering
\labellist
\small \hair 2pt \pinlabel {Before R1} at 75 62 \pinlabel {After R1}
at 248 62
\endlabellist
\includegraphics[scale=0.5]{positive}
\caption{The R1 move.}
\label{r1}
\end{figure}
Consider an R1 move which undoes a positive kink; see
Figure~\ref{r1}. Write $C_*^{before}$ for the complex before undoing
the kink and $C_*^{after}$ for the complex after performing
R1. Without regard to the differential, $C_*^{before}$ is the sum of
three pieces: $C_*^{before} = C_{0,+} \oplus C_{0,-} \oplus C_1$,
where the $0$ or $1$ indicates the resolution at the crossing in
question and the $+$ (resp. $-$) denotes those generators represented
by monomials not containing (resp. containing) the isolated small
circle in the $0$-resolution. In each of the three local pictures in
question, label the non-closed component as $a$ and the isolated small
circle (in the $0$-pictures) as $b$. Denote the differential from
$C_{0,\pm}$ to $C_1$ by $d_\pm$.
In fact, $d_+$ maps generators of $C_{0,+}$ bijectively to generators
of $C_1$. We would like to cancel all components of $d_+$, leaving
ourselves with $C_{0,-}$ and an induced differential on this
summand. There are some obvious components of this differential,
namely those coming from components internal to $C_{0,-}$ in the whole
complex $C_*^{before}$. These components correspond to almost all of
the differential on $C_*^{after}$ under the bijection sending a
generator $p$ of $C_*^{after}$ to $bp$ in $C_{0,-}$. The only things
missing are the vertical differentials from marks on $b$. We want to
show that the induced differential from cancellation precisely adds in
these missing components.
Indeed, any induced components come from compositions
\[
\xymatrix{C_{0,-} \ar[r]^{d_-} & C_1 \ar[r]^{d_+^{-1}} & C_{0,+}
\ar[r]^{d_{v,b}} & C_{0,-}}
\]
where the final map comes from the component of $d_v$ associated to
marks on $b$. The map $d_-$ (a join) is nonzero only on elements of
the form $bp$, where $p$ does not contain $a$. We have $d_-(bp) = \pm
ap$, but applying $d_+^{-1}$, the sign cancels and we get $ap \in
C_{0,+}$. Applying $d_{v,b}$ gives us $w \cdot bap$, where $w$ is the
sum of the weights of marks on $b$. This was precisely the component
we were looking for, and invariance under R1 follows.
\section{Relation with odd Khovanov homology.}\label{movemarksec}
Jaeger shows that for knots, his (reduced) complex actually computes
reduced Khovanov homology. We would like to do the same with reduced
odd Khovanov homology. Suppose $L$ is actually a knot $K$ with
basepoint $p$. Following \cite{Jaeger}, the main point is that we can
move marks past crossings without changing the isomorphism type of the
twisted complex. Consider a mark $m$ near a crossing $c$ of $D$, with
local picture \marktop. Let $D'$ be the marked diagram with this local
picture replaced by \markbot. We may assume that $c$ is oriented as in
Figure~\ref{crossing} and that the crossing orientations on $D'$ are
the same as $D$.
\begin{figure}
\centering
\includegraphics[scale=0.3]{crossing}
\caption{Orientation at the crossing $c$.}
\label{crossing}
\end{figure}
\begin{theorem}\label{movemark}
The twisted complexes associated to $D$ and to $D'$ are isomorphic.
\end{theorem}
An analogous statement holds when sliding a mark over a crossing,
rather than under. As in \cite{Jaeger}, Theorem~\ref{movemark} (plus
the analogous statement) immediately implies that the reduced twisted
complex computes reduced odd Khovanov homology for knots. (More
precisely, it computes Khovanov homology tensored with
$\Z[x_i]$). Indeed, one can simply move all the marks to the same edge
as the basepoint, effectively killing $d_v$ and leaving only $d_{odd}$
in the differential. We will now prove Theorem~\ref{movemark}.
\begin{proof}
In \cite{Jaeger}, with coefficients taken modulo 2, this theorem is
a purely local computation. Unfortunately, in odd Khovanov homology,
signs on maps are not determined entirely by local data, so we must
work a bit harder. Still, Jaeger's chain map (with appropriate
signs) will work in our situation.
To define the map, it will be convenient to follow \cite{Jaeger} and
use local pictures. In this notation, the complex of $D$ will be
written $\vid \oplus \hid$, where each summand actually represents
all summands of the total complex of $D$ whose resolution at $c$ is
as depicted. There is one such summand for each ``outer''
resolution, i.e. each resolution $\rho$ of all crossings except
$c$. The complex of $D'$ will also be written as $\vid \oplus \hid$,
with the same interpretation.
Fix an outer resolution $\rho$. With respect to the summands \vid
and \hid, the isomorphism of complexes is given by $F_{\rho}
:= \begin{pmatrix} \vid & \pm \htov \\ & \hid \end{pmatrix}: \vid
\oplus \hid \to \vid \oplus \hid$. The sign in this formula will
depend on $\rho$ in a way that will be specified below. Regardless
of the chosen sign, it is clear that each $F_{\rho}$ is invertible;
the inverse is the same map with opposite sign. Set $F = \sum_{\rho}
F_{\rho}$. Then $F$ is invertible, and we only need show that $F$
commutes with $d_v$ and $d_{odd}$. We may write $F_{\rho} = \id +
H_{\rho}$, and this will be useful later.
To specify the sign on $H_{\rho}$, consider
$(C_*^{odd}(D),d_{odd})$, which is the same as
$(C_*^{odd}(D'),d_{odd})$ since the only difference between $D$ and
$D'$ is the placement of a mark. There is a component $d_{c,\rho}$
of $d_{odd}$ coming from $c$. It has a naive sign from the
orientation on $c$; write $\sigma(d_{c,\rho}) = 0$ if the actual
sign agrees with the naive sign and $\sigma(d_{c,\rho}) = 1$
otherwise. In the process of defining $d_v$ above, we also put signs
on certain vertices of the complete cube of resolutions. There are
two such vertices associated to $\rho$; call them $(\rho,0)$ and
$(\rho,1)$ where the $0$ or $1$ denotes the resolution of $c$. Write
$\sigma(\rho,i) = 0$ if we did not flip the sign on
$d_{v,(\rho,i)}$, and write $\sigma(\rho,i) = 1$ if we did. Define
the sign on
|
$H_{\rho}$ to be
\begin{equation}\label{hsign}
\sigma(H_{\rho}) := \sigma(d_{c,\rho}) \cdot \sigma(\rho,1).
\end{equation}
To show $F$ is a chain map, we will first consider those components
of $d_v$ and $d_{odd}$ which correspond to the mark $m$ and the
crossing in the local picture \marktop (or \markbot). These are the
components Jaeger deals with in \cite{Jaeger}. The relevant
commutative diagram in his paper also works in our situation, once
it is suitably interpreted, and once signs are added.
\begin{center}
\scalebox{0.85}{
$\xymatrix@R=1.5cm@C=3cm{ \vid \oplus \hid \ar[r]^{\begin{pmatrix} w \, \vld \\
\vtoh & w \, \htd \end{pmatrix}}
\ar@<0.3ex>[d]_{\begin{pmatrix} \vid & w \htov \\ &
\hid \end{pmatrix}} & \vid \oplus \hid
\ar@<0.3ex>[d]^{\begin{pmatrix} \vid & w \htov \\ &
\hid \end{pmatrix}}
\\
\vid \oplus \hid \ar[r]_{\begin{pmatrix} w \, \vrd \\ \vtoh &
w \, \hbd \end{pmatrix}} & \vid \oplus \hid }$
$\xymatrix@R=1.5cm@C=3cm{ \vid \oplus \hid \ar[r]^{\begin{pmatrix} w \, \vld \\
-\vtoh & w \, \htd \end{pmatrix}}
\ar@<0.3ex>[d]_{\begin{pmatrix} \vid & -w \htov \\ &
\hid \end{pmatrix}} & \vid \oplus \hid
\ar@<0.3ex>[d]^{\begin{pmatrix} \vid & - w \htov \\ &
\hid \end{pmatrix}}
\\
\vid \oplus \hid \ar[r]_{\begin{pmatrix} w \, \vrd \\ -\vtoh &
w \, \hbd \end{pmatrix}} & \vid \oplus \hid }$}
\end{center}
\begin{center}
\scalebox{0.8}{
$\xymatrix@R=1.5cm@C=3cm{ \vid \oplus \hid \ar[r]^{\begin{pmatrix} w \, \vld \\
\vtoh & -w \, \htd \end{pmatrix}}
\ar@<0.3ex>[d]_{\begin{pmatrix} \vid & -w \htov \\ &
\hid \end{pmatrix}} & \vid \oplus \hid
\ar@<0.3ex>[d]^{\begin{pmatrix} \vid & -w \htov \\ &
\hid \end{pmatrix}}
\\
\vid \oplus \hid \ar[r]_{\begin{pmatrix} w \, \vrd \\ \vtoh &
-w \, \hbd \end{pmatrix}} & \vid \oplus \hid }$
$\xymatrix@R=1.5cm@C=3cm{ \vid \oplus \hid \ar[r]^{\begin{pmatrix} w \, \vld \\
-\vtoh & -w \, \htd \end{pmatrix}}
\ar@<0.3ex>[d]_{\begin{pmatrix} \vid & w \htov \\ &
\hid \end{pmatrix}} & \vid \oplus \hid
\ar@<0.3ex>[d]^{\begin{pmatrix} \vid & w \htov \\ &
\hid \end{pmatrix}}
\\
\vid \oplus \hid \ar[r]_{\begin{pmatrix} w \, \vrd \\ -\vtoh &
-w \, \hbd \end{pmatrix}} & \vid \oplus \hid }$ }
\end{center}
Above are four copies of the diagram from \cite{Jaeger} with signs
added. They cover the possible cases when $\sigma(\rho,0) = 0$ (in
other words, all cases in which the upper-left entries of the
horizontal maps have a $+$ sign). It turns out that, once we
consider the case $\sigma(\rho,0) = 0$, the case $\sigma(\rho,0) =
1$ involves the same set of matrix multiplications, up to an overall
sign. The diagrams on the left have $\sigma(d_{c,\rho}) = 0$ and the
ones on the right have $\sigma(d_{c,\rho}) = 1$. In the top
diagrams, $\sigma(\rho,1) = 0$ (so the crossing change at $c$ is a
split), and in the bottom ones, $\sigma(\rho,1) = 1$ (so the
crossing change is a join). On the horizontal maps, a dot represents
left multiplication by whichever circle contains the dot. The
vertical maps ($H_{\rho}$) have signs as specified in
Equation~\eqref{hsign}.
The reader may check, by multiplying matrices, that the diagrams
above do commute. A few relations will be needed. First, \vtoh \,
$\circ$ \htov \, = \htd \, $-$ \hbd, and this holds regardless of
whether the crossing change is a split or a join. (One sees this by
explicitly writing down the maps when the first cobordism is a split
and when it is a join). Analogously, \htov \,$\circ$ \vtoh \, = \vld
\,$-$ \vrd.
There are also relations depending on whether the crossing change at
$c$ is a split (as in the top two diagrams) or a join (as in the
bottom two diagrams). When the crossing change is a split, we have
\vrd \,$\circ$ \htov \, = \htov \,$\circ$ \htd, as well as \vld \, =
\vrd since the dots are on the same circle. When it is a join, we
have the relations \vrd \, $\circ$ \htov \,$=$ $-$\htov \, $\circ$
\htd and \hbd \, = \htd. With these relations, one can see that the
four diagrams above do commute. There are four more diagrams to
consider, with $\sigma(\rho,0) = 1$, but these computations follow
from the same set of matrix multiplications. The only difference is
that some matrices pick up an overall factor of $-1$.
Next we want to show that $F$ commutes with those components of
$d_v$ corresponding to marks outside the local crossing picture. We
may fix an outer resolution $\rho$. Writing $F_{\rho} = \id +
H_{\rho}$, we can restrict attention to $H_{\rho}$, because $\id$
commutes with everything outside the local picture. There are
several cases to consider, and each is an easy algebraic
computation. We will consider the cases when the crossing change at
$c$ is a join; the split case is very similar.
First of all, note that since $\rho$ is fixed, it does not matter
here which sign was assigned to $H_{\rho}$, so we may assume the
sign is positive.
The crossing change at $c$ is a join or a split. Assuming it is a
join, by our earlier construction, we applied a sign change either
to $d'_v$ before the crossing change or after (not both); in other
words, $\sigma(\rho,0) \neq \sigma(\rho,1)$. Hence we want to show
that $H_{\rho}$ anticommutes with the relevant components of $d'_v$
(recall that the $'$ indicates the naive signs). The domain of
$H_{\rho}$ is the resolution $(\rho,1)$. This has one circle, say
$a$, which intersects the local crossing picture, and possibly
several other circles (say $\{p_i\}$). $H_{\rho}$ splits $a$ into
two circles, say $b$ and $c$ (chosen so that $H_{\rho}(a) =
bc$). Let $\pi$ be a monomial in the $p_i$. Let $m$ be a mark
outside the local picture; $m$ may lie on any circle $q$ of the
resolution (possibly $a$). Let $w$ denote the variable associated to
$m$, and let $d'_{v,q}$ be the component of $d'_v$ coming from $q$.
If $q$ is not $a$, then
\[
d'_{v,q}(H_{\rho}(\pi)) = d'_{v,q}((b-c)\pi) = wq(b-c)\pi,
\]
while
\[
H_{\rho}(d'_{v,q}(\pi)) = H_{\rho}(wq\pi) = w(b-c)q\pi,
\]
which equals $-d'_{v,q}(H_{\rho}(\pi))$ as desired. Similarly,
\[
d'_{v,q}(H_{\rho}(a\pi)) = d'_{v,q}(bc\pi) = wqbc\pi,
\]
while
\[
H_{\rho}(d'_{v,q}(a\pi)) = H_{\rho}(wqa\pi) = -H_{\rho}(waq\pi) =
-wbcq\pi,
\]
which is $-d'_{v,q}(H_{\rho}(a\pi))$ as desired.
Now, suppose $q = a$ and the mark $m$ lies on $b$ after the
split. We have
\[
d'_{v,q}(H_{\rho}(\pi)) = d'_{v,q}((b-c)\pi) = -wbc\pi,
\]
while
\[
H_{\rho}(d'_{v,q})(\pi) = H_{\rho}(wa\pi) = wbc\pi,
\]
which is $-d'_{v,q}(H_{\rho}(\pi))$. For $a\pi$, the composition
either way gives zero. If the mark insteads lies on $c$ after the
split, the computation is exactly analogous, so we have finished
with the case where the crossing change at $c$ is a join.
The split case is very similar. We may assume no sign changes were
applied to $d'_v$ before or after the crossing; since
$\sigma(\rho,0) = \sigma(\rho,1)$, we may assume both are zero. We
now want to show $H$ commutes with $d'_v$. We must consider marks on
the top local circle, on the bottom local circle, and disjoint from
either circle. After doing the computations in each case, we see
that $H$ commutes with $d_v$.
Our final task is to show $F$ (or equivalently $H$) commutes with
$d_{odd}$, and we need only consider components of $d_{odd}$ coming
from crossings outside our local picture. Consider an external
crossing $e$, and fix a resolution of all the other external
crossings. We get two outer resolutions $\rho$ and $\rho'$, where in
$\rho$ we take the $0$-resolution at $e$ and in $\rho'$ we take the
$1$-resolution. Let $\Delta\sigma(H) = \sigma(H_{\rho}) -
\sigma(H_{\rho'})$ (taken modulo 2). Let $\Delta\sigma(e) =
\sigma(d_{e,\rho}) - \sigma(d_{e,\rho'})$ with notation similar to
above. We may define $\Delta\sigma(c)$ analogously.
Next consider the two-dimensional configuration generated by $c$ and
$e$ (with all other crossings resolved as we decided above). Call
this $f$, the ``forward'' configuration. There is a square
associated to $f$; its sides come from $d'_c$ and $d'_e$ (without
sign corrections). Write $a_f = 0$ if it commutes and $a_f = 1$ if
it anti-commutes; if it does both, use the convention specified in
Section~\ref{construction}.
From $f$ we can obtain a ``backward'' configuration by resolving $c$
and replacing the corresponding oriented arc by one rotated 90
degrees counter-clockwise. Call this configuration
$b$. Figure~\ref{bubbles} has many examples of configurations and
their backwards partners. For the backward configuration, we again
get a square which either commutes (set $a_b = 0$) or anti-commutes
(set $a_b = 1$). The sides of this square come from $d'_e$ and $H$
(taken with the ``naive'' positive sign), since $H$ amounts to doing
the crossing change at $c$ ``backwards.''
What we want to show is that $\Delta\sigma(H) + \Delta\sigma(e) +
a_b = 0$ modulo $2$. The following lemma allows us to do this:
\begin{lemma}\label{configs}
For a two-dimensional configuration associated to oriented
crossings $c$ and $e$ in a diagram, define $f$, $b$, $a_f$, and
$a_b$ as above ($a_f$ and $a_b$ have values modulo 2).
\begin{enumerate}
\item If the arc associated to $e$ is a split in the backward
configuration $b$, then $a_f = a_b + 1$.
\item If it is a join, then $a_f = a_b$.
\end{enumerate}
\end{lemma}
\begin{proof}
There are several cases to be considered; in fact, a diagram is
more useful than words here. Figure~\ref{bubbles} depicts the
relevant 2-dimensional configurations. The left column shows the
forward and backward configurations such that $e$ is a split in
the backwards configuration. The right column does the same for
configurations where $e$ is a join in the backwards
configuration. All configurations are labeled with ``comm.'' if
the corresponding square commutes and ``anti.'' if it does
not. The content of the lemma is that these labels are correct
(each is a simple verification), plus the fact that in the left
column, a configuration and its backwards partner have opposite
labels while in the right column they have the same labels. Note
that our choice of convention for the configurations $X$ and $Y$
of Figure~\ref{XandY} is needed for the lemma to hold.
\end{proof}
\begin{figure}
\labellist
\small \hair 2pt \pinlabel Anti. at 38 733 \pinlabel Anti. at 38
631 \pinlabel Anti. at 38 550 \pinlabel Comm. at 10 416 \pinlabel
Comm. at 10 350 \pinlabel Comm. at 10 290 \pinlabel Anti. at 10
226 \pinlabel Comm. at 10 155 \pinlabel Comm. at 182 733 \pinlabel
Comm. at 182 631 \pinlabel Comm. at 182 550 \pinlabel Anti. at 228
447 \pinlabel Anti. at 160 350 \pinlabel Anti. at 160 290
\pinlabel Comm. at 160 226 \pinlabel Anti. at 160 155 \pinlabel
Comm. at 326 769 \pinlabel Comm. at 326 664 \pinlabel Comm. at 326
562 \pinlabel Comm. at 325 458 \pinlabel Anti. at 324 315
\pinlabel Comm. at 324 240 \pinlabel Comm. at 324 165 \pinlabel
Comm. at 324 95 \pinlabel Comm. at 565 760 \pinlabel Comm. at 565
668 \pinlabel Comm. at 565 568 \pinlabel Comm. at 550 458
\pinlabel Anti. at 550 315 \pinlabel Comm. at 570 217 \pinlabel
Comm. at 570 142 \pinlabel Comm. at 570 75 \pinlabel Forward at 88
810 \pinlabel Backward at 228 810 \pinlabel Forward at 363 810
\pinlabel Backward at 492 810
\endlabellist
\begin{center}
\includegraphics[scale=0.7]{bubbles}
\end{center}
\caption{The cases needed for Lemma~\ref{configs}.}
\label{bubbles}
\end{figure}
Lemma~\ref{configs} immediately finishes the proof of
Theorem~\ref{movemark}. Indeed, we know $\Delta\sigma(e) +
\Delta\sigma(c) + a_f = 1$ modulo 2, since odd Khovanov homology
satisfies $d^2 = 0$. Note that $\Delta\sigma(H) = \Delta\sigma(c)$
when we are in the first case of Lemma~\ref{configs}, and
$\Delta\sigma(H) = \Delta\sigma(c) + 1$ otherwise. Hence in either
case we can conclude $\Delta\sigma(H) + \Delta\sigma(e) + a_b = 0$,
as desired.
\end{proof}
\section{Spanning trees.}\label{spanningtrees}
As in \cite{Jaeger} and \cite{Roberts}, after inverting some of $R =
\Z[x_i]$, one can cancel the vertical differentials in the reduced
twisted complex to obtain a spanning-tree complex computing reduced
odd Khovanov homology. The situation will not be quite as nice as in
characteristic $2$, since the lack of a way to canonically determine
signs in odd Khovanov homology will lead to a sign ambiguity in the
spanning-tree differential. Still, we will discuss the situation
briefly.
Let $S$ denote the ring obtained from $R$ by inverting all products of
sums of the form $x_{i_1} + \ldots + x_{i_l}$ where $i_1, \ldots, i_l$
index some subset of the marks. Form the twisted complex with
coefficients in $S$. Consider a complete resolution $\rho$, with
circles $a_1, \ldots, a_k$. Let $w_i$ denote the sum of the variables
corresponding to marks on $a_i$. The complex $(V_{\rho}, d_{v,\rho})$
is actually the Koszul complex associated to the elements $w_1,
\ldots, w_k$ of $S$. We want to show this complex is acyclic.
In fact, the Koszul complex would already be acyclic over $R$, except
in the lowest degree (since the $w_i$ form a regular sequence in
$\Z[x_i]$). After tensoring with $S$, all the $w_i$ become invertible,
and the lowest homology of the complex (namely $S$ modulo the $w_i$)
is also trivial.
So after cancellation of $d_v$, we get a complex where the only
contributions come from connected resolutions, i.e. spanning trees of
the Tait graph of $D$. Consider two connected resolutions $T$ and
$\tilde{T}$, differing at only two crossings. There are two
intermediate two-component resolutions $\rho$ and $\rho'$ between
them. In each, one circle contains the basepoint. Let $w$ (resp. $w'$)
denote the sum of the variables of the marks on the circle without the
basepoint in $\rho$ (resp. $\rho'$). Then the contribution to
$\partial T$ in the spanning-tree complex, if we used the naive maps,
would be the sum of $T \rightarrow \rho \stackrel{1/w}{\rightarrow}
\rho \rightarrow \tilde{T}$ and $T \rightarrow \rho'
\stackrel{1/w'}{\rightarrow} \rho' \rightarrow \tilde{T}$. To put in
the actual signs, note that both $\rho$ and $\rho'$ come from
splitting circles in $T$. Hence the sign-corrections to $d_{v,\rho}$
and $d_{v,\rho'}$, and hence to $1/w$ and $1/w'$, are the same. The
other four maps, though, need to form an anticommuting square in the
cube of resolutions. The maps come from a 2-dimensional configuration
which is either $X$ or $Y$, depending on the orientations of the
relevant crossings. Recall that our convention was that $X$
anticommutes and $Y$ commutes. Hence if the configuration is $Y$, one
or three of the four maps must pick up a sign. Because of this sign,
the coefficient of the differential from $T$ to $\tilde{T}$ is $\pm
(1/w - 1/w')$. On the other hand, if the configuration is $X$, the
coefficient from $T$ to $\tilde{T}$ is $\pm (1/w + 1/w')$. The author
does not know a good way to decide between the $+$ and $-$ signs on
the outside without actually making explicit sign assignments in the
cube of resolutions.
\bibliographystyle{plain}
|
\section{Introduction and Background} \label{intro}
The most widely used method of shuffling cards is riffle
shuffling. Roughly speaking, one cuts the deck of cards into two piles of
approximately equal size and then riffles the two piles together. A precise
mathematical model of riffle shuffles is the Gilbert-Shannon-Reeds (or GSR)
shuffle, found independently by Gilbert $\cite{Gilbert}$ and Reeds
$\cite{Reeds}$. This model says to first cut the $n$ card deck into two
packs of size $m$ and $n-m$ with probability $\frac{{n \choose
m}}{2^n}$. Then drop cards from these packs one at a time, such that if
pack 1 has $A_1$ cards and pack 2 has $A_2$ cards, the next card is dropped
from pack 1 with probability $\frac{A_1}{A_1+A_2}$ and from pack 2 with
probability $\frac{A_2}{A_1+A_2}$.
Before defining biased shuffles, let us recall the notion of
the descent set of a permutation. An element $\pi \in S_n$ is said to
have a descent at position $i$ if $\pi(i)>\pi(i+1)$. By convention we
say that all $\pi \in S_n$ have a descent at position $n$. The descent
set of $\pi$ is the set of positions at which $\pi$ has a descent.
This paper analyzes a notion of biased riffle shuffles which
generalizes the GSR shuffle (the GSR shuffle will correspond to the case
$a=2,p_1=p_2=\frac{1}{2}$). These biased shuffles seem to have first been
considered on pages 153-4 of Diaconis, Fill, and Pitman $\cite{DiFiPi}$. We
now give four descriptions of these biased riffle shuffles. These
descriptions generalize the descriptions of the GSR shuffle in Bayer and
Diaconis $\cite{Bayer}$. It is elementary to prove that these descriptions
are equivalent.
\begin{center}
Descriptions of Biased $a$-shuffles
\end{center}
\begin{enumerate}
\item Cut the $n$ card deck into $a$ piles by picking pile sizes
according to the $mult(a;\vec{p})$ law, where $p=(p_1,\cdots,p_a)$. In
other words, choose $b_1,\cdots,b_a$ with probability:
\[ {n \choose b_1 \cdots b_a} \prod_{i=1}^a p_i^{b_i} \]
Then choose uniformly one of the ${n \choose b_1 \cdots b_a}$ ways
of interleaving these packets, leaving the cards in each packet in their
original relative order. (In the language of descents, choose uniformly one
of the ${n \choose b_1 \cdots b_a}$ permutations whose inverse has descent
set contained in $\{b_1,b_1+b_2,\cdots,b_1+\cdots+b_a=n\}$).
\item As in Description 1, cut the $n$ card deck into $a$ piles
according to the $mult(a;\vec{p})$ law. Now drop cards from the $a$
packets one at a time, according to the rule that if the $i$th packet
has $A_i$ cards, then the next card is dropped from the $i$th packet
with probability $\frac{A_i}{A_1+\cdots+A_a}$.
\item Drop $n$ points in $[0,1]$ according to the following
procedure. Break the unit interval into $a$ sub-intervals of length
$\frac{1}{a}$. Pick the $i$th interval with probability proportional
to $p_i$. Then drop uniformly in this interval. Label the points
$x_1,\cdots,x_n$ in order of smallest to largest. The map $x \mapsto
ax$ (mod 1) reorders these points. The induced measure on $S_n$ is the
same as in Descriptions 1 and 2.
\item The inverse of a biased $a$-shuffle has the following
description. Start with an ordered deck of $n$ cards face
down. Successively and independently, cards are turned face up and
dealt into a random pile $i$ with probability proportional to
$p_i$. After all the cards have been distributed, the piles are
assembled from left to right and the deck is turned face down.
\end{enumerate}
We denote the measure on $S_n$ defined by Descriptions 1-4 by
$P_{n,a;\vec{p}}$. For example, one can check that the measure
$P_{3,2;p_1,1-p_1}$ assigns to permutations in cycle form the following
masses:
\begin{eqnarray*}
(1)(2)(3) & & \ \ \ \ p_1^3+p_1^2p_2+p_1p_2^2+p_2^3\\
(1)(23) & & \ \ \ \ p_1^2p_2\\
(2)(13) & & \ \ \ \ 0\\
(3)(12) & & \ \ \ \ p_1p_2^2\\
(123) & & \ \ \ \ p_1p_2^2\\
(132) & & \ \ \ \ p_1^2p_2
\end{eqnarray*}
If $\vec{p}=(p_1,\cdots,p_a)$ and $\vec{p'}=(p_1',\cdots,p_b')$,
define the product:
\[ \vec{p} \otimes \vec{p'}=
(p_1p_1',\cdots,p_1p_b',\cdots,p_ap_1',\cdots,p_ap_b') \]
The following fact, which shows that biased riffle shuffles
convolve well, is stated without proof in Diaconis, Fill, and Pitman
$\cite{DiFiPi}$.
\begin{prop} \label{convol} The convolution of $P_{n,a;\vec{p}}$ and
$P_{n,b;\vec{p'}}$ is $P_{n,ab;\vec{p} \otimes \vec{p'}}$.
\end{prop}
\begin{proof}
This follows from the inverse description of card
shuffling. Lexicographically combining the pile assignments from an inverse
a-shuffle and an inverse b-shuffles gives uniform and independent pile
assignments for an inverse ab-shuffle.
\end{proof}
Proposition $\ref{convol}$ is the starting point for this
paper. Little seems to be known about biased riffle shuffles. The
Gilbert-Shannon-Reeds shuffle (the case of equal $p_i$), however, has been
fairly well studied (e.g. Bayer and Diaconis $\cite{Bayer}$ or Diaconis,
McGrath, and Pitman $\cite{DiMcPi}$).
\section{Bounding the Time to Uniform}
This section uses the concept of a strong uniform time to upper bound the
time for biased riffle shuffles to get close to the uniform
distribution. The bounds obtained are of the same order as lower bounds due to
Lalley \cite{Lalley}.
Recall that the total variation
distance between two probability distributions $P_1$ and $P_2$ on a set $X$
is defined as:
\[ \| P_1-P_2 \| = \frac{1}{2} \sum_{x \in X} |P_1(x)-P_2(x)| \]
Let $P^{*k}$ denote the $k$-fold convolution of $P$. Let $U$ be the
uniform distribution on $S_n$.
\begin{theorem} \label{upper}
\[ \| P_{n,a;\vec{p}}^{*k}-U \| \leq {n \choose 2} [p_1^2+\cdots+p_a^2]^k \]
\end{theorem}
\begin{proof}
For each $k$, let $A^k$ be a random $n*k$ matrix formed by letting
each entry equal $i$ with probability $p_i$. Note that the random matrix
$A^k$ corresponds to a random permutation under the measure
$P^{*k}_{n,a;\vec{p}}$. To see this, recall Description 4 of biased riffle
shuffles (the inverse description). A single inverse $a$ shuffle
corresponds to a column of $A^k$ by letting the $i$th entry in the column
of $A^k$ equal the pile into which card $i$ is placed.
Let $T$ be the first time that the rows of $A^k$ are
distinct. It is not hard to see that $T$ is a strong uniform time for
$P^{*k}_{n,a;\vec{p}}$ in the sense of Sections 4B-4D of Diaconis
$\cite{Diac}$. Namely, the permutation associated to the matrix $A^T$
is uniform. This is because, as in Proposition $\ref{convol}$, the
inverse of the $k$ fold convolution of $a$-shuffles may be viewed as
inverse sorting into $a^k$ piles, and at time $T$ each pile has at
most 1 card. Symmetry implies that these cards are in uniform random
order. It is proved on page 76 of Diaconis $\cite{Diac}$ that:
\[ |P^{*k}_{n,a;\vec{p}}-U| \leq Prob(T > k) \]
Let $V_{ij}$ be the event that rows $i$ and $j$ of $A^k$ are
the same. The probability that $V_{ij}$ occurs is
$[p_1^2+\cdots+p_a^2]^k$. The theorem follows since:
\begin{eqnarray*}
Prob(T>k) & = & Prob (\cup_{1 \leq i < j \leq n}) A_{ij}\\
& \leq & \sum_{1 \leq i < j \leq n} Prob(A_{ij})\\
& = & {n \choose 2} [p_1^2+\cdots+p_a^2]^k
\end{eqnarray*}
\end{proof}
{\bf Remarks}
\begin{enumerate}
\item Theorem $\ref{upper}$ shows that $k=2log_{\frac{1}{\sum_{i=1}^a p_i^2}}n$
steps suffice to get close to the uniform distribution (in the case
$a=2,p_1=p_2=\frac{1}{2}$ this is $2log_2n$).
\item Lalley \cite{Lalley} proved that there exists an open neighborhood of
$p_1=\frac{1}{2}$ such that for all $p_1$ in this neighborhood,
a $P_{n,2;p_1,p_2}$ shuffle takes at least
\[ \frac{3+\theta}{4} log_{\frac{1}{p_1^2+p_2^2}}n \]
steps to get close to the uniform distribution. Here $\theta=\theta_{p_1}$ is
the unique real number such that
\[ p_1^{\theta}+p_2^{\theta} = (p_1^2+p_2^2)^2 \]
Note that when $p_1=p_2=\frac{1}{2}$ this bound is $\frac{3}{2}log_2 n$, which
is of the same order as the $2 log_2 n$ bound of Theorem \ref{upper}, and agrees
exactly with the more refined analysis of Bayer and Diaconis \cite{Bayer} for the
GSR shuffles.
\end{enumerate}
\section{Gessel's Bijection and Cycle Structure} \label{bijec}
This section begins by describing a bijection of Gessel
$\cite{Gessel}$. This requires some preliminary notation and
concepts. Recall that a permutation $\pi \in S_n$ is said to have a descent
at position $i$ if $\pi(i) > \pi(i+1)$. We adopt the convention that all
$\pi \in S_n$ have a descent at position $n$. Define a necklace on an
alphabet to be a sequence of cyclically arranged letters of the alphabet. A
necklace is said to be primitive if it is not equal to any of its
non-trivial cyclic shifts. For example, the necklace $(a\ a\ b\ b)$ is
primitive, but the necklace $(a\ b\ a\ b)$ is not.
Given a word $w$ of length $n$ on an ordered alphabet, the 2-row
form of the standard permutation $st(w) \in S_n$ is defined as
follows. Write $w$ under $1 \cdots n$ and then write under each letter of
$w$ its lexicographic order in $w$, where if two letters of $w$ are the
same, the one to the left is considered smaller. For example (page 195 of
Gessel and Reutenauer $\cite{Gessel}$):
\[ \begin{array}{c c c c c c c c c c c c c c}
& & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12\\
w & = & b & b & a & a & b & c & c & c & b & c & b & b\\
st(w) &= & 3 & 4 & 1 & 2 & 5 & 9 & 10 & 11 & 6 & 12 & 7 & 8
\end{array} \]
For a finite ordered alphabet $A$, Gessel and Reutenauer
$\cite{Gessel}$ give a bijection $U$ from the set of length $n$ words $w$
of onto the set of finite multisets of necklaces of total size $n$, such
that the cycle structure of $st(w)$ is equal to the cycle structure of
$U(w)$. To define $U(w)$, one replaces each number in the necklace of
$st(w)$ by the letter above it. In the example, the necklace of $st(w)$ is
$(1\ 3), (2\ 4), (5), (6\ 9), (7\ 11\ 8\ 12\ 10)$. This gives the following
multiset of necklaces on $A$:
\[ (a\ b)(a\ b)(b)(b\ c)(b\ c\ b\ c\ c) \]
Theorem $\ref{Gesbij}$, one of the main results of this section,
will follow from this bijection.
\begin{theorem} \label{Gesbij} Fix $r_1,\cdots,r_a \geq 0$ such that
$\sum_{i=1}^a r_i=n$. The bijection $U$ defines by restriction a
cycle-structure preserving bijection $\bar{U}$ from elements of $S_n$ with
descent set contained in $\{r_1, r_1+r_2 , \cdots, r_1+\cdots+r_a=n\}$ to
multisets of primitive necklaces on the alphabet $\{1,\cdots,a\}$ formed
from a total of $r_i$ $i$'s. \end{theorem}
\begin{proof}
Restrict $U$ to the set of words with $r_i$ $i's$. It is clear
that an element $\pi$ of $S_n$ can arise as the standard permutation
of at most one word with $r_i$ $i's$. Also, the $\pi$ which arise are
precisely those $\pi$ such that the descent set of $\pi^{-1}$ is
contained in $\{r_1, r_1+r_2 , \cdots, r_1+\cdots+r_a=n\}$. This
proves the theorem.
\end{proof}
Corollary $\ref{translate}$ will translate Theorem $\ref{Gesbij}$
into the language of generating functions. This uses some further
notation. Define the quantity $M(r_1,\cdots,r_a)$ as:
\[ M(r_1,\cdots,r_a) = \frac{1}{n} \sum_{d|n,r_1,\cdots,r_a} \mu(d)
\frac{\frac{n}{d}!}{\frac{r_1}{d}! \cdots \frac{r_a}{d}!} \]
One easily proves by Moebius inversion (e.g. page 172 of Hall
$\cite{Hall}$) that $M(r_1,\cdots,r_a)$ is the number of primitive
circular words from an alphabet $\{1,\cdots,a\}$ in which the letter
$i$ appears $r_i$ times.
Recall that we are using the convention that all permutations
in $S_n$ have a descent at position $n$. For $b_i,n_i \geq 0$, let
$\vec{b}=(b_1,\cdots,b_a)$ and $\vec{n}=(n_1,n_2,\cdots)$. Let
$A_{\vec{b},\vec{n}}$ be the number of permutations on $b_1 + \cdots +
b_a$ letters with descent set contained in $\{b_1,b_1+b_2,\cdots,
b_1+\cdots+b_a\}$ and $n_i$ $i$-cycles.
\begin{cor} \label{translate} For all $a \geq 1$,
\[ \sum_{\vec{b},\vec{n}} A_{\vec{b},\vec{n}} \prod_{i=1}^a z_i^{b_i}
\prod_j x_j^{n_j} = \prod_{i=1}^{\infty} \prod_{r_1,\cdots,r_a \geq 0
\atop r_1+\cdots+r_a=i} (\frac{1}{1-z_1^{r_1}\cdots
z_a^{r_a}x_i})^{M(r_1,\cdots,r_a)} \]
\end{cor}
\begin{proof}
The coefficient of $\prod_{i=1}^a z_i^{b_i} \prod_j x_j^{n_j}$
on the left hand side is equal to $A_{\vec{b},\vec{n}}$, the number of
permutations on $b_1+\cdots+b_a$ letters with descent set contained in
$\{b_1,b_1+b_2,\cdots, b_1+\cdots+b_a\}$ and $n_j$ $j$-cycles. Theorem
$\ref{Gesbij}$ says that this is the number of multisets of necklaces
on the alphabet $\{1,\cdots,a\}$ with $b_i$ $i$'s and $n_j$
$j$-cycles. The corollary now follows from the interpretation of
$M(r_1,\cdots,r_a)$ as the number of primitive circular words of
length $n$ from an alphabet $\{1,\cdots,a\}$ in which the letter $i$
appears $r_i$ times.
\end{proof}
Corollary $\ref{translate}$ will be used to study the cycle structure
of a permutation under the measure $P_{n,a,\vec{p}}$. Let $E_{n,a,\vec{p}}$
denote expectation with respect to the measure
$P_{n,a,\vec{p}}$, and $N_i$ denote the random variable on $S_n$ such that
$N_i(\pi)$ is the number of $i$-cycles of $\pi$. The case of Theorem
$\ref{cycle}$ with all $p_i=\frac{1}{a}$ is known from Diaconis, McGrath,
and Pitman $\cite{DiMcPi}$.
\begin{theorem} \label{cycle}
\[ \sum_{n=0}^{\infty} u^n E_{n,a,\vec{p}} \prod_{i=1}^N x_i^{N_i} =
\prod_{i=1}^{\infty} \prod_{r_1,\cdots,r_a \geq 0 \atop r_1+\cdots+r_a=i}
(\frac{1}{1-p_1^{r_1} \cdots p_a^{r_a} u^i x_i})^{M(r_1,\cdots,r_a)} \]
\end{theorem}
\begin{proof}
Corollary $\ref{translate}$ and elementary manipulations imply
that:
\begin{eqnarray*}
\prod_{i=1}^{\infty} \prod_{r_1,\cdots,r_a \geq 0 \atop r_1+\cdots+r_a=i}
(\frac{1}{1-p_1^{r_1} \cdots p_a^{r_a} u^i x_i})^{M(r_1,\cdots,r_a)} & = &
\sum_{n=0}^{\infty} u^n \sum_{b_1+\cdots+b_a=n \atop \vec{n}: \sum in_i=n} A_{\vec{b},\vec{n}} \prod_{i=1}^a p_i^{b_i} \prod_j x
|
_j^{n_j}\\
& = & \sum_{n=0}^{\infty} u^n \sum_{b_1+\cdots+b_a=n \atop \vec{n}: \sum in_i=n} [{n
\choose b_1 \cdots b_a} \prod_{i=1}^a p_i^{b_i}]
[\frac{A_{\vec{b},\vec{n}}}{{n \choose b_1 \cdots b_a}}] \prod_j x_j^{n_j}
\end{eqnarray*}
We give a probabilistic interpretation to:
\[ \sum_{n=0}^{\infty} u^n \sum_{b_1+\cdots+b_a=n \atop \vec{n}: \sum
in_i=n} [{n \choose b_1 \cdots b_a} \prod_{i=1}^a p_i^{b_i}]
[\frac{A_{\vec{b},\vec{n}}}{{n \choose b_1 \cdots b_a}}] \prod_j
x_j^{n_j} \]
The first term in square brackets is the chance that a deck cut
according to the $mult(n,\vec{p})$ distribution is cut into packets of size
$b_1,\cdots,b_a$. To interpret the second term in square brackets, use the
fact from page 17 of Stanley $\cite{Stanley}$ that the total number of
permutations on $n=b_1+\cdots+b_a$ letters with descent set contained in
$\{b_1,b_1+b_2,\cdots,b_1+\cdots+b_a\}$ is the multinomial coefficient ${n
\choose b_1 \cdots b_a}$. Thus the second term is equal to the chance that
choosing uniformly among permutations on $n$ letters whose inverse has
descent set contained in $\{b_1,b_1+b_2,\cdots,b_1+\cdots b_a\}$ gives a
permutation with $n_i$ $i$-cycles. This proves the theorem.
\end{proof}
As an example of an application of Theorem $\ref{cycle}$, one obtains
an expression for the expected number of fixed points after a $k$-fold
convolution of the measure $P_{n,a,\vec{p}}$.
\begin{cor} \label{fixpoint} The expected number of fixed points of a
permutation under the $k$-fold convolution of $P_{n,a,\vec{p}}$ is:
\[ \sum_{j=1}^n [p_1^j+\cdots+p_a^j]^k \]
\end{cor}
\begin{proof}
Recall from the introductory section that the $k$-fold
convolution of an $a$-shuffle with parameters $(p_1,\cdots,p_a)$ is
equivalent to an $a^k$ shuffle with parameters equal to the $a^k$
possible products $p_{s_1} \cdots p_{s_k}$ where each $s_i \in
\{1,\cdots,a\}$ and repetition is allowed. Thus it suffices to prove
the corollary in the case $k=1$.
In the generating function of Theorem $\ref{cycle}$, one wants
to set $x_1=x$, $x_i=1$ for $i \geq 2$, then differentiate with
respect to $x$, set $x=1$, and finally take the coefficient of $y^n$.
Setting $x_1=x$, $x_i=1$ for $i \geq 2$ in the generating
function of Theorem $\ref{cycle}$ gives:
\[ \frac{1}{1-y} \frac{1-p_1y}{1-p_1xy} \cdots
\frac{1-p_ay}{1-p_axy} \]
because the $x_1=x$ term contributes $\frac{1}{\prod_{i=1}^a
(1-p_ixy)}$ and the $x_i=1$ for $i \geq 2$ term contributes
$\frac{\prod_{i=1}^a (1-p_iy)}{1-y}$. The corollary now follows by
easy algebra.
\end{proof}
{\bf Remarks}
\begin{enumerate}
\item In the case of $p_i=\frac{1}{a}$, Corollary $\ref{fixpoint}$ shows
that the expected number of fixed points after $k$ $a$-shuffles is:
\[ \sum_{j=1}^n \frac{1}{a^{(j-1)k}} \]
which is known from Diaconis, McGrath, and Pitman $\cite{DiMcPi}$. In fact
Holder's inequality gives:
\[ \frac{1}{a^{j-1}} \leq p_1^j+\cdots+p_a^j \]
so that the expected number of fixed points is smallest for unbiased
riffle shuffles.
\item It turns out that for $\frac{1}{(p_1^2+\cdots+p_a^2)^k} \gg 1$, the
number of fixed points is close to its Poisson(1) limit. In fact fixed
points (and more generally other functionals of cycle structure) approach
their limit distribution more quickly than $P_{n,a,\vec{p}}$ approaches its
uniform limit.
\end{enumerate}
\section{Enumerative Applications of Gessel's Bijection}
This section considers some enumerative applications of Theorem
$\ref{Gesbij}$. To begin, formulas will be found for
the chance that an
$n$-cycle in $S_n$ has a given descent set $J$. Recall that all
permutations in $S_n$ are considered to have a descent at position $n$. We also
use the notation that if
$J=\{j_1<j_2< \cdots j_d=n \}$ and $j_0=0$, then $C(J)$, the composition of
the descent set $J$, is equal to $(j_1-j_0,\cdots,j_d-j_{d-1})$.
Stanley $\cite{Stanley}$ gives two formulas for the number of
permutations with descent set $J$. These will both turn out to have analogs
for the case of $n$-cycles.
\begin{prop} \label{Stan1} (Page 69 of Stanley $\cite{Stanley}$) The number of
elements of $S_n$ with descent set $J$ is:
\[ \sum_{K \subseteq J} (-1)^{|J|-|K|} {n \choose C(K)} \]
\end{prop}
This carries over to $n$-cycles as follows, where
$M(r_1,\cdots,r_a)$ is defined as in Section $\ref{bijec}$.
\begin{cor} \label{ncyc1} The number of $n$-cycles with descent set $J$ is:
\[ \sum_{K \subseteq J} (-1)^{|J|-|K|} M(C(K)) \]
\end{cor}
\begin{proof}
By Moebius inversion on the power set of $\{1,\cdots,n\}$, it
suffices to show that the number of $n$ cycles with descent set
contained in $K$ is $M(C(K))$. This follows from Theorem
$\ref{Gesbij}$.
\end{proof}
There is also a determinantal formula for the number of
permutations with descent set $J$. Suppose that the elements of $J$ are $1
\leq j_1 \leq j_2 ... \leq j_k \leq n-1$. Define $j_0=0$ and $j_{k+1}=n$.
\begin{prop} \label{Stan2} (Page 69 of Stanley $\cite{Stanley}$) The number of
elements of $S_n$ with descent set $J$ is the determinant of a $k+1$ by
$k+1$ matrix, where $(l,m) \in [0,k] \times [0,k]$:
\[ det {n-j_l \choose j_{m+1}-j_l} \]
\end{prop}
This can be generalized to $n$-cycles. Given $J$, a subset of
$\{1,...,n-1\}$, let $J^d$ be the subset of $J$ consisting of all numbers
divisible by $d$. If $J$ is non-empty, label these elements $1 \leq j^d_1
\leq j^d_2 \cdots \leq j^d_{|J^d|} \leq n-1$. Define $j^d_0=0$ and
$j^d_{|J^d|+1}=n$.
\begin{theorem} \label{ncycdet} The number of $n$-cycles
with descent set $J$ is:
\[ \frac{1}{n} \sum_{d|n} \mu (d) (-1)^{|J|-|J^d|} det
{\frac{n}{d}-\frac{j^d_l}{d} \choose
\frac{j^d_{m+1}}{d}-\frac{j^d_l}{d}} \]
\end{theorem}
\begin{proof}
From Theorem $\ref{ncyc1}$, the number of $n$-cycles with
descent set $J$ is:
\begin{eqnarray*}
\sum_{K \subseteq J} (-1)^{|J|-|K|} M(C(K)) & = & \frac{1}{n} \sum_{K
\subseteq J} (-1)^{|J|-|K|} \sum_{d: K \subseteq J^d} \mu(d) {\frac{n}{d} \choose
C(\frac{K}{d})}\\
& = & \frac{1}{n} \sum_{d|n} \mu(d) \sum_{K \subseteq J^d} (-1)^{|J|-|K|}
{\frac{n}{d} \choose C(\frac{K}{d})}\\ & = & \frac{1}{n} \sum_{d|n} \mu(d)
(-1)^{|J|-|J^d|} \sum_{K \subseteq J^d} (-1)^{|J^d|-|K|} {\frac{n}{d} \choose
C(\frac{K}{d})}
\end{eqnarray*}
Proposition $\ref{Stan1}$ shows that $\sum_{K \subseteq J^d}
(-1)^{|J^d|-|K|} {\frac{n}{d} \choose C(\frac{K}{d})}$ is the
number of permutations on $\frac{n}{d}$ symbols with descent set
$\frac{J^d}{d}$. The theorem then follows from Proposition
$\ref{Stan2}$.
\end{proof}
The enumeration of matrices with fixed row and column sums is
related to some problems in statistics (see for instance the work of
Diaconis and Sturmfels $\cite{DiaSturm}$). Proposition $\ref{invol}$
relates the theory of such matrices to the theory of descents in
involutions.
\begin{prop} \label{invol} The number of involutions in $S_n$ with descent
set contained in $K=\{k_1,...,k_r=n\}$ is equal to the number of symmetric $r*r$
matrices with non-negative integer entries and with $i$th row sum
$k_i-k_{i-1}$, where by convention $k_0=0$. \end{prop}
\begin{proof}
Theorem $\ref{Gesbij}$ shows that it suffices to count the number
of multisets of primitive necklaces on an alphabet of $k_i-k_{i-1}$ $i$'s,
where each necklace has length 1 or 2. Note that a primitive necklace of
length 2 consists of a pair of distinct elements. So for $i \neq j$, let
$X_{ij}$ be the number of pairs of letter i with letter j, and
let $X_{ii}$ be the number of singleton $i$'s. The matrix $(X_{ij})$ has
all the desired properties.
\end{proof}
\section{Inversion and Descent Structure After a Shuffle}
It is natural to study the inversion and descent structure of a
permutation obtained after a biased riffle shuffle. Recall that $\pi$ is said to
invert the pair $(i,j)$ with $i<j$ if
$\pi(i)>\pi(j)$. The number of inversions of $\pi$ is the number of pairs
which $\pi$ inverts and will be denoted $Inv(\pi)$. It is easy to see that
$Inv(\pi)=Inv(\pi^{-1})$ and that $Inv(\pi)$ is the length of $\pi$ in
terms of the generators $\{(i,i+1):1 \leq i \leq n-1 \}$. Theorem
$\ref{invgen}$ will give a $q$-exponential generating function for $Inv$
after a biased riffle shuffle. This uses the notation:
\[ [n]! = \prod_{i=0}^{n-1} (1+q+\cdots+q^i) \]
\[ \qb{n}{k} = \frac{[n]!}{[k]![n-k]!} \]
As usual, $E_{n,a,\vec{p}}$ denotes expectation with respect to the
measure $P_{n,a,\vec{p}}$. As will be explained in the course of the proof,
the second equality in Theorem $\ref{invgen}$ is purely formal in the sense
that it only holds if $|q|<1$, and thus only the first equality should be used for
the purpose of computing moments.
\begin{theorem} \label{invgen}
\begin{eqnarray*}
\sum_{n=0}^{\infty} \frac{u^n}{[n]!}
E_{n,a,\vec{p}} \ q^{Inv} & = & \prod_{i=1}^a [\sum_{j=0}^{\infty}
\frac{(up_i)^j}{[j]!}] \\
& = & \prod_{i=1}^a \prod_{j=0}^{\infty} \frac{1}{1-up_i(1-q)q^j}
\end{eqnarray*}
\end{theorem}
\begin{proof}
The following identity is clear from elementary manipulations and
the definition of $q$-multinomial coefficients:
\[ \sum_{n=0}^{\infty} \sum_{b_i \geq 0 \atop b_1+\cdots+b_a=n}
\frac{p_1^{b_1}\cdots p_a^{b_a} \qb{n}{b_1 \cdots b_a} u^n}{[n]!} =
\prod_{i=1}^a [\sum_{j=0}^{\infty} \frac{(up_i)^j}{[j]!}] \]
The left-hand side can be rewritten as:
\[ \sum_{n=0}^{\infty} \frac{u^n}{[n]!} \sum_{b_i \geq 0 \atop
b_1+\cdots+b_a=n} [ {n \choose b_1 \cdots b_a} \prod_{i=1}^a p_i^{b_i}]
\frac{\qb{n}{b_1 \cdots b_a}}{ {n \choose b_1 \cdots b_a}} \]
Since $Inv(\pi)$ is equal to $Inv(\pi^{-1})$, it is sufficient to
analyze the number of inversions in the inverse of a permutation chosen
from the measure $P_{n,a;\vec{p}}$. Recalling the first description of
biased riffle shuffling in Section $\ref{intro}$, note that the term in
brackets corresponds to picking the packet sizes $b_1,\cdots,b_a$ according
to the $mult(a;\vec{p})$ law. From pages 22 and 70 of Stanley
$\cite{Stanley}$, it is known that $\qb{n}{b_1 \cdots b_a}$ is the sum of
$q^{Inv(\pi)}$ over all $\pi$ in $S_n$ with descent set contained in
$\{b_1,b_1+b_2,\cdots,b_1+\cdots+b_a=n\}$ and that $ {n \choose b_1 \cdots
b_a}$ is the number of permutations with descent set contained in
$\{b_1,b_1+b_2,\cdots,b_1+\cdots+b_a=n\}$. These observations prove the
first equality of the theorem.
The second equality follows from a famous identity of Euler, which
is true if $|x|,|q|<1$:
\[ \prod_{j=0}^{\infty} \frac{1}{1-xq^n} = \sum_{j=0}^{\infty}
\frac{x^j}{(1-q) \cdots (1-q^j)} \]
\end{proof}
Theorem $\ref{invgen}$ can be used to compute the expected number
of inversions after a $k$-fold convolution of a $P_{n,a;\vec{p}}$
shuffle. However, we prefer the following direct probabilistic argument.
\begin{theorem} \label{expecinversion}
The expected number of inversions under the $k$-fold convolution
of $P_{n,a;\vec{p}}$ is:
\[ \frac{{n \choose 2}}{2} [1-(p_1^2+\cdots+p_a^2)^k] \]
\end{theorem}
\begin{proof}
For $1 \leq i<j \leq n$, define a random variable $X_{i,j}$ as follows. In the
inverse model of card shuffling, let $X_{i,j}=1$ if card $i$ goes to a pile to
the right of card $j$, and let $X_{i,j}=0$ otherwise. It is easy to see that if
$\pi$ is the permutation obtained after the shuffle, then $\pi(i)>\pi(j)$ exactly
when $X_{i,j}=1$. Thus,
\[ Inv = \sum_{1 \leq i<j \leq n} X_{i,j}. \]
It is clear that each $X_{i,j}$ has expected value $\frac{1-(p_1^2+\cdots+
p_a^2)^k}{2}$, because this is one half the chance that cards $i$ and $j$ fall in
different piles. The theorem now follows by linearity of expectation.
\end{proof}
{\bf Remarks}
\begin{enumerate}
\item Note that a uniformly chosen element of $S_n$ has on average $\frac{{n
\choose 2}}{2}$ inversions. In fact the distribution for inversions on $S_n$ is
the sum $X_1+\cdots+X_n$ where the $X_i$ are independent and uniform on $[0,i-1]$.
\item By Holder's inequality, the expected number of inversions is maximum
for $k$ unbiased $a$ shuffles (which is the same as an $a^k$ shuffle), and
in this case is $\frac{{n \choose 2}}{2} [1-\frac{1}{a^k}]$. For instance,
a $1$ shuffle of a sorted deck gives no inversions, and a 2 shuffle of a
sorted deck gives a permutation which has on average one half as many
inversions as a random permutation.
\item It would be interesting to use Theorem $\ref{invgen}$ to study the
asymptotics of inversions after a biased riffle shuffle. Even for the case
$a=2,p_1=p_2=\frac{1}{2}$, it is not known if the $n \rightarrow \infty$
limit distribution is normal.
\item The same technique used in Theorem \ref{expecinversion} can be used to
study the distribution of $Des(\pi)$, the number of descents of a permutation
$\pi$ after a biased riffle shuffle. For example, using the convention that all
elements of $S_n$ have a descent at position $n$, the expected number of descents
would be
\[ 1 + \frac{n-1}{2} [1-(p_1^2+\cdots+p_a^2)^k] \]
It is perhaps surprising that these moments can be computed so easily. One
reason to be surprised is that in the case of unbiased shuffles, Bayer and
Diaconis \cite{Bayer} showed that $Des(\pi^{-1})$ is a sufficient statistic for
the random walk. Nevertheless, computing the moments of $Des(\pi^{-1})$ is
more difficult than computing the moments of $Des(\pi)$, as a glance at the work
of Mann \cite{Mann} will make clear.
\end{enumerate}
\section{Acknowledgements} The author is grateful to Persi Diaconis for numerous
conversations about the theory of descents and shuffling cards. Ira Gessel was
also kind enough to share his thoughts on descents. This research was done under
the support of the National Defense Science and Engineering Graduate Fellowship
(grant no. DAAH04-93-G-0270) and the Alfred P. Sloan Foundation Dissertation
Fellowship.
|
\section{Introduction}
For more than 40 years, \cite{BC66,BC67,BC68} (see also \cite{Witham}),
various ways of solving the non-linear equation
\begin{equation}
\ddot{z}(1+z'^2) - z''(1-\dot{z}^2) = 2\dot{z} z' \dot{z}'
\label{BI}
\end{equation}
are known.
Recent work on higher dimensional time-like zero mean curvature
hypersurfaces includes \cite{Christo,Milbredt,Ho08,BNO08}
Here we show that (\ref{BI}) can develop singularities
in finite time, starting from finite initial data. The structure
of these singularities is described by the self-similar ansatz
\begin{equation}
z(t,x) = z_0 - \hat{t} + \hat{t}^{\alpha}h
\left(\frac{x}{\hat{t}^{\beta}}\right) + \dots
\label{selfs}
\end{equation}
where $\hat{t}:=t_0-t\rightarrow 0$ (the dots are indicating lower
order terms). Inserting (\ref{selfs}) into (\ref{BI}) one finds
the above ansatz to be consistent provided
$\beta=(1+\alpha)/2 > 1$, and
\begin{equation}
h''\left(2\alpha h - \frac{(\alpha+1)^2}{4}\xi^2\right) =
(\alpha-1)\left[h'^2 + \alpha h -
\frac{3}{4}(\alpha+1)\xi h'\right]\; ,
\label{sim_equ}
\end{equation}
for (\ref{selfs}) to be an asymptotic solution of (\ref{BI}).
For consistency with a finite outer solution of (\ref{BI}),
the profile $h$ must satisfy
\begin{equation}
h(\xi)\propto A_{\pm}\xi^{\frac{2\alpha}{\alpha+1}} \quad
\mbox{for} \quad \xi\rightarrow\pm\infty
\label{growth}
\end{equation}
(for a general discussion of matching self-similar solutions to the
exterior see \cite{EF08}).
The ansatz (\ref{selfs}) is formally consistent for a continuum
of similarity exponents $\alpha\ge 1$ and for any solution of the similarity
equation (\ref{sim_equ}). However, by considering the regularity of solutions
of (\ref{sim_equ}) in the origin $\xi=0$ the similarity
exponent must be one of the sequence
\begin{equation}
\alpha = \alpha_n = \frac{n+1}{n}, \quad n\in \mathbb{N},
\label{exp}
\end{equation}
certainly if $h(0)=0=h'(0)$, and presumably in general
(i.e. {\it all} relevant solutions of (\ref{sim_equ})).
Of this infinite sequence, we believe that only $\alpha = 2$ is
realized for generic initial data; indeed, in this case
\begin{equation}
\xi = \zeta + c\zeta^3/3 , \quad h(\xi) = \zeta^2/2 + c\zeta^4/4,
\label{sim_sol}
\end{equation}
which we will deduce from a parametric string solution corresponding to
(\ref{BI}).
The importance of the similarity solution (\ref{selfs}) lies in the
fact that it can be generalized to arbitrary dimensions, in particular
to membranes. We find that the same type of singular solution
is observed in any dimension, even having the same spatial structure
(\ref{sim_sol}).
\section{The similarity equation}
A way of satisfying (\ref{sim_equ}) is to demand
\begin{equation}
L^2:=h'^2 + 2\alpha h - (\alpha + 1)\xi h' = 0.
\label{sim_r}
\end{equation}
(differentiating e.g. $(1+\alpha)\xi = h' + 2\alpha h/h'$
one can eliminate $h''$, reducing (\ref{sim_equ}) to an identity,
as long as $h' \ne 1$).
The transformation
\begin{equation}
h(\xi) = \xi^2 g(\xi) = \xi^2\left(\frac{(1+\alpha)^2}{8\alpha}
-\frac{v^2}{2\alpha}\right)
\label{trans}
\end{equation}
yields
\begin{equation}
-\frac{d\xi}{\xi} =
|
\frac{vdv}{v^2\pm\alpha v+ (\alpha^2-1)/4} = \frac{1}{2}
\left(\frac{\alpha+1}{v\mp\frac{\alpha+1}{2}} -
\frac{\alpha-1}{v\mp\frac{\alpha-1}{2}}\right)dv,
\label{solve1}
\end{equation}
i.e. (choosing the upper sign)
\begin{equation}
\frac{\left|v-(\alpha+1)/2\right|^{\alpha+1}}
{\left|v-(\alpha-1)/2\right|^{\alpha-1}} = \frac{E}{\xi^2}.
\label{solve2}
\end{equation}
This yields solutions $v\in [(\alpha-1)/2,(\alpha+1)/2)$,
\begin{eqnarray}
\label{lim}
&&v\approx\frac{\alpha-1}{2} + \left(\frac{\xi^2}{E}\right)^
{\frac{1}{\alpha-1}} + \dots\quad \mbox{as}\quad \xi\rightarrow 0 \\ \nonumber
&&v\approx\frac{\alpha+1}{2} - \left(\frac{\xi^2}{E}\right)^
{\frac{1}{\alpha+1}} + \dots\quad \mbox{as}\quad \xi\rightarrow \pm\infty,
\end{eqnarray}
i.e.
\begin{eqnarray}
\label{limits}
&&h(\xi)\ge 0, h(0)=0, \\ \nonumber
&&h(\xi)\propto \xi^2/2 \quad \mbox{as}\quad \xi\rightarrow 0 \\
&&h(\xi)\propto \frac{1+\alpha}{2\alpha}
\xi^{\frac{2\alpha}{1+\alpha}} \nonumber \quad \mbox{as}\quad
\xi\rightarrow \pm\infty. \nonumber
\end{eqnarray}
Note that these solutions are consistent with the growth conditions
(\ref{growth}).
To solve the {\it second order equation} (\ref{sim_equ}) we note that
$\tilde{h}(\xi):=c\; h(\xi/\sqrt{c})$ solves (\ref{sim_equ}), if $h$ does,
and that
\begin{equation}
\frac{h'}{\xi} - \frac{2h}{\xi^2} = \frac{1}{\alpha}
f\left(\sqrt{\frac{(\alpha+1)^2}{4}-2\alpha\frac{h(\xi)}{\xi^2}}\right)
\quad \equiv \left(\frac{1}{\alpha}f(v)\right)
\label{reduce}
\end{equation}
reduces (\ref{sim_equ}) to
\begin{equation}
-\left(v^2-\frac{(\alpha+1)^2}{4}\right)
\left(v^2-\frac{(\alpha-1)^2}{4}\right) =
f\left(\alpha vf' - (\alpha-1)f-(\alpha+2)v^2+(\alpha^2-1)(\alpha-2)/4\right),
\label{reduce1}
\end{equation}
and
\begin{equation}
\frac{d\xi}{\xi} = -\frac{vdv}{f(v)}=\frac{\alpha dg}{f}.
\label{reduce2}
\end{equation}
The growth condition (\ref{growth})
implies that $h$ grows less than quadratically at infinity.
Thus we deduce from (\ref{reduce}) that $f$ vanishes at
$(\alpha+1)/2$. Furthermore, from a direct calculation using the growth
exponent (\ref{growth}) we find the first derivative, yielding
the initial conditions
\begin{equation}
f\left(\frac{\alpha+1}{2}\right) = 0, \quad
f'\left(\frac{\alpha+1}{2}\right) = 1 .
\label{growth1}
\end{equation}
Using (\ref{growth1}), (\ref{reduce1}) yields a polynomial solution
\begin{equation}
f(v) = \left(v-\frac{(\alpha+1)}{2}\right)
\left(v-\frac{(\alpha-1)}{2}\right) = v^2-\alpha v + \frac{\alpha^2-1}{4},
\label{reduce3}
\end{equation}
(i.e.) (\ref{solve2}), which corresponds to the first order
equation (\ref{sim_r}), but also an infinity of other solutions
(a Taylor expansion
|
}$'s ``energy representation''.
Just as before, $f\of{E}$ will approach a function of $E$ without any delta-function contributions in the thermodynamic limit.
It is normalized to unity, $\sInt{0}{\infty}{E} f\sof{E} = \sBK{\ensuremath{\psi_\text{d}}}{\ensuremath{\psi_\text{d}}} = 1$.
(Remember that the lowest possible energy is fixed at zero.)
In the mathematical literature \cite{Marchetti2012-0}, it is known as the Hamiltonian's spectral measure associated with the state $\sKet{\ensuremath{\psi_\text{d}}}$.
If $\sKet{\ensuremath{\psi_\text{d}}}$ is an atomic orbital state or some other spatially localized wave function, $f\sof{E}$ can also be identified with the local DOS \cite{Busch1998-0,Li2001-0,Yeganegi2014-0}.
The advantage of the MSDOS is that matrix elements of the Hamiltonian or related operators can be represented as integrals over all energies, in particular the return amplitudes:
\begin{equation}
u_n
=
\BAK{\ensuremath{\psi_\text{d}}}{e^{-in\frac{\tau\ensuremath{\hat{H}}}{\hbar}}}{\ensuremath{\psi_\text{d}}}
=
\Int{0}{\infty}{E} e^{-in\frac{\tau E}{\hbar}} f\sof{E}
.
\label{eq:RA}
\end{equation}
The return amplitude and the MSDOS are a Fourier transform pair.
Consider the case, when the system at hand is invariant under a certain symmetry transformation.
Then one may be able to find a set of base states $\sKet{\tilde{\chi}_n}$, such that $\sBAK{\tilde{\chi}_n}{\ensuremath{\hat{H}}}{\tilde{\chi}_n} = \sBAK{\tilde{\chi}_{n'}}{\ensuremath{\hat{H}}}{\tilde{\chi}_{n'}}$, for any pair $n$,$n'$.
In this case $N^{-1} \sTrace{\delta\sof{E - \ensuremath{\hat{H}}}} = \sBAK{\tilde{\chi}_n}{\delta\sof{E-\ensuremath{\hat{H}}}}{\tilde{\chi}_n}$ for any $n$.
This means that the MSDOS associated with $\sKet{\tilde{\chi}_n}$ {\em can be identified with the DOS.}
This is exactly the situation in the tight-binding model, for the special case when $\sKet{\ensuremath{\psi_\text{d}}}$ is a lattice site eigenstate $\sKet{\V{x}_\text{d}}$:
By translational invariance, the matrix elements $\sBAK{\V{x}}{\ensuremath{\hat{H}}}{\V{y}}$ only depend on the distance $\V{y}-\V{x}$.
In one dimension, one obtains with Eq.~\eqref{eq:TBDOS}:
\begin{equation}
f\sof{E} = \rho\sof{E}
=
\frac{1}{\pi}
\frac{1}{\sqrt{E\sof{4\gamma-E}}}
\label{eq:TBSM}
\end{equation}
Consequently, the van Hove singularities and the spectral dimension can be found in $f\sof{E}$ as well.
In general, $f\sof{E}$ and $\rho\sof{E}$ are not equal!
As a counterexample, consider the detection state $\sKet{\ensuremath{\psi_\text{d}}} = \sbr{\sKet{a} + \sKet{-a}}/\sqrt{2}$ and compute its MSDOS from the Brillouin zone integral with $\sBK{\pm a}{k} = \sqrt{a/(2\pi)} e^{\pm i a k}$:
\begin{align}
f\sof{E}
= & \nonumber
\frac{1}{2}
\Int{-\frac{\pi}{a}}{\frac{\pi}{a}}{k}
\delta\sof{E - E\sof{k}}
\br{\Bra{a} + \Bra{-a}}
\KB{k}{k}
\br{\Ket{a} + \Ket{-a}}
\\ = &
\frac{a}{4\pi} \Int{-\frac{\pi}{a}}{\frac{\pi}{a}}{k}
\delta\sof{E - E\sof{k}}
4 \cos^2\sof{ak}
=
\frac{1}{\pi} \frac{\br{1-\frac{E}{2\gamma}}^2}{\sqrt{E\sbr{4\gamma-E}}}
.
\label{eq:TBSMCounterEx1}
\end{align}
Thus, here the MSDOS has an additional factor relative to the DOS.
Although the MSDOS and the DOS are different for this detection state, we note that they both feature a one-over-square-root divergence at $E=0$ and at $E=4\gamma$.
That is, the non-analytic points of $f\sof{E}$ are the van Hove singularities and the spectral dimension $d_S$ is the same as the one found in the DOS $d_S^\text{DOS}$.
As we have shown, $f\sof{E}$ and $\rho\sof{E}$ are not equal for a general choice of detection state.
Nevertheless, $f\sof{E}$ may have $L'$ non-analytic points $E^*_l$, just like $\rho\sof{E}$ has the van Hove singularities.
We assume that $f\sof{E}$ admits the following asymptotic expansion around these points:
\begin{equation}
f\sof{E^*_l\pm\epsilon}
\sim
\tilde{f}_l\sof{\pm\epsilon}
+
A^\pm_l \epsilon^{\frac{d_S}{2}-1}
,
\label{eq:SpecMeasAssump}
\end{equation}
where the coefficients $A^\pm_l$ depend on the particular point $E^*_l$ and on the direction of the approach.
Since the singularity may be present only in a derivative, we introduce the ``analytic remainder'', which is nothing else but the Taylor expansion up to order $d_S/2-1$:
\begin{equation}
\tilde{f}_l\sof{\pm\epsilon}
=
\Sum{0\le n < \tfrac{d_S}{2}-1}{}
\frac{f^{(n)}\sof{E^*_l}}{n!} \sbr{\pm\epsilon}^n
.
\label{eq:DefAnalRemain}
\end{equation}
Clearly, an expansion like Eq.~\eqref{eq:SpecMeasAssump} is always possible.
However, the identification of $f\sof{E}$'s singularities with those of $\rho\sof{E}$ is not always possible.
We consider a detection state as ``ordinary'' when the corresponding MSDOS's singularities are located where the van Hove singularities are, and when its spectral dimension coincides with $d_S^\text{DOS}$.
For the 1d tight-binding model with $\sKet{\ensuremath{\psi_\text{d}}} = \sKet{x_\text{d}}$, we see from Eq.~\eqref{eq:TBSM} that
\begin{align}
f\sof{0+\epsilon} = & f\sof{4\gamma - \epsilon} = \frac{1}{\pi\sqrt{4\gamma}} \epsilon^{\frac{1}{2}-1} \\
f\sof{0-\epsilon} = & f\sof{4\gamma + \epsilon} = 0.
\end{align}
This identifies the spectral dimension as unity and $A_0^- = A_1^+ = 0$ as well as $A_0^+ = A_1^- = (4 \pi^2 \gamma)^{-1/2}$ and $\tilde{f}_l\sof{\pm\epsilon} = 0$.
Table~\ref{tab:Constants} lists all the constants used throughout the manuscript.
As mentioned, when the detection state is chosen as a lattice site eigenstate, $f\sof{E}$ and $\rho\sof{E}$ will coincide in the tight-binding model in any dimension.
\begin{table}
\begin{tabular}{c||c|c|c|c}
& TB & FP & L\'evy & Def. \\
\hline \hline
$L$ & $d+1$ & $1$ & $1$ & Sec.~\ref{sec:SpecMeas} \\
$E^*_l$ & $4 l \gamma$ & $0$ & $0$ & Eq.~\eqref{eq:SpecMeasAssump} \\
$d_S$ & $d$ & $d$ & $2\frac{d}{\alpha}$ & Eq.~\eqref{eq:SpecMeasAssump} \\
$A^+_l$ & $\frac{\sbr{-1}^{\frac{l}{2}}\binom{d}{l}}{\sGma{\frac{d}{2}}\sbr{4\pi\gamma}^\frac{d}{2}}$ & $\frac{E_0^{-\frac{d}{2}}}{\Gma{\frac{d}{2}}} $ & $\frac{2}{\alpha}\frac{E^{-\frac{d}{\alpha}}_0}{\Gma{\frac{d}{2}}} $ & Eq.~\eqref{eq:SpecMeasAssump} \\
$A^-_l$ & $\frac{\sbr{-1}^{\frac{d-l}{2}}\binom{d}{l}}{\sGma{\frac{d}{2}}\sbr{4\pi\gamma}^\frac{d}{2}}$ & $0$ & $0$ & Eq.~\eqref{eq:SpecMeasAssump} \\
$C_l$ & $i^l \binom{d}{l} \br{\frac{-i\hbar}{4\pi\gamma\tau}}^{\frac{d}{2}}$ & $\br{\frac{-i\hbar}{E_0\tau}}^{\frac{d}{2}}$ & $\frac{\Gma{1+\frac{d}{\alpha}}}{\Gma{1+\frac{d}{2}}} \br{\frac{-i\hbar}{E_0\tau}}^{\frac{d}{\alpha}}$ & Eq.~\eqref{eq:DefC} \\
\hline
Eq. & \eqref{eq:DefCTB} & \eqref{eq:NRA} & \eqref{eq:LevyA} &
\end{tabular}
\caption{
Table of the different coefficients in the tight-binding model (TB), the free particle (FP) and the L\'evy particle (L\'evy).
In the tight-binding model the detection state is a lattice site eigenstate $\sKet{\ensuremath{\psi_\text{d}}} = \sKet{\V{x}_\text{d}}$.
For the other two models the detection state is a Heisenberg state given by Eq.~\eqref{eq:NRDefPsi}.
$0 < \alpha < 2$ is the L\'evy parameter, see Eq.~\eqref{eq:LevyDisp}.
The last column lists where to find the definition of the quantity.
The last row lists, where the specific result is found in the main text.
In the continuous space models, we used Eq.~\eqref{eq:DefC} to compute the constants $C_l$ from the $A^\pm$'s.
In the tight-binding model, we also used Eq.~\eqref{eq:DefC} to obtain the $A^\pm$'s.\footnote{
Eq.~\eqref{eq:DefC} alone is not sufficient to determine both $A^+_l$ and $A^-_l$ from $C_l$, because it is one equation for two variables.
We employed the additional condition, that $A^\pm_l$ must be real from which we inferred that $A^\pm_l$ vanishes for some $l$.
A rigorous computation involves the Mellin transform of the MSDOS around one of its singular points and uses its representation as an integral over the Brillouin zone \cite{Arnold2012-1}:
\begin{equation*}
\mathcal{M}_l^\pm\sofff{f;s}
:=
\Int{0}{\infty}{\epsilon}
\epsilon^{s-1}
\Int{\ensuremath{\mathbb{B}}}{}{\V{k}}
\delta\sof{E^*_l \pm \epsilon - E\sof{\V{k}}}
\sAbs{\ensuremath{\psi_\text{d}}\sof{\V{k}}}^2
,
\end{equation*}
where $\ensuremath{\psi_\text{d}}\sof{\V{k}}$ is the momentum representation of $\sKet{\ensuremath{\psi_\text{d}}}$.
The delta function is easily resolved, and the remaining integrand is expanded up to second order around the critical points $\V{k}^*$ {\em of the energy surface} $E\sof{\V{k}}$ that correspond to the singular energy $E^*_l$, i.e. $E\sof{\V{k}^*} = E^*_l$.
The Mellin transform has several poles in the complex $s$-plane.
The pole with the largest real part lies at $s=d_S/2-1$ and determines the small $\epsilon$ behavior of the MSDOS.
The coefficients $A^\pm_l$, as well as possible logarithmic factors can be extracted from this pole.
This is possible for any translationally invariant system.
A full derivation will be carried out in another publication.
}
The tight-binding entry for $A^+_l$ is valid for even $l$ and zero otherwise.
The tight-binding entry for $A^-_l$ is valid for even $d-l$ and zero otherwise.
In even dimensions $A^-_0$ and $A^+_d$ vanish and furthermore logarithmic factors appear.
For the free particle and the L\'evy particle there is only one singular point with $l=0$.
\label{tab:Constants}
}
\end{table}
In this special case the return amplitudes are Bessel functions of the first kind.
This can again be seen from an integral over the Brillouin zone using the dispersion relation Eq.~\eqref{eq:TBDispersion}:
\begin{equation}
u_n
=
\frac{a}{2\pi}
\Int{-\frac{\pi}{a}}{\frac{\pi}{a}}{k}
e^{-i 2n\tfrac{\gamma\tau}{\hbar}\br{1 - \cos\sof{ak}}}
=
e^{-i2n\tfrac{\gamma\tau}{\hbar}}
\BesselJ{0}{2n\tfrac{\gamma\tau}{\hbar}}
,
\label{eq:RATB1D}
\end{equation}
where we first used the replacement $2\sin^2\sof{x/2} = 1 - \cos x$, and then the integral representation of the Bessel function.
In higher dimensions of the tight-binding model, the integral over the Brillouin zone factorizes and we find:
\begin{equation}
u_n
=
\Prod{j=1}{d}
\frac{a}{2\pi}
\Int{-\frac{\pi}{a}}{\frac{\pi}{a}}{k_j}
e^{-i n\tfrac{4\gamma\tau}{\hbar}\sin^2\tfrac{ak_j}{2}}
=
\brr{
e^{-in\tfrac{2\gamma\tau}{\hbar}}
\BesselJ{0}{\tfrac{2\gamma\tau}{\hbar}n}}^d
\label{eq:RATB}
\end{equation}
An alternative integral representation of the Bessel function [Eq.~8.411(10) of \cite{Gradshteyn2007-0}] reveals the Bessel function as the Fourier transform of the arcsin law.
This allows us to use Eq.~\eqref{eq:TBSM} directly in Eq.~\eqref{eq:RA} in the 1d case:
\begin{equation}
u_n
=
\frac{1}{\pi}
\Int{0}{4\gamma}{E}
\frac{e^{-in\frac{\tau E}{\hbar}}}{\sqrt{\sbr{4\gamma - E}E}}
=
e^{-i\tfrac{2\gamma\tau}{\hbar}n}
\BesselJ{0}{\tfrac{2\gamma\tau}{\hbar}n}
,
\label{eq:}
\end{equation}
where the variable change $E = 2\gamma(1+x)$ has to be used to recover the reference's formula.
We plot the MSDOS of the tight-binding model for two and three dimensions in Fig.~\ref{fig:SM}(a-b).
\subsection{The wrapped MSDOS}
In Eq.~\eqref{eq:RA}, we expressed the return amplitudes in terms of $f\sof{E}$.
The same can be done to the resolvent [using Eq.~\eqref{eq:RA}, Eq.~\eqref{eq:DefResolv} and the geometric series]:
\begin{equation}
u\sof{z}
=
\Int{0}{\infty}{E}
\frac{f\sof{E}}{1 - z e^{-i \frac{\tau E}{\hbar}}}
=
\frac{1}{2\pi} \Int{0}{2\pi}{\lambda'}
\frac{\mu\sof{\lambda'}}{1 - ze^{-i\lambda'}}
.
\label{eq:DefResolvWMSDOS}
\end{equation}
Since the complex exponential in the denominator is periodic, it makes sense to gather all contributions of $f\sof{E}$ with the same phase.
The result is the ``wrapped MSDOS'' (WMSDOS):
\begin{equation}
\mu\of{\lambda}
:=
\frac{2\pi \hbar}{\tau}
\Sum{m=-\infty}{\infty}
f\of{\tfrac{\hbar}{\tau}\sbrr{\lambda+2\pi m}}
.
\label{eq:DefWMSDOS}
\end{equation}
$\mu\sof{\lambda}$ can be understood as ``$f\sof{E}$ wrapped around the unit circle''.
It is actually the spectral measure of the evolution operator associated with $\sKet{\ensuremath{\psi_\text{d}}}$.
It is normalized according to $(2\pi)^{-1}\sInt{0}{2\pi}{\lambda}\mu\sof{\lambda} = \sBK{\ensuremath{\psi_\text{d}}}{\ensuremath{\psi_\text{d}}} = 1$.
Example plots of $\mu\sof{\lambda}$ can be found in the insets of Fig.~\ref{fig:SM}.
In the mathematical literature, Eq.~\eqref{eq:DefResolvWMSDOS} is called the Cauchy transform of the measure $\mu\of{\lambda}\mathrm{d} \lambda$ \cite{Cima2006-0}.
In contrast to the series definition, Eq.~\eqref{eq:DefResolv}, the integral representation is also valid for $\sAbs{z} > 1$.
A system with an infinite energy band, like a free particle in continuous space, will always have infinitely many terms in the sum of Eq.~\eqref{eq:DefWMSDOS}.
For a system with a finite energy band, like the tight-binding model, most of the terms in Eq.~\eqref{eq:DefWMSDOS} will be zero, because they are outside the support of $f\sof{E}$.
The support of $f\sof{E}$ gets stretched or compressed by a factor $\tau/\hbar$ before it is wrapped onto the interval $[0,2\pi]$.
At certain critical values of $\tau$ a new term will appear in the sum of Eq.~\eqref{eq:DefWMSDOS}.
To better understand Eq.~\eqref{eq:DefWMSDOS}, consider the one dimensional tight-binding model again.
For very small values of $\tau$, smaller than the critical value:
\begin{equation}
\tau_c
:=
\frac{2\pi\hbar}{4\gamma}
=
\frac{\pi\hbar}{2\gamma}
,
\label{eq:TBDefCritTau}
\end{equation}
which is set by the width of the energy band, $4\gamma$ [see Eq.~\eqref{eq:TBDispersion}], the support of $\mu\sof{\lambda}$ is actually smaller than $2\pi$ and $\mu\sof{\lambda}$ is just a rescaled version of $f\sof{E}$.
Additional terms appear in $\mu\sof{\lambda}$, as soon as $\tau$ surpasses a multiple of the critical value $\tau_c$.
Assume that $\sbr{n-1}\tau_c < \tau < n\tau_c$ for some positive integer $n$, then we obtain by combining Eq.~\eqref{eq:DefWMSDOS} and Eq.~\eqref{eq:TBSM}:
\begin{equation}
\mu\sof{\lambda}
=
\Sum{m=0}{n}
\frac{2}{\sqrt{
\sbr{\lambda+2\pi m}
\br{\tfrac{4 \gamma\tau}{\hbar} - \lambda - 2\pi m}
}}
.
\label{eq:TBWMSDOS}
\end{equation}
We see that $f\sof{E}$'s singularities are inherited by $\mu\sof{\lambda}$.
The critical behavior of $\mu\sof{\lambda}$ around its singular points translates to the behavior of the resolvent $u\sof{re^{i\lambda}}$ close to the unit circle, i.e. in the limit $r\to1^-$.
As we show in Appendix~\ref{app:Plemelj}, the chain of definitions for $f\sof{E}$ from Eq.~\eqref{eq:SpecMeasAssump} can be traced forward to $u\sof{e^{i\lambda}}$, in order to find the singularities in the resolvent.
We summarize this behavior in the following equation:
\begin{equation}
u\sof{e^{i\sbr{\Lambda^*_l \pm\epsilon}}}
\sim
\tilde{u}_l\sof{\pm\epsilon}
+
B^\pm_l\epsilon^{\frac{d_S}{2}-1}
,
\label{eq:AsymResolv}
\end{equation}
where the notation is similar to Eq.~\eqref{eq:SpecMeasAssump}.
The constants $B^\pm_l$ and $A^\pm_l$ are related, as we will show later in Eqs.~\eqref{eq:DefC} and \eqref{eq:DefCTB}, where both are computed from information about $u_n$.
The particular way how one obtains these constants -- from $f\sof{E}$, or from $u_n$ -- is a matter of convenience.
The wrapping procedure Eq.~\eqref{eq:DefWMSDOS} shifts the positions of the singularities from $E^*_l$ to:
\begin{equation}
\Lambda^*_l
:=
\frac{E^*_l \tau}{\hbar}
\mod 2\pi
.
\label{eq:DefSingLambdas}
\end{equation}
For the 1d tight-binding model with localized detection state, these are the points
\begin{equation}
\Lambda^*_0 = 0, \qquad \Lambda^*_1 = \frac{4\gamma\tau}{\hbar} \mod 2\pi
.
\label{eq:}
\end{equation}
For special choices of $\tau$, two singular energies $E^*_l$ and $E^*_{l'}$ become equivalent:
\begin{equation}
\tau_c^{(l,l')}
:=
\frac{2 \pi \hbar}{\sAbs{E^*_l - E^*_{l'}}}
.
\label{eq:DefCritTau}
\end{equation}
These are the critical sampling periods \cite{Thiel2018-0}, and we will later show that such choices of $\tau$ yield special behavior of the first detection probabilities.
In the tight-binding model these are:
\begin{equation}
\tau_c^{(l)}
=
\frac{\pi\hbar}{2l\gamma}
,
\label{eq:TBDefCritTaus}
\end{equation}
for $l \in \sbrrr{1,\hdots,d}$.
At these critical detection periods, two or more singularities of $f\sof{E}$ get mapped to {\em one} singularity of $\mu\sof{\lambda}$.
The number $L$ of singularities of $\mu\sof{\lambda}$ is then smaller than $L'$, which is the number of singularities in $f\sof{E}$.
In the one dimensional tight-binding model, this is the already encountered critical value from Eq.~\eqref{eq:TBDefCritTau}.
However, $\mu\sof{\lambda}$'s singularities exhibit exactly the same power laws as $f\sof{E}$.
\subsection{Singularities in the generating function}
We conclude this section with identifying the singularities in the generating function of the detection amplitudes evaluated on the unit circle.
This is done by taking the limit $r\to1^-$ in Eq.~\eqref{eq:GenFunc} and using Eq.~\eqref{eq:AsymResolv}.
The singularities of $\varphi\sof{e^{i\lambda}}$ are the points $\Lambda^*_l$ defined by Eq.~\eqref{eq:DefSingLambdas}.
Close to these points, we have:
\begin{equation}
\varphi\sof{e^{i\sbr{\Lambda^*_l+\epsilon}}}
\sim
1 -
\frac{1}{
\tilde{u}_l\sof{\pm\epsilon}
+
B^\pm_l\epsilon^{\frac{d_S}{2}-1}
}
.
\end{equation}
We find a competition of terms in the denominator.
Depending on the value of the spectral dimension, either the power term or the analytic remainder dominates.
For $d_S\le2$ the analytic remainder $\tilde{u}_l$ is zero, while for $d_S >2$ it constitutes the leading order.
Performing the small-$\epsilon$ expansion, we find that $\varphi\sof{e^{i\lambda}}$'s singularity is always in one of its derivatives.
There is a crossover at the critical dimension $d_S = 2$:
\begin{equation}
\varphi\sof{e^{i\sbr{\Lambda^*_l\pm\epsilon}}}
\sim
\left\{ \begin{aligned}
1 - \frac{
\epsilon^{2-\frac{d_S}{2}-1}
}{
B^\pm_l
}, & \qquad d_S < 2, \\
\tilde{\varphi}_l\sof{\pm\epsilon}
+
\frac{
B^\pm_l
}{
\brr{ u\sof{e^{i\Lambda^*_l}} }^2
}
\epsilon^{\frac{d_S}{2}-1}, & \qquad d_S > 2
\end{aligned} \right.
.
\label{eq:AsymIntegrand}
\end{equation}
When $d_S$ is an even integer, logarithmic corrections appear.
This case is discussed in Appendix \ref{app:Even}.
An analytical remainder of $\varphi\sof{e^{i\lambda}}$ is always present (although it is trivial for small dimensions).
Interestingly, we see that, in higher dimensions, additional constants appear.
They are the derived from the return amplitudes:
\begin{equation}
u\sof{e^{i\Lambda^*_l}}
=
\Sum{n=0}{\infty}
e^{in\Lambda^*_l}
u_n
.
\label{eq:DefAddConstants}
\end{equation}
These series converge for $d_S>2$.
As we have mentioned, $u_n$ and $f\sof{E}$ as well as $\varphi_n$ and $\varphi\sof{e^{i\lambda}}$ are Fourier pairs.
In the next section, we apply an asymptotic formula for Fourier transforms to relate Eq.~\eqref{eq:AsymIntegrand} with the large $n$ behavior of $\varphi_n$.
After that, we apply the same formula to Eq.~\eqref{eq:SpecMeasAssump}.
\section{Using a Fourier-Tauber formula}
\label{sec:LargeN}
According to Refs.~\cite{Erdelyi1956-0,Gamkrelidze1989-0}, the singular points of $\varphi\sof{e^{i\lambda}}$ and the power-law behavior around these points determine the large $n$ asymptotics of its Fourier coefficients, which are the first detection amplitudes $\varphi_n$.
This is basically the Fourier analogue to the Tauberian theorems for the Laplace transform well-known in the theory of random walks \cite{Feller1971-0,Klafter2011-0}.
The notable difference is that in the classical setup, there usually is only one singular point at vanishing Laplace variable with the consequence that the first passage probability decays monotonically in the long-time limit.
This condition is violated in our case, as there are in general multiple singularities in $\varphi\sof{e^{i\lambda}}$.
This fact is clearly related to the presence of quantum interference.
Nevertheless, each of them can be isolated and the Tauberian theorem can be applied from both sides of the singular points.
This leads to a sum of different power-law terms accompanied with a complex exponential factor in $n$.
A derivation of the formula is given in Appendix~\ref{app:Tauber}, while rigorous proofs are found in the above cited references.
The main statement is the following:
Let $h\sof{x}$ be a function with $L$ singularities at $x_l^*$, each admitting an expansion like Eq.~\eqref{eq:SpecMeasAssump}:
\begin{equation}
h\sof{x^*_l\pm\epsilon}
\sim
\tilde{h}_l\sof{\pm\epsilon}
+
H^\pm_l \epsilon^{\nu-1}
,
\label{eq:HAssump}
\end{equation}
for some $l$-independent $\nu>0$.
Then, its Fourier transform behaves for large $n$ like:
\begin{equation}
\frac{1}{2\pi}\Int{0}{2\pi}{x} e^{-inx} h\sof{x}
\sim
\frac{\Gma{\nu}}{2\pi n^{\nu}} \Sum{l=0}{L-1} e^{-inx^*_l} \brr{
\frac{H^+_l}{i^{\nu}}
+
\frac{H^-_l}{\sbr{-i}^{\nu}}
}
.
\label{eq:HResult}
\end{equation}
In our prior publication \cite{Friedman2017-0,Friedman2017-1,Thiel2018-0} the large $n$ behavior was inferred from integrals along branch cuts in the complex plane.
The same procedure would be viable here.
In fact, each singular point $e^{i\Lambda^*_l}$ corresponds to a branch point of $\varphi\sof{z}$.
Instead of writing $\varphi_n$ as a Fourier transform of $\varphi\sof{e^{i\lambda}}$ one could put the branch cuts of $\varphi\sof{z}$ along rays to complex infinity and integrate around them.
However, the connection to the energy properties is clearer using the Fourier-Tauber theorem.
Since $\varphi\sof{e^{i\lambda}}$ admits the expansion Eq.~\eqref{eq:AsymIntegrand} around each of the singular points $\Lambda^*_l$, one finds that $\varphi_n$ behaves for large $n$ like:
\begin{equation}
\varphi_n
\sim
\Sum{l=0}{L-1}
\frac{e^{-i n \Lambda^*_l}}{2\pi}
\times
\left\{ \begin{aligned}
\tfrac{\sGma{2-\frac{d_S}{2}}}{n^{2-\frac{d_S}{2}}}
\brr{
\tfrac{i^{\frac{d_S}{2}}}{B^+_l}
+
\tfrac{\sbr{-i}^{\frac{d_S}{2}}}{B^-_l}
}
, & \; d_S < 2 \\
\tfrac{\sGma{\frac{d_S}{2}}}{\sbrr{u\sof{e^{i\Lambda^*_l}}}^2 n^{\frac{d_S}{2}}}
\brr{
\tfrac{B^+_l}{i^{\frac{d_S}{2}}}
+
\tfrac{B^-_l}{\sbr{-i}^{\frac{d_S}{2}}}
}
, & \; d_S > 2
\end{aligned} \right.
\label{eq:AsymFDA1}
\end{equation}
The squared absolute value of this expression is the desired first detection probability and reproduces Eq.~\eqref{eq:AsymFDP}.
There, we hid most of the constants in the complex numbers $F_{l,d_S}$ which are now made explicit.
Eq.~\eqref{eq:AsymFDP} [and Eq.~\eqref{eq:AsymFDA1}] is our main result and conveys the following qualitative properties:
The first detection probability decays like a power law that only depends on the spectral dimension.
The decay exponent exhibits a crossover at the critical dimension two and it is exactly double the exponent from the first passage probability of classical random walks \cite{Redner2007-0}.
The frequencies of the oscillations are determined by the positions $E^*_l$ of the singularities of the MSDOS via Eq.~\eqref{eq:DefSingLambdas}.
Eq.~\eqref{eq:AsymFDP} is valid for systems with a continuous energy spectrum.
In the classical theory, the spectral dimension can always be identified with the exponent $d_S^\text{DOS}$ in $\rho\sof{E}$.
It is hence a property of the Hamiltonian alone, independent of the initial or detection state.
Such an identification is possible for ordinary detection states whose MSDOS $f\sof{E}$ behaves sufficiently smoothly around the van Hove singularities and also does not vanish at these points.
This is basically a condition on the overlap of $\sKet{\ensuremath{\psi_\text{d}}}$ with certain energy eigenstates, and has been used in a modified form also in \cite{Li2017-0}.
A notable class of exceptions are those states that have no overlap with the eigenstates of the singular energies.
We refer to them as ``insufficiently populated'' states.
Consider the 1d tight-binding model with the detection state $\sKet{\ensuremath{\psi_\text{d}}} = \sbr{\sKet{a} - \sKet{-a}}/\sqrt{2}$.
Repeating the computations that led to Eq.~\eqref{eq:TBSMCounterEx1}, we find the MSDOS for this state to be:
\begin{equation}
f\sof{E}
=
\frac{1}{4\pi\gamma^2}\sqrt{E\sbr{4\gamma-E}}
.
\label{eq:TBSMCounterEx2}
\end{equation}
Although the density of states is an arcsin law, $f\sof{E}$ is a semicircle law.
The spectral dimension in this case is $d_S = 3$ whereas $d_S^\text{DOS} = 1$!
The reason is that $\sKet{\ensuremath{\psi_\text{d}}}$ was chosen to have no momentum components with $k=0$ and $k=\pi/a$, which correspond to the singular energies.
Hence, this $\sKet{\ensuremath{\psi_\text{d}}}$ is insufficiently populated around the singular energies, leading to a discrepancy between $\rho\of{E}$'s and $f\sof{E}$'s spectral dimension.
Another choice would be a state with wave vector representation supported only in the interval $[\pi/(2a),3\pi/(4a)]$.
This example would even have different singular energies, because the support of $f\sof{E}$ lies in $[2\gamma,4\gamma\sin^2\sof{3\pi/8}]$.
Yet another exception is a heavy-tailed detection state that decays like $\sBK{x}{\ensuremath{\psi_\text{d}}} \sim \sAbs{x}^{-1-\nu}$, for some $\nu \in [0,2]$.
In momentum space, such a detection state will behave like $\sAbs{k}^\nu$ around the origin.
This leads to a spectral dimension $d_S = 1+\nu$ as opposed to $d_S^\text{DOS} = 1$.
All these cases show that the spectral dimension and the singular points, that we use in this article, are strictly speaking properties of $f\sof{E}$ and not of $\rho\sof{E}$.
In many important cases, however, the positions of the singularities coincide in both, as do the power-law exponents.
(Although the prefactors $A^\pm_l$ found in $f\sof{E}$ may be different from those found in $\rho\sof{E}$.)
This is what we called ordinary.
The classical theory does not know of insufficiently populated states.
All eigenstates of the Hamiltonian/Laplacian are extended, due to the continuous nature of the spectrum, and have support in every lattice site.
Hence, each lattice site has overlap with every energy, in particular with the singular ones.
The only possible exception is a non-ergodic system that splits up into separate pieces.
As a superposition of lattice states with negative interference in some energy is our of question due to positivity of probabilities, it is not possible to construct an insufficiently populated state in the classical first passage theory.
The coefficients $B^\pm_l$ are often not easy to obtain.
Knowledge about the return amplitudes $u_n$ opens up an alternate way to compute $\varphi_n$.
We remember that $f\sof{E}$ and $u_n$ are also a Fourier pair and that we have access to $f\sof{E}$'s singularities in Eq.~\eqref{eq:SpecMeasAssump}.
Therefore, the Fourier-Tauber theorem is now applied to Eq.~\eqref{eq:RA} using Eq.~\eqref{eq:SpecMeasAssump}.
The result is:
\begin{equation}
u_n
\sim
\frac{1}{n^{\frac{d_S}{2}}}
\Sum{l=0}{L-1}
C_l
e^{-in\Lambda^*_l}
,
\label{eq:RAAsym}
\end{equation}
where
\begin{equation}
C_l
:=
\Gma{\frac{d_S}{2}}
\br{\frac{\hbar}{\tau}}^{\tfrac{d_S}{2}}
\Sum{l'\sim l}{}
\brr{
\frac{
A^+_{l'}
}{
i^{\frac{d_S}{2}}
}
+
\frac{
A^-_{l'}
}{
\sbr{-i}^{\frac{d_S}{2}}
}
}
.
\label{eq:DefC}
\end{equation}
In Eq.~\eqref{eq:RAAsym} we have already identified the critical angles $\Lambda^*_l = \tau E^*_l/\hbar$ from Eq.~\eqref{eq:DefSingLambdas}.
The sum in Eq.~\eqref{eq:DefC} runs over all equivalent energies, i.e. $e^{i\tau E^*_{l'}/\hbar} = e^{i\Lambda^*_l}$.
Remarkably, the return amplitudes oscillate with the same frequencies as do the detection amplitudes, which is also true in the above discussed exceptions, because $u_n$ and $\varphi_n$ are related to $f\sof{E}$ (and not to $\rho\sof{E}$).
The relation between the MSDOS and the time decay of the return amplitudes is known in the literature \cite{Marchetti2012-0}.
In Refs.~\cite{Muelken2006-0,Muelken2011-0} a similar argument was used to relate the spectral dimension to the decay of the return amplitudes.
However, in these references the DOS was used instead of the MSDOS.
Also the role of the van Hove singularities was under-appreciated.
Therefore the authors were only able to predict the decay of the envelope of $u_n$ but not the oscillations.
Often the matrix elements of the evolution operator are much more accessible than the MSDOS.
Consequently the coefficients $C_l$ may be more easily available than the $B^\pm_l$'s.
Therefore, we provide this alternative approach to derive $\varphi_n$.
For example: In Eq.~\eqref{eq:RATB}, we found the return amplitudes for the tight-binding model in arbitrary dimension.
Application of the asymptotic formula $\sBesselJ{0}{x} \sim \cos\sof{x - \pi/4} \sqrt{2/(\pi x)}$, yields:
\begin{align}
u_n
\sim & \nonumber
e^{-i\frac{2d\gamma\tau}{\hbar}n}
\br{\frac{\hbar}{\pi \gamma\tau n}}^{\frac{d}{2}}
\cos^d\of{\frac{2\gamma\tau}{\hbar}n - \frac{\pi}{4}}
\\ = &
\br{\frac{\hbar}{4\pi \gamma\tau n}}^{\frac{d}{2}}
\Sum{l=0}{d}
\binom{d}{l}
e^{-i\frac{4l\gamma\tau}{\hbar}n + i \sbr{2l-d}\frac{\pi}{4}}
.
\end{align}
From here, one identifies:
\begin{equation}
C_l
=
\br{\frac{\hbar}{4\pi\gamma\tau}}^{\frac{d}{2}}
\binom{d}{l}
e^{i\sbr{2l-d}\frac{\pi}{4}}
, \quad
\Lambda^*_l
=
\frac{4\gamma\tau}{\hbar} l
,
\label{eq:DefCTB}
\end{equation}
with $l\in\sbrrr{0,1,\hdots,d}$ for the tight-binding model.
The above equation is correct, provided that $\tau$ does not assume a critical value.
In that case, two singular points merge and the corresponding $C_l$'s have to be added.
The points $4\gamma l$ are exactly the van Hove singularities of the density of states, see Fig.~\ref{fig:SM}(a,b).
Multiplication of Eq.~\eqref{eq:RAAsym} with $r^n e^{i\lambda n}$, summing over $n$, and taking the limit $r\to1^-$, gives us the resolvent $u\sof{e^{i\lambda}}$.
Close to the critical points $\Lambda^*_l$ it behaves as:
\begin{align}
u\of{e^{i\sbr{\Lambda^*_l\pm\epsilon}}}
\sim & \nonumber
\tilde{u}_l\sof{\pm\epsilon}
+
\Gma{1-\tfrac{d_S}{2}}
C_l
\br{1-e^{\pm i\epsilon}}^{\frac{d_S}{2}-1}
\\ \sim &
\tilde{u}_l\sof{\pm\epsilon}
+
\Gma{1-\tfrac{d_S}{2}}
C_l
\sbr{\mp i\epsilon}^{\frac{d_S}{2}-1}
.
\label{eq:}
\end{align}
The last line was obtained by taking $\sbr{1-e^{ix}}^\nu \sim \sbr{-ix}^\nu$.
Comparing this equation with \eqref{eq:AsymResolv} allows us to relate the different coefficients:
\begin{equation}
B^\pm_l
=
\Gma{1-\tfrac{d_S}{2}}
e^{\mp i \tfrac{\pi\sbr{d_S-2}}{4}}
C_l
.
\label{eq:Coeffs1}
\end{equation}
$B^\pm_l$ and $A^\pm_l$ can be related via Eq.~\eqref{eq:DefC}.
Plugging the result into Eq.~\eqref{eq:AsymFDA1}, we can express the detection amplitudes in terms of the decay behavior of the return amplitudes (i.e. in terms of $C_l$):
\begin{equation}
\varphi_n
\sim
\left\{ \begin{aligned}
\frac{\br{1-\frac{d_S}{2}}\sin\of{\frac{\pi d_S}{2}}}{\pi n^{2-\frac{d_S}{2}}}
\Sum{l=0}{L-1}
\frac{e^{-i n \Lambda^*_l}}{C_l}
, & \quad d_S < 2 \\
\frac{1}{n^{\frac{d_S}{2}}}
\Sum{l=0}{L-1}
\frac{C_l}{\sbrr{u\sof{e^{i\Lambda^*_l}}}^2}
e^{-i n \Lambda^*_l}
, & \quad d_S > 2
\end{aligned} \right.
.
\label{eq:AsymFDA2}
\end{equation}
We used $\Gma{x}\Gma{1-x} = \pi/\sin\sof{x\pi}$.
Alternatively, one could express the detection amplitudes in terms of the MSDOS $f\sof{E}$ (that is in terms of $A^\pm_l$) using Eq.~\eqref{eq:DefC}.
For $d_S < 2$ one obtains:
\begin{equation}
\varphi_n
\sim
-\frac{1}{\pi^2n^2} \br{\frac{n\tau}{\hbar}}^{\frac{d_S}{2}}
\Sum{l=0}{L-1}
\frac{
\Gma{2-\tfrac{d_S}{2}}
\sin\of{\tfrac{\pi d_S}{2}}
e^{-i n \Lambda^*_l}
}{
\sSum{l'\sim l}{}
A^+_{l'} \sbr{-i}^{\frac{d_S}{2}}
+ A^-_{l'} i^{\frac{d_S}{2}}
}
.
\label{eq:AsymFDA3a}
\end{equation}
For $d_S > 2$ one obtains:
\begin{equation}
\varphi_n
\sim
\frac{\Gma{\frac{d_S}{2}}}{\br{\frac{n\tau}{\hbar}}^{\frac{d_S}{2}}}
\Sum{l'=0}{L'-1} \frac{e^{-i n\frac{\tau E^*_{l'}}{\hbar}}}{u^2\sof{e^{i\frac{\tau E^*_{l'}}{\hbar}}}}
\brr{
A^+_{l'}\sbr{-i}^{\frac{d_S}{2}}
+
A^-_{l'} i^{\frac{d_S}{2}}
}
.
\label{eq:AsymFDA3b}
\end{equation}
Eqs.~(\ref{eq:AsymFDA2}-\ref{eq:AsymFDA3b}) complement Eq.~\eqref{eq:AsymFDA1}.
They are useful when neither the MSDOS $f\sof{E}$, nor an expansion of the return amplitudes, are available.
Using Eq.~\eqref{eq:DefCTB} in Eq.~\eqref{eq:AsymFDA2} one obtains the detection amplitudes for the tight-binding model for dimensions larger than $2$:
\begin{equation}
\varphi_n
\sim
\br{\frac{\hbar}{4\pi \gamma\tau n}}^{\frac{d}{2}}
\Sum{l=0}{d}
\binom{d}{l}
\frac{
e^{-i\frac{4l\gamma\tau}{\hbar} n + i \sbr{2l-d}\tfrac{\pi}{4}}
}{
\sbrr{u\sof{e^{i\frac{4l\gamma\tau}{\hbar}}}}^2
}
.
\label{eq:TBFDA}
\end{equation}
The modulus squared of this expression is the first detection probability:
\begin{equation}
F_n
\sim
\br{\frac{\hbar}{4\pi \gamma\tau n}}^{d}
\Abs{
\Sum{l=0}{d}
\binom{d}{l}
\frac{
e^{i\sbr{d-2l}\br{\frac{2\gamma\tau}{\hbar} n -\tfrac{\pi}{4}} - 2i\arg{u\sof{e^{i\frac{4l\gamma\tau}{\hbar}}}}}
}{
\sAbs{u\sof{e^{i\frac{4l\gamma\tau}{\hbar}}}}^2
}
}^2
.
\label{eq:TBFDPHighD}
\end{equation}
From the expression of $u_n$, Eq.~\eqref{eq:RATB}, and the definition of $u\sof{z}$, one can infer that $u\sof{e^{i4l\gamma\tau/\hbar}}$ is the complex conjugate of $u\sof{e^{i
|
4\sbr{d-l}\gamma\tau/\hbar}}$.
Hence, for each term in the sum of Eq.~\eqref{eq:TBFDPHighD}, its complex conjugate appears as well.
The complex exponentials can actually be replaced by cosines and half of the terms in the sum can be dropped.
The constants $u\sof{e^{i4l\gamma\tau/\hbar}}$ are defined by Eq.~\eqref{eq:DefAddConstants}, no closed form is known to us, and so they have to be computed numerically.
Eq.~\eqref{eq:TBFDPHighD} holds for all dimensions larger than two.
The one dimensional tight-binding model has already been discussed extensively in Refs.~\cite{Friedman2017-0,Friedman2017-1,Thiel2018-0}.
There, the following formula was reported:
\begin{equation}
F_n
\sim
\frac{4\gamma\tau}{\hbar \pi n^3}
\cos^2\of{\frac{2\gamma\tau}{\hbar} n + \frac{\pi}{4}}
,
\label{eq:TBFDP1D}
\end{equation}
which is perfect accordance with Eq.~\eqref{eq:AsymFDA2}.
Curiously, in both the one- and three dimensional case, one finds a power law with exponent $-3$.
In the two dimensional case logarithmic corrections to the power law appear.
This case is discussed in Appendix~\ref{app:Even}.
We simply state here the result for the 2d tight-binding model:
\begin{equation}
F_n
\sim
\br{ \frac{ 4\pi\gamma\tau}{\hbar n \ln^2 n}}^2
\Abs{
\frac{1}{2}
-
2\sin\of{\frac{4\gamma\tau}{\hbar} n}
}^2
.
\label{eq:TBFDP2D}
\end{equation}
In Eq.~\eqref{eq:TBFDP1D} and Eq.~\eqref{eq:TBFDP2D} the Zeno effect is visible.
When the limit $\tau\to0$ is taken, our asymptotic result vanishes.
From Eq.~\eqref{eq:QuantumRenewal} and Eq.~\eqref{eq:RATB}, one finds that actually $F_n \to \delta_{n,1}$.
The physical meaning is that the particle is found at the detection site, immediately after the experiment commences.
Hence the long-time asymptotics of $F_n$ vanish.
The expression in Eq.~\eqref{eq:TBFDPHighD}, however, diverges for fixed $n$ in the limit $\tau\to0$!
(This can be seen from numerical evaluation of $u\sof{e^{i4l\gamma\tau/\hbar}}$.)
In the numerical simulations of $F_n$, an intermediary regime appears in $F_n$ that is not described by our asymptotic formula.
As $\tau$ decreases, this intermediary region grows in size.
The $F_n$ values in that regime go to zero as $\tau$ decreases.
At the same time, $F_1$ goes to unity, so that in total $F_n\to \delta_{n,1}$, and the Zeno effect is restored.
It can not be observed in our asymptotic formula, though.
Numerical simulations were performed by using Eq.~\eqref{eq:RATB} in Eq.~\eqref{eq:QuantumRenewal} and solving the resulting system of linear equations for $\varphi_n$.
In Fig.~\ref{fig:TB_FDP}, we present numerical simulations for the tight-binding model in dimensions two and three for $\tau = 0.25\hbar/\gamma$.
In the two-dimensional case, we fitted the envelope of the asymptotic result \eqref{eq:TBFDP2D}.
Due to the slow logarithmic convergence it is necessary to replace $\ln^2n$ by $\sbr{\ln n + x}^2$.
$x$ is determined from a fit.
Both expressions are equivalent in the asymptotic limit.
The figure depicts the fit.
One clearly sees the expected dimension dependent power-law decay of Eq.~\eqref{eq:AsymFDP} in the envelope of $F_n$ as well as the oscillations, and an overall good agreement with our prediction.
\begin{figure*}
\includegraphics[width=0.99\textwidth]{TBFDP.pdf}
\caption{
Probability of first detected return for the tight-binding model in a semi-logarithmic plot.
Blues squares are numerical results, orange circles are the predictions of Eq.~\eqref{eq:TBFDP2D} and Eq.~\eqref{eq:TBFDPHighD}.
(a): Two dimensional case with $\tau = 0.25\hbar/\gamma$.
The $n^{-2}$ power-law and the oscillations are clearly visible.
The logarithmic factor was adjusted to $(\pi + \ln n)^{-4}$, see Appendix~\ref{app:Even}.
Inset: Two dimensional case for the critical detection period $\tau = \tau_c = (\pi/2)\hbar/\gamma$.
No oscillations are present here, because $\mu\sof{\lambda}$ has only one singular point.
We find a $n^{-2}$ power law with logarithmic factor $(x + \ln n)^{-4}$, where $x \approx 10.531$, see Appendix~\ref{app:Even}.
(b): Three dimensional case for $\tau = 0.25 \hbar/\gamma$.
Inset: $d=3$ with $\tau = \tau_c = \sbr{\pi/2}\hbar/\gamma$.
Oscillations are only present for non-critical $\tau$.
No fitting was applied to the right-hand side plots.
\label{fig:TB_FDP}
}
\end{figure*}
As discussed, the frequency of the oscillations is given by Eq.~\eqref{eq:DefSingLambdas} and can be controlled via the detection period $\tau$.
This is demonstrated in Fig.~\ref{fig:TB_FFT}, where we plotted the power spectrum of $n^3 F_n$ for $d=3$ and $\tau =0.15\hbar/\gamma$.
(The power spectrum is the modulus squared of the discrete Fourier transform of $n^3 F_n$.)
This makes it possible to visualize the characteristic frequencies.
The frequencies $\Lambda^*_l = 0.6 l$, $l\in\sbrrr{0,1,2,3}$ are visible as peaks in the power spectrum of $n^3 F_n$.
\begin{figure}
\includegraphics[width=0.99\columnwidth]{TBFFT.pdf}
\caption{
Power spectrum of $n^3 F_n$ for $d=3$ and $\tau=0.15\hbar/\gamma$.
Stripping off the power law off of $F_n$ by the multiplication with $n^3$ exposes the oscillatory terms.
The frequencies of the oscillations correspond to peaks in the Fourier transform.
Compare the position of the peaks with the prediction of equation Eq.~\eqref{eq:DefSingLambdas}, $\Lambda^*_l = 0.6 l$, $l=0, 1, 2, 3$.
\label{fig:TB_FFT}
}
\end{figure}
As another demonstration, we present the case of a critical detection amplitude, $\tau_c = (\pi/2) \hbar/\gamma$ from Eq.~\eqref{eq:TBDefCritTau}.
For this value of $\tau$, all critical energies become equivalent and get mapped to the same critical point $\Lambda^*$.
As a consequence the oscillations in $F_n$ disappear and only the power law remains.
This is shown in the insets of Fig.~\ref{fig:TB_FDP}.
Finally, we want to illustrate our discussion on insufficiently populated states and we want to showcase the dependence of the spectral dimension on the detection state.
Therefore, we consider the two dimensional tight-binding model with the detection state:
\begin{equation}
\sKet{\ensuremath{\psi_\text{d}}}
=
\frac{1}{2} \brr{
\sKet{\sbr{a,a}}
+
\sKet{\sbr{-a,-a}}
-
\sKet{\sbr{a,-a}}
-
\sKet{\sbr{-a,a}}
}
.
\label{eq:PsiDetSpecial}
\end{equation}
By construction it has no overlap with the singular energies $E^*_l = 0, 4\gamma, 8\gamma$.
The corresponding MSDOS is a convolution of Eq.~\eqref{eq:TBSMCounterEx2} with itself.
This is a convolution of two MSDOS's with $d_S = 3$ each, resulting in $d_S = 6$.
$f\sof{E}$ is presented in Fig.~\ref{fig:TBSpecial}(a) and does not resemble Fig.~\ref{fig:SM}(a) at all.
For this choice, the return amplitudes are given by:
\begin{equation}
u_n
=
e^{-i\frac{4\gamma\tau}{\hbar}n}
\brr{
\BesselJ{0}{\frac{2\gamma\tau}{\hbar}}
+
\BesselJ{2}{\frac{2\gamma\tau}{\hbar}}
}^2
\label{eq:}
\end{equation}
and decay like $n^{-3}$.
The spectral dimension in $f\sof{E}$ is $d_S = 6$ instead of $d_S^\text{DOS}=2$ found in the DOS!
The simulations depicted in Fig.~\ref{fig:TBSpecial}(b) reflect this fact nicely.
\begin{figure*}
\includegraphics[width=0.99\textwidth]{TBSpecial.pdf}
\caption{
Two dimensional tight-binding model with the special detection state of Eq.~\eqref{eq:PsiDetSpecial}.
(a) The MSDOS $f\sof{E}$.
It is wildly different from the DOS $\rho\sof{E}$ depicted in the inset or in detail in Fig.~\ref{fig:SM}(a).
The spectral dimension found in $f\sof{E}$ is six rather than two, which is obtained from the DOS.
The singularities of $f\sof{E}$ are in the in its second derivative, but they can be found in the same positions as the singularities of $\rho\sof{E}$.
(b) The first detection probabilities $F_n$ for this case.
In contrast to a ordinary choice of the detection state, we do not find a $n^{-2}\ln^{-4} n$ decay of $F_n$ but rather a $n^{-6}$ decay in accordance with Eq.~\eqref{eq:AsymFDP} for large spectral dimensions.
\label{fig:TBSpecial}
}
\end{figure*}
In the next two sections we discuss two off-lattice models.
The first is the free particle in continuous space and the second is the L\'evy particle.
\section{The free particle in continuous space}
\label{sec:FreePart}
\begin{figure*}
\includegraphics[width=0.99\textwidth]{FPFDP.pdf}
\caption{
First detection probabilities for the free particle and the L\'evy particle.
Blue squares: $d=1$, Orange circles: $d=3$, Black lines: theoretical prediction of Eq.~\eqref{eq:NRFDP} and Eq.~\eqref{eq:LevyFDP}
(a): Free particle. $\tau$ is measured in units of $M\sigma^2/\hbar$.
As $\tau$ decreases, $F_1$ moves closer to unity and a plateau appears for small $n$.
The plateau extends to the right as $\tau$ becomes smaller and the plateau value converges to zero.
This is the Zeno effect $F_n\to\delta_{n,1}$.
Our asymptotic solution only describes the non-plateau regime of $F_n$.
(b) L\'evy particle with $\tau = 1 \hbar/E_0$, $\alpha=0.8$ and different $d$.
Prediction and simulations agree nicely.
By tuning $\alpha$ arbitrary power law exponents can be observed.
\label{fig:FreePart}
}
\end{figure*}
We now consider a free non-relativistic particle with mass $M$ in continuous $d$-dimensional space.
The Hamiltonian is given by the kinetic energy
\begin{equation}
\ensuremath{\hat{H}} := \frac{\hbar^2}{2M} \hat{\V{k}}^2
,
\label{eq:FPHamiltonian}
\end{equation}
where we write its momentum in terms of the wave vector $\hbar \V{k}$.
The first difficulty one faces here is the definition of the detection state.
In contrast to the lattice system, position eigenstates have zero width, and projection onto such states can be tricky.
We assume instead that our detector has a finite accuracy $\sigma$ and projects the wave function to a Heisenberg state with minimum uncertainty.
In momentum representation this state is defined as:
\begin{equation}
\psi_{\V{0}}\of{\V{k}}
:=
\br{\frac{2\sigma^2}{\pi}}^{\frac{d}{4}}
\exp\off{-\sigma^2\V{k}^2}
,
\label{eq:NRDefPsi}
\end{equation}
where the wave vector/momentum $\V{k} = \V{p}/\hbar$ is a real $d$-dimensional vector.
Such a wave function has $\sEA{\V{r}} = \V{0}$, $\sEA{\V{p}}=\V{0}$, furthermore: $\sEA{\sbr{\V{r}-\V{x}_0}^2} = d \sigma^2$, and $\sEA{\V{p}^2} = d \hbar^2/(4 \sigma^2)$.
In particular it has minimum uncertainty between the momentum and position coordinates.
We choose this state as initial and detection state: $\sKet{\ensuremath{\psi_\text{d}}} = \sKet{\ensuremath{\psi_\text{in}}} = \sKet{\psi_{\V{0}}}$.
For the free particle the dispersion relation is the usual kinetic energy:
\begin{equation}
E\of{\V{k}}
=
\frac{\sbr{\hbar\V{k}}^2}{2M}
.
\label{eq:NRDispersion}
\end{equation}
As an abbreviation, we define the detection state's energy per degree of freedom: $E_0 := \EA{\V{p}^2/(2m)}/(2d) = \hbar^2/(4m\sigma^2)$.
Using the momentum representation of the evolution operator, we can obtain the return amplitudes from an integral over all wave vectors:
\begin{align}
\sBAK{\psi_{\V{0}}}{\ensuremath{\hat{U}}\of{n\tau}}{\psi_{\V{0}}}
= &
\Int{\ensuremath{\mathbb{R}} ^d}{}{\V{k}}
\sAbs{\psi_{\V{0}}\sof{\V{k}}}^2 e^{-i \frac{n\tau E\sof{\V{k}}}{\hbar}}
=
\frac{1}{\br{1 + i \frac{E_0 \tau}{\hbar}n}^{\frac{d}{2}}}
\label{eq:NRTEO}
.
\end{align}
The expression has two regimes for $n$:
When $1 \gg n E_0\tau/\hbar = (n \hbar \tau) /(4M\sigma^2) $, the return amplitude is approximately unity.
For fixed $n$, this is the case when either $\tau$ is very small or $\sigma$ is very large.
In the opposite case, the return amplitude decays like a power law, which reveals the spectral dimension as equal to the Euclidean one,$d_S =d$.
The crossover between the two regimes appears at $\tilde{n} \approx \hbar/(E_0\tau) = 4M\sigma^2/(\hbar\tau)$ and marks the time at which the momentum-induced dispersion of the wave packet becomes comparable to the initial width of the wave packet.
In the limit $\sigma\to0$, the return amplitudes vanish.
This signals to us that an infinitely precise position measurement is not physically meaningful.
Since the dispersion relation \eqref{eq:NRDispersion} is a parabola, there is only one critical point on the energy surface, which is located at zero momentum.
This shows there will be no oscillations in the first detection probabilities in accord with the simulations depicted in Fig.~\ref{fig:FreePart}(a).
The resolvent at the critical value on the unit circle can be expressed via the Hurwitz zeta function $\zeta\sof{s;a} := \sSum{n=0}{\infty} \sbr{n+a}^{-s}$.
\begin{equation}
u\sof{z=1}
=
\br{\frac{-i\hbar}{E_0\tau}}^{\frac{d}{2}}
\zeta\sof{\tfrac{d}{2};-i\tfrac{\hbar}{E_0\tau}}
,
\label{eq:NRResolv}
\end{equation}
From the momentum presentation and the dispersion relation it is also easy to find the MSDOS.
\begin{equation}
f\sof{E}
=
\Int{\ensuremath{\mathbb{R}} ^d}{}{\V{k}} \sAbs{\psi_{\V{0}}\sof{\V{k}}}^2
\delta\sof{E - \tfrac{\sbr{\hbar \V{k}}^2}{2m}}
=
\frac{e^{-\frac{E}{E_0}}}{\Gma{\tfrac{d}{2}} E_0}
\br{\tfrac{E}{E_0}}^{\frac{d}{2}-1}
.
\label{eq:NRSpecMeas}
\end{equation}
Of course, $f\sof{E} = 0$ for $E<0$.
Therefore, one can easily identify the constants $A^\pm$ of the MSDOS's singularity at $E^*=0$:
\begin{equation}
A^+ := \frac{1}{\Gma{\tfrac{d}{2}} E_0^{\frac{d}{2}} }, \quad A^- = 0, \quad d_S = d
\label{eq:NRA}
\end{equation}
Using Eq.~\eqref{eq:NRResolv} and \eqref{eq:NRA} in Eqs.~\eqref{eq:AsymFDA3a} and \eqref{eq:AsymFDA3b}, and squaring the result, one obtains:
\begin{equation}
F_n
\sim
\left\{ \begin{aligned}
\frac{E_0\tau}{4 \pi^2 \hbar n^3}
, & \quad d = 1 \\
\br{\frac{E_0\tau}{\hbar n \ln^2 n}}^2
, & \quad d = 2 \\
\Abs{\zeta\sof{\tfrac{d}{2};-i\tfrac{\hbar}{E_0\tau}}}^{-4}
\br{\frac{E_0 \tau}{\hbar n}}^{d}
, & \quad d > 2
\end{aligned} \right.
.
\label{eq:NRFDP}
\end{equation}
(For the 2d case see Appendix~\ref{app:Even}.)
Fig.~\ref{fig:FreePart} shows excellent agreement between simulations and Eq.~\eqref{eq:NRFDP}.
To our surprise, the Zeno effect is not visible in our equation for $d>3$!
Using the formula $\sAbs{\zeta\sof{s;-i/x}} \sim x^{s-1}$, as $x\to0$, we find that $F_n \propto \tau^{4-d}$.
In our simulations, however, we still find $F_n\to\delta_{n,1}$ as $\tau\to0$.
As $\tau$ decreases a plateau forms in $F_n$ for small $n$ that increases in size and decreases in height.
In the same time $F_1$ moves closer to unity.
The asymptotic formula is valid only in the non-plateau region.
This is similar to the tight-binding case.
\section{Free L\'evy-particle}
\label{sec:Levy}
Instead of the regular dispersion relation \eqref{eq:NRDispersion}, one can also impose an anomalous energy-momentum relation:
\begin{equation}
E\sof{\V{k}}
=
C \sAbs{\V{k}}^\alpha
,
\label{eq:LevyDisp}
\end{equation}
with $0<\alpha<2$, and some constant $C$ with suitable units.
This can be viewed as a continuous interpolation between a non-relativistic and a relativistic dispersion relation, the latter being attained by putting $\alpha=1$.
Such a dispersion relation can be obtained, when the Laplacian in the Hamiltonian is replaced by a fractional Laplacian (a Riesz-Feller derivative) of order $\alpha/2$, which is obviously very different than the canonical approach that leads to the Dirac equation.
Detection and preparation state are again taken to be the Gaussian, Eq.~\eqref{eq:NRDefPsi}.
We define the energy constant $E_0 := C (2\sigma^2)^{-\alpha/2}$ which is related to the energy of the state $\sKet{\psi_{\V{0}}}$.
Using Eq.~\eqref{eq:NRDefPsi} and the dispersion relation, we can write the return amplitude in terms of an integral.
We use spherical coordinates and change variables to $y = 2\sigma^2 k^2$, to obtain:
\begin{equation}
\sBAK{\psi_{\V{0}}}{\ensuremath{\hat{U}}\sof{t}}{\psi_{\V{0}}}
=
\frac{1}{\Gma{\frac{d}{2}}}
\Int{0}{\infty}{y}
y^{\frac{d}{2}-1}
e^{-y - i \frac{E_0 t}{\hbar} y^{\frac{\alpha}{2}}}
.
\label{eq:LevyTEO}
\end{equation}
For large $t$, expanding the $e^{-y}$ term leads to an asymptotic series in inverse powers of $t$.
The leading order is $t^{-d/\alpha}$.
Therefore the spectral dimension is $d_S = 2d/\alpha$ and can be tuned to any real number larger than $d$ by adjusting $\alpha$.
The same result is obtained when computing $f\sof{E}$ from the Brillouin zone:
\begin{equation}
f\sof{E}
=
\frac{2}{\alpha}\frac{e^{-\br{\frac{E}{E_0}}^{\frac{2}{\alpha}}}}{\Gma{\tfrac{d}{2}}E_0}
\br{\frac{E}{E_0}}^{\frac{d}{\alpha}-1}
.
\label{eq:LevySpecMeas}
\end{equation}
From here we identify the coefficients $A^\pm$:
\begin{equation}
A^+
:=
\frac{2}{\alpha}\frac{1}{\Gma{\tfrac{d}{2}}}
E_0^{-\frac{d}{\alpha}}
, \quad A^- = 0
, \quad d_S = \frac{2}{\alpha} d,
\label{eq:LevyA}
\end{equation}
and write down the first detection probabilities:
\begin{equation}
F_n
\sim
\left\{ \begin{aligned}
\frac{\br{1-\frac{d}{\alpha}}^2}{\pi^2 n^{4 - \frac{2d}{\alpha}}}
\br{\frac{\Gma{\frac{d}{2}+1}}{\Gma{\frac{d}{\alpha}+1}}}^2
\br{ \frac{\tau E_0}{\hbar}}^{\frac{2d}{\alpha}}
, & \quad d < \alpha \\
\br{\frac{\tau E_0\Gma{1+\tfrac{d}{2}}}{\hbar n \ln^2 n} }^2
, & \quad d = \alpha \\
\br{\frac{\Gma{\frac{d}{\alpha}+1}}{\Gma{\frac{d}{2}+1}}}^2
\frac{1}{\sAbs{u\sof{z=1}}^4}
\br{ \frac{\hbar}{n \tau E_0} }^{\frac{2d}{\alpha}}
, & \quad d > \alpha
\end{aligned} \right.
.
\label{eq:LevyFDP}
\end{equation}
The remaining constant can be computed via a numeric integral:
\begin{equation}
u\sof{z=1}
=
\frac{1}{\Gma{\frac{d}{2}}}
\Int{0}{\infty}{y}
\frac{y^{\frac{d}{2}-1} e^{-y}}{
1 - e^{-i\frac{E_0\tau}{\hbar} y^{\frac{\alpha}{2}}}
}
\label{eq:}
\end{equation}
Simulations of such a process are possible by numerically evaluating the integral of Eq.~\eqref{eq:LevyTEO}.
Results are depicted in Fig.~\ref{fig:FreePart}(b).
This simple but important example shows that Eq.~\eqref{eq:AsymFDP} also holds for {\em fractional} values of the spectral dimension.
\section{Discussion}
\label{sec:Disc}
This article exposed the relation between the quantum first detection probability and the MSDOS $f\sof{E}$.
Qualitative features like the power law decay and the frequencies of the oscillations have been obtained from $f\sof{E}$, in particular from its singular points and its behavior in their vicinity.
In the ordinary case these properties can also be inferred from the DOS $\rho\sof{E}$.
We stress that our main results from Eq.~\eqref{eq:AsymFDP}, also hold for the problem of the first detected {\em arrival}, i.e. when $\sKet{\ensuremath{\psi_\text{in}}} \ne \sKet{\ensuremath{\psi_\text{d}}}$.
The additional steps are carried out in Appendix~\ref{app:Arrival}.
In the main text, we only restricted ourselves to the problem of first detected return for notational economy.
The frequencies of the oscillations in $F_n$ can be found in the asymptotic decay of the return amplitudes.
This is clear from comparison of Eq.~\eqref{eq:RAAsym} and Eq.~\eqref{eq:AsymFDA1}.
The reason is that they both are related via $f\sof{E}$.
The $\tau$-dependence of the frequencies gives rise to the existence of critical detection periods, when the number of different frequencies changes abruptly.
In the ordinary case, the frequencies are determined by the van Hove singularities.
The power law exponents of Eq.~\eqref{eq:AsymFDP} are the exact double of the classical exponents of the first passage problem for random walks.
The hand-waving argument is that Eq.~\eqref{eq:QuantumRenewal} is an equation for amplitudes, whereas its classical analogue Eq.~\eqref{eq:ClassicalRenewal} is an equation for probabilities.
The necessary squaring operation brings the additional factor of two in the exponent.
This exponent doubling when going from the classical to the quantum problem was also reported in Refs.~\cite{Muelken2006-0,Muelken2011-0} for the return probability, and also in Ref.~\cite{Boettcher2015-0} where it manifested in a halving of the walk dimension for certain discrete-time quantum walks.
However, only for ordinary states the relevant spectral dimension $d_S$ agrees with the one found in the DOS, $d_S^\text{DOS}$.
As a consequence of the large decay exponents of $F_n$, it assumes substantial values only for small $n$.
The larger exponent comes with the price that the quantum system is not {\em almost surely detectable}, in the sense that the total probability of detection, $P_\text{det} = \sSum{n=1}{\infty} F_n$, is always smaller than unity.
This holds unless $\mu\sof{\lambda}$ is a sum of delta functions \cite{Gruenbaum2013-0}.
Hence, in our infinite space models, featuring a continuous energy spectrum, there is always a non-zero probability that the particle escapes the detector.
Such a deficit in the total return probability is also known in the classical problem, where it marks the dichotomy between recurrent and transient random walks \cite{Polya1921-0}.
Random walks with a spectral dimension smaller than the critical dimension two will eventually return to their initial position with probability one and therefore are considered recurrent.
For transient random walks with a spectral dimension larger than two there is a finite probability that they never return to their initial site.
Although the total detection probability is smaller than unity, one can compute the average first detection time: $\sEA{n} = \sSum{n=1}{\infty} F_n n / P_\text{det}$ {\em under the condition} that the system was detected at all.
As a consequence of the larger exponents we found (the slowest decay is $n^{-2}\ln^{-4}n$), this expectation is always finite, contrasting the classical situation.
The conditional variance of the first detection time is finite only in dimensions larger than three.
We found that $d_S = 2$ is a critical dimension, which also plays an important role for quantum search algorithms.
For a coined quantum search algorithm in dimensions larger than two, the Grover efficiency $\sLandau{\sqrt{N}}$ can be attained \cite{Aaronson2003-0}.
In coinless, oracle quantum searches, the critical dimension is four \cite{Childs2004-0,Li2017-0}.
However, the last reference shows that the decisive quantity is exactly the {\em spectral} dimension, just as in our problem.
Finally, our main assumption for the identification of the spectral dimensions found in the DOS and in the MSDOS, is that neither $\sKet{\ensuremath{\psi_\text{d}}}$ nor $\sKet{\ensuremath{\psi_\text{in}}}$ are insufficiently populated, i.e. they both overlap with the critical van Hove energies.
Exceptions have been discussed in the main text and show that the first detection probabilities subtly depend on the initial and detection state.
This is in sharp contrast to the classical problem, where the particular choice of initial state often only changes the transient, but not the long-time behavior of the first passage probability.
This new dependence on the detection state is encoded in the MSDOS $f\sof{E}$ associated with the detection state.
As we have shown in the main text this important quantity is related to, but ultimately different from the DOS.
It requires a high degree of symmetry in the system and a special choice of states for the DOS and the MSDOS to coincide.
We felt that $f\sof{E}$ is rarely known outside the mathematical community, hence we focused on ``ordinary'' situations where the spectral dimension and the singularities of $f\sof{E}$ can also be found in the well-known density of states.
Still, this state-dependence can lead to very counter-intuitive results, if one does not carefully confirm the overlap condition.
Many open questions remain for the future.
These include the details of the arrival problem, for instance the dependence of the amplitudes $F_{l,d_S}$ from Eq.~\eqref{eq:AsymFDP} on the distance between initial an detector position.
We do not have a clear intuition on the situation when the initial state is insufficiently populated, but the detection state is not.
Also unclear is what happens when the energy spectrum is neither completely discrete nor completely continuous, that is when it is singular continuous as in the Aubrey-Harper model \cite{Lahiri2017-0}.
It is possible to extend our arguments to coined quantum walks, by discussing the WMSDOS $\mu\sof{\lambda}$ of their evolution operator.
Our most surprising finding is the sensitivity to the initial and detection states for out-of-the-ordinary states, which is a purely quantum phenomenon related to interference.
This necessitates an adjustment of the concept of spectral dimension from $d_S^\text{DOS}$, defined by the DOS, to $d_S$ defined by the MSDOS.
We are confident, that this work will lift the quantum first detection theory closer to the level of its classical brother, the first passage theory of random walks.
\acknowledgements
The authors acknowledge support from the Israeli Science Foundation under Grant No. 1898/17.
FT is funded by the Deutsche Forschungsgemeinschaft under grant TH-2192/1-1.
|
\section{Introduction}
The gamma-ray luminosity of star-forming galaxies originates from the large-scale population of cosmic rays (CRs) interacting with the interstellar medium (ISM) and from the ensemble of discrete high-energy sources, such as supernova remnants, pulsars, and their nebulae, most of which result from the evolution of the short-lived and most massive stars. Additional contributions may come from an active galaxy nucleus or, more hypothetically, from the decay or annihilation of dark matter particles.
Studying the gamma rays of a galaxy whose emission arises predominantly from its star-formation activity can inform us about the acceleration of CRs in powerful objects and its transport through the ISM. Comparing different galaxies can then be a test of our understanding of these processes by revealing how global properties such as star formation rate (SFR), gas content and metallicity, or galaxy size affect the population of high-energy objects and CRs.
In the Milky Way (MW), diffuse interstellar emission dominates the gamma-ray output and has proven to be a rich source of information, even beyond the physics of CRs \citep[see the recent review by][]{Grenier15}. Besides the MW, \textit{seven} external star-forming galaxies have been firmly detected in gamma rays with the \textit{Fermi} Large Area Telescope (LAT), including the Large Magellanic Cloud \citep[LMC;][]{LMC10,LMC16}, the Small Magellanic Cloud \citep{SMC10}, the Andromeda galaxy M31 \citep[][hereafter Paper I]{M312010}, starburst galaxies M82 and NGC~253 \citep{M8210}, NGC~2146 \citep{NGC2146tang} and Arp 220 \citep{arp220Peng16,arp220Griffin16}. In Paper I, based on a subset of these detections, a correlation was suggested between gamma-ray luminosity and SFR; it was later strengthened by a large systematic study of more than 60 galaxies \citep{Ackermann12} and now appears as a possible constraint on the origin and transport of CRs \citep{Martin14}. More recently, a deep study of the LMC has shown that discrete sources can make up a significant contribution to the global gamma-ray output, especially the most exceptional ones \citep{LMCpsr15,corbet16}, and revealed extended emission with unexpected properties \citep{LMC16}; both findings confirm the need for more studies of external star-forming galaxies.
With their relatively high gas masses, star formation activities, and small distances to Earth, M31 and M33 have long been predicted to be gamma-ray sources. Earlier gamma-ray observations of M31 and M33 involved COS-B \citep{M31cosb} and EGRET \citep{M31EGRET1,M31EGRET2}, but only upper limits (ULs) were derived. Using 2 yr of LAT observations, Paper I reported a $5.3\sigma$ detection of M31 and a marginal spatial extension ($\sim 1.8\sigma$); at the same time, M33 was not detected, but it was suggested to be detectable within years if its gamma-ray luminosity obeys the above-mentioned correlation with SFR. As the only other large spiral in the Local Group of galaxies besides the MW, M31 is a highly relevant target for a comparative study. Moreover, with an angular size over 3$^\circ$, it is one of the rare nearby galaxies holding potential for a resolved analysis.
In this paper we revisited the gamma-ray emission from M31 and M33 using more than 7 yr of Pass 8 observations, which is the latest version of LAT data and has overall improved performance over previous Pass 7 data \citep{pass8Atwood}. The paper is organized as follows. We briefly introduce the \textit{Fermi}-LAT instrument and Pass 8 data in Section 2 and present in detail the morphological and spectral analysis in Section 3. We discuss possible interpretations of our findings in Section 4 and summarize our results in Section 5.
\section{Data set and analysis methods}
The LAT is a pair-conversion telescope comprising a $4 \times 4$ array of silicon strip trackers and cesium iodide calorimeters covered by a segmented anti-coincidence detector to reject charged-particle background events. The LAT covers the energy range from 20 MeV to more than 300 GeV with a field of view of 2.4 sr. It operates predominantly in survey mode and observes the entire sky every two orbits (3 hr) by rocking north and south about the orbital plane on alternate orbits \citep{LAT09}.
We used SOURCE class events, converting in both the front and back sections of the LAT, but excluding those with a zenith angle larger than $90^\circ$ or collected when the LAT's rocking angle was larger than $52^\circ$ to avoid the Earth limb contamination. We considered events with reconstructed energies in the energy range 0.1$-$100 GeV and with reconstructed directions within a $14^\circ \times 14^\circ$ region of interest (ROI). For the analysis of M31, we selected 88 months of Pass 8 data collected between 2008 August 4 and 2015 December 1, with an ROI center at $(\alpha,\delta)=(10\fdg6847,41\fdg2687)$. The data set used for the analysis of M33 spans 85 months, with an ROI centered on $(\alpha,\delta)=(23\fdg4621,30\fdg6599)$. The coordinates for both galaxies were taken from the SIMBAD\footnote{ http://simbad.u-strasbg.fr/simbad/} database and correspond to the J2000 epoch.
For each ROI, a complete spatial and spectral source model was built. We used the latest model gll\_iem\_v06.fits for the Galactic interstellar emission and the isotropic emission spectrum iso\_P8R2\_SOURCE\_V6\_v06.txt for the extragalactic emission and residual instrumental background. Point sources within $20^\circ$ around M31 or M33 in the LAT Third Source Catalog \citep[3FGL;][]{3FGL} were included in the model (except 3FGL~J0042.5+4117, which is M31), with spectral parameters set free to vary for sources within $5^\circ$ around M31 or M33. This source model not including M31 or M33 will hereafter be referred to as the background model. On top of this background model, we explored several possibilities for the morphology and spectrum of M31 or M33.
Each source model was fitted to the data following a maximum likelihood approach for binned data and Poisson statistics \citep{mattox96}. Unless otherwise stated, we used a $0\fdg1 \times 0\fdg1$ pixel size and four logarithmic energy bins per decade. The analysis was performed using the P8R2\_SOURCE\_V6 Instrument Response Functions and the {\it Fermi}\ Science Tools version 10-01-01 available from the {\it Fermi} Science Support Center\footnote{ http://fermi.gsfc.nasa.gov/ssc/}. The significance of model components for M31 or M33 is quantified with the test statistic (TS), which is expressed as TS $=2(\log \mathcal{L}-\log \mathcal{L}_{0})$, where $\log \mathcal{L}$ and $\log \mathcal{L}_{0}$ are the logarithms of the maximum likelihood of the complete source model and of the background model (i.e. the source model without M31 or M33 included), respectively. The significance of the spatial extension of M31 or M33 is quantified by TS$_{\rm ext}$, which is twice the difference between the $\log \mathcal{L}$ obtained with an extended source model and that obtained with a point-like source model at its best-fit position.
As a potentially extended gamma-ray source, and one possibly shining because of interstellar processes, M31 requires some caution in the use of the Galactic interstellar emission model. This model is developed from radio and infrared tracers of interstellar gas \citep{catSNR16}. In particular, it is based on the Leiden$-$Argentine$-$Bonn 1.4\,GHz observations of atomic gas \citep{LAB05} and on a dust reddening map \citep{Schlegel98}, both of which are all-sky data in which M31 appears. M31 was removed from these maps in developing the interstellar emission model for the MW. Otherwise, any emission from M31 would be erroneously absorbed in the fitting of the Galactic interstellar emission model. In the case of the 1.4\,GHz data, M31 was removed by applying the following two cuts in the $(l,b,v_{\rm LSR})$ data space: (1) $l$ from $119^\circ$ to $123^\circ$, $b$ from $-23\fdg5$ to $-19\fdg5$, $v_{\rm LSR}$ up to $-120$\,km\,s$^{-1}$; and (2) $l$ from $121^\circ$ to $124^\circ$, $b$ from $-22^\circ$ to $-19\fdg5$, $v_{\rm LSR}$ from $-120$ to $-50$\,km\,s$^{-1}$. Examination of higher-resolution observations of this region from the Effelsberg$-$Bonn H\,{\sc i} survey \citep{Winkel16} confirms that such cuts effectively remove the great majority of the disk of M31 from the data. For $-30$\,km\,s$^{-1}$ $ > v_{\rm LSR} > $ $-50$\,km\,s$^{-1}$ (i.e. data not cut out), foreground emission from the MW blends with remaining signal from M31 at the northeastern tip of M31. We estimated that, on some lines of sight in this direction, up to $\sim40$\% of the signal from M31 might have been incorporated in the maps used in the Galactic interstellar emission model. Yet, this confusion happens over a very restricted region compared to the full extent of M31, and at a distance of $1\fdg25$ from the center of the galaxy, such that any gamma-ray emission correlated with the disk of M31 should safely be recovered. Another possible source of bias in the study of extended sources is that they may be part of the large-scale residuals reinjected into the final model \citep[see][for details]{catSNR16}, and we checked that it is not the case for the region around M31.
\section{Data analysis results}
\subsection{M31}
\subsubsection{Morphological analysis}
Figure \ref{cmap_m31} shows the LAT counts map (left) and residual counts map after background subtraction (right) in the 1$-$100 GeV energy range. M31 is clearly visible in the counts map and appears more prominently in the residual map. The gamma-ray morphology of M31 is characterized using the $\mathtt{pointlike}$ tool \citep{Kerr10} on a data set restricted to energies above 1 GeV to benefit from the better angular resolution. We explored different geometrical models, such as point source, disk, elliptical disk, Gaussian, or elliptical Gaussian. We also considered spatial templates from observations at other wavelengths: \textit{Herschel}/PACS map at 160 $\mu m$, \textit{Spitzer}/IRAC map at 3.6 $\mu m$, and an atomic gas column density $N_{\rm H}$ map from \cite{Braun09}, uncorrected for self-opacity. The latter models are intended to test the spatial correlation of the gamma-ray emission with star formation sites, the old stellar population, or interstellar gas, respectively. We also tested two-component models such as a point source at the center of M31 and an extended component around it.
The spectrum of M31 was initially modeled by a simple power law (PL), an assumption that we revisited once a satisfactory spatial model is identified.
Fit results are reported in Table \ref{spatptlikeM31}.
Starting from a simple point-source model located at the center of M31, we found that optimizing the position of the point source provides a limited improvement with significance $< 2\sigma$, but allowing for an extension improves the fit with a significance below $3\sigma$ for the uniform-brightness disk model. Allowing for an offset of the disk center with respect to the center of M31 results in a slightly more significant extension and an offset from the center of M31 that is not significant ($<2\sigma$). Using a 2D Gaussian intensity distribution instead of a uniform-brightness disk degrades the fit likelihood by a negligible amount. Similarly, allowing for some elongations of the signal in some directions with elliptical disk or elliptical Gaussian models does not significantly improve the fit. Two-component models consisting of a point source and a disk or 2D Gaussian component around it, all centered at the M31 center, also led to very marginal improvements. These two-component models are therefore not required, especially since in each case the point-source component is not significantly detected.
Among template map models, the $N_{\rm H}$ map yields the fit with the lowest likelihood of all tested models. For the same number of degrees of freedom, the \textit{Herschel}/PACS or \textit{Spitzer}/IRAC maps are not favored compared to a simple point source at the center of M31, but the \textit{Spitzer}/IRAC map provides a slightly better fit to the data than the \textit{Herschel}/PACS map. These results are consistent with those obtained with geometrical models because the $N_{\rm H}$ map, and to a lesser extent the \textit{Herschel}/PACS map, are dominated by the relatively extended disk of M31, while the \textit{Spitzer}/IRAC map is dominated by its bulge.
We retained the uniform-brightness disk with radius of $0\fdg38\pm 0\fdg05$ as the best-fit morphological model for M31 because it is the simplest of the best-fitting models. The different tests summarized above indicate that the emission is consistent with being symmetric around the center of the galaxy. Yet, we emphasize that, based on the data currently at our disposal, we cannot reject nonuniform-brightness distributions or multicomponent models. With such a disk model, M31 has TS $=51$ and TS$_{\rm ext}=7.6$ from an analysis in the 1$-$100 GeV band. Including lower-energy events down to 100 MeV results in a more significant detection and extension with TS $=95$ and TS $_{\rm ext}=16$ (and a source extension consistent with that obtained from the 1$-$100 GeV data analysis). In including lower-energy events, we verified that source 3FGL~J0040.3+4049 does not influence the results because of its proximity to M31 (the source lies within the optical or infrared disk of M31; see Figure \ref{cmap_m31}). In analyzing 1$-$100 GeV events, 3FGL~J0040.3+4049 is well resolved from M31 because it has a hard spectrum with photon index 1.3, but the poor angular resolution of the LAT below 1 GeV may introduce some cross-talk between both sources. Fixing the spectral parameters of 3FGL~J0040.3+4049 (either the spectral index only or both the index and the prefactor) to the values determined from the 1 to 100 GeV analysis yields TS $=97-98$ for M31, similar to the value obtained when parameters for 3FGL~J0040.3+4049 are left free in the fit, confirming that the source has little impact on the properties derived for M31.
Figure \ref{tsmap_m31} (left panel) shows the TS map for the background model, in a $3\fdg5 \times 3\fdg5$ region around M31 (that is, adding a point-source model to the background model and testing it over a grid of positions). Contours and shapes for the best-fit spatial models tested here are overlaid. This plot illustrates that the gamma-ray emission is clearly extended but over an area much smaller than the full extent of M31. The flux appears confined to the central parts of the galaxy and does not fill the disk or extend far from it. To investigate whether there are unmodeled emission components around M31 or whether multiple sources are necessary to account for the total emission in the direction of M31, we computed the TS map for a source model including M31. Figure \ref{tsmap_m31} (right panel) shows two residual point-like excesses to the east and northwest of M31. These were dubbed Excess1 and Excess2\footnote{Excess2 is spatially coincident with a source in the \textit{Fermi}-LAT Collaboration internal 7 yr source list, with an angular separation of $4.^{\prime}0$. }, and their optimal positions are given in Table \ref{spatptlikeM31}. They were added to our source model to evaluate their impact on the fit. With TS values of 8 and 12 for Excess1 and Excess2, respectively, both were below the standard detection threshold of 25. Comparing the $\log \mathcal{L}$ values of fits with and without components modeling the two excesses indicates a fit improvement with a significance $<3\sigma$, so we decided not to consider these components further in the analysis. We note, however, that including these sources in the background model results in the extension of M31 being smaller and less significant, suggesting that the emission might actually be even more confined to the inner regions than discussed above.
\newpage
\subsubsection{Spectral analysis}
For the spectral analysis of M31, we performed a binned maximum likelihood fitting in the 0.1$-$100 GeV energy range, using the $\mathtt{gtlike}$ tool provided in the \textit{Fermi} Science Tools, with 30 logarithmic energy bins in total. To avoid possible cross-talk, the spectral index of the background source mentioned above (3FGL~J0040.3+4049) was fixed to the value determined in the 1$-$100 GeV analysis, but we checked that leaving it free in this broadband analysis has a negligible impact. Using the best-fit disk model described above, we first compared a simple PL, a PL with exponential cutoff (PLEC), and a log-parabola (LP) for M31. The addition of a curvature in the spectrum does not significantly improve the fit ($<3\sigma$). The flux from M31 is satisfactorily described by a PL with photon index $\Gamma=2.4\pm0.1_{\rm stat+syst}$ and a 0.1$-$100 GeV energy flux of $(5.6\pm0.6_{\rm stat+syst})\times 10^{-12}\rm \,erg\, cm^{-2}\,s^{-1}$ (see below for the computation of systematic uncertainties). This is reported in Table \ref{specgtlike} and illustrated in Figure \ref{sed_m31}, where the best-fit PL model is plotted together with spectral points. The latter were determined by performing a maximum likelihood analysis in 10 logarithmically spaced energy bins over 0.1$-$100 GeV. Within each bin, the spectrum of M31 was modeled as a simple PL with fixed index $\Gamma=2$, and the normalization of M31 was allowed to vary while all other sources were fixed to their best-fit parameters obtained from the broadband analysis. ULs on the flux at 95\% confidence level were derived using the Bayesian method when M31 has TS $<4$ ($2\sigma$) in a given bin. For the spectral points and spectral parameters, systematic uncertainties in the LAT effective area were estimated by refitting the data using as a scaling functions to ``bracket'' the effective area, following the recommendations of the {\it Fermi} Science Support Center and using as a scaling function $\pm 5\%$ over 0.1$-$100 GeV.
We checked that using a different photon index within each bin, e.g., 2.4 instead of 2.0, or setting normalizations free for diffuse components and sources within $2^\circ$ of M31, has an insignificant impact. In the latter test, the spectral parameters of 3FGL~J0040.3+4049 were fixed to those determined in the 1$-$100 GeV analysis, which tends to underestimate the uncertainties on the low-energy flux from M31 because the poor angular resolution of the LAT at low energies would have allowed some cross-talk between both sources. This choice is, however, justified by the point-like nature and hard spectrum of 3FGL~J0040.3+4049.
To help pinpoint the possible origin of the emission from M31, we also tested more physically motivated spectral models (Table \ref{specgtlike}). We considered interstellar gamma-ray emission spectra from a GALPROP model of the MW \citep{Strong10}, selecting the plain diffusion model for a halo height of 4\,kpc, and an average spectrum of observed millisecond pulsars (MSPs) in the MW \citep{Cholis14}, which is a PLEC model with a photon index of 1.6 and a cutoff energy of 4 GeV. Still using the best-fit disk model, we found that the emission from M31 has a spectrum that is consistent with that of
|
the total interstellar emission from the MW or with its pion-decay component (the $\log \mathcal{L}$ difference compared to the best-fit PL model is negligible); it is comparatively less consistent with an average MSP spectrum and with the inverse-Compton (IC) component of the interstellar emission from the MW (because the latter is too flat), but the differences in terms of $\log \mathcal{L}$ are modest.
\subsubsection{Flux variability}
To examine the variability of the gamma-ray flux from M31, we computed a long-term light curve with a 90-day binning, for events in the energy range 0.1$-$100 GeV. In each time bin, all sources (including M31) within $5^\circ$ of the nominal position of M31 had spectra fixed to the shapes obtained from the full data set analysis, and only normalizations were allowed to vary. ULs at 95\% confidence level were calculated when M31 had TS $<1$ in a given time bin. The result is shown in Figure \ref{lc_M31M33} (left panel). Using the 90-day binning, we quantified the variability significance following the same method used in \cite{3FGL} and obtained $1.4\sigma$ (for 28 degrees of freedom). The emission is therefore consistent with being steady, at least down to the scale of a few months.
\subsection{M33}
\subsubsection{Morphological analysis}
We repeated the procedure used for M31 to characterize the gamma-ray morphology of M33. Using SOURCE class data and event energies $>1$ GeV, weak excess emission appears in the direction of M33, but at a level that seems comparable to other positive fluctuations in the field. The excess is consistent with a point-like source with TS $=8$, at a position that is slightly offset from M33. To establish whether this weak source may be spatially associated with M33, we restricted the data set to the PSF3 subclass events, which have the most accurately reconstructed directions. Figure \ref{cmap_m33} shows the corresponding residual counts map after background subtraction.
We explored different spatial models for M33, and the result is that the gamma-ray emission in the direction of M33 is consistent with a point-like source at position $(\alpha,\delta) =(23\fdg625\pm0\fdg047, 30\fdg509\pm0\fdg043)$, $0\fdg2$ offset from the center of M33 (Table \ref{spatptlikeM33}). The TS of the source is 23 for an analysis in the 1$-$100 GeV range, and 28 when including lower-energy events down to 100 MeV. Spatial models consisting of a \textit{Herschel}/PACS map at 160 $\mu m$ or of a point source at the center of M33 can be excluded at the $\geq 3\sigma$ confidence level.
Figure \ref{tsmap_m33} shows a TS map for the background model and events energies $>1$ GeV.
Overlaid are the position of the center of M33, the best-fit point-source position and the \textit{Herschel}/PACS map contours.
The plot illustrates that the gamma-ray emission is most likely not connected to M33. The source may be a background active galaxy nucleus. To evaluate this possibility, we searched for variability of the signal but found nothing significant (see Section 3.2.3).
\subsubsection{Spectral analysis}
We performed a spectral analysis using $\mathtt{gtlike}$, for all SOURCE events in the energy range 0.1$-$100 GeV.
Since we concluded above that the weak gamma-ray emission in the direction of M33 is not connected to the latter, we computed a flux UL for M33 over the entire energy band. In addition to the point source offset from M33, M33 was included in the source model either as a point source or as the \textit{Herschel}/PACS map template, and we assumed a PL spectrum with a fixed index of 2.2 (the typical value found for other detected star-forming galaxies; see \citealt{Ackermann12}). Using the \textit{Herschel}/PACS template, we obtained a photon flux UL of $0.3\times 10^{-8}\rm \,ph\, cm^{-2}\,s^{-1}\,$ and an integrated energy flux UL of $2.0\times 10^{-12}\rm \,erg\, cm^{-2}\,s^{-1}\,$ (95\% confidence level).
\subsubsection{Flux variability}
To examine the variability of the gamma-ray flux from the source in the direction of (but offset from) M33, we computed a long-term light curve with a 90-day binning, for all SOURCE events in the energy range 0.1$-$100 GeV. We followed a procedure similar to that used for M31 in Section 3.1.3 and the result is shown in Figure \ref{lc_M31M33} (right panel). For 27 degrees of freedom, the gamma-ray emission from the source is consistent with a constant signal, with a variability significance of 1.2$\sigma$.
\section{Discussion}
\subsection{Update to the $L_{\gamma}-$SFR correlation}
We derived for M31 and M33 the gamma-ray luminosity $L_{\gamma}=4\pi d^2 F_{100}$, where $d$ is the distance to the galaxy and $F_{100}$ is the photon flux above 100\,MeV. Assuming that the 0.1$-$100 GeV gamma-ray emission is dominated by CRs interacting with interstellar gas (via pion production and decay and to a smaller extent bremsstrahlung), one can compute an average emissivity per hydrogen atom as $\overline{q}_\gamma=L_{\gamma}/N$, where $N=1.19 \times 10^{57} \times (M_{\rm HI}+M_{\rm H_2})$ is the total number of hydrogen atoms in a galaxy, with $M_{\rm HI}$ and $M_{\rm H_2}$ being the mass of neutral and molecular hydrogen, respectively, in $\rm M_\odot$ units, and the conversion factor is in H-atom$\rm/ M_\odot$. The input parameters and results are summarized in Table \ref{emissM31M33}.
For M31, our $F_{100}$, $L_{\gamma}$, and $\overline{q}$ estimates are completely consistent with those reported in Paper I. For M33, we get ULs on $L_{\gamma}$ and $\overline{q}$ that are $\sim40\%$ lower than those in Paper I. The mildly improved constraints on M33 do not challenge the observed $L_{\gamma}-$to$-$SFR correlation \citep{Ackermann12}, especially because M33 was lying on the upper side of the estimated intrinsic dispersion that spans a bit less than an order of magnitude in $L_{\gamma}$ \citep[such a large scatter is expected from modeling of the interstellar emission from star-forming galaxies; see][]{Martin14}. Assuming that their gamma-ray emission results from CR$-$gas interactions, the inferred emissivities suggest that M31 and M33 have an average CR density that is at most half that of the MW.
Yet, in the case of M31, this assumption of emission being from CR$-$gas interactions can now be questioned. The observed emission is concentrated within a $0\fdg4$ angular radius, which translates into a physical radius of about 5\,kpc at the distance of M31. Most of the atomic and molecular gas in M31 actually lies beyond this radius, in a ring located at 10\,kpc and beyond it (see, e.g., \citealt{Smith12}). This extended gas ring is also where most of the star formation occurs \citep{Ford13}, and consequently where most sources of CRs such as supernova remnants are supposed to be. Yet, we did not detect such an extended contribution to the signal. Using the best-fit disk model, we derived a 95\% confidence level UL of $0.5\times 10^{-8}\rm \,ph\, cm^{-2}\,s^{-1}\,$ for additional 0.1$-$100 GeV emission correlated with the gas disk of M31 (as traced by the $N_{\rm H}$ map from \citealt{Braun09}). The average gas-related contribution to the emission from M31 thus has to be smaller than 50\% of the currently estimated flux of the galaxy. Depending on the nature of this observed central emission in M31 (interstellar or not; see below), the total interstellar luminosity could therefore be up to 50\% higher or more than 50\% lower than previously assumed. In either case, this does not challenge the observed $L_{\gamma}-$to$-$SFR correlation for the reasons given above. This is illustrated in Figure \ref{Lg_LIR}, where we used the correlation from Figure 4 of \cite{Ackermann12}, along with our updated measurements for M31 and M33 and the revised estimate for the diffuse emission from the LMC presented in \cite{LMC16}.
\subsection{Interstellar emission}
If the observed central emission of M31 is interstellar in origin, it can be accounted for in at least two ways.
A first possibility is that the emission is gas related and the low gas content of the area subtended by the gamma-ray emission is compensated by a higher CR density, and hence gas emissivity, in the inner regions of M31. Yet, this relatively high density of CRs would be found several kiloparsecs away from the main sites of current or recent star formation \citep{Ford13}, while these sites of star formation do not shine in gamma rays at a detectable level despite being gas-rich. This is reminiscent of a discussion of the gamma-ray emission of the LMC \citep{LMC16}, in which areas relatively devoid of gas and star formation were found to be sites of significant gamma-ray production.
A second possibility is that the emission is dominated by IC scattering of a population of energetic electrons in the dense radiation field in the inner regions of M31 resulting from the large concentration of stars. In Section 3.1.2, we showed that the measured gamma-ray spectrum is slightly more consistent with a pion-decay spectrum than with an IC spectrum; on the other hand, we used models of the MW for these spectral fits, and M31 may have a different IC spectrum because of a different interstellar radiation field. The required central population of energetic electrons is not dominating in a radio synchrotron map of M31 (see Figure 2 of \citealt{Taba13}), where strong synchrotron emerges from the center of the galaxy but on a much smaller scale than that of the observed gamma-ray emission. Yet, synchrotron emission also depends on the distribution of the magnetic field in the galaxy, so the same population of energetic electrons may show up differently in synchrotron and IC. A puzzling fact is that IC emission would thus dominate the gas-related emission. In MW models, IC amounts to at most 45\% of the luminosity of the gas-related components, and such a high fraction implies a large confinement volume (see Table 2 of \citealt{Strong10}); it is not straightforward to figure out why a large galaxy like M31 would exhibit the opposite relation, i.e., gas-related emission being at most 50\% of the IC emission.
A possible solution to these apparent discrepancies is that those high-energy particles responsible for the gamma-ray emission in the inner regions of M31 are not CRs resulting from recent star formation activity. The latter is thought to be the dominant source of non-thermal particles in the MW (energetically speaking), and this is assumed in the MW models referred to. In M31, however, because of its 10 times lower SFR compared to the MW \citep{Ford13}, the population of energetic particles in the inner regions may be contributed for the most part by another source, an old stellar population (see below) or the central supermassive black hole, for instance.
\subsection{Unresolved source population}
An alternative scenario is that the emission is not interstellar in origin but comes from a population of unresolved objects. The lack of correlation with the distribution of star formation sites does not favor sources related to short-lived massive stars, such as supernova remnants or normal pulsars and their nebulae. Instead, the location of the emission in the inner regions of M31, where a significant fraction of the old stellar population can be found \citep{Barmby06}, and where the largest concentration of X-ray sources is \citep{Voss07,Stiele10}, supports low-mass X-ray binaries and/or MSPs as possible sources of the signal.
Such a situation is reminiscent of discussions about the nature of the so-called Galactic center (GC) excess \citep[see e.g.,][]{Abazajian12,Gordon13,Mirabal13,Yuan14}. In particular, \cite{Brandt15} suggested that a population of MSPs deposited in the MW inner regions by the disruption of globular clusters can account for all observed properties of the GC excess. This scenario implies a deposited stellar mass of about $5 \times 10^8 \, \rm M_\odot$, in a central region extending out to a galactocentric radius of 10 kpc \citep{Gnedin10}, associated with an average flux at 2 GeV of $2 \times 10^{-15} \rm \,GeV\, cm^{-2}\,s^{-1}\,$ per unit deposited stellar mass, at a distance of 8.3\,kpc \citep{Brandt15}. This translates into a total flux of $10^{-6} \rm \,GeV\, cm^{-2}\,s^{-1}\,$. For comparison, the flux at 2 GeV from M31 translated to a distance of 8.3\,kpc is $4 \times 10^{-6} \rm \,GeV\, cm^{-2}\,s^{-1}\,$ and the bulk of the emission comes from within a radius of 5\,kpc. The $\sim$4 times higher flux in M31 can be attributed to the number of globular clusters being 3--4 times greater in M31 than in the MW \citep{Galleti07}, which could result from a proportionately higher initial mass in globular clusters that subsequently dissolved in the disk and bulge of M31.
In Section 3.1.2, we showed that an average MSP spectrum is almost as good a fit to the data as a PL with free parameters, so the interpretation of the emission from M31 being due to populations of MSPs cannot be rejected from spectral arguments. Moreover, the lack of a significant curvature in the observed spectrum can be the result of a possible additional contribution to the signal from IC emission by the pairs released by the pulsars in the ISM \citep{Petrovic15}. The observed PL spectrum for M31 differs from that inferred for the GC excess; it is flat in the 0.1$-$1 GeV range, while the GC excess spectrum seems to cut off below 1 GeV. Yet, at these energies, the point-spread function of the LAT becomes relatively large (68\% containment radius above 1$^\circ$), and the derivation of the GC excess spectrum is affected by large uncertainties \citep[][submitted]{GCE_fermi}. In that respect, our external vantage point on M31 may provide a cleaner view of the central emission from a grand-design spiral galaxy, and in particular the contribution of old stellar populations: M31 has a 10 times lower SFR than the MW \citep{Ford13}, which should decrease the disk emission, while its bulge is $5-6$ times more massive \citep{Tamm12,Licquia15}, which could enhance any contribution from old objects.
\subsection{Dark matter}
Another possible interpretation of the central, extended, and seemingly symmetric emission from M31 is that it results from the decay or annihilation of dark matter particles. To evaluate whether such an interpretation is likely, we made a naive estimate of the expected signal from dark matter and compared it to the measured value. The calculation involves so-called \textit{J}-factors that were computed for Navarro$-$Frenk$-$White distributions of the smooth dark matter halo component \citep{Navarro97}.
Using the GC excess as a reference (but emphasizing that the interpretation of the latter in terms of dark matter is far from obvious), a flux at 2 GeV of $2\times10^{-7} \rm \,GeV\, cm^{-2}\,s^{-1}$ is measured, and the \textit{J}-factor over the studied region is $2\times 10^{22} \rm \,GeV^2\, cm^{-5}$ \citep[][submitted]{GCE_fermi}. In M31, the \textit{J}-factor integrated over the extent of the detected gamma-ray signal is $8\times 10^{18} \rm \,GeV^2\, cm^{-5}$ \citep[][in preparation]{DM_fermiHAWC}, but this value should be considered as uncertain by a factor of a few because the lack of rotation curve data within 7 kpc of the center of M31 results in large uncertainties in the central density distribution \citep{Tamm12}. From the ratio of \textit{J}-factors, one would expect a flux at 2 GeV from dark matter annihilation or decay in M31 of $8\times 10^{-11} \rm \,GeV\, cm^{-2}\,s^{-1}$, which is a factor 5 below the observed value. Because of the uncertainties in the \textit{J}-factor estimates, we cannot exclude from simple photometric arguments the possibility that dark matter accounts for a significant fraction of the observed signal. A dedicated analysis to characterize the dark matter contribution to the gamma-ray signal of M31 is beyond the scope of this paper and will be presented elsewhere \citep[][in preparation]{DM_fermiHAWC}.
\section{Conclusion}
We have analyzed more than 7 yr of \textit{Fermi}-LAT Pass 8 0.1$-$100 GeV observations of the Local Group galaxies M31 and M33. M33 is still undetected, and the flux UL we derived is $\sim40\%$ lower than that determined in 2010 from 2 yr of Pass 6 data. In contrast, M31 is detected with a significance of nearly $10\sigma$. The main improvement compared to our previous analysis is that gamma-ray emission from M31 is now detected as extended. This extension, however, is rather limited, and consequently its significance remains modest, at the $4\sigma$ level. The spatial distribution of the signal is consistent with a uniform-brightness disk with an angular radius of $0\fdg4$, 5\,kpc at the distance of M31, and no offset from the center of the galaxy, but nonuniform or multicomponent intensity distributions cannot be dismissed based on the current observations. The small extent of the source seems to exclude emission coming from the main gas ring and from the dominant star formation sites, contrary to expectations for typical interstellar emission. Possible and nonexclusive interpretations include a population of unresolved sources, energetic particles originating in sources not related to massive star formation, or dark matter. This result should be helpful in clarifying the origin of the excess gamma-ray emission observed in the inner regions of our Galaxy.
\vspace{0.3cm}
The \textit{Fermi} LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT, as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States; the Commissariat \`a l'Energie Atomiqueand and the Centre National de la Recherche Scientifique/Institut National de Physique Nucl\'eaire et de Physique des Particules in France; the Agenzia Spaziale Italiana
and the Istituto Nazionale di Fisica Nucleare in Italy; the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK), and Japan Aerospace Exploration Agency (JAXA) in Japan; and the K.~A.~Wallenberg Foundation, the Swedish Research Council, and the Swedish National Space Board in Sweden. Additional support for science analysis during the operations phase is gratefully acknowledged from the Istituto Nazionale di Astrofisica in Italy and the Centre National d'\'Etudes Spatiales in France.
X.H. is supported by the National Natural Science Foundation of China through grant 11503078 and by the Ministry of Science and Technology of the Republic of China (Taiwan) through grant 104-2811-M-007-059. A.K.H.K. is supported by the Ministry of Science and Technology of the Republic of China (Taiwan) through grant 103-2628-M-007-003-MY3. J.C.W. and M.Z. are supported by the National Natural Science Foundation of China through grant 11573060.
The authors thank Pauline Barmby for providing the IRAC 3.6 $\mu$m data and Annie Hughes and Robert Braun for providing the gas column density data. P.M. thanks Amaury Fau for his early works on the subject.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
\bibliographystyle{apj}
|
\section{Introduction}\label{sec-intro}
We consider a simple random walk $(S_n)_{n\in \mathbb{N}}$ on $\mathbb{Z}^d$ starting
from the origin. The range of the walk between
two times $k,n$ with $k\le n$, is denoted as
$\mathcal{R}[k,n]:=\{S_k,\dots,S_n\}$ with the shortcut $\mathcal{R}_n=\mathcal{R}[0,n]$.
Its Newtonian capacity, denoted $\cc{\mathcal{R}_n}$,
can be seen as the hitting probability of $\mathcal{R}_n$
by an independent random walk starting from
{\it far away} and properly normalized. Equivalently, using reversibility,
it can be expressed as the sum of
escape probabilities from $\mathcal{R}_n$ by an independent random walk
starting along the range.
In other words, $\cc{\mathcal{R}_n}$ is random and has the following representations:
\begin{equation}\label{def-cap}
\cc{\mathcal{R}_n}\ =\ \lim_{z\to\infty}\,
\frac{\mathbb{P}_{0,z} (\widetilde H_{\mathcal{R}_n}<\infty\ |\ S)}{G(z)}=
\sum_{x\in \mathcal{R}_n} \mathbb{P}_{0,x} \big(\widetilde H^+_{\mathcal{R}_n}=\infty\ |\ S\big),
\end{equation}
where $\mathbb{P}_{0,z}$ is the law of two independent
walks $S$ and $\widetilde S$ starting at $0$ and $z$ respectively, $G(\cdot)$ is Green's function, and
$\widetilde H_{\Lambda}$ (resp. $\widetilde H^+_{\Lambda}$)
stands for the hitting (resp. return) time of $\Lambda$
by the walk $\widetilde S$.
In view of \reff{def-cap}, the study of the capacity of the range
is intimately related to the question of estimating probabilities of
intersection of random walks. This chapter has grown
quite large, with several motivations from
statistical mechanics keeping the interest alive (see Lawler's
celebrated monograph \cite{Law91}). The last decade has witnessed revival interests both
after a link between uniform spanning trees and loop erased random walks
was discovered
(see \cite{LawSW}, \cite{Hut} for recent results) and
after the introduction of random interlacements
by Sznitman in \cite{S10} which mimic a random walk
confined in a region of volume comparable to its time span.
The study of the capacity of the range of a random walk has
a long history. Jain and Orey \cite{JO} show that in
any dimension $d\ge 3$,
there exists a constant $\gamma_d\in [0,\infty)$, such that almost surely
\begin{equation}\label{cap.limit}
\lim_{n\to\infty}\frac 1n \mathrm{Cap}(\mathcal{R}_n) = \ \gamma_d,\quad\text{and}\quad
\gamma_d>0,\ \text{if and only if,}\ d\ge 5.
\end{equation}
The first order asymptotics is obtained in dimension $3$ in \cite{Chang},
where $\mathrm{Cap}(\mathcal{R}_n)$ scales like $\sqrt n$. Dimension $4$ is {\it the critical dimension},
and a central limit theorem with a non-gaussian limit is established in \cite{ASS19}.
In higher dimensions, a central limit theorem is proved
in \cite{Sch19} for $d=5$, and in \cite{ASS18a} for $d\ge 6$.
Here, we mainly study the downward deviations for
the capacity of the range in dimension $d\ge 5$,
in the moderate and large deviations regimes.
We also establish a large deviations
principle in the upward direction.
Our analysis is, as in our previous works \cite{AS,AS19}, related to the celebrated large
deviation analysis of the volume of the Wiener sausage
by van den Berg, Bolthausen and den Hollander \cite{BBH01}.
The folding of the Wiener sausage, under squeezing its volume, became
a paradigm of {\it folding}, with localization in a domain with holes
of order one (the picture of a Swiss Cheese popularized in \cite{BBH01}).
The variational formula for the rate function
was shown to have minimizers of different nature in $d=3$ and in $d\ge 5$
suggesting dimension-dependent optimal scenarii to achieve the deviation.
For the discrete analogue of the Wiener sausage, we established in
\cite{AS, AS20a}
some path properties confirming some observations of \cite{BBH01}.
\paragraph{Main results}
Our first result concerns the large and moderate deviations in dimension $7$ and higher. In this case, we obtain upper and lower bounds which are of the same order (on a logarithmic scale), and we cover (almost) the whole set of possible moderate deviations in the non-Gaussian regime.
\begin{theorem}\label{thm:d6}
Assume $d\ge 7$. There exist positive constants $\varepsilon$,
$\underline \kappa$
and $\overline \kappa$ (only depending on the dimension),
such that for any $n^{\frac{d-2}{d}} \cdot \log n \le \zeta \le \varepsilon n$, and for $n$ large enough,
\begin{equation}\label{dev.nongauss.7}
\exp\left(- {\underline\kappa}\cdot \zeta^{1-\frac{2}{d-2}} \right)\ \le\
\mathbb{P}\left( \cc{\mathcal{R}_n}-\mathbb{E}[ \cc{\mathcal{R}_n}]\le -\zeta\right)\ \le\
\exp\left(- {\overline\kappa}\cdot \zeta^{1-\frac{2}{d-2}} \right).
\end{equation}
\end{theorem}
Recall that a central limit theorem is proved in
\cite{ASS18a}, where we show in particular that $\operatorname{var}(\cc{\mathcal{R}_n}) \sim \sigma^2 n$, for some constant $\sigma>0$.
Our next result proves now a Moderate Deviation Principle in the Gaussian regime.
\begin{theorem}\label{theo.Gaussian}
Assume $d\ge 7$. For any sequence $\{\zeta_n\}_{n\ge 0}$,
satisfying $\lim_{n\to \infty} \zeta_n/\sqrt n = \infty$,
and $\lim_{n\to \infty} \zeta_n (\log n) / n^{\frac{d-2}d} = 0$, we have
\begin{equation}\label{limit-TCL}
\lim_{n\to \infty} \frac{n}{\zeta_n^2} \cdot
\log \mathbb{P}\left(\pm(\cc{\mathcal{R}_n}-\mathbb{E}[\cc{\mathcal{R}_n}])> \zeta_n\right)
=-\frac{1}{2\sigma^2}.
\end{equation}
\end{theorem}
In dimension $5$, we obtain similar estimates, but we do not reach the Gaussian regime:
\begin{theorem}\label{thm:d5}
Assume $d=5$. There exist positive constants $\varepsilon$,
$\underline \kappa$ and $\overline \kappa$,
such that for any $ n^{5/7}\cdot \log n \le \zeta\le \varepsilon n$, and $n$ large enough,
\begin{equation*}
\exp\left(- {\underline\kappa}\cdot(\frac{\zeta^2}{n})^{1/3} \right)\ \le\
\mathbb{P}\left( \cc{\mathcal{R}_n}-\mathbb{E}[ \cc{\mathcal{R}_n}]\le -\zeta\right)\ \le\
\exp\left(- {\overline\kappa}\cdot (\frac{\zeta^2}{n})^{1/3} \right).
\end{equation*}
\end{theorem}
\begin{remark}\label{rem-dev-d5}
\emph{
In $d=5$, the variance of $\mathrm{Cap}(\mathcal{R}_n)$ is of order $n\log n$,
\cite{Sch19}. Thus, the moderate deviations should go
from a gaussian regime with a speed
of order $\zeta^2/(n\log n)$, to a large deviation regime with
a speed of order $(\zeta^2/n)^{1/3}$, and
with a transition occurring for $\zeta$ of order $\sqrt {n}(\log n)^{3/4}$.
For an explanation of the exponent $5/7$ which limits us here,
see Remark \ref{rem.5}.
Note that in the case of the volume of the range,
a similar transition has been established
by Chen \cite{Chen} in dimension $3$, and by the authors in $d\ge 5$ in
the companion paper \cite{AS19}.}
\end{remark}
\begin{remark}
\emph{In dimension $6$ our result is less precise. One can only show that
\begin{equation*}
\exp\left(- {\underline\kappa} \cdot \zeta^{1/2} \right)\ \le\
\mathbb{P}\left( \cc{\mathcal{R}_n}-\mathbb{E}[ \cc{\mathcal{R}_n}]\le -\zeta\right)\ \le\
\exp\left(- \frac{{\overline\kappa}}
{\log (n/\zeta)}\cdot \zeta^{1/2} \right).
\end{equation*}
}
\end{remark}
Our next result provide path properties of the trajectory
under the constraint of moderate deviations. To state it,
one needs more notation.
For $r>0$, and $x\in \mathbb{Z}^d$, set
$$
Q(x,r):=[x-\frac r2,x+\frac r2)^d\cap \mathbb{Z}^d.
$$
Given $\Lambda\subseteq \mathbb{Z}^d$, and $n\ge 0$,
let $\ell_n(\Lambda)$ be the
time spent in $\Lambda$ before time $n$.
For $\rho \in (0,1]$, and $r,n$ positive integers, we let
\begin{equation}\label{def-CV}
{\mathcal C}_n(r,\rho):= \{x\in r\mathbb{Z}^d\, :\, \ell_n(Q(x,r))\ge \rho r^d\},
\qquad\text{and }\qquad
\mathcal{V}_n(r,\rho):=\bigcup_{x\in {\mathcal C}_n(r,\rho)} Q(x,r).
\end{equation}
Define also
for a sequence of values of deviation $(\zeta_n)_{n\ge 1}$,
\begin{eqnarray*}
\rho_{\textrm{typ}} := \left\{
\begin{array}{ll}
\zeta_n^{5/3}/n^{7/3} & \text{if }d=5\\
\zeta_n^{-2/(d-2)} & \text{if }d\ge 7,
\end{array}
\right.
\quad
\tau_{\textrm{typ}}:= \left\{
\begin{array}{ll}
n & \text{if }d=5\\
\zeta_n & \text{if }d\ge 7,
\end{array}
\right.
\quad \text{and}\quad
\chi_d:= \left\{
\begin{array}{ll}
5/7 & \text{if }d=5\\
\frac {d-2}{d} & \text{if }d\ge 7.
\end{array}
\right.
\end{eqnarray*}
\begin{theorem}\label{thm:scen-5}
Assume $d=5$, or $d\ge 7$.
There are positive constants $\alpha$, $\beta$, $\varepsilon$ and $C_0$, such that for any sequence $(\zeta_n)_{n\ge 1}$, satisfying
$$n^{\chi_d}\cdot \log n \le \zeta_n\le \epsilon n,$$
defining $(r_n)_{n\ge 1}$ by
$$r_n^{d-2}\rho_{\textrm{typ}} = C_0\log n,$$
one has
\begin{equation}\label{result.path}
\lim_{n\to\infty}
\mathbb \mathbb{P}\left(\ell_n(\mathcal{V}_n(r_n,\beta\rho_{\textrm{typ}}))
\ge \alpha\, \tau_{\textrm{typ}} \mid \cc{\mathcal{R}_n}-\mathbb{E}[ \cc{\mathcal{R}_n}]
\le -\zeta_n\right) = 1.
\end{equation}
Moreover, there exists $A>0$, such that
\begin{equation}\label{result.capacite}
\lim_{n\to\infty}
\mathbb \mathbb{P}\left(\mathrm{Cap}(\mathcal{V}_n(r_n,\beta \rho_{\textrm{typ}})) \le A|\mathcal{V}_n(r_n,\beta \rho_{\textrm{typ}})|^{1-2/d} \mid \cc{\mathcal{R}_n}-\mathbb{E}[ \cc{\mathcal{R}_n}]
\le -\zeta_n\right) = 1.
\end{equation}
\end{theorem}
Theorem~\ref{thm:scen-5} provides some information on
the density the random walk has to realize in order to achieve
the deviation. We obtain that $\mathcal{V}_n(r_n,\beta \rho_{\textrm{typ}})$ is typically ball-like,
in the sense that its capacity is of the order of its volume
to the power $1-2/d$, as it is the case for Euclidean balls.
The final result concerns the upward deviations.
Our decomposition \reff{decomp-ASS} allows us
to adapt the argument of Hamana and Kesten, \cite{HK},
written for the volume of the range of a random walk.
\begin{theorem}\label{theo:upward}
Assume $d\ge 5$. The following limit exists for all $x>0$:
\begin{equation*}
\psi_d(x):=-\lim_{n\to\infty} \ \frac{1}{n}\log \mathbb{P}\big(
\cc{\mathcal{R}_n}\ge n\cdot x\big).
\end{equation*}
Furthermore, there exists a constant $\gamma_d^*>\gamma_d$ (defined in \reff{cap.limit}),
such that the function $\psi_d$ is continuous and convex on $[0,\gamma_d^*]$, increasing on $[\gamma_d,\gamma_d^*]$, and satisfies
\begin{equation*}
\psi_d(x) \left\{\begin{array}{ll}
= 0 & \text{if }x\le \gamma_d \\
\in (0, \infty) & \text{if } x\in (\gamma_d,\gamma_d^*]\\
=\infty & \text{if }x>\gamma_d^*.
\end{array}
\right.
\end{equation*}
\end{theorem}
We also obtain Gaussian upper bounds (up to a logarithmic factor)
in the regime of moderate deviations, see Proposition \ref{cor:upward}.
\paragraph{Our approach to downward deviations.}
The cornerstone of our approach is a decomposition formula obtained
in \cite{ASS18b}:
\begin{equation}\label{decomp-ASS}
\forall A,B\quad\text{finite sets of }\mathbb{Z}^d,\qquad\quad
\cc{A\cup B}=\cc{A}+\cc{B}-\chi_\mathcal{C}(A,B),
\end{equation}
where $\chi_\mathcal{C}(A,B)$ called {\it the cross-term} has a nice expression.
In this work, the decomposition \reff{decomp-ASS} allows us to
follow a simple approach devised in \cite{AS}, and later improved in \cite{AS19}, to study
downward deviations for the volume of the range in dimensions $d\ge 3$.
We partition the time-period of length $n$ into
intervals of length $T\le n$, and by iterating \eqref{decomp-ASS} appropriately one can write our functional of the range,
$\cc{\mathcal{R}_n}$, as a sum of i.i.d. terms minus a certain sum of cross-terms
of the form $\chi_{\mathcal C}(\mathcal{R}_{iT},\mathcal{R}[iT,(i+1)T])$, with $i$ going from $1$ to $\lfloor n/T\rfloor $.
The so-called corrector, is the sum of these cross-terms that we integrate over $\mathcal{R}[iT,(i+1)T]$.
We then show that for some appropriate
time-scale $T$ it is this corrector which is responsible for (most of)
the deviations.
The final step is to estimate the cost for such deviations. This analysis is similar to the corresponding one
for the volume of the range that we performed in \cite{AS19},
but it also requires some new ingredients, in particular Lemmas \ref{lem-HD}, \ref{lem-asympt} and \ref{lem:rear}.
On the other hand the proof of Theorem \ref{theo.Gaussian} relies on the following estimate, similar to the result for the intersection of two ranges that was obtained in \cite{AS20c}:
first we observe that $\chi_{\mathcal C}(A,B)$ is bounded above by (twice)
another functional $\widetilde {\chi}(A,B)$, defined for any
$A,B\subseteq \mathbb{Z}^d$ by
$$\widetilde{\chi}(A,B) := \sum_{x\in A}\sum_{y\in B}
\mathbb{P}_x\big(H^+_{A}=\infty \big)\cdot G(x-y)\cdot
\mathbb{P}_y\big(H^+_{B}=\infty \big). $$
We then show that for some $\kappa>0$,
if $\mathcal{R}_\infty$ and $\widetilde \mathcal{R}_\infty$ are the ranges of
two independent walks,
\begin{equation}\label{stretch-chi}
\mathbb{E}\left[\exp\left(\kappa\cdot \widetilde
\chi(\mathcal{R}_\infty, \widetilde \mathcal{R}_\infty)^{1-\frac{2}{d-2}}\right)\right] <\infty.
\end{equation}
\paragraph{Heuristics.}
We use the sign $\approx$ to express
that two quantities are {\it of the comparable order} (which here will have a deliberately vague meaning, and precise statements come later).
As already mentioned, the first
step in this work is a simple decomposition
for the capacity of a union of sets in term of a cross-term
\begin{equation}\label{heur-1}
\chi_\mathcal{C}(A,B)\ \approx\ 2\!\sum_{x\in A}\sum_{y\in B}
\mathbb{P}_x\big(H^+_{A}=\infty \big)\cdot G(x-y)\cdot
\mathbb{P}_y\big(H^+_{B}=\infty \big),
\end{equation}
see \reff{decomp-1} and \reff{decomp-2} for
a precise expression.
The key phenomenon responsible for producing a small
capacity for the range of a random walk is {\it an increase
of the cross-term on an appropriate scale}. In other words, the
walk {\it folds} into a ball-like domain in order to increase
some {\it self-interaction} captured by the cross-term.
Now to be more concrete,
let us divide the range $\mathcal{R}[0,2n]$ into two subsets
$\mathcal{R}[0,n]$ and $\mathcal{R}[n,2n]$. Let us call, for simplicity $\mathcal{R}_n^1=\mathcal{R}[0,n]-S_n$,
and $\mathcal{R}_n^2=\mathcal{R}[n,2n]-S_n$ the two subranges translated by $S_n$ so that they
become independent. By \eqref{heur-1} and translation invariance of the capacity we see that
\[
\cc{\mathcal{R}[0,2n]}=\cc{\mathcal{R}_n^1}+\cc{\mathcal{R}_n^2}-\chi_\mathcal{C}(\mathcal{R}_n^1,\mathcal{R}_n^2).
\]
Now, assume that both walks stay inside a ball of radius $R$ a time
of order $\tau\le n$, and are unconstrained afterward.
Thus, under the strategy we mentioned,
\begin{equation}\label{heur-4}
\begin{split}
\chi_\mathcal{C}(\mathcal{R}_n^1,\mathcal{R}_n^2)\ \approx\ & G(R)\times \cc{\mathcal{R}_\tau^1}
\times \cc{\mathcal{R}_\tau^2}+\mathcal{O}\big(G(\sqrt n)n^2\big)\\
\ \approx\ & G(R)\big( \min(\tau,R^{d-2})\big)^2+\mathcal{O}\big(n^{\frac{6-d}{2}}\big).
\end{split}
\end{equation}
The term $\mathcal{O}\big(G(\sqrt n)n^2\big)$ appears if $\tau$ is smaller than $n$,
and accounts for the unconstrained contribution to the cross-term.
In obtaining \reff{heur-4}, we have used that if $\mathcal{R}_\tau^1$ and $\mathcal{R}_\tau^2$ are inside
a ball of radius $R$, then their capacity is bounded by the
capacity of the ball, which is of order $R^{d-2}$, as well as by
their volume bounded by $\tau$.
Thus, it is useless to consider $\tau$ larger than $R^{d-2}$,
since then $\tau$ no more affects the cross-term and increasing $\tau$ (or decreasing $R$ below $\tau$) only makes the strategy more costly.
Now a deviation of order $\zeta$ is reached if
\begin{equation}\label{heur-2}
\frac{1}{R^{d-2}} \tau^2\approx \zeta.
\end{equation}
Recall that the cost of being localized a time $\tau$ in a ball of radius $R$ is of order $\exp(-\tau/R^2)$ (up to a constant in the exponential).
So we need to find a choice of $(\tau,R)$ which minimizes this cost under the
constraint \reff{heur-2}. In other words one needs to maximize $\sqrt \zeta\cdot R^{(d-6)/2}$.
This leads to two regimes.
\begin{itemize}
\item When $d=5$, $R$ (and then $\tau$) is as large as possible. So, $\tau=n$ and
$R^{d-2}=n^2/\zeta$ by \reff{heur-2}. The strategy is time homogeneous for any
$\zeta$!
\item When $d\ge 7$, then $\tau$ is as small as possible,
that is $\tau=R^{d-2}=\zeta$. The strategy is time-inhomogeneous.
\end{itemize}
When $d=6$, the strategy remains unknown, but the cost should
be of order $\exp(-\sqrt \zeta)$.
\paragraph{Application to a polymer melt.}
The model of random interlacements, introduced by Sznitman \cite{S10}, is roughly speaking
the union of the ranges of trajectories obtained by a Poisson
point process on the space of doubly infinite trajectories, and is such that
the probability of avoiding a set $K$ is $\exp(-u\cdot \cc{K})$, where
$u>0$ is a fixed parameter.
With this in mind, let us consider the following model of polymer
among a polymer melt interacting by exclusion.
We distinguish one polymer, a simple random walk, interacting with
a cloud of other random walk trajectories modeled by
random interlacements which we call for short {\it the melt}.
The interaction is through exclusion:
the walk and the melt do not intersect. When integrating
over the interlacements law, the measure on the walk with the effective
interaction has a density proportional to $\exp(-u \cdot \cc{\mathcal{R}_n})$, with respect to the law of a simple random walk.
As a corollary of our deviation estimates, one can address some issues
on this polymer.
Since this follows in the same way as the study of the Gibbs
measure tilted by the volume of the range was a corollary of \cite{AS}, we
repeat neither the statements corresponding to Theorem 1.8 of \cite{AS},
nor the proofs here.
The simplest and most notable difference with the latter theorem is that
the proper scaling of the intensity parameter $u$ which provides a phase
transition is when it is of order $n^{-2/(d-2)}$ in dimension $d\ge 5$.
Thus, one would consider the polymer partition function as a function
of $u\in \mathbb{R}^+$
\[
Z_n(u)=\mathbb{E}\Big[\exp\big(-\frac{u}{n^{2/(d-2)}}(
\cc{\mathcal{R}_n}-\mathbb{E}[\cc{\mathcal{R}_n}])\ \big)\Big].
\]
Theorem 1.8 of \cite{AS} is true, here also after the drop in dimension
is performed, and establishes the existence of a phase transition
as one tunes $u$.
On the other hand, considering the quenched model, where the random interlacements is given a typical realization,
is an interesting open problem, beyond
the reach of the present techniques.
\paragraph{Organization.} The paper is organized as follows. In the next section, we recall some basic estimates on the random walk, the capacity, and the range that we will need. Section~\ref{sec-martin}, and more precisely Proposition~\ref{prop.corrector}
makes the link between downward deviations for the capacity and upward deviations of a corrector.
The corrector itself
is studied in Section~\ref{sec-UB}, where we prove the upper bounds in
Theorems~\ref{thm:d6} and \ref{thm:d5}, as well as Theorem \ref{thm:scen-5}.
In Section \ref{sec-LB}, we prove the lower
bounds in Theorems~\ref{thm:d6} and~\ref{thm:d5}.
The proof of Theorem \ref{theo.Gaussian} is done in Section \ref{sec.Gaussian}. Finally, we prove Theorem~\ref{theo:upward} concerning
the upward deviations in Section~\ref{sec-HK}.
\section{Preliminaries}\label{sec-prelim}
\subsection{Further notation}
For $z\in \mathbb{Z}^d$, $d\ge 5$, we denote by $\mathbb{P}_z$ the law of the simple random walk starting from $z$, and simply write it $\mathbb{P}$ when $z=0$. We let
$$G(z):=\mathbb E\left[\sum_{n=0}^\infty {\bf 1}\{S_n=z\}\right],$$
be the Green's function.
It is known (see \cite{Law91}) that for some positive constants $c$ and $C$,
\begin{equation}\label{Green}
\frac{c}{\|z\|^{d-2}+1}\ \le \ G(z)\ \le \ \frac{C}{\|z\|^{d-2}+1},\quad \text{for all }z\in \mathbb{Z}^d,
\end{equation}
with $\|\cdot \|$ the Euclidean norm.
We also consider for $T>0$, and $z\in \mathbb{Z}^d$,
$$G_T(z):=\mathbb E\left[\sum_{n=0}^T {\bf 1}\{S_n=z\}\right].$$
In particular for any $z\in \mathbb{Z}^d$, and $T\ge 1$,
\begin{equation}\label{inGT}
\mathbb P(z\in \mathcal{R}_T) \le G_T(z).
\end{equation}
For $A\subset \mathbb{Z}^d$, we denote by $|A|$ the cardinality of $A$, and by
$$H_A:=\inf\{n\ge 0\, :\, S_n\in A\},\quad \text{and}\quad H_A^+:= \inf\{n\ge 1\, :\, S_n\in A\},$$
respectively the hitting time of $A$ and the first return time to $A$.
We also need the following well known fact, see \cite{Law91}.
There exists a constant $C>0$, such that for any $R>0$ and $z\in \mathbb{Z}^d$,
\begin{equation}\label{hit}
\mathbb{P}_z\left(\inf_{k\ge 0}\|S_k\|\le R\right) \le C\cdot \left(\frac{R}{\|z\|}\right)^{d-2}.
\end{equation}
\subsection{On the capacity}\label{sec-capa}
The capacity of a finite subset $A\subset \mathbb{Z}^d$, with $d\ge 3$, is defined by
\begin{equation}\label{cap.def}
\mathrm{Cap}(A):=\lim_{\|z\|\to \infty} \, \frac{1}{G(z)} \mathbb{P}_z(H_A<\infty).
\end{equation}
It is well known, see Proposition 2.2.1 of \cite{Law91},
that the capacity is monotone for inclusion:
\begin{equation}\label{cap.mon}
\mathrm{Cap}(A)\le \mathrm{Cap}(B), \quad \text{for any }A\subset B,
\end{equation}
and satisfies the sub-additivity relation
\begin{equation}\label{cap.subadd}
\mathrm{Cap}(A\cup B)\le \mathrm{Cap}(A) +\mathrm{Cap} (B) - \mathrm{Cap}(A\cap B),\quad \text{for all }A,B\subset \mathbb{Z}^d.
\end{equation}
Another equivalent definition of the capacity is the following
(see (2.12) of \cite{Law91}).
\begin{equation}\label{cap.def2}
\mathrm{Cap}(A) =\sum_{x\in A} \mathbb{P}_x(H_{A^+}=\infty).
\end{equation}
In particular it implies that
\begin{equation}\label{cap.card}
\mathrm{Cap}(A) \le |A|, \quad \text{for all }A\subset \mathbb{Z}^d.
\end{equation}
The starting point for our decomposition is the definition \eqref{cap.def}
of the capacity in terms of a hitting time.
It implies that for any two finite subsets $A,B\subset \mathbb{Z}^d$,
\begin{equation}\label{decomp-1}
\cc{A\cup B}=\cc{A}+\cc{B}-\chi_{\mathcal C}(A,B),
\end{equation}
with
\begin{equation*}
\chi_{\mathcal C}(A,B):=\lim_{z\to\infty} \frac{1}{G(z)}
\mathbb{P}_z\big(\{H_{A}<\infty\}\cap \{H_{B}<\infty\}\big).
\end{equation*}
In particular by \eqref{cap.def} and the latter formula, one has
\begin{equation}\label{borne.cap.AUB}
0\le \chi_{\mathcal C}(A,B) \le \min(\mathrm{Cap}(A),\mathrm{Cap}(B)).
\end{equation}
Now, we have shown in \cite{ASS19} that
\begin{equation}\label{decomp.chiC}
\chi_{\mathcal C}(A,B)=\chi(A,B) + \chi(B,A) -\varepsilon(A,B),
\end{equation}
with
\begin{equation}\label{decomp-2}
\chi(A,B)=\sum_{x\in A}\sum_{y\in B}
\mathbb{P}_x\big(H^+_{A\cup B}=\infty \big)\cdot G(x-y)\cdot
\mathbb{P}_y\big(H^+_{B}=\infty \big),
\end{equation}
and,
\begin{equation}\label{decomp-2bis}
0\le \varepsilon(A,B)\le \cc{A\cap B}\le |A\cap B|,
\end{equation}
where the last inequality follows from \eqref{cap.card}.
We will need some control on the speed of convergence in \eqref{cap.limit}.
\begin{lemma}\label{lem.exp.cap}
Assume $d\ge 5$. One has
\begin{equation*}
\left| \mathbb E[\mathrm{Cap}(\mathcal{R}_n)] - \gamma_d n \right| = \mathcal O(\psi_d(n)),
\end{equation*}
with
\begin{equation*}
\psi_d(n) = \left\{
\begin{array}{ll}
\sqrt n & \text{if }d=5\\
\log n & \text{if }d=6\\
1 & \text{if }d\ge 7.
\end{array}
\right.
\end{equation*}
\end{lemma}
\begin{proof}
By \eqref{decomp-1}, \eqref{decomp.chiC}, \eqref{decomp-2}, and \eqref{decomp-2bis} one has the rough lower bound:
\begin{equation}\label{decomp.lemma}
\mathrm{Cap}(\mathcal{R}_{n+m}) \ge \mathrm{Cap}(\mathcal{R}_n) + \mathrm{Cap}(\mathcal{R}[n,n+m]) - 2\sum_{k=0}^n \sum_{\ell = n}^{n+m} G(S_k -S_\ell),
\end{equation}
for any integers $n,m\ge 1$ (a better inequality will be used later, but this one is enough here).
Then one concludes exactly as in \cite{AS2}, using \eqref{cap.subadd}, Hammersley's lemma and Lemma 3.2 in \cite{ASS18a},
which controls the moments of the error term in the right-hand side of \eqref{decomp.lemma}.
For the details, we refer to the proof of (1.13) in \cite{AS2}.
\end{proof}
The next result provides some useful bounds on the variance of the capacity of the range, which were obtained in \cite{Sch19} in case of dimension $5$, and in \cite{ASS18a} in higher dimension.
\begin{proposition}\label{lem.var.cap}
One has,
\begin{eqnarray*}
\operatorname{var}(\mathrm{Cap}(\mathcal{R}_n)) =\left\{
\begin{array}{ll}
\mathcal{O}( n \log n) & \text{if }d=5\\
\mathcal{O}(n) & \text{if }d\ge 6.
\end{array}
\right.
\end{eqnarray*}
\end{proposition}
\begin{remark}
\emph{
Actually sharp asymptotics are known: in dimension $5$, one has $\operatorname{var}(\mathrm{Cap}(\mathcal{R}_n))\sim \sigma_5 n\log n$, and in higher dimension $\operatorname{var}(\mathrm{Cap}(\mathcal{R}_n))\sim \sigma_dn$, for some positive constant $(\sigma_d)_{d\ge 5}$, see respectively \cite{Sch19} and \cite{ASS18a}. }
\end{remark}
As a consequence of the previous results one can obtain Gaussian type upper bounds for the moderate deviations in the upward deviations.
\begin{proposition}\label{cor:upward}
There exist positive constants $(c_d)_{d\ge 5}$, such that for any $n\ge 2$, and $\zeta>0$,
\begin{eqnarray*}
\mathbb{P}\big(\cc{\mathcal{R}_n} - \mathbb{E}[\cc{\mathcal{R}_n}]\ge \zeta \big) \le \left\{
\begin{array}{lll}
\exp\left(-c_5\cdot \frac{\zeta^2}{n(\log n)^3}\right) & \text{if }d=5\\
\exp\left(-c_6\cdot \frac{\zeta^2}{n(\log \log n)^2}\right) & \text{if }d=6\\
\exp\left(-c_d\cdot \frac{\zeta^2}n\right) & \text{if }d\ge 7.
\end{array}
\right.
\end{eqnarray*}
\end{proposition}
\begin{proof}
For simplicity let us concentrate on the proof when $d=5$.
We will explain at the end the necessary modifications to the proof when $d\ge 6$.
Note first that one can always assume that $\zeta$ is smaller than $n/2$.
We use now \eqref{decomp-ASS} repeatedly along a dyadic decomposition of $\{0,\dots,n\}$. This gives for $L\ge 1$, with $m_L:=\lfloor n/2^L\rfloor$,
$$\cc{\mathcal{R}_n} = \sum_{i=1}^{2^L} \cc{\mathcal{R}^{(i)}_{m_L}} - \sum_{\ell = 1}^L \Sigma_\ell,$$
where the $\mathcal{R}^{(i)}_{m_L}$ are consecutive pieces of the range of length either $m_L$ or $m_L+1$, and
$$\Sigma_\ell := \sum_{j=1}^{2^{\ell - 1}} \chi_C(\mathcal{R}^{(2j-1)}_{m_\ell},\mathcal{R}^{(2j)}_{m_\ell}),$$
with similar notation as above, in particular $m_\ell = \lfloor n/2^\ell\rfloor$.
Thus,
\begin{align}\label{proof.upward}
\nonumber \mathbb{P}\big(\cc{\mathcal{R}_n}-\mathbb{E}[\cc{\mathcal{R}_n} ]> \zeta \big) \le\ &
\mathbb{P}\left(\sum_{i=1}^{2^L} \cc{\mathcal{R}^{(i)}_{m_L}}-\mathbb{E}[\cc{\mathcal{R}^{(i)}_{m_L}}] >\frac \zeta 2\right)\\
& + \sum_{\ell = 1}^{L} \mathbb{P}\left( \mathbb{E}[\Sigma_\ell] - \Sigma_\ell > \frac{\zeta}{2L}\right).
\end{align}
We fix now $L$, such that $n/\zeta \le m_L \le 2n/\zeta$.
The first term in \eqref{proof.upward} is ruled out using Bernstein's inequality and Proposition \ref{lem.var.cap}, which give
for some constant $c>0$.
\begin{eqnarray}\label{proof.upward2}
\mathbb{P}\left(\sum_{i=1}^{2^L} \cc{\mathcal{R}^{(i)}_{m_L}}-\mathbb{E}[\cc{\mathcal{R}^{(i)}_{m_L}}] >\zeta/2\right) \le \exp\left(-c \frac{\zeta^2}{n \log m_L}\right).
\end{eqnarray}
Concerning the sum in \eqref{proof.upward}, note first that by Lemma \ref{lem.exp.cap}, one has
$$\mathbb{E}[\Sigma_\ell] = \mathcal{O}(2^{\ell/2} \sqrt n) ,$$
for any $\ell \ge 1$. Therefore, one can assume that $\ell$ is such that $2^{\ell/2} \sqrt{n} > c\zeta/L$, for some constant $c>0$, for otherwise the corresponding probability is zero.
For such $\ell$ one has by using standard concentration results (see Theorem 4.4. in \cite{CL}):
$$\mathbb{P}\left( \mathbb{E}[\Sigma_\ell] - \Sigma_\ell > \frac{\zeta}{2L}\right) \le \exp\left(-c\frac{(\zeta/L)}{ \sqrt{m_\ell} +L n (\log m_\ell)/\zeta}\right) \le \exp\left(-\frac{c\zeta^2}{n(\log n)^3}\right),$$
which completes the proof in case $d=5$. In case $d\ge 6$, the variance is linear. So first, the term $\log m_L$ can be removed in \eqref{proof.upward2}.
Moreover, in case $d=6$, one has $\mathbb{E}[\Sigma_\ell ] = \mathcal{O}(2^\ell \log n)$, and thus only the $\ell$'s such that $\zeta \ge 2^\ell \ge c\zeta/\log n$ need to be considered.
There are order $\log \log n$ such integers, and for each of them one has by the same argument as above,
$$ \mathbb{P}\left( \mathbb{E}[\Sigma_\ell] - \Sigma_\ell > \frac{\zeta}{C\log \log n}\right)\le \exp(-c\zeta^2/(\log \log n)^2),$$
which concludes the proof in case $d=6$. The case $d\ge 7$ is similar, since this time $\mathbb{E}[\Sigma_\ell ] = \mathcal{O}(2^\ell)$, and thus there are only a bounded number of integers $\ell$'s that need to be considered.
\end{proof}
\section{Transfer of downward deviations to the corrector}\label{sec-martin}
The possibility of establishing the heuristic picture
described in the introduction stems from writing
the capacity of a union of sets as a sum of capacities
and a cross-term. The latter though typically small is nonetheless
responsible for the fluctuations.
Iterating this decomposition leads to an expression of the capacity of the range as a sum of i.i.d. terms minus a sum of cross-terms.
The so-called corrector is obtained by summing appropriate conditional expectations of these cross-terms.
Our first result in this section, Lemma \ref{lem.corrector}, provides an explicit expression for (what turns out to be an upper bound for) this corrector in terms of a sum of convoluted
Green's functions taken along the trajectory and
weighted by escape probability terms. We then recall a result from \cite{AS19}, which relates the deviations of the capacity to those of the corrector, which we state here as Proposition \ref{prop.corrector}.
Thus, the strategy is similar to the one used to treat
downward deviations for the range
developed in \cite{AS19}. However
the form of the corrector is slightly different. Roughly it involves a convolution of Green's function with itself together multiplied by escape
probability terms, where in \cite{AS} only Green's function appeared.
A detailed analysis of this corrector is
carried out in Sections \ref{sec-d7} and \ref{sec-d5}.
Before we can state precisely the result,
some preliminary notation is required.
For $I\subset \mathbb{N} $, we write $\mathcal{R}(I):=\{S_k,\ k\in I\}$,
for the set of visited sites during times $k\in I$.
Since for any two intervals $I,J\subset \mathbb{N}$, one has $\mathcal{R}(I\cup J)=\mathcal{R}(I)\cup \mathcal{R}(J)$, \reff{decomp-1} gives
\begin{equation}\label{decomp-3}
\cc{\mathcal{R}(I\cup J)}= \cc{\mathcal{R}(I)}+\cc{\mathcal{R}(J)}-\chi_{\mathcal C}\big(\mathcal{R}(I),\mathcal{R}(J)\big).
\end{equation}
Next, given two sets $A$ and $B$, their symmetric difference is defined as $A\Delta B:=(A\cap B^c) \cup (B\cap A^c)$.
Note in particular that for any $I,J\subset \mathbb{N}$, one has $\mathcal{R}(I)\Delta \mathcal{R}(J)\subset \mathcal{R}(I\Delta J)$.
Moreover, it follows from \eqref{cap.mon}, \eqref{cap.subadd} and \eqref{cap.card} that for any $A,B\subset \mathbb{Z}^d$,
$$|\mathrm{Cap}(A)-\mathrm{Cap}(B)|\le \mathrm{Cap}(A\Delta B)\le |A\Delta B|. $$
Applying this inequality to ranges on some intervals $I$ and $J$, we get
\begin{equation}\label{diff.sym.ranges}
|\mathrm{Cap}(\mathcal{R}(I)) - \mathrm{Cap}(\mathcal{R}(J))|\le |I\Delta J |.
\end{equation}
Now given some integer $T\le n$, we define for $j\ge 0$ and $\ell \ge 1$,
$$I_{j,\ell}:=[j+(\ell-1)T,j+\ell T], \quad \text{and}\quad \widetilde I_{j,\ell}:=I_{j,1}\cup \dots\cup I_{j,\ell}.$$
It follows from \eqref{diff.sym.ranges} that almost surely
\begin{equation}\label{decomp-6}
|\mathrm{Cap}(\mathcal{R}_n)- \frac{1}{T}\sum_{j=0}^{T-1}
\mathrm{Cap}(\mathcal{R}(\widetilde I_{j,\lfloor n/T\rfloor} )) |\le T.
\end{equation}
On the other hand, applying \eqref{decomp-3} recursively we obtain for any $j=0,\dots,T-1$,
\begin{equation}\label{decomp-7}
\mathrm{Cap}(\mathcal{R}(\widetilde I_{j,\lfloor n/T\rfloor} ) ) = \sum_{\ell=1}^{ \lfloor n/T\rfloor}
\mathrm{Cap} (\mathcal{R}(I_{j,\ell})) - \sum_{\ell=1}^{\lfloor n/T\rfloor -1}
\chi_{\mathcal C} \big(\mathcal{R}(\widetilde I_{j,\ell}),\mathcal{R}(I_{j,\ell+1})\big) .
\end{equation}
Define now
$$\chi_n(T):= \frac{1}{T}\sum_{j=0}^{T-1} \sum_{\ell=1}^{\lfloor n/T\rfloor -1}
\chi_{\mathcal C} \big(\mathcal{R}(\widetilde I_{j,\ell}),\mathcal{R}(I_{j,\ell+1})\big), $$
and note that \eqref{decomp-6} and \eqref{decomp-7} give for any $T\le \zeta/2$,
\begin{align}\label{dev-decomp}
\nonumber & \mathbb{P}\left(\mathrm{Cap}(\mathcal{R}_n)- \mathbb E[\mathrm{Cap}(\mathcal{R}_n)] \le -\zeta \right) \le \mathbb{P}\left( \frac{1}{T}\sum_{j=0}^{T-1} \mathrm{Cap}(\mathcal{R}(\widetilde I_{j,\lfloor n/T\rfloor} )) - \mathbb E[\mathrm{Cap}(\mathcal{R}(\widetilde I_{j,\lfloor n/T\rfloor} ))] \le -\frac {\zeta}{2}\right) \\
& \le \mathbb{P}\left( \frac{1}{T}\sum_{j=0}^{T-1} \sum_{\ell=1}^{ \lfloor n/T\rfloor}
\mathrm{Cap} (\mathcal{R}(I_{j,\ell})) - \mathbb E[\mathrm{Cap} (\mathcal{R}(I_{j,\ell}))] \le -\frac {\zeta}{4}\right) + \mathbb{P}\left(\chi_n(T) \ge \frac{\zeta}{4}\right).
\end{align}
The first term on the right-hand side of \eqref{dev-decomp} is dealt with Bernstein's inequality and Proposition \ref{lem.var.cap}, which show that for any $\zeta> \frac{n\log n}{T}$,
for some constant $c>0$.
\begin{align}\label{dev-decomp.2}
\nonumber & \mathbb{P}\left( \frac{1}{T}\sum_{j=0}^{T-1} \sum_{\ell=1}^{ \lfloor n/T\rfloor}
\mathrm{Cap} (\mathcal{R}(I_{j,\ell}))- \mathbb E[\mathrm{Cap} (\mathcal{R}(I_{j,\ell}))] \le -\frac {\zeta}{4}\right) \\
& \le T \max_{0\le j\le T-1}
\mathbb{P}\left( \sum_{\ell=1}^{ \lfloor n/T\rfloor}
\mathrm{Cap} (\mathcal{R}(I_{j,\ell}))- \mathbb E[\mathrm{Cap} (\mathcal{R}(I_{j,\ell}))]
\le -\frac {\zeta}{4}\right) \le T \exp(-c\frac{\zeta}{T}).
\end{align}
For the second term in the right-hand side of \eqref{dev-decomp}, we will use a general result of \cite{AS19}, which allows to compare the (moderate) deviations of $\chi_n(T)$ to those of its compensator, defined by
\begin{equation}\label{def-xin*}
\xi_n^*(T):= \frac 1T \sum_{j=0}^{T-1} \sum_{\ell = 1}^{\lfloor n/T\rfloor-1} \mathbb{E}\left[\chi_{\mathcal C} \big(\mathcal{R}(\widetilde I_{j,\ell}),\mathcal{R}(I_{j,\ell+1})\big)\mid \mathcal F_{j+\ell T}\right].
\end{equation}
More specifically, Proposition 4.1 in \cite{AS19} (see also the proof of Corollary 4.2 there) shows that for some constant $c>0$, for any $\zeta>0$,
\begin{equation}\label{dev-chinT}
\mathbb{P}(\chi_n(T) \ge \frac {\zeta}{4}) \le \exp(-c\frac{\zeta}{T}) + \mathbb{P}(\xi_n^*(T) \ge c\zeta ) ,
\end{equation}
(where here we use also that by \eqref{cap.card} and \eqref{borne.cap.AUB}, each term of the sum in the definition of $\chi_n(T)$ is bounded by $T$).
We next define
\begin{equation}\label{def-xin}
\xi_n(T):= \sum_{k=0}^n \sum_{x\in \mathcal{R}_k}
\mathbb{P}_x\big(H^+_{\mathcal{R}_k}=\infty\big)\cdot\frac{G\star G_T(x-S_k)}{T}.
\end{equation}
\begin{lemma}\label{lem.corrector}
One has, for any $n\ge 1$ and $1\le T\le n$,
$$\xi_n^*(T)\le 2\xi_n(T).$$
\end{lemma}
\begin{proof} By \eqref{decomp-2}, for any sets $A$ and $B$,
$$\chi(A,B)\le \widetilde \chi(A,B):= \sum_{x\in A}\sum_{y\in B}
\mathbb{P}_x\big(H^+_A=\infty \big)\cdot G(x-y)\cdot
\mathbb{P}_y\big(H^+_{B}=\infty \big). $$
Note that $\widetilde \chi$ is symmetric
in the sense that $\widetilde \chi(A,B) = \widetilde \chi(B,A)$, for any $A,B$.
Bounding the last probability term by one, we get
$$\chi_{\mathcal C}(A,B)\stackrel{\eqref{decomp.chiC}}{\le} \chi(A,B) + \chi(B,A) \le 2 \overline \chi(A,B),
\quad\text{with }\quad
\overline \chi(A,B):=\sum_{x\in A} \sum_{y\in B} \mathbb{P}_x\big(H^+_A=\infty \big)\cdot G(x-y).$$
Now for any $j,\ell$, the Markov property and translation invariance of the simple random walk give
\begin{align*}
\mathbb E\left[\overline \chi \big(\mathcal{R}(\widetilde I_{j,\ell}),\mathcal{R}(I_{j,\ell+1})\big)
\mid \mathcal F_{j+\ell T}\right]
& =\ \sum_{x\in \mathcal{R}(\widetilde I_{j,\ell}) } \mathbb P_x(H_{\mathcal{R}(\widetilde I_{j,\ell})}^+=\infty) \sum_{y\in \mathbb{Z}^d} G(x-y) \cdot
\mathbb P\big( y \in \mathcal{R} (I_{j,\ell+1}) \mid \mathcal F_{j+\ell T}\big) \\
& \stackrel{\eqref{inGT}}{\le} \, \sum_{x\in \mathcal{R}(\widetilde I_{j,\ell})} \mathbb P_x(H_{\mathcal{R}(\widetilde I_{j,\ell})}^+=\infty) \cdot G\star G_T(x-S_{j+\ell T}),
\end{align*}
and the lemma follows from the definition \eqref{def-xin} and \eqref{def-xin*} of $\xi_n(T)$ and $\xi_n^*(T)$ respectively.
\end{proof}
Combining \eqref{dev-decomp}, \eqref{dev-decomp.2}, \eqref{dev-chinT}, and Lemma \ref{lem.corrector} we obtain the main result of this section.
\begin{proposition}\label{prop.corrector}
There exists a positive constant $c$, such that for any $n\ge 2$, $\zeta>0$, and $T\ge 1$ satisfying $T\le \zeta/2$, and $\zeta \ge \frac{n\log n}{T}$,
$$\mathbb{P}\left(\mathrm{Cap}(\mathcal{R}_n)- \mathbb E[\mathrm{Cap}(\mathcal{R}_n)] \le -\zeta \right) \le 2T\exp(-c\frac{\zeta}{T}) + \mathbb{P}\left(\xi_n(T) \ge c\zeta\right).$$
\end{proposition}
\begin{remark}\label{rem.5}
\emph{In dimension $5$, the mean of $\xi_n(T)$ is of order $n/\sqrt T$. So the upper deviations for $\xi_n(T)$ start to decay only for
$\zeta>n/\sqrt T$, and since on the other hand one needs to take $T$ at most of order $(\zeta n)^{1/3}$, to ensure the term $\exp(-c\zeta/T)$ to have the
right decay, this imposes the condition $\zeta>n^{5/7}$. In particular the approach we have here has no chance to work up to the Gaussian regime.
On the other hand in dimension $7$ and higher, the mean of $\xi_n(T)$ is of order $n/T$, and $T$ can be chosen of order $\zeta^{2/(d-2)}$,
which only imposes the a priori condition $\zeta>n^{(d-2)/d}$, leaving a chance to cover entirely the non-Gaussian regime.}
\end{remark}
\section{Upper Bounds}\label{sec-UB}
We prove here the upper bounds in \eqref{dev.nongauss.7} and in Theorem \ref{thm:d5}, as well as Theorem \ref{thm:scen-5}.
We start by some preliminaries, which shall be used as well in Section \ref{sec.Gaussian}, concerning the Gaussian regime.
\subsection{Basic estimates}\label{sec-ineq}
For $r>0$, and $x\in \mathbb{R}^d$, we recall that
$ Q(x,r):=[x-r/2,x+r/2)^d\cap \mathbb{Z}^d,$
and for simplicity $Q(r):=Q(0,r)$.
\begin{lemma}\label{lem-HD}
Assume that $d\ge 5$.
There exists a constant $C_1>0$, such that for any $r\ge 1$, and any
$\Lambda\subset Q(r)$,
\begin{equation}\label{lem-HD.1}
\sum_{x\in \Lambda} \frac{1}{\|x\|^{d-4}+1}
\cdot \mathbb{P}_x(H^+_\Lambda=\infty)\, \le\, C_1\, r^2.
\end{equation}
\end{lemma}
\begin{proof}
Without loss of generality, one can assume $r\ge 2$. For $i\ge 0$, write
$$\Lambda_i := \Lambda\cap \left( Q( r2^{-i})\backslash Q(r2^{-i-1})\right),$$
and define
$L:=\lfloor \log_2(r) \rfloor$. Then, for some positive constants $C_0$ and $C_1$,
\begin{equation*}
\begin{split}
\sum_{x\in \Lambda} \frac{1}{\|x\|^{d-4}+1}
\cdot \mathbb{P}_x(H^+_\Lambda=\infty) \ &
\le \ \sum_{i=0}^L \sum_{x\in \Lambda_i}
\frac{1}{\|x\|^{d-4}+1}\cdot \mathbb{P}_x(H^+_\Lambda=\infty)\\
& \le \ \sum_{i=0}^L \big(\frac{2^{i+1}}{r}\big)^{d-4}\cc{\Lambda_i }
\le\ \sum_{i=0}^L \big(\frac{2^{i+1}}{r}\big)^{d-4}\cc{Q(\frac{r}{2^i})}\\
&\le C_0\, \sum_{i=0}^L \big(\frac{2^{i+1}}{r}\big)^{d-4}\cdot
\big(\frac{r}{2^{i}}\big)^{d-2} \le\ C_1 \, r^2.
\end{split}
\end{equation*}
\end{proof}
The second result we need is the following.
\begin{lemma}\label{lem-asympt}
Assume $d\ge 5$. There exists a constant $C_2>0$, such that for any $x\in \mathbb{Z}^d$, and any $T\ge 1$,
\begin{equation*}
\varphi_T(x):=\frac{G\star G_T(x)}{T}\ \le\ C_2\cdot \min\left(\frac{1}{1+\|x\|^{d-2}},
\frac{1}{T(1+\|x\|^{d-4})}\right).
\end{equation*}
\end{lemma}
\begin{proof}
First $G_T\le G$, so that $G\star G_T\le G\star G$, and
an elementary computation gives that $G\star G(x)\le C_2/ (1+\|x\|^{d-4})$, for all $x\in \mathbb{Z}^d$, and some $C_2>0$. This already proves one of the two desired bounds.
For the other one write, by definition of $G_T$,
\begin{equation}\label{GstarGS}
G\star G_T(x) = \sum_{y\in \mathbb{Z}^d} G(x-y)G_T(y) = \sum_{k=1}^T \mathbb E[G(x-S_k)].
\end{equation}
Let $\tau$ be the hitting time of the cube $Q(x,2)$ for the walk starting at 0,
and note that one can assume $\|x\|\ge 4$.
Since $G$ is harmonic on $\mathbb{Z}^d\backslash \{0\}$, we have for any $k\ge 0$,
$\mathbb{E} [G(x-S_{k\wedge \tau})]=G(x)$. This entails
\[
G(x)=\mathbb{E} [{\text{\Large $\mathfrak 1$}}\{\tau\ge k\}G(x-S_{k})]+
\mathbb{E} [{\text{\Large $\mathfrak 1$}}\{\tau<k\}G(x-S_{\tau})]\ge
\mathbb{E} [G(x-S_k)]-\mathbb{E}[{\text{\Large $\mathfrak 1$}}\{\tau<k\} G(x-S_k)].
\]
Now, we use that $G(x)$ is bounded by $G(0)$, so that the previous inequality gives
\[
\mathbb{E} [G(x-S_k)] \le G(x)+G(0) \mathbb{P}(\tau<\infty) \stackrel{\eqref{hit}}{\le} (1+CG(0))\cdot G(x),
\]
for some constant $C>0$.
Injecting this in \eqref{GstarGS} and using \eqref{Green}, proves the second inequality.
\end{proof}
Our last estimate requires some new notation.
For a (deterministic) function $S:\mathbb{N} \to \mathbb{Z}^d$ (not necessarily to the nearest neighbor),
and for any $\mathcal{K}\subset \mathbb{N}$, we define for any $\Lambda\subset \mathbb{Z}^d$,
$$\ell_\mathcal{K}(\Lambda):=\sum_{k\in \mathcal{K}} {\bf 1}\{S(k) \in \Lambda\}.$$
\begin{lemma}\label{lem:rear}
Assume $d\ge 3$. Let $S:\mathbb{N} \to \mathbb{Z}^d$, and $\mathcal{K}\subset \mathbb{N}$, be such that for some $\rho\in (0,1)$ and $r\ge 1$,
$$\ell_\mathcal{K}(Q(x,r)) \le \rho r^d, \qquad \text{for all }x\in r\mathbb{Z}^d.$$
There exists a constant $C_3>0$ (independent of $\rho$, $r$, $S$, and $\mathcal{K}$), such that for any $z\in \mathbb{Z}^d$,
\begin{equation}\label{lem:rear.2}
\sum_{k\in \mathcal{K}} \frac{{\text{\Large $\mathfrak 1$}}(\|S(k)-z\|\ge 2r)}{\|S(k)-z\|^{d-2}} \ \le\ C_3 \, \rho^{1-\frac 2d} \, |\mathcal{K}|^{2/d}.
\end{equation}
\end{lemma}
\begin{proof}
We start by proving that for any $R\ge 2r$, and any $z\in \mathbb{Z}^d$,
\begin{equation}\label{lem:rear.1}
\sum_{k\in \mathcal{K}} \frac{{\text{\Large $\mathfrak 1$}}(2r \le \|S(k)-z\|\le R)}{\|S(k)-z\|^{d-2}} \ \le\ C_3 \, \rho \, R^2.
\end{equation}
Consider a covering of the cube $Q(z,R)$ by a partition made of smaller cubes which are translates of $Q(r)$, with centers in the set $z+r\mathbb{Z}^d$.
For each $x\in z+r\mathbb{Z}^d$, with $x\neq z$, the contribution of the points $S(k)$ lying in $Q(x,r)$ to the sum we need to bound, is upper bounded (up to some constant) by $\rho r^d \cdot \|x-z\|^{2-d}$, and \eqref{lem:rear.1} follows as we observe that, for some constant $C>0$,
$$\sum_{x\in z+r\mathbb{Z}^d} \frac{{\text{\Large $\mathfrak 1$}}\{r\le \|z-x\|\le R\}}{\|z-x\|^{d-2}} \le C \frac{R^2}{r^d}.$$
We then deduce \eqref{lem:rear.2}, by observing that by rearranging the points $(S(k))_{k\in \mathcal{K}}$, one can only increase the sum (at least up to a multiplicative constant) by assuming they are all in $Q(z,2(\frac{|\mathcal{K}|}{\rho})^{1/d})$, and still satisfy the hypothesis of the lemma.
\end{proof}
\subsection{The sets $\mathcal K_n$}\label{sec.KnAn}
We recall here our main tools from \cite{AS19}, which require some new notation. For $n\ge 0$, and $\Lambda\subseteq \mathbb{Z}^d$, define
the time spent in $\Lambda$ by the walk up to time $n$ as
$$
\ell_n(\Lambda):= \sum_{k=0}^n {\text{\Large $\mathfrak 1$}}\{S_k\in \Lambda\}.
$$
Then given $\rho>0 $, $r\ge 1$, and $n\ge 1$, set
\begin{equation}\label{def-K}
\mathcal{K}_n(r,\rho):=\{k\le n:\ \ell_n(Q(S_k,r))\ge \rho r^d\}.
\end{equation}
The following result is Theorem 1.5 and Proposition 3.1 from \cite{AS19}.
\begin{theorem}[\cite{AS19}]
\label{lem-AS}
There exist positive constants $C_0$ and $\kappa$,
such that for any $\rho>0$, $r\ge 1$, and $n\ge 1$, satisfying
\begin{equation}\label{hyp.r}
\rho\, r^{d-2} \ge C_0\log n,
\end{equation}
one has for any $L\ge 1$,
\begin{equation*}
\mathbb P\big(|\mathcal{K}_n(r,\rho)|\ge L\big)\,
\le \, C_0 \exp\left(-\kappa\, \rho^{\frac 2d}\, L^{1-\frac 2d} \right).
\end{equation*}
Furthermore, for any $A>0$, there exists $\alpha>0$, such that
\begin{equation*}
\mathbb P\big(|\mathcal{K}_n(r,\rho)|\ge L, \, \ell_n(\mathcal{V}_n(r,2^{-d}\rho)) \le \alpha L\big)\,
\le \, C_0 \exp\left(-A \rho^{\frac 2d}\, L^{1-\frac 2d} \right).
\end{equation*}
\end{theorem}
\subsection{Dimension seven and larger}\label{sec-d7}
We assume here that $d\ge 7$, and fix the value of $T$ as
\begin{equation}\label{def.T.7}
T:= \lceil \gamma \cdot \zeta^{\frac{2}{d-2}} \rceil,
\end{equation}
for some constant $\gamma\in (0,1)$ (depending on dimension $d$) that will be fixed later (in the proof of Theorem \ref{thm:scen-5} below).
Under the event of moderate deviations considered here (when the capacity of the range up to time $n$ is reduced by an amount $\zeta$ from its mean value),
the walk typically folds its trajectory a time of order $\zeta$, in a region of volume $\zeta^{d/(d-2)}$. Thus the typical density of the range in the folding region is
$$\overline \rho := \zeta^{-\frac{2}{d-2}}.$$
Define $\rho_i$, $r_i$, and $L_i$, for $i\in \mathbb{Z}$, by
$$\rho_i := 2^{-i} \cdot \overline \rho, \qquad r_i^{d-2}\cdot\rho_i = C_0\log n, \qquad \text{and}\qquad L_i:= \zeta \cdot 2^{\frac{2i}{d-2}},$$
with $C_0$ as in Theorem \ref{lem-AS}. Define
$$N:=\lceil \frac{d-2}{2}\cdot \log_2(n/\zeta)\rceil, \quad \text{and} \quad M:= \lceil \log_2(1 /\overline \rho )\rceil,$$
so that $n \le L_N \le 2 n$, and $1\le \rho_{-M} \le 2$.
For $-M\le i\le N$, set
\begin{equation*}
\widehat \mathcal{K}_i : = \mathcal{K}_n(r_i,\rho_i)\setminus \bigcup_{-M\le j<i}
\mathcal{K}_n(r_j,\rho_j),
\end{equation*}
with the convention that $\widehat \mathcal{K}_{-M} = \mathcal{K}_n(r_{-M},\rho_{-M})$.
Finally for $A>0$, $\delta>0$, and $I< \min(M,N)$, define
\begin{equation*}
\mathcal E(A,\delta, I):=\left( \bigcap_{-I\le i\le I} \left\{|\widehat \mathcal{K}_i|\le \delta L_i\right\}\right)\cap \left( \bigcap_{I<i\le N} \left\{|\widehat \mathcal{K}_i|\le A L_i\right\}\right)\cap \left( \bigcap_{-M\le i< -I} \left\{|\widehat \mathcal{K}_i|\le A L_i\right\}\right).
\end{equation*}
Our main result here is the following proposition.
\begin{proposition}\label{prop.Anxin7}
For any $A>0$, there exist $\delta>0$ and $I\ge 0$, such that for any $n\ge 2$, and $n^{\frac{d-2}{d}}\cdot \log n \le \zeta\le n$,
$$\mathcal E(A,\delta, I) \ \subseteq \ \{\xi_n(T) \le \zeta\}.$$
\end{proposition}
Before we give the proof of this proposition, let us show how it implies the upper bound in Theorem \ref{thm:d6}, as well as Theorem \ref{thm:scen-5} for dimension $7$ and higher, assuming for a moment the lower bound in Theorem \ref{thm:d6} (which will be proved later and independently in Section \ref{sec-LB}).
\begin{proof}[Proof of Theorem \ref{thm:d6}: the upper bound]
Note first that Proposition \ref{prop.Anxin7} and Theorem \ref{lem-AS} give
$$\mathbb{P}(\xi_n(T) > \zeta) \le \mathbb{P}( \mathcal E(1,\delta, I)^c) \le C\exp(-c\zeta^{1-\frac{2}{d-2}}),$$
for some constant $c>0$, where $\delta$ and $I$ are those given by Proposition \ref{prop.Anxin7}, associated to $A=1$. Note also that by definition $T$ is of order $\zeta^{2/(d-2)}$, see \eqref{def.T.7}, and thus the above estimate together with
Proposition \ref{prop.corrector} prove the upper bound in Theorem \ref{thm:d6}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:scen-5}]
Assume the lower bound in Theorem \ref{thm:d6}, and let us start with the proof of \eqref{result.path}.
First choose $\gamma$ small enough in the definition \eqref{def.T.7} of $T$, so that
conditionally on the event of moderate deviations
$MD(n,\zeta):=\{\mathrm{Cap}(\mathcal{R}_n)-\mathbb{E}[\mathrm{Cap}(\mathcal{R}_n)]\le -\zeta\}$, the probability of the event $\{\xi_n(T) \le c\zeta\}$ goes to zero, with $c$ some appropriately chosen constant. Note that this is possible thanks to (the proof of) Proposition \ref{prop.corrector}. Then choose $A$ large enough, so that conditionally on $MD(n,\zeta)$,
the probability of any of the events $\{|\widehat \mathcal{K}_i|>AL_i\}$, for $i\in \mathbb{Z}$, goes to zero (which is always possible thanks to Theorem \ref{lem-AS}), where implicitly $\zeta$ is replaced by $c\zeta$ in the definition of these events.
Then Propositions \ref{prop.corrector} and \ref{prop.Anxin7} show that conditionally on $MD(n,\zeta)$, one of the events
$\{|\widehat \mathcal{K}_i|>\delta L_i\}$, with $-I\le i \le I$, holds with probability going to $1$ (where $\delta$ and $I$ are given by Proposition \ref{prop.Anxin7}),
and \eqref{result.path} follows from the second part of Theorem \ref{lem-AS}.
Finally, the characterization of the capacity in \reff{result.capacite},
is a simple consequence of a general result of \cite{AS20b}, namely
(1.15) of Theorem 1.5, once we know \reff{result.path}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop.Anxin7}]
Let $\widehat \mathcal{K}_{N+1}$ be such that
\begin{equation}\label{eq.Ki}
\widehat \mathcal{K}_{N+1}:=\{0,\dots,n\}\backslash \bigcup_{-M\le i \le N} \widehat \mathcal{K}_i.
\end{equation}
Now, we decompose $\xi_n(T)$ over the various $\widehat \mathcal{K}_i$.
By \eqref{def-xin}, for any $I\le \min(-M,N)$,
$$
\xi_n(T) \le \Sigma_1 + \Sigma_2 + 2 \Sigma_3 + 2 \Sigma_4+2\Sigma_5,
$$
where (note that $r_i\le \sqrt T$ when $i\le 0$, and
$\varphi_T(z)=\frac 1T G\star G_T(z)$ is defined in Lemma \ref{lem-asympt})
$$
\Sigma_1 : = \sum_{i=-I}^{N+1} \sum_{k\in \widehat \mathcal{K}_i} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \mathbb{P}_x(H^+_{\mathcal{R}_k}=\infty)\cdot {\text{\Large $\mathfrak 1$}}\{x\in Q(S_k,r_{i-1})\},$$
$$
\Sigma_2 : = \sum_{i=-M}^{-I} \sum_{k\in \widehat \mathcal{K}_i} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \mathbb{P}_x(H^+_{\mathcal{R}_k}=\infty)\cdot {\text{\Large $\mathfrak 1$}}\{x\in Q(S_k,\sqrt T)\},$$
$$
\Sigma_3:= \sum_{i=-M}^{N+1} \sum_{j=i}^{N+1} \sum_{k\in \widehat \mathcal{K}_i} \sum_{k'\in \widehat \mathcal{K}_j} \varphi_T(S_{k'}-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{S_{k'}\in Q(S_k,r_{j-1})\setminus Q(S_k,r_{i-1})\},$$
$$
\Sigma_4:= \sum_{i=-M}^{0} \sum_{j=i}^{0} \sum_{k\in \widehat \mathcal{K}_i} \sum_{k'\in \widehat \mathcal{K}_j} \varphi_T(S_{k'}-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{S_{k'}\notin Q(S_k,\sqrt T)\},$$
$$
\Sigma_5:= \sum_{i=-M}^{N+1} \sum_{j=\max(i,0)}^{N+1} \sum_{k\in \widehat \mathcal{K}_i} \sum_{k'\in \widehat \mathcal{K}_j} \varphi_T(S_{k'}-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{S_{k'}\notin Q(S_k,r_{j-1})\},$$
Note that the third term $\Sigma_3$ is not included in $\Sigma_1$ and $\Sigma_2$, since in these last two terms we sum over points of the space, not over time indices. This is important since one important tool used to control them is Lemma \ref{lem-HD}.
Now assume that $\mathcal E(A,\delta, I)$ holds, and let us bound $\Sigma_1$ first.
For $-I\le i\le N+1$, define
$$J(i) = - i + \lfloor \frac{d-4}{2} \log_2(T)-\frac d2\log_2 (\log n) - \frac{d-2}{2} h\rfloor , $$
with $h$ some positive constant to be chosen later, so that for any $-I\le i\le N+1$,
$$\frac {L_i \cdot r_{J(i)}^2}{T} \le C 2^{-h} \cdot \frac{\zeta}{\log n},$$
for some constant $C>0$ (that might change from line to line). Note here that since $-M\le \log_2(\gamma) - \log_2(T)$, by choosing $\gamma$ small enough (once $h$ is fixed), one can always assume that $J(N+1)\ge -M$, which we will do now.
Then Lemmas \ref{lem-HD} and \ref{lem-asympt} show that for any $-I\le i\le N+1$, on $\mathcal E(A,\delta, I)$,
$$ \sum_{k\in \widehat \mathcal{K}_i} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \mathbb{P}_x(H^+_{\mathcal{R}_k}=\infty)\cdot {\text{\Large $\mathfrak 1$}}\{x\in Q(S_k,r_{J(i)})\}\le C|\widehat \mathcal{K}_i| \frac{r_{J(i)}^2}{T} \le CL_i\frac{r_{J(i)}^2}{T} \le C2^{-h}\frac{\zeta}{\log n}. $$
On the other hand, for $i$ such that $r_{J(i)} < r_{i-1}$, and $k\in \widehat \mathcal{K}_i$, we use that by definition the time spent on concentric shells around $S_k$ is bounded, up to distance $r_{i-1}$.
This gives for such $i$, using again Lemma \ref{lem-asympt},
\begin{align*}
& \sum_{k\in \widehat \mathcal{K}_i} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{x\in Q(S_k,r_{i-1})\setminus Q(S_k,r_{J(i)})\} \\
& \le C|\widehat \mathcal{K}_i| \sum_{J(i) \le j \le i-1} \frac{\rho_j r_j^d}{T r_j^{d-4}} \le C|\widehat \mathcal{K}_i| \sum_{J(i) \le j \le i-1} \frac{\log n}{T r_j^{d-6}} \\
& \le C|\widehat \mathcal{K}_i| \frac{\log n}{T}.
\end{align*}
Moreover, by hypothesis on $\zeta$, one has $\frac{n \log n}{T} \le C\zeta \cdot (\log n)^{-\frac{2}{d-2}}$, and by \eqref{eq.Ki} it also holds $\sum_i |\widehat \mathcal{K}_i| = n$.
Therefore, by fixing now the constant $h$ large enough, we get for all $n$ large enough,
$$\Sigma_1\le C\left\{(N-M)2^{-h}\frac{\zeta}{\log n} + \frac{n\log n}{T}\right\} \le \frac{\zeta}{8}.$$
Similarly, using Lemmas \ref{lem-HD} and \ref{lem-asympt}, we get by choosing $I$ large enough,
$$\Sigma_2 \le C \sum_{-M\le i \le -I} |\widehat \mathcal{K}_i| \le C 2^{-\frac{2I}{d-2}} \cdot \zeta\le \frac{\zeta}{8}.$$
We consider the term $\Sigma_3$.
Note that for any $k$, by definition of $\widehat \mathcal{K}_j$, there are at most $C\rho_j r_j^d$ indices $k'\in \widehat \mathcal{K}_j$,
such that $S_{k'}\in Q(S_k,r_{j-1})$.
Therefore, using Lemma \ref{lem-asympt}, we get for $n$ large enough,
\begin{align*}
\Sigma_3 & \le C\sum_{-M\le i\le N+1} |\widehat \mathcal{K}_i| \sum_{i\le j\le N+1} \frac{\rho_j r_j^d}{Tr_{i-1}^{d-4}}
\le C \sum_{-M\le i\le N+1} \zeta\cdot \frac{r_{i-1}^2}{T^{\frac 2{d-2}}}\sum_{i\le j\le N+1} \frac{r_j^2 \log n}{Tr_{i-1}^{d-4}} \\
& \le C\, \frac{\zeta\cdot \log n}{T} \sum_{M\le i\le N+1} \frac{r_{N+1}^2}{r_{i-1}^{d-6}T^{\frac 2{d-2}}}
\le C\frac{n\log n}{T} \le \frac{\zeta}{8}.
\end{align*}
Next, using simply Lemma \ref{lem-asympt}, we obtain
(choosing first $I$ large enough, and then $\delta$ small enough)
\[
\Sigma_4 \le C \sum_{-M\le i\le 0}
\sum_{-M\le j\le 0} \frac{|\widehat \mathcal{K}_i| \cdot |\widehat \mathcal{K}_j|}{T^{\frac{d-2}{2}}}
\le C \zeta \sum_{-M\le i\le -I} A 2^{\frac{2i}{d-2}} + C\delta\zeta
\sum_{-I\le i \le 0} 2^{\frac{2i}{d-2}} \le \frac{\zeta}{8}.
\]
By Lemma \ref{lem:rear}, one has for some $C>0$,
(choosing first $I$ large enough, and then $\delta$ small enough)
\begin{align*}
\Sigma_5&\le \sum_{-M\le i\le N+1} |\widehat \mathcal{K}_i| \sum_{j\ge \max(i,0)} |\widehat \mathcal{K}_j|^{2/d} \rho_j^{1-\frac 2d} \\
& \le C \sum_{-M\le i\le N+1} |\widehat \mathcal{K}_i| \sum_{j\ge \max(i,0)} 2^{\frac{4j}{d(d-2)} - j(1-\frac 2d)} \\
& \le C \sum_{-M\le i\le N+1} |\widehat \mathcal{K}_i| 2^{-\max(i,0)(1-\frac 2{d-2})} \\
& \le C\zeta \left\{\delta \sum_{-I\le i\le I} 2^{-i\frac{d-6}{d-2}} + A \sum_{i\ge I} 2^{-i\frac{d-6}{d-2}} +A\sum_{M\le i\le -I} 2^{\frac{2i}{d-2}} \right\} \le \frac{\zeta}{8},
\end{align*}
This concludes the proof of the proposition.
\end{proof}
\subsection{Dimension five}\label{sec-d5}
We assume here that $d=5$ and let
$$T:=\lceil \gamma \cdot (\zeta n)^{1/3}\rceil,$$
with $\gamma$ some constant (chosen similarly as in the previous subsection), and
$$\overline{\rho}:= \zeta^{5/3} \, n^{-7/3}.$$
Define next $\rho_i$, $r_i$, and $L_i$, for $i\in \mathbb{Z}$, by
$$\rho_i := 2^i \cdot \overline \rho, \qquad r_i^3\cdot\rho_i = C_0\log n, \qquad \text{and}\qquad L_i:= n \cdot 2^{-\frac{2i}{3}},$$
with $C_0$ as in Theorem \ref{lem-AS}. Let for $i\in \mathbb{Z}$,
$\widehat \mathcal{K}_i:=\mathcal{K}_n(r_i,\rho_i)\setminus
\bigcup_{ j>i} \mathcal{K}_n(r_j,\rho_j). $
Then, let $N$ be the smallest integer, such that $1\le r_N\le 2$, and for $A>0$, $\delta>0$, and $0\le I\le N$, let
\begin{equation*}
\mathcal E(A,\delta, I):=\left( \bigcap_{-I\le i\le I} \left\{|\widehat \mathcal{K}_i|\le \delta L_i\right\}\right)\cap \left( \bigcap_{I<i\le N} \left\{|\widehat \mathcal{K}_i|\le A L_i\right\}\right).
\end{equation*}
Our main result here is the following proposition, which implies both the upper bound in Theorem \ref{thm:d5}, as well as Theorem \ref{thm:scen-5} for $d=5$. Since this can be
done in exactly the same way as in dimension $7$ and higher, we will not repeat the arguments here.
\begin{proposition}\label{prop.Anxin5}
For any $A>0$, there exist $\delta>0$ and $I\ge 0$, such that for any $n\ge 2$, and $n^{5/7}\cdot \log n \le \zeta\le n$,
$$\mathcal E(A,\delta, I) \ \subseteq \ \{\xi_n(T) \le \zeta\}.$$
\end{proposition}
\begin{proof}
Given some $I\ge 0$, let
$$
\widetilde \mathcal{K}_0:= \{0,\dots,n\}\backslash \bigcup_{-I\le i\le N}\mathcal{K}_n(r_i,\rho_i).
$$
Note that for any $I$,
$$
\xi_n(T)\le 2\Sigma_1 + 2\Sigma_2 + \Sigma_3 + \Sigma_4,$$
where,
$$\Sigma_1 := \sum_{-I\le i\le N} \sum_{k\in \widehat \mathcal{K}_i} \sum_{k'=0}^n \varphi_T(S_{k'}-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{S_{k'}\in Q(S_k,r_{i+1})\},$$
$$\Sigma_2:= \sum_{-I\le i \le N} \sum_{k\in \widehat \mathcal{K}_i} \sum_{-I\le j\le i} \sum_{k'\in \widehat \mathcal{K}_j} \varphi_T(S_{k'}-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{S_{k'}\notin Q(S_k,r_{j+1})\},$$
$$\Sigma_3: = \sum_{k\in \widetilde \mathcal{K}_0} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{x\in Q(S_k,r_{-I})\},$$
$$\Sigma_4:= \sum_{k\in \widetilde \mathcal{K}_0} \sum_{x\in \mathcal{R}_k} \varphi_T(x-S_k) \cdot {\text{\Large $\mathfrak 1$}}\{x\notin Q(S_k,r_{-I})\},$$
Assume now that $\mathcal E(A,\delta, I)$ holds. Let $J$ be the smallest integer, such that $r_J\le \sqrt{T}$.
Using Lemma \ref{lem-asympt}, and the bound $\sum_i |\widehat \mathcal{K}_i|\le n$,
we get for some $C>0$, and $n$ large enough,
\begin{align*}
\Sigma_1&\le C \sum_{-I\le i \le N} |\widehat \mathcal{K}_i| \left(\sum_{i\le j\le J} \frac{\rho_jr_j^5}{r_j^3} + \sum_{j>\max(i,J)} \frac{\rho_jr_j^5}{Tr_j}\right) \\
& \le C\log n \sum_{-I\le i \le N} |\widehat \mathcal{K}_i| \left(\sum_{i\le j\le J} \frac{1}{r_j} + \sum_{j>\max(i,J)} \frac{r_j}{T}\right)
\le C\frac{n\log n}{\sqrt T} \le \frac{\zeta}{8},
\end{align*}
using also the hypothesis on $\zeta$ for the last inequality. The same argument gives as well $\Sigma_3\le \zeta/4$.
Using in addition Lemma \ref{lem:rear}, we get
taking first $I$ large enough, and then $\delta$ small enough.
\begin{align*}
\Sigma_2 &\le C \sum_{-I\le i \le N} |\widehat \mathcal{K}_i | \sum_{-I\le j\le i} |\widehat \mathcal{K}_j|^{2/5} \rho_j^{3/5}
\le C \frac{\zeta}{n}\cdot \sum_{-I\le i\le N } |\widehat \mathcal{K}_i | \sum_{-I\le j\le i} 2^{j(-\frac{4}{15}+\frac 35)} \\
& \le C \delta\zeta\sum_{-I\le i\le I} 2^{-i/3} + C\cdot \zeta\cdot A
\sum_{i\ge I} 2^{-i/3} \le \frac{\zeta}{8},
\end{align*}
The same argument gives as well $\Sigma_4\le \zeta/4$,
concluding the proof.
\end{proof}
\section{Lower Bounds}\label{sec-LB}
We prove here the lower bounds in Theorems \ref{thm:d6} and \ref{thm:d5}.
In fact in dimension $5$ the result covers a larger range of possible values for $\zeta$.
\begin{proposition} \label{prop.lower5}
Assume $d=5$.
There exist positive constants $\varepsilon_0$ and $\underline{\kappa}$, such that for any $n\ge 2$, and any $\sqrt n (\log n)^3 \le \zeta \le \epsilon_0 n$,
one has
\begin{eqnarray*}
\mathbb{P}\left(\cc{\mathcal{R}_n}-\mathbb{E}[\cc{\mathcal{R}_n}] \le - \zeta\right) \ge \exp(- \underline{\kappa} \cdot (\frac{\zeta^2}n)^{1/3}).
\end{eqnarray*}
\end{proposition}
\begin{proof}
The proof of \eqref{decomp-2} in \cite{ASS19} reveals that for any finite $A,B\subset \mathbb{Z}^d$, one has also
\begin{equation}\label{decomp.lower}
\mathrm{Cap}(A\cup B) \le \mathrm{Cap}(A) + \mathrm{Cap}(B) - \chi_0(A,B),
\end{equation}
with
$$\chi_0(A,B):=\sum_{x\in A\setminus B}\sum_{y\in B} \mathbb{P}_x(H_{A\cup B}^+ = \infty) G(y-x) \mathbb{P}_y(H_B^+ = \infty).$$
Now given $n\ge 1$, set $\ell = \lfloor \frac{n}{10} \rfloor $, and $m=n-\ell$.
We apply \eqref{decomp.lower} with $A = \mathcal{R}_m$ and $B=\mathcal{R}[m, n]$.
Fix $\varepsilon_0>0$ (later chosen small enough), and define
$$E:=\left\{\ccc{\mathcal{R}_n} \ge -\varepsilon_0 n\right\},$$
where we use the notation $\ccc{\mathcal{R}_n}$ for the centered capacity.
Using \eqref{decomp.lower}, Lemma \ref{lem.exp.cap}, and Proposition \ref{cor:upward}, we deduce that
for some constant $c>0$,
\begin{equation}\label{lower5.1}
\begin{split}
\mathbb{P}\left(-\varepsilon_0n\le \ccc{\mathcal{R}_n}\le - \zeta\right)
& \ge \ \mathbb{P}(E, \, \chi_0(\mathcal{R}_m,\mathcal{R}[m,n]) \ge 4\zeta) - \mathbb{P}(\ccc{\mathcal{R}_m}\ge \zeta) \\
&\qquad - \mathbb{P}(\ccc{\mathcal{R}[m,n]} \ge \zeta) \\
& \ge\ \mathbb{P}(E, \, \chi_0(\mathcal{R}_m,\mathcal{R}[m,n])\ge 4\zeta) - 2\exp(-c\frac{\zeta^2}{n(\log n)^3}).
\end{split}
\end{equation}
Note that when $\zeta\ge \sqrt n (\log n)^3$, then $\zeta^2/(n(\log n)^3) \ge (\log n) (\zeta^2/n)^{1/3}$, and
therefore the last term above is negligible.
Now, let $\rho>0$ be some small constant (to be fixed later) and consider the event
$$F:= \{\|S_k\| \le \rho\cdot n^{2/3}\, \zeta^{-1/3}, \quad \text{for all } k\le n\}.$$
Note that by \eqref{Green} and \eqref{cap.def2}, on the event $F$,
\begin{equation}\label{chi0Rm}
\chi_0(\mathcal{R}_m,\mathcal{R}[m,n])\ \ge\ c_\rho\cdot \frac{\zeta}{n^2}\cdot \mathrm{Cap}(\mathcal{R}[m,n]) \cdot \left(\mathrm{Cap}(\mathcal{R}_n) - \mathrm{Cap}(\mathcal{R}[m,n])\right),
\end{equation}
for some constant $c_\rho>0$, going to infinity as $\rho$ goes to zero.
Furthermore, by \eqref{cap.subadd}, one has
$$\mathrm{Cap}(\mathcal{R}_n) \le \mathrm{Cap}(\mathcal{R}_m) + \mathrm{Cap}(\mathcal{R}[m,n]),$$
and thus by Lemma \ref{lem.exp.cap} and Proposition \ref{cor:upward}, by taking $\varepsilon_0$ small enough, we get for $n$ large enough,
\begin{align*}
\mathbb{P}\left(\mathrm{Cap}(\mathcal{R}[m,n]) \le \gamma_5 \frac{\ell}{2},\, E\right) \ &
\le\ \mathbb{P}\left( \mathrm{Cap}(\mathcal{R}_m) \ge \gamma_5 (m +\ell/3) \right)\\
&\le \
\mathbb{P}\left( \ccc{\mathcal{R}_m} \ge \gamma_5 \frac{\ell}{10} \right)\le \ \exp\left(- c' \frac{n}{(\log n)^3}\right),
\end{align*}
for some constant $c'>0$, and with $\gamma_5$ as in \eqref{cap.limit}.
Similarly one has for some possibly smaller constant $c'>0$,
$$
\mathbb{P}\left(\mathrm{Cap}(\mathcal{R}_n) - \mathrm{Cap}(\mathcal{R}[m,n])\le \gamma_5 \frac n 4,\, E\right)\
\le\ \mathbb{P}\left(\ccc{\mathcal{R}[m,n]}\ge \gamma_5\ell \right) \le \ \exp\left(- c' \frac{n}{(\log n)^3}\right).
$$
Then \eqref{chi0Rm} gives
\begin{align*}
\mathbb{P}( \chi_0(\mathcal{R}_m,\mathcal{R}[m,n]) \ge \frac{c_\rho \gamma_5^2}{100}\cdot \zeta, \, E)\
& \ge\ \mathbb{P}( \chi_0(\mathcal{R}_m,\mathcal{R}[m,n]) \ge \frac{c_\rho \gamma_5^2}{100}\cdot \zeta, \, E \cap F) \\
& \ge \ \mathbb{P}(E\cap F) - 2 \exp\left(- c' \frac{n}{(\log n)^3}\right)\\
&\ge \ \mathbb{P}(F) - \mathbb{P}(E^c) - 2\exp\left(- c' \frac{n}{(\log n)^3}\right).
\end{align*}
Coming back to \eqref{lower5.1}, and choosing $\rho$, such that $c_\rho \ge 300/\gamma_5^2$, we deduce that
\begin{align*}
\mathbb{P}\left( \ccc{\mathcal{R}_n} \le - \zeta\right) \ & = \ \mathbb{P}\left(-\varepsilon_0n
\le \ccc{\mathcal{R}_n}\le - \zeta\right)+ \mathbb{P}(E^c)\\
& \ge \ \mathbb{P}(F) - 2\exp\left(- c' \frac{n}{(\log n)^3}\right) - 2\exp(-c\, (\log n)\cdot \zeta^{2/3} n^{-1/3}).
\end{align*}
Moreover, it is well known that for any $\rho>0$, there exists $\kappa>0$, such that
$$\mathbb{P}(F) \ \ge\ \exp(-\kappa \cdot \zeta^{2/3}n^{-1/3}),$$
and this concludes the proof.
\end{proof}
In dimension $6$ and more the result reads as follows.
\begin{proposition}\label{prop.lower6}
Assume $d\ge 7$. There exist positive constants $\varepsilon_0$, $K$ and $\underline{\kappa}$,
such that for any $n\ge 2$ and any $K n^{\frac{d-2}{d}} \le \zeta \le \varepsilon_0\, n$, one has
$$
\mathbb{P}\left(\cc{\mathcal{R}_n} - \mathbb{E}[\cc{\mathcal{R}_n}]\le -\zeta\right)
\ \ge\ \exp\left(- \underline{\kappa} \cdot \zeta^{1-\frac{2}{d-2}} \right).$$
In dimension $d=6$, the same result holds for $n^{\frac{d-2}{d}}(\log \log n)^2 \le \zeta \le \varepsilon_0\, n$.
\end{proposition}
\begin{proof}
We prove the result for $d\ge 7$ to keep notation simple, but the same argument works as well for $d=6$.
Set $\ell : = \lfloor 5\zeta/\gamma_d \rfloor$.
Using \eqref{cap.subadd}, Lemma \ref{lem.exp.cap}, and Proposition \ref{cor:upward},
we obtain that for some constant $c>0$,
\begin{align*}
\mathbb{P}\left(\ccc{\mathcal{R}_n}\le - \zeta\right) & \ge \mathbb{P}\left(\ccc{\mathcal{R}_\ell} \le -3\zeta\right)
- \mathbb{P}\left(\ccc{\mathcal{R}[\ell,n]} \ge \zeta\right)\\
& \ge \mathbb{P}\left(\cc{\mathcal{R}_\ell}\le \zeta\right) - \exp(-c \cdot \frac{\zeta^2}{n}),
\end{align*}
at least provided $\zeta$ is large enough, which one can always assume.
Now the hypothesis on $\zeta$ implies that the last term is negligible, provided $K$ is chosen large enough, and
by the same argument as in the proof of Proposition \ref{prop.lower5},
one can see that the first term on the right-hand side is of the right order (which is of the order of the event $F$ where the walk stays confined in a ball of radius $c'\zeta^{1/(d-2)}$, with $c'>0$ small enough, during the whole time $\ell$). This concludes the proof of the proposition.
\end{proof}
\section{The Gaussian regime} \label{sec.Gaussian}
The starting point to proving Theorem \ref{theo.Gaussian} is a
standard dyadic decomposition which follows from using \eqref{decomp-1}
repeatedly along a dyadic scheme. For any $L\ge 1$, and $n\ge 2^L$,
\begin{equation}\label{UBd5-1}
\mathrm{Cap}(\mathcal{R}_n)-\mathbb{E}[\mathrm{Cap}(\mathcal{R}_n)]= \sum_{i=1}^{2^L} \left( \mathrm{Cap}(\mathcal{R}_i^L) - \mathbb{E}[ \mathrm{Cap}(\mathcal{R}_i^L)] \right)
-\sum_{\ell=1}^{L}\sum_{i=1}^{2^{\ell-1}} Y_i^\ell,
\end{equation}
where $Y_i^\ell:= \chi_{\mathcal C}(\mathcal{R}_{2i-1}^\ell, \mathcal{R}_{2i}^\ell ) -
\mathbb{E}[\chi_{\mathcal C}(\mathcal{R}_{2i-1}^\ell, \mathcal{R}_{2i}^\ell )]$,
and the $\{\mathcal{R}_i^\ell\}_{i=1,\dots,2^\ell}$, are independent
ranges of length $n2^{-\ell}$ (the time-length is not
exactly equal for each of them since we do not suppose that $n$
is of the form $n=2^K$, for some $K\ge 1$, but they
differ by at most one unit).
A gaussian-type fluctuation is due to the sum of the $2^L$ self-similar
terms in \reff{UBd5-1}, after $L$ is chosen appropriatly. It is classical
(see \cite{Chen})
to use G\"artner-Ellis' Theorem after we show that the contribution
of the $Y_i^\ell$ is negligible.
Thus, the main technical novelty of this section is the
stretched exponential moment bound \reff{stretch-chi}, which is performed
in Section \ref{subsec.2.gauss}.
After recalling some well-known results
in Section \ref{subsec.1.gauss},
we conclude the proof of Theorem
\ref{theo.Gaussian} in Section \ref{subsec.3.gauss}.
\subsection{Preliminary results}\label{subsec.1.gauss}
We first state an instance
of G\"artner-Ellis' Theorem (see Theorem 2.3.6 in \cite{DZ}).
\begin{theorem}[G\"artner-Ellis]\label{theo-GE}
Let $\{X_n\}_{n\ge 0}$ be a sequence of real random variables.
Let $\{b_n\}_{n\ge 0}$ going to infinity, and
for any $\theta\in \mathbb{R}$,
\begin{equation*}
\text{If $\forall\theta\in \mathbb{R}$}\quad
\lim_{n\to\infty} \frac{1}{b_n} \log \mathbb{E}[\exp(\theta b_n\cdot X_n)]=
\frac{\sigma^2}{2}\cdot \theta^2,\quad
\text{then, $\forall\lambda>0$,}\quad
\lim_{n\to\infty} \frac{1}{b_n}
\log \mathbb{P}(X_n> \lambda)=-\frac{\lambda^2}{2\sigma^2}.
\end{equation*}
\end{theorem}
We recall now a large deviation estimates for variables
with stretched exponential moment.
\begin{theorem}[A. Nagaev \cite{anagaev}]\label{theo-nagaev}
Let $\{Y_n\}_{n\ge 0}$ be a sequence of centered random variables,
such that
$\mathbb{E}[\exp(\kappa |Y_1|^\alpha)]<\infty$, for some constants $\kappa>0$, and $\alpha\in (0,1]$. Then
there are positive constants $c$ and $C$, such that for any $n\ge 1$ and any $t>n^{\frac 1{2-\alpha}}$,
\begin{equation*}
\mathbb{P}\big(Y_1+\dots+Y_n>t\big)\le C\exp(-c t^\alpha).
\end{equation*}
\end{theorem}
\subsection{Stretched exponential moment of the cross term}\label{subsec.2.gauss}
The heart of the proof of Theorem \ref{theo.Gaussian}
use Theorem \ref{theo.stretched.exp} below which is more general
than \reff{stretch-chi}, and has interest of its own.
It is analogous to the arguments of \cite{AS20c}.
Define for any subsets $A,B\subseteq \mathbb{Z}^d$,
$$\Gamma(A,B) = \sum_{x\in A} \sum_{y\in B} G(y-x) \mathbb{P}_y(H_B^+ =\infty). $$
Recall that $0\le \chi
|
_{\mathcal C}(A,B) \le 2 \Gamma(A,B)$, for any $A,B\subseteq \mathbb{Z}^d$.
\begin{theorem}\label{theo.stretched.exp}
Let $\mathcal{R}_\infty$ and $\widetilde \mathcal{R}_\infty$ be the ranges of two independent random walks on $\mathbb{Z}^d$, with $d\ge 7$. There exist positive constants $c_1,c_2$, such that
for all $t$ large enough,
$$\exp(-c_1t^{1-\frac 2{d-2}} ) \le \mathbb{P}(\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}_\infty ) > t) \le \exp(- c_2 t^{1-\frac 2{d-2}}). $$
\end{theorem}
Let us notice that in the definition of $\Gamma$ it is fundamental to keep the escape probabilities, in other words one cannot simply bound them by one. Indeed one could show that the tail distribution of
$\Gamma'(\mathcal{R}_\infty,\widetilde \mathcal{R}_\infty):= \sum_{x\in \mathcal{R}_\infty} \sum_{y\in \widetilde \mathcal{R}_\infty} G(x-y)$ obeys a different decay at infinity.
\begin{proof}[Proof of Theorem \ref{theo.stretched.exp}]
We start with the lower bound. Observe that $\Gamma(\cdot,\cdot)$ is increasing in both arguments for the inclusion of sets, thus for any $n\ge 1$,
$$\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}_\infty ) \ge \Gamma (\widetilde \mathcal{R}_n, \mathcal{R}_n).$$
Therefore the lower bound is obtained by forcing the two walks to stay confined in a ball of radius $t^{\frac{1}{d-2}}$ for a time $Ct$, with $C>0$ large enough, exactly as in the proof of Proposition \ref{prop.lower5}.
We now move to the upper bound. The proof is obtained in three steps.
In the first step, we reduce the time window of one walk to a finite interval, as follows. Observe that for any integer $n\ge 1$,
\begin{align*}
& \mathbb{E}[\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}[n,\infty))] \le \mathbb{E}\left[\sum_{k=0}^\infty \sum_{\ell = n}^\infty G(S_k-\widetilde S_\ell)\right]= \sum_{k=0}^\infty \sum_{\ell = n}^\infty\mathbb{E}[G(S_{k+\ell})] \\
& = \sum_{k=n}^\infty (k+1-n)\mathbb{E}[G(S_k)]\le C\sum_{k=n}^\infty\frac{k+1-n}{k^{\frac{d-2}{2}}} \le \frac{C}{n^{\frac{d-6}{2}}},
\end{align*}
for some constant $C>0$. Therefore if we let $n :=\exp(t^{1-\frac{2}{d-2}})$, then by Markov's inequality,
$$\mathbb{P}(\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}[n,\infty))\ge 1) \le \mathbb{E}[\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}[n,\infty))] \le C\exp(- \frac{d-6}{2}\cdot t^{1-\frac{2}{d-2}}), $$
and thus, due to the inequality
$$\Gamma(\widetilde \mathcal{R}_\infty, \mathcal{R}_\infty) \le \Gamma (\widetilde \mathcal{R}_\infty, \mathcal{R}_n) + \Gamma (\widetilde \mathcal{R}_\infty, \mathcal{R}[n,\infty)), $$
it just remains to bound the first term on the right-hand side.
In a second step we claim that for any subset $\Lambda\subseteq \mathbb{Z}^d$, and any $t\ge 1$,
\begin{equation}\label{Claim.Gamma}
\mathbb{P}(\Gamma(\widetilde \mathcal{R}_\infty, \Lambda) > t ) \le \exp\left(-\frac{t\cdot \log 2}{2\sup_{x\in \mathbb{Z}^d} \mathbb{E}_x[\Gamma(\widetilde \mathcal{R}_\infty, \Lambda)]}\right).
\end{equation}
To see this, we use again that for any $A,B\subseteq \mathbb{Z}^d$, one has
$\Gamma(A\cup B,\Lambda) \le \Gamma(A,\Lambda) + \Gamma(B,\Lambda)$. Thus the Markov property and Markov's inequality show that the random variable
$\frac{\Gamma(\widetilde \mathcal{R}_\infty, \Lambda)}{2\sup_{x\in \mathbb{Z}^d} \mathbb{E}_x[\Gamma(\widetilde \mathcal{R}_\infty, \Lambda)]}$ is stochastically bounded by a Geometric random variable with mean $2$, from which \eqref{Claim.Gamma} follows immediately. Note also that for any $x$,
$$ \mathbb{E}_x[\Gamma(\widetilde \mathcal{R}_\infty, \Lambda)] \le \sum_{z\in \Lambda} G\star G(z-x)\cdot \mathbb{P}_z(H_{\Lambda}^+ = \infty) =:\mathcal F(\Lambda - x),$$
where we recall that $G\star G$ is the convolution of $G$ with itself, and
$$\mathcal F(\Lambda) := \sum_{z\in \Lambda} G\star G(z) \cdot \mathbb{P}_z(H_\Lambda^+=\infty). $$
Thus it amounts to show that for some positive constants $c$ and $C$, one has
\begin{equation}\label{Claim.2.F}
\mathbb{P}\left(\sup_{x\in \mathbb{Z}^d} \mathcal F(\mathcal{R}_n-x) > Ct^{\frac 2{d-2}}\right) \le C\exp(-ct^{1-\frac 2{d-2}}), \quad \text{with }n=\exp(t^{1-\frac{2}{d-2}}),
\end{equation}
which is our third and last step. Note that $\mathcal F$ is also subadditive in the sense that for any $A,B\subseteq \mathbb{Z}^d$,
$\mathcal F(A\cup B) \le \mathcal F(A) + \mathcal F(B)$. This allows to partition the range into different pieces, according to the occupation density
in a certain neighborhood, and then bound $\mathcal F$ on each of them.
To be more precise, set $\rho_0 := t^{-\frac 2{d-2}}$, and then for $i\ge 0$, define $\rho_i$ and $r_i$ by
$$\rho_i := 2^{-i}\rho_0,\quad \text{and} \quad \rho_i r_i^{d-2}= C_0 \log n,$$
with $C_0$ as in \eqref{hyp.r}.
Then let $\mathcal{R}_n(r_i,\rho_i):= \{S_k,\ k\in \mathcal{K}_n(r_i,\rho_i)\}$, and
$$
\Lambda_i := \mathcal{R}_n(r_i,\rho_i) \backslash
\bigcup_{0\le j< i} \mathcal{R}_n(r_j,\rho_j) ,\qquad \Lambda_i^*:
= \mathcal{R}_n \backslash \bigcup_{0\le j< i}\Lambda_i.
$$
By Theorem \ref{lem-AS}, one has for any $i\ge 0$,
$$\mathbb{P}(|\Lambda_i|\ge 2^{\frac{2i}{d-2}} t ) \le C\exp(-\kappa t^{1-\frac 2{d-2}}),$$
for some positive constants $C$ and $\kappa$, and in fact for $i>\frac{d-2}{2} \log_2(n+1)$, the above probability is zero, since by definition $|\Lambda_i|\le n+1$.
Therefore, if we let
$$\mathcal E := \left\{|\Lambda_i|\le 2^{\frac{2i}{d-2}} t, \ \text{for all }i\ge 0\right\},$$
then the above discussion shows that
$$\mathbb{P}(\mathcal E^c) \le C\exp(-(\kappa/2)\cdot t^{1-\frac 2{d-2}}),$$
at least for $t$ large enough. We now show that for some constant $C>0$,
\begin{equation}\label{claim.E.F}
\mathcal E \subseteq \left\{\sup_{x\in \mathbb{Z}^d} \mathcal F(\mathcal{R}_n-x) \le Ct^{\frac 2{d-2}}\right\},
\end{equation}
which will conclude the proof of the theorem. To simplify notation we only bound $\mathcal F(\mathcal{R}_n-x)$ for $x=0$, but it should be clear from the proof that all
our estimates are uniform with respect to $x$. We partition space into shells $(\mathcal S_k)_{k\ge 0}$, defined by $\mathcal S_0 := Q(0,r_0)$, and $\mathcal S_k:= Q(0,r_k)\backslash Q(0,r_{k-1})$ for $k\ge 1$. By subadditivity, one has
$$\mathcal F(\mathcal{R}_n) \le \sum_{k\ge 0} \mathcal F(\mathcal S_k \cap \mathcal{R}_n). $$
The proof of Lemma \ref{lem-asympt} shows that $G\star G(z) \le C \|z\|^{4- d}$, and thus Lemma \ref{lem-HD} gives
$$\mathcal F(\mathcal S_0\cap \mathcal{R}_n) \le \mathcal F(\mathcal S_0) \le Cr_0^2 \le Ct^{\frac{2}{d-2}}. $$
Then for $k\ge 1$, we write
$$\mathcal F(\mathcal S_k\cap \mathcal{R}_n) \le \sum_{i=0}^k \mathcal F(\mathcal S_k\cap \Lambda_i) + \mathcal F(\mathcal S_k\cap \Lambda_{k+1}^*).$$
On one hand one has on the event $\mathcal E$,
$$\mathcal F(\Lambda_0 \cap \mathcal S_0^c) \le C\frac{|\Lambda_0|}{r_0^{d-4}} \le Ct^{\frac 2{d-2}}. $$
On the other hand, for any $i\ge 1$,
\begin{align*}
\sum_{k\ge i } \mathcal F(\mathcal S_k\cap \Lambda_i) \le \sum_{z\in \Lambda_i \cap Q(0,r_{i-1})^c} G\star G(z)\le \sum_{z\in \Lambda_i \cap Q(0,r_{i-1})^c} \frac{C}{1+\|z\|^{d-4}} \le C\rho_i^{1-\frac{4}{d}} |\Lambda_i|^{4/d},
\end{align*}
using the same argument as in the proof of Lemma \ref{lem:rear} for the last inequality. Thus on the event $\mathcal E$, we get
$$\sum_{k\ge i } \mathcal F(\mathcal S_k\cap \Lambda_i) \le C 2^{-i\frac{d-6}{d-2}} t^{\frac 2{d-2}}. $$
It follows that on $\mathcal E$,
$$\sum_{i\ge 1} \sum_{k\ge i} \mathcal F(\mathcal S_k\cap \Lambda_i) \le C 2^{-i\frac{d-6}{d-2}} t^{\frac 2{d-2}} \le Ct^{\frac 2{d-2}} . $$
Similarly, one has
$$\sum_{k\ge 1} \mathcal F(\mathcal S_k\cap \Lambda_{k+1}^*) \le C\sum_{k\ge 1} \frac{\rho_k r_k^d}{r_{k-1}^{d-4}} \le C\frac{\log n}{r_0^{d-6}} \le Ct^{\frac 2{d-2}}. $$
Altogether this proves \eqref{claim.E.F}, and concludes the proof of the theorem.
\end{proof}
\subsection{Proof of Theorem \ref{theo.Gaussian}}\label{subsec.3.gauss}
Let $\{\zeta_n\}_{n\ge 0}$ be a sequence as in the statement of Theorem \ref{theo.Gaussian}, and let $L$ be the integer such that
$2^{L-1} \le \zeta_n <2^L$.
We first show that the cross terms appearing in
\eqref{UBd5-1} are negligible.
Applying Theorems \ref{theo-nagaev} and \ref{theo.stretched.exp}, we get that for any $\delta>0$, and any $\ell \le L$,
$$\limsup_{n\to \infty} \frac{n}{\zeta_n^2}\cdot \log \mathbb{P}\left(\pm \sum_{i=1}^{2^\ell} Y_i^\ell \ge \frac{\delta\zeta_n}{L} \right) = - \infty. $$
By using a union bound we also deduce
$$\limsup_{n\to \infty} \frac{n}{\zeta_n^2}\cdot \log \mathbb{P}\left(\pm \sum_{\ell = 1}^L \sum_{i=1}^{2^\ell} Y_i^\ell \ge \delta\zeta_n \right) = - \infty. $$
Thus indeed the cross terms in \eqref{UBd5-1} can be ignored, and we focus now on proving the Moderate Deviation Principle for the first sum.
For simplicity, let $Z_i:= |\mathcal{R}_i^L| -\mathbb{E}[ |\mathcal{R}_i^L|]$. We apply Theorem \ref{theo-GE} with $X_n:= \frac{\pm 1}{\zeta_n}\sum_{i=1}^{2^L}Z_i$, and
$b_n:=\zeta_n^2/n$. One has using independence, and the fact that $\frac{\zeta_n}{n}\cdot |Z_1|$ is bounded,
\begin{equation*}
\mathbb{E}[\exp(\theta b_n X_n]=
\Big(\mathbb{E}[\exp(\theta \frac{\zeta_n}{n} Z_1] \Big)^{2^L}\\
= \Big(1+\frac{\theta^2}{2}\big(\frac{\zeta_n}{n}\big)^2\cdot
\mathbb{E}[Z_1^2]+
\mathcal O\big(\big(\frac{\zeta_n}{n}\big)^3\cdot \mathbb{E}[|Z_1|^3]\big) \Big)^{2^L}.
\end{equation*}
Note that $2^L\cdot \mathbb{E}[Z_1^2]/n$ converges to $\sigma^2>0$,
and that the fourth centered moment of $\mathrm{Cap}(\mathcal{R}_n)$
is $\mathcal O(n^2 (\log n)^2)$. This can be seen as for the volume of the range, following the same proof as in \cite{LG86}.
Thus, using that $\mathbb{E}[|Z_1|^3]\le \mathbb{E}[Z_1^4]^{3/4}$, we have
\[
\big(\frac{\zeta_n}{n}\big)^3\cdot \mathbb{E}[|Z_1|^3]\le C
\big(\frac{\zeta_n\log n}{n}\big)^{3/2}.
\]
It follows that for any $\theta\in \mathbb{R}$,
\begin{equation*}
\lim_{n\to\infty} \frac{n}{\zeta_n^2} \log \mathbb{E}[
\exp\Big(\theta\frac{\zeta_n}{n} X_n\Big)=
\frac{\sigma^2}{2} \theta^2,
\end{equation*}
and one can then apply G\"artner--Ellis' Theorem, which concludes the proof of Theorem \ref{theo.Gaussian}.
\section{Upward Deviations}\label{sec-HK}
We prove here Theorem~\ref{theo:upward}.
Thanks to our decomposition \reff{decomp-1}, we can
adapt the approach of Hamana and Kesten \cite{HK},
who proved a similar result for the size of the range.
The approach of Hamana and Kesten is based on first proving an approximate subadditivity relation for the probability of upward deviations, that is the existence of some constants
$\chi \in (0,1)$, $c>0$, and $C>0$, such that for any $m,n\ge 1$ integers, and
$y,z$ positive reals,
\begin{equation}\label{KH-main}
\mathbb{P}\big(|\mathcal{R}_{m+n}|\ge y+z - C a(m,n) \big)\ \ge\
c\, \chi^{a(m,n)}\, \mathbb{P}\big(|\mathcal{R}_n|\ge y\big)\mathbb{P}\big(|\mathcal{R}_m|\ge z\big),
\end{equation}
with
\begin{equation*}
a(m,n):=(n\cdot m)^{\frac{1}{d+1}}.
\end{equation*}
The second step, which is general
and only based on \eqref{KH-main} and the fact that (when $d\ge 2$) one has $\lim_{m,n\to\infty} \frac{a(m,n)}{n\vee m}=0$,
shows that the following limit exists,
\[
\psi(x):=- \lim_{n\to\infty} \frac{1}{n}
\log \mathbb{P}\big(|\mathcal{R}_n|\ge x\cdot n \big), \quad \text{for all }x>0,
\]
and that $\psi$ is continuous and convex on $[0,1]$.
Here we prove an analogous result as \eqref{KH-main}, and use their general argument to conclude.
\begin{proof}[Proof of Theorem \ref{theo:upward}]
We first prove an analogous result as \eqref{KH-main}, but with $a(m,n)$ replaced by the function:
\[
\widetilde a(m,n)=(n\cdot m)^{\frac{1}{d-1}}.
\]
In other words we establish the following inequality. There exists
$\chi \in (0,1)$, and $C>0$, such that for any $m,n$ integers and $y,z$ positive reals,
\begin{equation}\label{amin-1}
\mathbb{P}\left(\cc{\mathcal{R}_{m+n}}\ge y+z - C\, \widetilde a(m,n) \right)\ge
\frac 12 \chi^{\widetilde a(m,n)} \mathbb{P}\left(\cc{\mathcal{R}_n}\ge y\right)
\mathbb{P}\left(\cc{\mathcal{R}_m}\ge z\right).
\end{equation}
The first step is to obtain the analogue of the following simple
deterministic bound used in \cite{HK}: if $\mathcal{R}_n$ and $\widetilde \mathcal{R}_m$ are two independent
copies of the range, there is a positive constant $C$, such that
for any $r\ge 1$
\[
\frac{1}{|Q(r)|}\sum_{z\in Q(r)}
|(z+\mathcal{R}_n)\cap \widetilde \mathcal{R}_m|
\le C\, \frac{n\cdot m}{r^d}.
\]
The corresponding bound in our context reads as follows:
\begin{equation}\label{amin-2}
\frac{1}{|Q(r)|}\sum_{z\in Q(r)} \sum_{x\in \mathcal{R}_n}
\sum_{y\in \widetilde \mathcal{R}_m} G(x-y+z)\le C \, \frac{n\cdot m}
{r^{d-2}},
\end{equation}
and is a direct consequence of \eqref{Green}
and the fact that for any $x\in \mathbb{Z}^d$, and for some constant $C>0$,
$$
\sum_{z\in Q(r)} \frac{1}{1+\|z-x\|^{d-2}} \le C\, r^2.
$$
Now to lighten notation, we simply write $a=\widetilde a(m,n)=\lfloor (mn)^{\frac{1}{d-1}}\rfloor$. Using that the capacity is translation-invariant, we deduce
\begin{equation}\label{amin-3}
\begin{split}
\cc{\mathcal{R}_{m+n+a}}&\stackrel{\eqref{cap.mon}}{\ge}\cc{\mathcal{R}_n\cup \mathcal{R}[n+a,n+m+a]}\\
&\stackrel{\eqref{decomp-1}}{=} \cc{\overline \mathcal{R}_n} +\cc{\widetilde \mathcal{R}_m} -\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+S'_a),
\end{split}
\end{equation}
with $\overline \mathcal{R}_n:= \mathcal{R}_n - S_n$, $S'_a := S_{n+a}- S_n$, and $\widetilde \mathcal{R}_m:=\mathcal{R}[n+a,n+m+a]-S_{n+a}$.
The Markov property implies that
$\overline \mathcal{R}_n$ and $\widetilde R_m$ are independent, and distributed as $\mathcal{R}_n$ and $\mathcal{R}_m$ respectively.
Furthermore,
\begin{equation}\label{amin-4}
\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+ S'_a)\stackrel{\eqref{decomp-2}}{\le} \sum_{x\in \overline \mathcal{R}_n} \sum_{y\in \widetilde \mathcal{R}_m} G(x-y- S'_a).
\end{equation}
Now, one idea of Hamana and Kesten \cite{HK} is to bound the law of $S_a'$ by a uniform
law on the cube $Q(a/d)$. Indeed for any $x\in Q(a/d)$, for which $\mathbb{P}(S_a=x)\neq 0$, one has
\begin{equation}\label{inf.hk}
\mathbb{P}(S_a'=x)\ge \frac{1}{(2d)^a},
\end{equation}
since there is at least one path of length $a$ going from $0$ to $x$.
Write $\overline Q(a/d)$ for the set of sites $x\in Q(a/d)$, for which $\mathbb{P}(S_a=x)\neq 0$. Then for any $x\in \overline Q(a/d)$, and any $\alpha>0$,
\begin{equation*}
\mathbb{P} \left( \mathrm{Cap}(\mathcal{R}_{m+n+a})\ge z+y-\frac{\alpha}{2} \right) \stackrel{\eqref{amin-3}}{\ge}
\mathbb{P}(S_a'=x)\cdot \mathbb{P}\left(\mathrm{Cap}(\overline \mathcal{R}_n)\ge z,\mathrm{Cap}(\widetilde \mathcal{R}_m)\ge y,
\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+x)\le \frac{\alpha}{2}\right).
\end{equation*}
Integrating with respect to the uniform measure on $\overline Q(a/d)$, we get
\begin{equation}\label{KH-2}
\begin{split}
\mathbb{P}\big( \mathrm{Cap}(\mathcal{R}_{m+n+a}) & \ge z+y-\frac{\alpha}{2}\big) \ \stackrel{\eqref{inf.hk}}{\ge} \
\frac{1}{(2d)^a} \\
& \times \frac{1}{|\overline Q(a/d)|}\sum_{x\in \overline Q(a/d)}
\mathbb{P}\left(\mathrm{Cap}(\overline \mathcal{R}_n)\ge z,\mathrm{Cap}(\widetilde \mathcal{R}_m)\ge y,
\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+x)\le \frac{\alpha}{2}\right).
\end{split}
\end{equation}
We need now to estimate the mean of $\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+\cdot)$
with respect to the uniform measure. According to \eqref{amin-2}, there
is a positive constant $C$, such that
\begin{equation}\label{amin-5}
\frac{1}{|\overline Q(a/d)|}\sum_{x\in \overline Q(a/d)} \chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+x)\le C \frac{m\cdot n}{a^{d-2}}\le Ca,
\end{equation}
where the last inequality follows from the definition of $a$.
Then by Chebychev's inequality, we obtain
\begin{equation}\label{cheb}
\frac{1}{|\overline Q(a/d)|}\sum_{x\in \overline Q(a/d)} {\text{\Large $\mathfrak 1$}}(\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+x)\le
2Ca)\ge \frac{1}{2}.
\end{equation}
As a consequence,
\begin{equation*}
\begin{split}
& \ \mathbb{P}\big(\mathrm{Cap}(\mathcal{R}_{m+n}) \ge z+y-a- 4Ca\big) \ \stackrel{\eqref{cap.subadd}, \eqref{cap.card}}{\ge} \
\mathbb{P}\big(\mathrm{Cap}(\mathcal{R}_{m+n+a})\ge z+y- 4Ca\big)\\
& \stackrel{\eqref{KH-2}}{\ge}\ \frac{1}{(2d)^a} \cdot \mathbb{E}\Big[{\text{\Large $\mathfrak 1$}}(\mathrm{Cap}(\overline \mathcal{R}_n)\ge z)\cdot
{\text{\Large $\mathfrak 1$}}(\mathrm{Cap}(\widetilde \mathcal{R}_m)\ge y)
\times \frac{1}{|\overline Q(a/d)|}\sum_{x\in \overline Q(a/d)} {\text{\Large $\mathfrak 1$}}(\chi_C(\overline \mathcal{R}_n,\widetilde \mathcal{R}_m+x)\le 2 Ca)\Big]\\
& \stackrel{\eqref{cheb}}{\ge}\ \frac{1}{2(2d)^a} \cdot \mathbb{P}\big(\cc{\mathcal{R}_n}\ge z\big)
\mathbb{P}\big(\cc{ \mathcal{R}_m}\ge y\big),
\end{split}
\end{equation*}
proving \eqref{amin-1}, with $\chi = 1/(2d)$.
It then follows from the general arguments of Hamana and Kesten, see Lemma 3 in \cite{HK}, that the following limit exists for all $x>0$:
$$\psi_d(x)\ := \ -\lim_{n\to \infty}\ \frac 1n \log \mathbb{P}\left(\mathrm{Cap}(\mathcal{R}_n) \ge nx\right).$$
We now prove that the range for which $\psi_d(x)$ is finite is not empty.
Define for $n\ge 0$,
\begin{equation}\label{cap.cn}
c_n:=\max_{\gamma:\{0,\dots,n\}\to \mathbb{Z}^d} \mathrm{Cap}(\{\gamma(0),\dots,\gamma(n)\}),
\end{equation}
where the max is taken over all nearest neighbor paths of length $n+1$.
By \eqref{cap.subadd}, it follows that $c_{n+m}\le c_n + c_m$, for all $n,m\ge 0$, so that by Fekete's lemma, the limit $\lim_{n\to \infty} c_n/n$ exists. Call $\gamma_d^*$ this limit.
Note that $\psi_d(x)$ is finite on $[\gamma_d,\gamma_d^*]$, since the probability that the simple random walk follows the path realizing the maximum in \eqref{cap.cn} is larger than or equal to $1/(2d)^{n+1}$, so that $\psi_d(x)\le \log(2d)$, for all $x\le \gamma_d^*$. Conversely, by definition of $c_n$, one has $\psi_d(x)=\infty$ for all $x> \gamma_d^*$.
Furthermore, it follows from Lemma 3 and Proposition 4 in \cite{HK}, that $\psi_d$ is continuous, and convex on $(0,\gamma_d^*]$.
Now Proposition \ref{cor:upward} and Lemma \ref{lem.exp.cap} show that when $d=5$, $\psi_d(x) \ge c(x-\gamma_5)^3$, for all $x\ge \gamma_d$.
Likewise, when $d\ge 6$, we get $\psi_d(x) \ge c(x-\gamma_d)^3$, for $\gamma_d \le x\le 1$.
Using convexity, this also shows that $\psi_d$ is increasing on $[\gamma_d,\gamma_d^*]$.
In addition one has $\psi_d(x) = 0$ for all $x<\gamma_d$, by definition of $\gamma_d$ as the limit of the (normalized) expected capacity, and using that by \eqref{cap.card}, $\mathrm{Cap}(\mathcal{R}_n)\le n$.
Finally we show that $\gamma_d^*>\gamma_d$.
Consider $\mathcal{D}_n$ the set of {\it no double backtrack at even times}
paths of length $n+1$ that we introduced in \cite{AS2}.
By definition, this is simply the set of paths $\gamma:\{0,\dots,n\}\to \mathbb{Z}^d$, such that for any even $k\le n$, one has $\gamma(k+2)\neq \gamma(k)$. The only important property we need is that from a
no-backtrack walk $\widetilde S$, and a sum of independent geometric
variables $\{\xi_i,\ i\in \mathbb{N}\}$, with parameter $1/(2d)^2$, we can build a simple random walk $S$
such that
\begin{equation*}
\mathcal{R}[0,n+2\sum_{i\le n/2} \xi_i]=\widetilde \mathcal{R}_n.
\end{equation*}
Thus, for any $\alpha>0$, we have by \eqref{cap.subadd} and \eqref{cap.card},
\begin{equation*}
\mathrm{Cap}(\widetilde \mathcal{R}_n)
\ge \mathrm{Cap}(\mathcal{R}_{(1+\alpha)n})-
{\text{\Large $\mathfrak 1$}}\left(\sum_{i\le n/2} \xi_i< \frac{\alpha n}{2}\right) \cdot (1+\alpha)(n+1).
\end{equation*}
By taking the maximum over $\mathcal{D}_n$ on the left hand side, and
then the expectation on the right hand side, we obtain
\begin{equation}\label{upw-const4}
c_n\ge \max_{\pi\in \mathcal{D}_n} \mathrm{Cap}(\pi) \ge \mathbb{E}[ \mathrm{Cap}(\mathcal{R}_{(1+\alpha)n})]-
(1+\alpha)(n+1)\cdot \mathbb{P}\left(\sum_{i\le n/2} \xi_i<\frac{\alpha n}{2}\right).
\end{equation}
Now take $\alpha< 1/(2d)^2$, and use Chebyshev's inequality, to see that the last term of \reff{upw-const4} is $\mathcal O(1)$.
Together with Lemma \ref{lem.exp.cap} it implies that
\begin{equation*}
c_n\ge \gamma_d(1+\alpha)n-\mathcal{O}(\sqrt n),
\end{equation*}
which indeed proves that $\gamma_d<\gamma_d^*$.
\end{proof}
\vspace{0.3cm}
\noindent{{\bf Acknowledgements:} Perla Sousi participated at an early stage
of the project and we thank her for stimulating discussions, and her
proof of Lemma~\ref{lem-HD}.
We also thank Quentin Berger and Julien Poisat for the idea of
considering the polymer melt. Finally, we thank two anonymous referees,
whose suggestions were crucial in clarifying the arguments.
The authors were partly supported by the French Agence Nationale
de la Recherche under grants ANR-17-CE40-0032 and ANR-16-CE93-0003. }
|
\section{Introduction}
Adapting deep learning methods to geometric data ({e.g.}, shapes) is a vibrant research area that has already produced state of the art algorithms for several geometric learning tasks ({e.g.}, \cite{qi2017pointnet,qi2017pointnet++,su2015multi}).
Two prominent approaches are: (i) mapping the geometric data to tensors ({e.g.}, images) and using off-the-shelf convolutional neural network (CNN) architectures and optimization techniques \cite{su2015multi,Wu_2015_CVPR,sinha2016deep,maron2017convolutional}; and (ii) developing novel architectures and optimization techniques that are tailored to the geometric data \cite{masci2015geodesic,qi2017pointnet,qi2017pointnet++}.
%
An important benefit of (i) is in reducing the geometric learning task to an image learning one, allowing to harness the \emph{huge} algorithmic progress of neural networks for images directly to geometric data.
Some previous attempts, following (i), to perform learning tasks on geometric data use projections to 2D planes, {e.g.}, by rendering the shapes \cite{su2015multi}. Such projections are not injective and suffer from occlusions, thus often require a collection of projections for a single shape. Other methods embed the shape in an encapsulating 3D grid \cite{Wu_2015_CVPR,maturana2015voxnet}; these methods require dealing with higher dimensional tensors and are usually less
|
robust to deformations. Other methods \cite{sinha2016deep,maron2017convolutional} try to find low distortion 2D mappings to an image domain. In this case the intrinsic dimensionality of the data is preserved, however, these maps suffer from high distortion and/or ignore the difference in the topologies of the surface (no boundary) and the image (with boundary).
In this paper, we advocate a novel 2D mapping method for representing sphere-type (genus zero, {e.g.}, the human model in Figure \ref{fig:E0}a, left) surfaces as images. The challenge in using an image to represent a surface has two aspects: geometrical and topological. Geometrically, a general curved surface cannot be mapped to a flat domain ({i.e.}, the image) without introducing a significant distortion. Topologically, an image has a boundary while sphere-type surfaces do not; hence, any mapping between the two will introduce cuts and discontinuities. Furthermore, a naive application of 2D convolution to the image would be ambiguous on the surface (see Figure \ref{fig:conv_on_flat_spheres} and Subsection \ref{ss:conv_on_flat_spheres}).
To address these challenges we think of the image as a periodic domain ({i.e.}, a torus) and relax the notion of a one-to-one mapping to that of a \emph{covering map} from the image domain onto the surface. That is, we construct a mapping from the image domain to the surface that covers the surface several times. For example, Figure \ref{
|
eqref{eq:Vdef}, and applying Lemma \ref{lem:TLyapunov}(iii) with $t=0$ and $\Delta = t$, Lemma \ref{lem:EphibTBbound} with $T = t$, and Lemma \ref{lem:expWbound} with $\lambda = C_V$ and $\Delta =t$, we see that for $t\in[0,\Delta]$,
\begin{align*}
\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}(t))\right]&=\mathbb{E}\left[\exp\left\{ r_1M_0(Z^{\alpha,x}(t))+r_2\norm{\mathcal{J}^{\alpha,\xi}(t)}_{B^\alpha}\right\}\right]\\
&\le V^\alpha(\xi)\mathbb{E}\left[\exp\left\{ C_V\norm{W}_t+r_2C_{\mathcal{J}}t\right\}\right]\\
&\le V^\alpha(\xi)\lb1+C_V\sqrt{\frac{8t}{\pi}}\right)^J\exp\left\{ (2J C_V^2+r_2C_{\mathcal{J}})t\right\}.
\end{align*}
Taking the supremum over $t\in[0,\Delta]$ on both sides completes the proof of the lemma.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:uniform}]
Fix a compact set $U_0$ in $U$ and a relatively compact set $A\subset\mathbb{X}$, and let $r_1,r_2\in(0,\infty)$, $\beta_0\in(0,1)$ and $\Delta_0,\eta_0,K_0\in(0,\infty)$ be as in Proposition \ref{prop:lyapunov}.
Let $\alpha\in U_0$ and $\xi\in A$.
By the Markov property for ${\Xi}^{\alpha,\xi}$ ({see Theorem \ref{thm:jointmarkov}}) and Proposition \ref{prop:lyapunov}, we have, for each $n\in\mathbb{N}$,
\begin{align*}
\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}(n\Delta_0))\right]\le\beta_0\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}((n-1)\Delta_0))\right]+K_0.
\end{align*}
Recursively applying the last display yields, for $n\in\mathbb{N}$,
\begin{align*}
\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}(n\Delta_0))\right]\le\beta_0^n V^\alpha(\xi)+\sum_{k=1}^{n-1}\beta_0^kK_0\le V^\alpha(\xi)+\frac{K_0}{1-\beta_0}.
\end{align*}
Another application of the Markov property for ${\Xi}^{\alpha,\xi}$, when combined with Lemma \ref{lem:Vbound0Delta}, shows that for all $t\in[n\Delta_0,(n+1)\Delta_0]$,
\begin{align*}
\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}(t))\right]&\le \mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi}(n\Delta_0))\right]\lb1+C_V\sqrt{\frac{8\Delta_0}{\pi}}\right)^J\exp\left\{\left( 2JC_V^2+r_2C_{\mathcal{J}}\right)\Delta_0\right\}\\
&\le\left( V^\alpha(\xi)+\frac{K_0}{1-\beta_0}\right)\lb1+C_V\sqrt{\frac{8\Delta_0}{\pi}}\right)^J\exp\left\{\left( 2J C_V^2+r_2C_{\mathcal{J}}\right) \Delta_0\right\}.
\end{align*}
To complete the proof, we
take the supremum over $t\in[n\Delta_0,(n+1)\Delta_0]$, $n\in\mathbb{N}_0$, $\alpha\in U_0$ and $\xi\in A$ on both sides of the previous inequality, and invoke the continuity of the map $(\alpha, \xi) \mapsto V^{\alpha}(\xi)$ and the compactness of
$U_0$ and $A$ to conclude the finiteness of the right-hand side.
\end{proof}
\section{Ergodicity of the joint process}
\label{sec:ergodic}
Throughout this section we assume the data $\{(d_i(\cdot),n_i),i\in\mathcal{I}\}$ satisfies Assumptions \ref{ass:independent}, \ref{ass:setB} and \ref{ass:projection}, and the coefficients $b(\cdot)$, $\sigma(\cdot)$ and $R(\cdot)$ satisfy Assumptions \ref{ass:holder} and \ref{ass:stable}.
\subsection{Uniqueness of the stationary distribution}
\label{sec:jointunique}
In this section we prove there is at most one stationary distribution for the joint process.
In the next section we prove existence of a stationary distribution.
The proof of uniqueness is nonstandard due to degeneracy of the joint process, which is a $2J$-dimensional process driven by a $J$-dimensional Brownian motion.
We use the asymptotic coupling method formalized by Hairer, Mattingly and Scheutzow in \cite{Hairer2011}.
In order to describe this method, we first need some notation.
Let ${\bf X}$ be a separable metric space with metric $\text{dist}(\cdot,\cdot)$.
Let $\mathcal{M}(\mathbb{D}({\bf X}))$ and $\mathcal{M}(\mathbb{D}({\bf X})\times\mathbb{D}({\bf X}))$ denote the set of probability measures on $(\mathbb{D}({\bf X}),\mathcal{B}(\mathbb{D}({\bf X})))$ and $(\mathbb{D}({\bf X})\times\mathbb{D}({\bf X}),\mathcal{B}(\mathbb{D}({\bf X}))\otimes\mathcal{B}(\mathbb{D}({\bf X})))$, respectively.
For $m_1,m_2\in\mathcal{M}(\mathbb{D}({\bf X}))$ let $\mathcal{C}(m_1,m_2)$ denote the set of \emph{couplings} of $m_1$ and $m_2$; that is,
$$\mathcal{C}(m_1,m_2):=\left\{\Upsilon\in\mathcal{M}(\mathbb{D}({\bf X})\times\mathbb{D}({\bf X})):\Upsilon(\cdot\times\mathbb{D}({\bf X}))=m_1(\cdot),\;\Upsilon(\mathbb{D}({\bf X})\times\cdot)=m_2(\cdot)\right\}.$$
Define the \emph{diagonal at infinity} by
$$\mathcal{D}:=\left\{ (\zeta_1,\zeta_2)\in\mathbb{D}({\bf X})\times\mathbb{D}({\bf X}):\lim_{t\to\infty}\text{dist}(\zeta_1(t),\zeta_2(t))=0\right\}.$$
We say $\Upsilon\in\mathcal{C}(m_1,m_2)$ is an \emph{asymptotic coupling} of $m_1$ and $m_2$ if $\Upsilon(\mathcal{D})=1$.
The following theorem is a continuous version of \cite[Theorem 1.1]{Hairer2011} (see, for example, \cite[Proposition 5.1]{AghRam19a} and \cite[Appendix C]{AghRam19b}), where we have also relaxed the requirement that ${\bf X}$ be \emph{complete}.
A careful examination of the proof of \cite[Theorem 1.1]{Hairer2011} reveals that the result still holds even if the metric space is not complete.
Also note that \cite[Theorem 1.1]{Hairer2011} proves a stronger result related to \emph{equivalent} asymptotic couplings, which we do not need here.
The version of the method we use is closely related to the \emph{asymptotic flatness} condition for the stochastic flow of a diffusion introduced by Basak and Bhattacharya \cite{Basak1992} to prove uniqueness of the stationary distribution for a degenerate diffusion.
Indeed, in the terminology of \cite{Basak1992}, we use a coupling construction to
show that the stochastic flow of the (degenerate) joint process is almost surely asymptotically flat;
see \eqref{eq:asymcoupled} in the proof of Theorem \ref{thm:unique} below.
\begin{theorem}\label{thm:coupling}
Let $\{\mathcal{P}_t\}=\{\mathcal{P}_t,t\ge0\}$ be a Markov transition semigroup on a separable metric space ${\bf X}$ admitting two stationary distributions $\mu_1$ and $\mu_2$.
For $i=1,2$ let $P_{\mu_i}$ denote the distribution of the Markov process with initial distribution $\mu_i$ and transition semigroup $\{\mathcal{P}_t\}$ on $(\mathbb{D}({\bf X}),\mathcal{B}(\mathbb{D}({\bf X})))$.
Suppose there is an asymptotic coupling of $P_{\mu_1}$ and $P_{\mu_2}$.
Then $\mu_1=\mu_2$.
\end{theorem}
With Theorem \ref{thm:coupling} in hand, we state and prove that there is at most one stationary distribution for the joint process.
\begin{theorem}
\label{thm:unique}
For each $\alpha\in U$ there is at most one stationary distribution for the joint process ${\Xi}^\alpha$.
\end{theorem}
\begin{proof}
Throughout this proof we fix $\alpha\in U$ and suppress the $\alpha$ dependence.
Suppose there are two stationary distributions $\mu_1$ and $\mu_2$ for the joint process.
For $i=1,2,$ let $P_{\mu_i}$ denote the distribution of the joint process with initial distribution $\mu_i$.
We construct the asymptotic coupling of $P_{\mu_1}$ and $P_{\mu_2}$ as follows.
Due to the uniqueness of the stationary distribution of the RBM stated in Theorem \ref{thm:RBMstable}, the first marginals of $\mu_1$ and $\mu_2$ must be equal in the sense that
\begin{equation}\label{eq:c1c2umarignal}\mu_1((A\times\mathbb{R}^J)\cap\mathbb{X})=\mu_2((A\times\mathbb{R}^J)\cap\mathbb{X}),\qquad A\in\mathcal{B}(G).\end{equation}
Let ${\Xi}_1=(Z_1,\mathcal{J}_1)$ and ${\Xi}_2=(Z_2,\mathcal{J}_2)$ denote the joint processes with respective initial distributions $\mu_1$ and $\mu_2$, and common driving Brownian motion $W$ such that $Z_1(0)$ and $Z_2(0)$ are independent of $W$.
Then $P_{\mu_i}(\cdot)=\P({\Xi}_i\in\cdot)$ for $i=1,2$.
In view of \eqref{eq:c1c2umarignal}, we can assume that ${\Xi}_1$ and ${\Xi}_2$ are built on the common probability space $(\Omega,\mathcal{F},\P)$ such that a.s.\ $Z_1(0)=Z_2(0)$.
Let $Y_1$ and $Y_2$ denote the respective constraining processes and set $\L_1(\cdot)=R^{-1}Y_1(\cdot)$ and $\L_2(\cdot)=R^{-1}Y_2(\cdot)$ as in \eqref{eq:Lalpha}.
By the pathwise uniqueness of RBMs (Theorem \ref{thm:rbm}) and Remark \ref{rmk:pathwiseunique}, a.s.\ $Z_1=Z_2$ and $\L_1=\L_2$.
Define the coupling $\Upsilon$ on $(\mathbb{D}(\mathbb{X})\times\mathbb{D}(\mathbb{X}),\mathcal{B}(\mathbb{D}(\mathbb{X}))\otimes\mathcal{B}(\mathbb{D}(\mathbb{X})))$ of the probability measures $P_{\mu_1}$ and $P_{\mu_2}$ on $(\mathbb{D}(\mathbb{X}),\mathcal{B}(\mathbb{D}(\mathbb{X})))$ by
$$\Upsilon\left(\mathcal{A}_1\times \mathcal{A}_2\right)=\P\left(({\Xi}_1,{\Xi}_2)\in \mathcal{A}_1\times \mathcal{A}_2\right),\qquad \mathcal{A}_1, \mathcal{A}_2\in\mathcal{B}(\mathbb{D}(\mathbb{X})).$$
For $i=1,2$ define $\mathcal{H}_i$ as in \eqref{eq:psib}, but with $\mathcal{H}_i$ and $\mathcal{J}_i$ in place of $\mathcal{H}$ and $\mathcal{J}$, respectively.
Since $\mathcal{H}_1$ and $\mathcal{H}_2$ are driven by the same
|
Brownian motion $W$ and a.s.\ $\L_1=\L_2$, it follows from \eqref{eq:psib} that
\begin{equation}\label{eq:psibdiffphiinfdiff}
\mathcal{H}_1(t)-\mathcal{H}_2(t)=\mathcal{J}_1(0)-\mathcal{J}_2(0),\qquad t\ge0.
\end{equation}
By Remark \ref{rmk:derivativeprocessbeta}, Definition \ref{def:dp}, the fact that a.s.\ $Z_1=Z_2$, and the linearity of the derivative map (Lemma \ref{lem:dmlinear}), a.s.
\begin{equation}\label{eq:phibdiffdmZpsibdiff}
\mathcal{J}_1-\mathcal{J}_2=\Lambda_{Z_1}(\mathcal{H}_1)-\Lambda_{Z_1}(\mathcal{H}_2)=\Lambda_{Z_1}(\mathcal{H}_1-\mathcal{H}_2).
\end{equation}
Due to \eqref{eq:phibdiffdmZpsibdiff}, \eqref{eq:psibdiffphiinfdiff}, the definition of $\tau_j(x)$ in \eqref{eq:tauj},
a repeated application of the bound in Proposition \ref{prop:dpcontract} with $S = \min(t,\tau_j(Z_1(0)))$ and $T = \min(t,\tau_{j+1}(Z_1(0)))$, Proposition \ref{prop:taujfinite} and Remark \ref{rmk:tau} show that we have a.s.
\begin{align}\label{eq:phicoupling}
\lim_{t\to\infty}\norm{\mathcal{J}_1(t)-\mathcal{J}_2(t)}_B&\leq\norm{\mathcal{J}_1(0)-\mathcal{J}_2(0)}_B\prod_{j=1}^\infty\delta_0^{1{\left\{\tau_j(Z_1(0))<\infty\right\}}}=0,
\end{align}
where $\delta_0 \in (0,1)$ is the contraction coefficient from Lemma \ref{lem:delta}.
Let
$$\mathcal{D}:=\left\{(\zeta_1,\zeta_2)\in\mathbb{D}(\mathbb{X})\times\mathbb{D}(\mathbb{X}):\lim_{t\to\infty}\norm{\zeta_1(t)-\zeta_2(t)}=0\right\}.$$
Since a.s.\ $Z_1=Z_2$, \eqref{eq:phicoupling} implies that
\begin{equation}\label{eq:asymcoupled}\Upsilon(\mathcal{D})=\P\left(\lim_{t\to\infty}\norm{{\Xi}_1(t)-{\Xi}_2(t)}=0\right)=\P\left(\lim_{t\to\infty}\norm{\mathcal{J}_1(t)-\mathcal{J}_2(t)}_{B^\alpha}=0\right)=1.\end{equation}
Therefore, $\Upsilon$ is an asymptotic coupling of $P_{\mu_1}$ and $P_{\mu_2}$, and Theorem \ref{thm:coupling} implies $\mu_1=\mu_2$.
\end{proof}
\subsection{Proof of Theorem \ref{thm:stable}}
\label{sec:proofstable}
Given $\alpha\in U$, $\xi\in\mathbb{X}$ and $t>0$, define the probability measure $Q_t^{\alpha,\xi}$ on $\mathbb{X}$ by
$$Q_t^{\alpha,\xi}(A):=\frac{1}{t}\int_0^tP_s^\alpha(\xi,A)ds,\qquad A\in\mathcal{B}(\mathbb{X}),$$
where $P_s^\alpha(\xi,A)$ is the transition function defined in \eqref{eq:Pt}.
With Corollary \ref{cor:uniform} in hand, the proof of existence of a stationary distribution follows a standard argument (see, e.g., the proof of \cite[Theorem 1.2]{Billingsley1999}), with the main difference being that the state space for the joint process $\mathbb{X}$ is not complete.
\begin{proof}[Proof of Theorem \ref{thm:stable}]
Fix $\xi_0\in\mathbb{X}$.
By Corollary \ref{cor:uniform},
\begin{equation}\label{eq:mdef}m:=\sup_{t\ge0}\mathbb{E}\left[ V^\alpha({\Xi}^{\alpha,\xi_0}(t))\right]<\infty.\end{equation}
Thus, by Markov's inequality, for all $t\ge0$ and $K<\infty$,
$$Q_t^{\alpha,\xi_0}\left(\left\{\xi\in\mathbb{X}:V^\alpha(\xi)\ge K\right\}\right)\le\frac{m}{K}.$$
Since $V^\alpha$ has compact level sets, it follows that the family of probability measures $\{Q_t^{\alpha,\xi_0}\}_{t\ge0}$ on the Polish space $G\times\mathbb{R}^J$ is tight.
Let $\mu$ denote any weak limit point.
By Theorem \ref{thm:RBMstable} and the fact that the renormalized occupation measures of the RBM converge to its unique stationary distribution (see, e.g., \cite[Chapter 4, Theorem 9.3]{EK2009}),
the first marginal of $\mu$ is the unique stationary distribution for the RBM, which, by \cite[Theorem 2]{Kang2014a}, is supported on $G^\circ$.
Thus, $\mu$ is supported on $G^\circ\times\mathbb{R}^J\subset\mathbb{X}$.
Let $s>0$ and $g:\mathbb{X}\mapsto\mathbb{R}$ be a bounded and continuous function.
Let $\varepsilon>0$.
Since $Q_t^{\alpha,\xi_0}$ converges to $\mu$ in the weak topology and $(P_s^\alpha g):\mathbb{X}\mapsto\mathbb{R}$ is a bounded and continuous function by the Feller continuity shown in Theorem \ref{thm:jointmarkov}, we can choose $t\ge 2s\norm{g}_\infty/\varepsilon$ sufficiently large so that
$$\abs{\int_\mathbb{X} (P_s^\alpha g)(\xi)\mu(d\xi)-\int_\mathbb{X} (P_s^\alpha g)(\xi)Q_t^{\alpha,\xi_0}(d\xi)}+\abs{\int_\mathbb{X} g(\xi)Q_t^{\alpha,\xi_0}(d\xi)-\int_\mathbb{X} g(\xi)\mu(d\xi)}<\varepsilon.$$
For such $t\ge 2s\norm{g}_\infty/\varepsilon$, we have
\begin{align*}
\abs{(\mu P_s^\alpha)(g)-\mu(g)}&\le\abs{\int_\mathbb{X} g(\xi)(\mu P_s^\alpha)(d\xi)-\int_\mathbb{X} g(\xi)(Q_t^{\alpha,\xi_0}P_s^\alpha)(d\xi)}\\
&\qquad+\abs{\int_\mathbb{X} g(\xi)(Q_t^{\alpha,\xi_0}P_s^\alpha)(d\xi)-\int_\mathbb{X} g(\xi)Q_t^{\alpha,\xi_0}(d\xi)}\\
&\qquad+\abs{\int_\mathbb{X} g(\xi)Q_t^{\alpha,\xi_0}(d\xi)-\int_\mathbb{X} g(\xi)\mu(d\xi)}\\
&\le\abs{\int_\mathbb{X} (P_s^\alpha g)(\xi)\mu(d\xi)-\int_\mathbb{X} (P_s^\alpha g)(\xi)Q_t^{\alpha,\xi_0}(d\xi)}\\
&\qquad+\frac{1}{t}\abs{\int_t^{t+s} (P_u^\alpha g)(\xi)du-\int_0^s (P_u^\alpha g)(\xi)du}\\
&\qquad+\abs{\int_\mathbb{X} g(\xi)Q_t^{\alpha,\xi_0}(d\xi)-\int_\mathbb{X} g(\xi)\mu(d\xi)}\\
&\le 2\varepsilon.
\end{align*}
Since $\varepsilon>0$ was arbitrary, it follows that $\mu$ is a stationary distribution for the joint process, which is unique by Theorem \ref{thm:unique}.
\end{proof}
\section{Sensitivities of the stationary distribution of an RBM}\label{sec:interchange}
In this section we prove Theorem \ref{thm:sensitivity}.
Throughout this section we assume the data $\{(d_i(\cdot),n_i),i\in\mathcal{I}\}$ satisfies Assumptions \ref{ass:independent}, \ref{ass:setB} and \ref{ass:projection}, and the coefficients $b(\cdot)$, $\sigma(\cdot)$ and $R(\cdot)$ satisfy Assumptions \ref{ass:holder} and \ref{ass:stable}.
Fix a continuous differentiable function $f:G\mapsto\mathbb{R}$ with bounded and continuous Jacobian $f':G\mapsto\mathbb{R}^{1\times J}$.
Let $x\in G$ and $\xi=(x,0)\in\mathbb{X}$.
For each $t>0$ define the function $\theta_t:U\mapsto\mathbb{R}$ by
$$\theta_t(\alpha):=\frac{1}{t}\int_0^t\mathbb{E}\left[ f(Z^{\alpha,x}(s))\right] ds,\qquad\alpha\in U.$$
By Corollary \ref{cor:pathwise}, for each $t\ge0$, $\theta_t(\cdot)$ is differentiable on $U$ with
$$\theta_t'(\alpha)=\frac{1}{t}\int_0^t\mathbb{E}\left[ f'(Z^{\alpha,x}(s))\mathcal{J}^{\alpha,\xi}(s)\right] ds,\qquad\alpha\in U.$$
Then by Theorem \ref{thm:stable} and Corollary \ref{cor:uniform},
\begin{align}
\label{eq:stable}
\lim_{t\to\infty}\theta_t(\alpha)=\mathbb{E}\left[ f(Z^{\alpha,x}(\infty))\right]\quad\text{and}\quad\lim_{t\to\infty}\theta_t'(\alpha)=\mathbb{E}\left[ f'(Z^{\alpha,x}(\infty))\mathcal{J}^{\alpha,\xi}(\infty)\right].
\end{align}
\begin{lem}
\label{lem:barh}
There exists a locally integrable function $\bar\theta:U\to[0,\infty)$ such that $|\theta_t'(\alpha)|\le\bar\theta(\alpha)$ for all $t\ge0$ and $\alpha\in U$.
\end{lem}
\begin{proof}
Define $\bar{\theta}:U\mapsto[0,\infty]$ by
$$\bar{\theta}(\alpha):=\sup_{t\ge0}|\theta_t'(\alpha)|,\qquad\alpha\in U.$$
Let $U_0$ be a compact subset of $U$.
By Corollary \ref{cor:uniform},
$$\sup_{\alpha\in U_0}\bar{\theta}(\alpha)\le\norm{f'}_\infty\sup_{\alpha\in U_0}\sup_{s\ge0}\mathbb{E}\left[\abs{\mathcal{J}^{\alpha,\xi}(s)}\right] ds<\infty,$$
where $\norm{f'}_\infty:=\sup_{x\in G}|f'(x)|<\infty$ since $f\in C_b^1(G)$.
This proves that $\bar\theta$ is locally bounded, and hence, locally integrable.
\end{proof}
Theorem \ref{thm:sensitivity} is now a simple consequence of this lemma.
\begin{proof}[Proof of Theorem \ref{thm:sensitivity}]
Let $-\infty<\alpha_1<\alpha_2<\infty$ be such that $[\alpha_1,\alpha_2]\subset U$.
By the Fundamental Theorem of Calculus,
$$\theta_t(\alpha_2)=\theta_t(\alpha_1)+\int_{\alpha_1}^{\alpha_2}\theta_t'(\alpha)d\alpha.$$
Letting $t\to\infty$ in the last display and using \eqref{eq:stable}, along with Lemma \ref{lem:barh} and the Lebesgue Dominated Convergence Theorem to interchange the limit and the integral, we obtain
\begin{align*}
F(\alpha_2)&=F(\alpha_1)+\lim_{t\to\infty}\int_{\alpha_1}^{\alpha_2}\theta_t'(\alpha) d\alpha\\
&=F(\alpha_1)+\int_{\alpha_1}^{\alpha_2}\mathbb{E}\left[ f'(Z^\alpha(\infty))\mathcal{J}^\alpha(\infty)\right] d\alpha.
\end{align*}
In particular, this implies that for almost every\ $\alpha\in[\alpha_1,\alpha_2]$, $F(\cdot)$ is differentiable at $\alpha$ and its derivative satisfies \eqref{eq:sensitivity}.
Since $[\alpha_1,\alpha_2]\subset U$ was arbitrary, this completes the proof.
\end{proof}
|
\section{Introduction}
Bitcoin~\cite{nakamoto2008bitcoin} is a digital currency that is gaining acceptance~\cite{soper2014paypal} and recognition~\cite{chowdhry2014google}, with an estimated market capitalization of over~4.5 billion US dollars, as of November~2014~\cite{blockchain2014marketCap}.
Bitcoin's security stems from a robust incentive system. Participants are required to provide expensive proofs of work, and they are rewarded according to their efforts. This architecture has proved both stable and scalable, and it is used by most contemporary digital currencies and related services, e.g.~\cite{litecoin2013site, dogecoin2013site, miller2014permacoin, namecoin2013site}. Our results apply to all such incentive systems, but we use Bitcoin terminology and examples since it serves as an active and prominent prototype.
Bitcoin implements its incentive systems with a data structure called the \emph{blockchain}.
The blockchain is a serialization of all money transactions in the system. It is a single global ledger maintained by an open distributed system.
Since anyone can join the open system and participate in maintaining the blockchain, Bitcoin uses a \emph{proof of work} mechanism to deter attacks: participation requires exerting significant compute resources.
A participant that proves she has exerted enough resources with a proof of work is allowed to take a step in the protocol by generating a block.
Participants are compensated for their efforts with newly minted Bitcoins. The process of creating a block is called \emph{mining}, and the participants~--- \emph{miners}.
In order to win the reward, many miners try to generate blocks.
The system automatically adjusts the \emph{difficulty} of block generation,
such that one block is added every~10 minutes to the blockchain.
This means that each miner seldom generates a block.
Although its revenue may be positive in expectation, a miner may have to wait for an extended period to create a block and earn the actual Bitcoins.
Therefore, miners form \emph{mining pools}, where all members mine concurrently and they share their revenue whenever one of them creates a block.
Pools are typically implemented as a \emph{pool manager} and a cohort of miners. The pool manager joins the Bitcoin system as a single miner.
Instead of generating proof of work, it outsources the work to the miners.
In order to evaluate the miners' efforts, the pool manager accepts partial proof of work and estimates each miner's \emph{power} according to the rate with which it submits such partial proof of work.
When a miner generates a full proof of work, it sends it to the pool manager which publishes this proof of work to the Bitcoin system.
The pool manager thus receives the full revenue of the block and distributes it fairly according to its members power.
Many of the pools are open~--- they allow any miner to join them using a public Internet interface.
Such open pools are susceptible to the classical \emph{block withholding attack}~\cite{rosenfeld2011analysis}, where a miner sends only partial proof of work to the pool manager and discards full proof of work.
Due to the partial proof of work it sends to the pool, the miner is considered a regular pool member and the pool can estimate its power.
Therefore, the attacker shares the revenue obtained by the other pool members, but does not contribute. It reduces the revenue of the other members, but also its own.
We provide necessary background on the Bitcoin protocol, pools and the classical block withholding attack in Section~\ref{sec:prelim}, and specify our model in Section~\ref{sec:model}.
In this work we analyze block withholding attacks among pools.
A pool that employs the \emph{pool block withholding attack} registers with the victim pool as a regular miner.
It receives tasks from the victim pool and transfers them to some of its own miners.
We call these \emph{infiltrating} miners, and the mining power spent by a pool the \emph{infiltration rate}.
When the attacking pool's infiltrating miners deliver partial proofs of work, the attacker transfers them to the victim pool, letting the attacked pool estimate their power.
When the infiltrating miners deliver a full proof of work, the attacking pool discards it.
This attack affects the revenues of the pools in several ways.
The victim pool's effective mining rate is unchanged, but its total revenue is divided among more miners.
The attacker's mining power is reduced, since some of its miners are used for block withholding, but it earns additional revenue through its infiltration of the other pool.
And finally, the total effective mining power in the system is reduced, causing the Bitcoin protocol to reduce the difficulty.
Taking all these factors into account, we observe that a pool might be able to increase its revenue by attacking other pools. Each pool therefore makes a choice of whether to attack each of the other pools in the system, and with what infiltration rate. This gives rise to the \emph{pool game}. We specify this game and provide initial analysis in Section~\ref{sec:poolGame}.
In Section~\ref{sec:twoPoolsOneAttacker} we analyze the scenario of exactly two pools where only one can attack the other.
Here, the attacker can always increase its revenue by attacking.
We conclude that in the general case, with any number of pools, no-pool-attacks is not a Nash equilibrium.
Next, Section~\ref{sec:twoPools} deals with the case of two pools, where each can attack the other. Here analysis becomes more complicated in two ways.
First, the revenue of each pool affects the revenue of the other through the infiltrating miners. We prove that for a static choice of infiltration rates the pool revenues converge.
Second, once one pool changes its infiltration rate of the other, the latter may prefer to change its infiltration rate of the former.
Therefore the game itself takes steps to converge.
We show analytically that the game has a single Nash Equilibrium and numerically study the equilibrium points for different pool sizes.
For pools smaller than $50\%$, at the equilibrium point both pools earn less than they would have in the non-equilibrium no-one-attacks strategy.
Since pools can decide to start or stop attacking at any point, this can be modeled as the \emph{miner's dilemma}~--- an instance of the iterative prisoner's dilemma. Attacking is the dominant strategy in each iteration, but if the pools can agree not to attack, both benefit in the long run.
Finally we address the case of an arbitrary number of identical pools in Section~\ref{sec:pPools}.
There exists a symmetric equilibrium point in which each pool attacks each other pool.
As in the minority two-pools scenario, here too at equilibrium all pools earn less than with the no-pool-attacks strategy.
Our results imply that block withholding by pools leads to an unfavorable equilibrium. Nevertheless, due to the anonymity of miners, a single pool might be tempted to attack, leading the other pools to attack as well. The implications might be devastating for open pools: If their revenues are reduced, miners will prefer to form closed pools that cannot be attacked in this manner.
Though this may be conceived as bad news for public mining pools, on the whole it may be good news to the Bitcoin system, which prefers small pools.
We discuss this and other issues pertaining to practice in Section~\ref{sec:discussion}.
In summary, our contributions are the following:
\begin{enumerate}
\item Definition of the pool game where pools in a proof-of-work secured system attack one another with a pool block withholding attack.
\item In the general case, no-pool-attacks is not an equilibrium.
\item With two minority pools, the only Nash Equilibrium is when the pools attack one another, and both earn less than if none had attacked.
Miners therefore face the miner's dilemma, an instance of the iterative prisoner's dilemma, repeatedly choosing between attack and no-attack.
\item With multiple pools of equal size there is a symmetric Nash equilibrium, where all pools earn less than if none had attacked.
\item For Bitcoin, inefficient equilibria for open pools may serve the system by reducing their attraction and pushing miners towards smaller closed pools.
\end{enumerate}
The classical block withholding attack is old as pools themselves, but its use by pools has not been suggested until recently. We overview related attacks and prior work in Section~\ref{sec:related}, and end with concluding remarks in Section~\ref{sec:conclusion}.
\section{Preliminaries --- Bitcoin and Pooled Mining} \label{sec:prelim}
Bitcoin is a distributed, decentralized digital currency~\cite{bitcoin2013protocol,bitcoin2013rules,nakamoto2008bitcoin,bitcoin2013source}.
Clients use the system by issuing transactions, and the system's only task is to serialize transactions in a single ledger and reject transactions that cannot be serialized due to conflicts with previous transactions.
Bitcoin transactions are protected with cryptographic techniques that ensure that only the rightful owner of a Bitcoin can transfer it.
The transaction ledger is stored in a data structure caller the \emph{blockchain}.
The blockchain is maintained by a network of \emph{miners}, which are compensated for their effort in Bitcoins. The miners are in charge of recording the transactions in the blockchain.
\subsection{Revenue for Proof Of Work}
The blockchain records the transactions in units of blocks.
Each block includes a unique ID, and the ID of the preceding block.
The first block, dubbed \emph{the genesis block}, is defined as part of the protocol.
A valid block contains the hash of the previous block, the hash of the transactions in the current block, and a Bitcoin address which is to be credited with a reward for generating the block.
Any miner may add a valid block to the chain by (probabilistically) proving that it has spent a certain amount of work and publishing the block with the proof over an overlay network to all other miners.
When a miner creates a block, it is compensated for its efforts with Bitcoins.
This compensation includes a per-transaction fee paid by the users whose transactions are included, and an amount of minted Bitcoins that are thus introduced into the system.
The rate at which the new Bitcoins are generated with each block is designed to slowly decrease towards zero, and will reach zero when~21 million Bitcoins are created. Then, the miners' revenue will be only from transaction fees.
The work which a miner is required to do is to repeatedly calculate a a hash function~--- specifically the SHA-256 of the SHA-256 of a block header. To indicate that he has performed this work, the miner provides a probabilistic proof as follows. The generated block has a nonce field, which can contain any value. The miner places different values in this field and calculates the hash for each value. If the result of the hash is smaller than a target value, the nonce is considered a solution, and the block is valid.
The number of attempts to find a single hash is therefore random with a geometric distribution, as each attempt is a Bernoulli trial with a success probability determined by the target value. At the existing huge hashing rates and target values, the time to find a single hash can be approximated by an exponential distribution. The average time for a miner to find a solution is therefore proportional to its hashing rate or \emph{mining power}.
To maintain a constant rate of Bitcoin generation, and as part of its defense against denial of service and other attacks, the system normalizes the rate of block generation. To achieve this, the protocol deterministically defines the target value for each block according to the time required to generate recent blocks. The target, or \emph{difficulty}, is updated once every~2016 blocks such that the average time for each block to be found is~10 minutes.
Note that the exponential distribution is memoryless. If all miners mine for block number $b$, once the block is found at time $t$, all miners switch to mine for the subsequent block $b+1$ at $t$ without changing their probability distribution of finding a block after~$t$. Therefore, the probability that a miner $i$ with mining power $m_i$ finds the next block is its ratio out of the \emph{total mining power} $m$ in the system.
\subsection*{Forks}
Block propagation in the overlay network takes seconds, whereas the average mining interval is~10 minutes. It is therefore possible for two miners to generate competing blocks, both of which list the same block as their predecessor. The system has mechanism to solve such situations, causing one of the blocks to be discarded. However, such bifurcations are rare and occur on average once every~60 blocks~\cite{decker2013propagation}, and we ignore them for the sake of simplicity. Since the choice of the discarded block on bifurcation is random, one may incorporate this event into the probability of finding a block, and consider instead the probability of finding a block that is not discarded.
\subsection{Pools}
As the value of Bitcoin rose, Bitcoin mining has become a rapidly advancing industry.
Technological advancements lead to ever more efficient hashing ASICs~\cite{taylor2013bespoke}, and mining datacenters are built around the world~\cite{popper2013mines}.
Mining is only profitable using dedicated cutting edge mining rigs, otherwise the energy costs exceed the expected revenue.
Although expected revenue from mining is proportional to the power of the mining rigs used, a single home miner using a small rig is unlikely to mine a block for years~\cite{swanson2013calculator}.
Consequently, miners often organize themselves into mining \emph{pools}.
Logically, a pool is a group of miners that share their revenues when one of them successfully mines a block. For each block found, the revenue is distributed among the pool members in proportion to their mining power\footnote{This is a simplification that is sufficient for our analysis. The intricacies of reward systems are explained in~\cite{rosenfeld2011analysis}.}.
The expected revenue of a pool member is therefore the same as its revenue had it mined \emph{solo}.
However, due to the large power of the pool, it finds blocks at a much higher rate, and so the frequency of revenue collection is higher, allowing for a stable daily or weekly income.
In practice, most pools are controlled by a pool manager.\footnote{A notable exception is P2Pool~\cite{forrsetv2011p2pool}, which we discuss in Section~\ref{sec:discussion}.}
Miners register with the pool manager and mine on its behalf: The pool manager generates tasks and the miners search for solutions based on these tasks that can serve as proof of work. Once they find a solution, they send it to the pool manager.
The pool manager behaves as a single miner in the Bitcoin system. Once it obtains a legitimate block from one of its miners, it publishes it.
The block transfers the revenue to the control of the pool manager.
The pool manager then distributes the revenue among the miners according to their mining power.
The architecture is illustrated in Figure~\ref{fig:threeHonestPools}
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{threeHonestPools.pdf}
\caption[.]{\protect
A system with 8 miners and~3 honest pools. Pool~1 has~3 registered miners, pools~2 and~3 have~2 registered miners each, and one miner mines solo.
}
\label{fig:threeHonestPools}
\end{figure}
In order to estimate the mining power of a miner, the pool manager sets a partial target for each member, much larger (i.e., easier) than the target of the Bitcoin system.
Each miner is required to send the pool manager blocks that are correct according to the partial target. The partial target is chosen to be large, such that partial solutions arrive frequently enough for the manager to accurately estimate the power of the miner, but small (hard) to reduce management overhead.
Pools often charge a small percentage of the revenue as fee. We discuss in Section~\ref{sec:discussion} the implications of such fees to our analysis.
Many pools are open and accept any interested miner.
Pool interface typically includes a web interface for registration and a miner interface for the mining software.
In order to mine for a pool, a miner registers with the web interface, supplies a Bitcoin address to receive its future shares of the revenue, and receives from the pool credentials for mining.
Then he feeds his credentials and the pool's address to its mining rig, which starts mining. The mining rig obtains its tasks from the pool and sends partial and full proof of work with the STRATUM protocol~\cite{btcWiki2014stratum}.
As it finds blocks, the pool manager credits the miner's account according to its share of the work, and transfers these funds either on request or automatically to the aforementioned Bitcoin address.
\subsection*{Too Big Pools}
Arguably in realistic scenarios of the Bitcoin system no pool controls a majority of the mining power. The reason is that the manager of a pool of this size can single-handedly take control of the Bitcoin system by generating the longest chain and ignoring blocks generated by other miners.
If the system reaches this situation it is severely unstable~\cite{andresen2014centralized} (and~\cite{eyal2013majority} warns that the system is unstable with even smaller pools). For one day in June~2014 a single pool called GHash.IO produced over $50\%$ of the blocks in the Bitcoin main chain. The Bitcoin community backlashed at the pool (which did nothing worse than being extremely successful). GHash.IO reduced its relative mining power and publicly committed to stay away from the $50\%$ limit.
\subsection{Block Withholding} \label{sec:classicalBWA}
Classical Block Withholding~\cite{rosenfeld2011analysis} is an attack performed by a pool member against the other pool members. The attacking miner registers with the pool and apparently starts mining honestly~--- it regularly sends the pool partial proof of work.
However, the attacking miner sends only partial proof of work. If it finds a full solution that constitutes a full proof of work it discards the solution, reducing the pool's total revenue. This attack is illustrated in Figure~\ref{fig:regularBWA}.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{regularBWA.pdf}
\caption[.]{\protect
Classical Block Withholding attack. A group of miners attack Pool~2 with a block withholding attack, denoted by a dashed red arrow.
}
\label{fig:regularBWA}
\end{figure}
The attacker does not change the pool's effective mining power, and does not affect directly the revenue of other pools. However, the attacked pool shares its revenue with the attacker. Therefore each miner earns less, as the same revenue is distributed among more miners.
Recall that the proof of work is only valid to a specific block, as it is the nonce with which the block's hash is smaller than the target. The attacking miner cannot use it.
Moreover, this attack reduces the attacker's revenue compared to solo mining or honest pool participation:
It suffers from the reduced revenue like the other pool participants, and its revenue is less than its share of the total mining power in the system.
This attack can therefore only be used for sabotage, at a cost to the attacker.
\subsubsection*{Detection}
Although a pool can detect that it is under a block withholding attack with good accuracy, it might not be able to detect which of its registered miners are the perpetrators. The reason is that, by design, the partial proof of work difficulty is much easier than the full proof of work difficulty.
A pool can estimate its expected mining power and its actual mining power by the rates of partial proofs of work and full proofs of work, respectively, supplied by its miners. A difference above a set confidence interval indicates an attack.
To detect whether a single miner is attacking it, the pool must use a similar technique, comparing the estimated mining power of the attacker based on its partial proof of work with the fact it never supplies a full proof of work. If the attacker has a small mining power, it will send frequent partial proofs of work, but the pool will only expect to see a full proof of work at very low frequency. Therefore, it cannot obtain statistically significant results that would indicate an attack.
An attacker can therefore use multiple small miners, rather than a single large one, and replace them frequently. For example, miners whose expected full proof of work frequency is yearly will see a non-negligible daily revenue ($\btc 25 / 12 / 31 \approx \btc 0.07$). Replacing them monthly will not allow a pool to confidently tag them as attackers without tagging legitimate miners as well.
\section{Model and Standard Operation} \label{sec:model}
Bitcoin is the first widely used system that rewards participants for proof of work at a dynamically normalized rate.
Its success demonstrates the strength of this architecture, and others were quick to follow. Most are digital currencies with a various proof of work algorithms; some have other uses, for example NameCoin~\cite{namecoin2013site}, which is a DNS replacement with no central authority.
This model, and therefore our results, apply to all such systems.
Nevertheless, since Bitcoin is both a prototype and a working example, we use the Bitcoin terminology.
We specify the basic model in which participants operate in Section~\ref{sec:baseModel}, proceed to describe how honest miners operate in this environment in Sections~\ref{sec:solo} and~\ref{sec:pools}, and how the classical block withholding attack is implemented with our model in Section~\ref{sec:bwMiner}.
\newcommand{\cmdNewTask}[1]{ \ensuremath{ \texttt{newTask}(#1) }}
\newcommand{\cmdPublish}[2]{ \ensuremath{ \texttt{publish}(#1, #2) }}
\newcommand{\cmdWork}[1]{ \ensuremath{ \texttt{work}(#1) }}
\newcommand{\send}[2]{ \ensuremath{ \texttt{send}(#1, #2) }}
\newcommand{\recv}[1]{ \ensuremath{ \texttt{recv}(#1) }}
\newcommand{\cmdTaskFromPool}[2]{ \ensuremath{ \texttt{taskFromPool}(#1, #2) }}
\newcommand{\cmdPoWToPool}[3]{ \ensuremath{ \texttt{PoWToPool}(#1, #2, #3) }}
\newcommand{\cmdPoWFromMiner}[1]{ \ensuremath{ \texttt{getPoW}(#1) }}
\newcommand{\cmdSendTask}[2]{ \ensuremath{ \texttt{sendTask}(#1, #2) }}
\newcommand{\cmdPay}[2]{ \ensuremath{ \texttt{pay}(#1, #2) }}
\newcommand{\cmdCollect}[2]{ \ensuremath{ \texttt{collect}(#1, #2) }}
\newcommand{\fPoW}{ \ensuremath{ \textit{fPoW} }}
\newcommand{\pPoW}{ \ensuremath{ \textit{pPoW} }}
\newcommand{\tasks}{ \ensuremath{ \textit{tasks} }}
\newcommand{\pPoWs}{ \ensuremath{ \textit{pPoWs} }}
\subsection{Model} \label{sec:baseModel}
The system is comprised of the Bitcoin network and nodes, and progresses in steps.
Each node has a unique ID \textit{id} and can generate \emph{tasks} by calling the $\cmdNewTask{\textit{id}}$ command. The task is associated with the given $\textit{id}$.
A node can work on a task for the duration of a step using the $\cmdWork{\textit{task}}$ command.
A node that works on tasks with \cmdWork{}\ is called a miner.
The command returns a set of partial proof of work and a set of full proofs of work.
The number of proofs in each set has a Poisson distribution, partial proofs with a large mean and full proofs with a small mean.
All nodes have identical power, and hence identical probabilities to generate proofs of work.
The Bitcoin network pays for full proofs of work. To acquire this payoff an entity publishes a task $\textit{task}$ and its corresponding proof of work $\textit{PoW}$ to the network by calling the $\cmdPublish{\textit{task}}{\textit{PoW}}$ command, which returns the amount earned. The payoff goes to the ID associated with \textit{task}.
The Bitcoin protocol normalizes revenue such that the average total revenue distributed in each step is a constant throughout the execution of the system.
A node can transact $b$ Bitcoins to another node with ID $w$ with the $\cmdPay{w}{b}$ command.
Apart from the \cmdWork{}\ command, all local operations, payments, message sending, propagation, and receipt are instantaneous.
We assume that the number of miners is large enough such that mining power can be split arbitrarily without resolution constraints. Denote the number of pools with $p$, the total number of mining power in the system with $m$ and the miners loyal to pool~$i$ ($1 \le i \le p$) with $m_i$.
We use a quasi-static analysis where miner loyalty to a pool does not change over time.
\subsection{Solo Mining} \label{sec:solo}
Nodes are defined by an implementation of a doStep function~--- their routine in a single step. As a first example we provide the behavior of a solo miner.
A solo miner is a node that generates its own tasks and publishes them to earn the payoff. The algorithm of a solo miner is given in Algorithm~\ref{alg:solo}.
\begin{algorithm}[t]
\caption{Solo Miner $w$}
\label{alg:solo}
\SetAlgoNoLine
\SetAlgoNoEnd
\DontPrintSemicolon
\SetNoFillComment
\KwFunction({$\text{doStep}$}){
$\textit{task} \gets \cmdNewTask{w}$ \;
$(\pPoW, \fPoW) \gets \cmdWork{\textit{task}}$ \;
$\cmdPublish{\textit{task}}{\fPoW}$ \;
}
\end{algorithm}
\subsection{Pools} \label{sec:pools}
Pools are nodes that serve as coordinators and multiple miners can register to a pool and work for it.
The pseudocode for a miner~$w$ working for a pool~$i$ is shown in Algorithm~\ref{alg:honest}.
The pool generates the tasks and sends a task to each miner.
The miner receives its task and works on it for the duration of the step.
Since full proofs of work are rarely found, the pool needs to reliably measure the miner's work in a different manner.
To facilitate pool operation, miners send the pool not only full proof of work, but also partial proofs of work.
The pool receives the proofs of work of all its miners, registers the partial proofs of work and publishes the full proofs.
It calculates its overall revenue, and proceeds to distribute it among its miners.
Each miner receives revenue proportional to its success in the current step, namely the ratio of its partial proofs of work out of all partial proofs of work the pool received. The pool pays miner $w$ its share $b$ of the revenue with $\cmdPay{w}{b}$, and notifies the miner of the payment with a separate message.
We assume that pools do not collect fees of the revenue. Pool fees and their implications on our analysis are discussed in Section~\ref{sec:discussion}.
\begin{algorithm}[t]
\caption{Honest miner $w$ at Pool $i$.}
\label{alg:honest}
\SetAlgoNoLine
\SetAlgoNoEnd
\DontPrintSemicolon
\SetNoFillComment
\KwTitle{Miner $w$} \;
\vspace{-0.5\baselineskip}
\KwState{
$\textit{revenue} \in \mathbb{R}$, initially 0
}
\BlankLine
\KwFunction({$\text{doStep}$}){
$\textit{task} \gets \recv{i}$ \;
$(\pPoW, \fPoW) \gets \cmdWork{\textit{task}}$ \;
\send{i}{(\pPoW, \fPoW)} \;
$\textit{revenue} \gets \textit{revenue} + \recv{i}$ \;
}
\BlankLine \BlankLine
\KwTitle{Pool $i$} \;
\vspace{-0.5\baselineskip}
\KwState{
$W \subset \textup{ID space}$ \;
$\tasks(\cdot): W \rightarrow \textup{Task space}$ \;
}
\BlankLine
\KwFunction({$\text{doStep}$}){
$\textit{stepRevenue} \gets 0$ \;
\ForEach{Registered Miner $w$} {
$\tasks(w) \gets \cmdNewTask{i}$ \;
$\send{w}{\tasks(w)}$ \;
}
\BlankLine
\KwIttayComment{Wait until end of step.}
\BlankLine
\ForEach{$w \in W$} {
$(\pPoW, \fPoW) \gets \recv{w}$ \;
$\textit{stepRevenue} \gets \textit{stepRevenue} + \cmdPublish{\textit{tasks}(w)}{\textit{fPoW}}$ \;
$\pPoWs(w) \gets pPoW$ \;
}
$\textit{stepPPow} \gets \sum_{\text{registered } w} \pPoWs(w)$ \;
\ForEach{$w \in W$} {
$\cmdPay{w}{\textit{stepRevenue} \times \pPoWs(w) / \textit{\textit{stepPPow}}}$ \;
$\send{w}{\textit{stepRevenue} \times \pPoWs(w) / \textit{\textit{stepPPow}}}$
}
}
\end{algorithm}
\subsection{Block Withholding Miner} \label{sec:bwMiner}
A miner registered at a pool can perform the classical block withholding attack, where it operates as if it worked for the pool, only it never sends its proof of work. The pseudocode is in Algorithm~\ref{alg:bwMiner}, for a miner interacting with a standard pool as in Algorithm~\ref{alg:honest}.
The pool registers the miner's partial proofs, but cannot distinguish between miners running the honest miner of Algorithm~\ref{alg:honest} and block withholding miners running Algorithm~\ref{alg:bwMiner}.
The implications are that a miner that engages in block withholding does not contribute to the pool's overall mining power, but still shares the pool's revenue according to its sent partial proofs of work.
\begin{algorithm}[t]
\caption{Block Withholding Miner $w$ at pool $i$.}
\label{alg:bwMiner}
\SetAlgoNoLine
\SetAlgoNoEnd
\DontPrintSemicolon
\SetNoFillComment
\KwFunction({$\text{doStep}$}){
$\textit{task} \gets \cmdTaskFromPool{i}{w}$ \;
$(\pPoW, \fPoW) \gets \cmdWork{\textit{task}}$ \;
\cmdPoWToPool{i}{\pPoW}{\emptyset} \;
$\cmdCollect{i}{w}$ \;
}
\end{algorithm}
To reason about a pool's efficiency we define its per-miner revenue as follows.
\begin{definition}[Revenue density]
The \emph{revenue density} of a pool is the ratio between the average revenue a pool member earns and the average revenue it would have earned as a solo miner.
\end{definition}
The revenue density of a solo miner, and that of a miner working with an unattacked pool are~1. If a pool is attacked with block withholding, its revenue density decreases.
\subsection{Continuous Analysis} \label{sec:continuous}
Because our analysis will be of the average revenue, we will consider proofs of work, both full and partial, as continuous deterministic sizes, according to their probability.
The $\cmdWork{}$ command will therefore return a deterministic fraction of proof of work.
\section{The Pool Game} \label{sec:poolGame}
\subsection{The Pool Block Withholding Attack}
Just as a miner can perform block withholding on a pool $j$, a pool~$i$ can use some of its miners to infiltrate a pool $j$ and perform a block withholding attack on~$j$. Denote the number of such infiltrating miners at step~$t$ by~$\xij{i}{j}(t)$. In this case, the infiltrating miners obtain their share of pool~$j$'s revenue, and transfer it back to pool~$i$. Infiltrators from~$i$ to~$j$, as well as any miners that honestly mine for pool~$i$, are \emph{loyal} to pool~$i$. Pool~$i$ distributes its revenue from mining and from its infiltrators evenly among all its registered miners, according to their partial proofs of work.
The pseudocode of a block withholding pool is shown in Algorithm~\ref{alg:bwa}. The pool's miners are oblivious to the change and their algorithm is identical to the one in Algorithm~\ref{alg:honest}.
\newcommand{\ensuremath{ \textit{prevRoundInfiltrationRevenue} }}{\ensuremath{ \textit{prevRoundInfiltrationRevenue} }}
\newcommand{\ensuremath{ \textit{inf} }}{\ensuremath{ \textit{inf} }}
\begin{algorithm*}[t]
\caption{Block Withholding Pool $i$.}
\label{alg:bwa}
\SetAlgoNoLine
\SetAlgoNoEnd
\DontPrintSemicolon
\SetNoFillComment
\KwState{
$\ensuremath{ \textit{prevRoundInfiltrationRevenue} } \in \mathbb{R}$, initially 0 \;
$W \subset \textup{ID space}$ \;
$\ensuremath{ \textit{inf} }(\cdot): W \rightarrow \textup{ID space} \cup {\bot}$ \;
$\tasks(\cdot): W \rightarrow \textup{Task space}$ \;
}
\BlankLine
\KwFunction({$\text{doStep}$}){
\ForEach{$w \in W$} {
\If({\hfill(infiltrator)}){$\ensuremath{ \textit{inf} }(w) \neq \bot$} {
$\tasks(w) \gets \cmdTaskFromPool{\ensuremath{ \textit{inf} }(w)}{i}$ \;
} \Else {
$\tasks(w) \gets \cmdNewTask{i}$ \;
}
$\cmdSendTask{w}{\tasks(w)}$ \;
}
\BlankLine
\KwIttayComment{Wait until end of round}
\BlankLine
$\textit{stepRevenue} \gets \ensuremath{ \textit{prevRoundInfiltrationRevenue} }$ \;
$\textit{stepPPow} \gets 0$ \;
\ForEach{$w \in W$} {
$(\pPoW, \fPoW) \gets \cmdPoWFromMiner{w}$ \;
$\textit{stepPPow} \gets \textit{stepPPow} + | \pPoW |$ \;
\If({\hfill(infiltrator)}){$\ensuremath{ \textit{inf} }(w) \neq \bot$} {
$\cmdPoWToPool{\ensuremath{ \textit{inf} }(w)}{\pPoW}{\emptyset}$ \;
} \Else {
$\textit{stepRevenue} \gets \textit{stepRevenue} + \cmdPublish{tasks(w)}{\fPoW}$ \;
}
}
\ForEach{$w \in W$} {
$\cmdPay{w}{\textit{stepRevenue} \cdot \pPoWs(w) / \textit{stepPPow}}$ \;
}
\BlankLine
$\ensuremath{ \textit{prevRoundInfiltrationRevenue} } \gets 0$ \;
\ForEach{$\{ w | w \in W \wedge \ensuremath{ \textit{inf} }(w) \neq \bot \}$} {
$\ensuremath{ \textit{prevRoundInfiltrationRevenue} } \gets \ensuremath{ \textit{prevRoundInfiltrationRevenue} } + \cmdCollect{\ensuremath{ \textit{inf} }(w)}{i}$ \;
}
}
\end{algorithm*}
\subsection{Block Withholding Recycling}
We assume that the infiltrating miners are loyal to the attacker.
However, some of the pool's members may be disloyal infiltrators. For example, pool~1 can use a loyal miner to infiltrate pool~2, and pool~2, thinking the miner is loyal to it, might use it to attack pool~1.
When sending disloyal miners to perform block withholding at other pools, an attacker takes a significant risk. The pool~2's perspective in the scenario described above. The disloyal miner can perform honest mining for pool~1, rather than withhold its blocks, and not return any revenue to pool~2. Moreover, it will take its share of pool~2's revenues (which thinks the miner is loyal to it) and deliver it back to pool~1.
To avoid such a risk, a pool needs a sufficient number of verified miners~--- miners that it knows to be loyal. In Bitcoin this happens in the common case where the pool owner has miners of his own.
\subsection{Revenue Convergence}
Note that pool~$j$ sends its revenue to infiltrators from pool~$i$ at the end of the step, and this revenue is calculated in pool~$i$ at the beginning of the subsequent step with the \ensuremath{ \textit{prevRoundInfiltrationRevenue} }\ variable. If there is a chain of pools of length $\ell$ where each pool infiltrates the next, the pool revenue will not be constant, since the revenue from infiltration takes one step to take each hop. If $\ell_{\max}$ is the longest chain in the system, the revenue stabilizes after $\ell_{\max}$ steps and remains constant. If there are loops in the infiltration graph, the system will converge to a certain revenue.
\begin{lemma}[Revenue convergence]
If infiltration rates are constant, the pool revenues converge.
\end{lemma}
\begin{proof}
Denote the revenue density of pool~$i$ at the end of step~$t$ by~$r_i(t)$, and define the revenue density vector
\[
\textbf{r(t)} \eqDef (r_1(t), \dots, r_p(t))^T \,\,\, .
\]
In every round, pool~$i$ uses its mining power of $m_1 - \sum_j \xij{1}{j}$ used for direct mining (and not attacking), and shares it among its $m_1 + \sum_j \xij{j}{1}$ members (all sums are over the range $1, \dots, p$), including malicious infiltrators.
Denote the direct mining revenue density of each pool (ignoring normalization, which is a constant factor) with the vector
\[
\textbf{m}
\eqDef
\left(
\frac{m_1 - \sum_j \xij{1}{j}}{m_1 + \sum_j \xij{j}{1}},
\dots,
\frac{m_p - \xij{p}{j}}{m_p + \sum_j \xij{j}{p}}
\right)^T \,\,\, .
\]
The revenue of Pool~$i$ in step $t$ taken through infiltration from pool~$j$'s revenue in step $t-1$ is $\xij{i}{j} r_j(t-1)$. Pool~i distributes this revenue among its $m_i + \sum_k \xij{k}{i}$ members~--- loyal and infiltrators.
Define the $p \times p$ \emph{infiltration matrix} whose $i, j$ element is
\[
\textbf{G}
\eqDef
\left[ \frac{\xij{i}{j}}{m_i + \sum_k \xij{k}{i}} \right]_{ij} \,\,\, .
\]
And the revenue vector at step $t$ is
\begin{equation} \label{eqn:rt}
\textbf{r}(t) = \textbf{m} + \textbf{G} \textbf{r}(t-1) \,\,\, .
\end{equation}
Since the row sums of the infiltration matrix are smaller than one, its largest eigenvalue is smaller than~1 according to the Perron-Frobenius theorem. Therefore, the revenues at all pools converges as follows for~$t \ge 1$:
\begin{equation} \label{eqn:rt}
\textbf{r}(t)
=
\left( \sum_{t'=0}^{t-1} G^t \right) \textbf{m} + G^t \textbf{r}(0)
\xrightarrow{t \rightarrow \infty}
(1 - \textbf{G})^{-1} \textbf{m}
\,\,\, .
\end{equation}
\end{proof}
\subsection{The Pool Game}
In the pool game pools try to optimize their infiltration rates of other pools to maximize their revenue. The overall number of miners and the number of miners loyal to each pool remain constant throughout the game.
Time progresses in rounds. Let $s$ be a constant integer large enough that revenue can be approximated as its convergence limit.
In each round the system takes~$s$ steps and then a single pool, picked with a round-robin policy, may change its infiltration rates of all other pools. The total revenue of each step is normalized to $1/s$, so the revenue per round is one.
\subsubsection*{Pool Knowledge}
The pool taking a step knows the rate of infiltrators attacking it (though not their identity) and the revenue rates of each of the other pools.
This knowledge is required to optimize a pool's revenue, as we explain in Section~\ref{sec:general}.
A pool can estimate the rate with which it is attacked by comparing the rates of partial and full proofs of work it receives from its miners, as explained in Section~\ref{sec:classicalBWA}.
In order to estimate the attack rates against each of the other pools, a pool can use one of two methods.
First, pools often publish this data to demonstrate their honesty to their miners~\cite{slush2014dashboard, ghash2014dashboard, discusfish2014dashboard}.
Second, a pool can infiltrate each of the other pools with some nominal probing mining power and measure the revenue density directly by monitoring the probe's rewards from the pool.
\subsection{General Analysis} \label{sec:general}
Recall that $m_i$ is the number of miners loyal to pool~$i$.
and $\xij{i}{j}(t)$ is the number of miners used by pool~$i$ to infiltrate pool~$j$ at step~$t$.
The mining rate of pool~$i$ is therefore the number of its loyal miners minus the miners it uses for infiltration.
This absolute mining rate denoted is divided by the total mining rate in the system, namely the number of all miners that do not engage in block withholding.
Denoted the direct mining rate at step~$t$ by
\begin{equation} \label{eqn:RiFull}
R_i
\eqDef
\frac{
m_i - \sum_{j = 1}^p \xij{i}{j}
}{
m - \sum_{j = 1}^p \sum_{k = 1}^p \xij{j}{k}
}
\end{equation}
The revenue density of pool $i$ at the end of step~$t$ is its revenue from direct mining together with its revenue from infiltrated pools, divided by the number of its loyal miners together with block-withholding infiltrators that attack it:
\begin{equation} \label{eqn:riFull}
r_i(t)
=
\frac{
R_i(t) + \sum_{j = 1}^{p} \xij{i}{j}(t) r_j(t)
}{
m_i + \sum_{j = 1}^{p} \xij{j}{i}(t)
}
\,\,\, .
\end{equation}
When pool~$i$ takes a step~$t$, it knows the revenue density of all other pools $r_j(t-1)$ and its total infiltration rate $\sum_{j = 1}^{p} \xij{j}{i}(t)$.
\subsection*{No attack}
If no pool engages in block withholding,
\[
\forall i, j: \xij{i}{j} = 0 \,\,\, ,
\]
we have at all times
\[
\forall i: r_i(t) = 1/m \,\,\, ,
\]
that is, each miner's revenue is proportional to its power, be it in a pool or working solo.
\section{One Attacker} \label{sec:twoPoolsOneAttacker}
We begin our analysis with a simplified game of two pools,~1 and~2, where pool~1 can infiltrate pool~2, but pool~2 cannot infiltrates pool~1.
The $m- m_1 - m_2$ miners outside both pools mine solo (or with closed pools that do not attack and cannot be attacked).
This scenario is illustrated in Figure~\ref{fig:twoPoolsOneAttackerIllustration}.
The dashed red arrow indicates that $\xij{1}{2}$ of pool~1's mining power infiltrates pool~2 with a block withholding attack.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{twoPoolsOneAttacker.pdf}
\caption[.]{\protect
The one-attacker scenario. Pool~1 attacks pool~2.
}
\label{fig:twoPoolsOneAttackerIllustration}
\end{figure}
Since Pool~2 does not engage in block withholding, all of its $m_2$ loyal miners work on its behalf. Pool~1, on the other hand does not employ $\xij{1}{2}$ of its loyal miners, and its direct mining power is only $m_1 - \xij{1}{2}$.
The Bitcoin system normalizes these rates by the total number of miners that publish full proofs, namely all miners but $\xij{1}{2}$.
The pools' direct revenues are therefore
\begin{equation} \label{eqn:oneAttacker:Rs}
\begin{aligned}
R_1 &= \frac{m_1 - \xij{1}{2}}{m - \xij{1}{2}} \\
R_2 &= \frac{m_2}{m - \xij{1}{2}} \,\,\, .
\end{aligned}
\end{equation}
Pool~2 divides its revenue among its loyal miners and the miners that infiltrated it. Its revenue density is therefore
\begin{equation} \label{eqn:oneAttacker:r2}
r_2 =
\frac{
R_2
}{
m_2 + \xij{1}{2}
}
\,\,\, .
\end{equation}
Pool~1 divides its revenue among its registered miners. The revenue includes both its direct mining revenue and the revenue its infiltrators obtained from pool~2, which is $r_2 \cdot \xij{1}{2}$. The revenue per loyal Pool~1 miner is therefore
\begin{equation} \label{eqn:oneAttacker:r1}
r_1 =
\frac{R_1 + \xij{1}{2} \cdot r_2}{m_1}
\,\,\, .
\end{equation}
We obtain the expression for~$r_1$ in Equation~\ref{eqn:oneAttacker:r1}
by substituting $r_2$ from Equation~\ref{eqn:oneAttacker:r2} and $R_1$ and $R_2$ from equation~\ref{eqn:oneAttacker:Rs}:
\[
r_1 =
\frac{
(\xij{1}{2})^2-m_1 (m_2+\xij{1}{2})
}{
m_1 (\xij{1}{2}-1) (m_2+\xij{1}{2})
}
\]
\subsection{Game Progress}
Pool~1 controls its infiltration rate of pool~2, namely $\xij{1}{2}$, and will choose the value that maximizes the \emph{revenue density} (per-miner revenue) $r_1$ on the first round of the pool game.
The value of $r_1$ is maximized at a single point in the feasible range $0 \le \xij{1}{2} \le m_1$.
Since pool~2 cannot not react to pool~1's attack, this point is the stable state of the system, and we denote the value of \xij{1}{2}\ there by
$
\xijbar{1}{2}
\eqDef
\arg \max_{\xij{1}{2}} r_1
\,\,\, ,
$
and the values of the corresponding revenues of the pools with $\bar{r}_1$ and $\bar{r}_2$.
Substituting the stable value $\xij{1}{2}$ we obtain the revenues of the two pools; all are given in Figure~\ref{fig:oneAttackerEqns}.
\begin{figure*}[t]
\begin{equation}
\begin{aligned}
& \xijbar{1}{2}
=
\frac{
m_2 - m_1 m_2 + \sqrt{-m_2^2 (-1 + m_1 + m_1 m_2)}
}{
-1 + m_1 + m_2
}
\\
& \bar{r}_1
=
\frac{
m_1 + (2 + m_1) m_2 + 2 \sqrt{-m_2^2 (-1 + m_1 + m_1 m_2)}
}{
m_1 (1 + m_2)^2
}
\\
& \bar{r}_2
=
\frac{
-m_2 (-1 + m_1 + m_2)^2
}{
\left( m_2^2 + \sqrt{-m_2^2 (-1 + m_1 + m_1 m_2)} \right)
\left( 1 - m_1 (1 + m_2) + \sqrt{-m_2^2 (-1 + m_1 + m_1 m_2)} \right)
}
\end{aligned}
\end{equation}
\caption{
Stable state where only pool~1 attacks pool~2.
}
\label{fig:oneAttackerEqns}
\end{figure*}
\subsection{Numerical Analysis}
\begin{figure*}[t]
\centering
|
\subfloat[\xij{1}{2}]{
\includegraphics[width=0.3\linewidth]{{twoPools02oneAttacker-m1-delta0.0100-accuracy0.00100-x1}.png}
\label{fig:twoPoolsOneAttacker:x12}
}
\hfil
\subfloat[$r_1$]{
\includegraphics[width=0.3\linewidth]{{twoPools02oneAttacker-m1-delta0.0100-accuracy0.00100-r1}.png}
\label{fig:twoPoolsOneAttacker:r1}
}
\hfil
\subfloat[$r_2$]{
\includegraphics[width=0.3\linewidth]{{twoPools02oneAttacker-m1-delta0.0100-accuracy0.00100-r2}.png}
\label{fig:twoPoolsOneAttacker:r2}
}
\caption[.]{\protect
Two pools where one infiltrates the other: Optimal infiltration rate $\xij{1}{2}$ and corresponding revenues~($r_1$ and $r_2$) as a function of pool sizes. The line in~\subref{fig:twoPoolsOneAttacker:x12} shows~$\xij{1}{2}=0$ and the lines in~\subref{fig:twoPoolsOneAttacker:r1} and~\subref{fig:twoPoolsOneAttacker:r2} show the revenue density of~1.
}
\label{fig:twoPoolsOneAttacker}
\end{figure*}
We analyze this game numerically by finding the $\xij{1}{2}$ that maximizes $r_1$ and substituting this value for $r_1$ and $r_2$. We vary the sizes of the pools through the entire possible range and depict the optimal $\xij{1}{2}$ and the corresponding revenues in Figure~\ref{fig:twoPoolsOneAttacker}. Each point in each graph therefore represents the equilibrium point of a game with the corresponding $m_1$ and $m_2$ sizes, where we normalize $m = 1$. The top right half of the range in all graphs is not feasible, as the sum of $m_1$ and $m_2$ is larger than~1. We use this range as a reference color, and we use a dashed line to show the bound between this value within the feasible range.
Figure~\ref{fig:twoPoolsOneAttacker:x12} shows the optimal infiltration rate. In the entire feasible range we see that pool~1 chooses a strictly positive value for~$\xij{1}{2}$. Indeed, the revenue of pool~1 is depicted in Figure~\ref{fig:twoPoolsOneAttacker:r1} and in the entire feasible region it is strictly larger than~1, which the pool would have gotten without attacking ($\xij{1}{2} = 0$). Figure~\ref{fig:twoPoolsOneAttacker:r2} depicts the revenue of Pool~2, which is strictly smaller than~1 in the entire range.
Note that the total system mining power is reduced when pool~1 chooses to infiltrate pool~2. Therefore, the revenue of third parties, miners not in either pool, increases from~$1/m$ to~$1/(m - \xij{1}{2})$. Pool~2 therefore pays for the increased revenue of its attacker and everyone else in the system.
\subsection{Implications to the general case}
Consider the case of $p$ pools. For any choice of the pools sizes $m_1, \dots, m_p$, at least one pool will choose to perform block withholding:
\begin{lemma}
In a system with~$p$ pools, the point $\forall j, k: x_j^k = 0$ is not an equilibrium.
\end{lemma}
\begin{proof}
Assume towards negation this is not the case, and $\forall j, k: x_j^k = 0$ is an equilibrium point. Now consider a setting with only pools~1 and~2, and treat the other pools as independent miners.
This is the setting analyzed above and we have seen there that pool~1 can increase its revenue by performing a block withholding attack on pool~2. Denote pool~1's infiltration rate by~$\tilde{x}_1^2 > 0$. Now, take this values back to the setting at hand with~$p$ pools. The revenue of pool~1 is better when
\[
x_1^2 = \tilde{x}_1^2, \forall (j, k) \neq (1, 2): x_j^k = 0 \,\,\, .
\]
Therefore, pool~1 can improve its revenue by attacking pool~2, and no-one-attacks is not an equilibrium point.
\end{proof}
~
\section{Two Pools} \label{sec:twoPools}
\begin{figure*}[t]
\centering
\subfloat[\xij{1}{2}]{
\includegraphics[width=0.4\linewidth]{{twoPools02-m1-delta0.0100-accuracy0.00100-x1}.png}
\label{fig:twoPools:x1}
}
\hfil
\subfloat[\xij{2}{1}]{
\includegraphics[width=0.4\linewidth]{{twoPools02-m1-delta0.0100-accuracy0.00100-x2}.png}
\label{fig:twoPools:x2}
}
\subfloat[$r_1$]{
\includegraphics[width=0.4\linewidth]{{twoPools02-m1-delta0.0100-accuracy0.00100-r1}.png}
\label{fig:twoPools:r1}
}
\hfil
\subfloat[$r_2$]{
\includegraphics[width=0.4\linewidth]{{twoPools02-m1-delta0.0100-accuracy0.00100-r2}.png}
\label{fig:twoPools:r2}
}
\caption[.]{\protect
Two attacking pools system: Optimal infiltration rates ($x_1$ and $x_2$) and corresponding revenues ($r_1$ and $r_2$) as a function of pool sizes. Lines in~\subref{fig:twoPools:x1} and~\subref{fig:twoPools:x2} are at $\xij{1}{2}=0$ and $\xij{2}{1}=0$, respectively. Lines in~\subref{fig:twoPools:r1} and~\subref{fig:twoPools:r2} are at $r_1=1$ and $r_2=1$, respectively.
}
\label{fig:twoPools}
\end{figure*}
We proceed to analyze the case where two pools may attack each other and the other miners mine solo. Again we have pool~1 of size~$m_1$ and pool~2 of size~$m_2$; pool~1 controls its infiltration rate $\xij{1}{2}$ of pool~2, but now pool~2 also controls its infiltration rate~$\xij{2}{1}$ of pool~1.
This scenario is illustrated in Figure~\ref{fig:twoPoolsIllustration}
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{twoPools.pdf}
\caption[.]{\protect
Two pools attacking each other.
}
\label{fig:twoPoolsIllustration}
\end{figure}
The total mining power in the system is $m - \xij{1}{2} - \xij{2}{1}$
The direct revenues $R_1$ and $R_2$ of the pools from mining are their effective mining rates, without infiltrating mining power, divided by the total mining rate.
\begin{equation}
\begin{aligned}
R_1 = \frac{m_1 - \xij{1}{2}}{m - \xij{1}{2} - \xij{2}{1}} &&\\
R_2 = \frac{m_2 - \xij{2}{1}}{m - \xij{1}{2} - \xij{2}{1}} && \,\,\, .
\end{aligned}
\end{equation}
The total revenue of each pool is its direct mining revenue, above, and the infiltration revenue from the previous round, which is the attacked pool's total revenue multiplied by its infiltration rate.
The pool's total revenue is divided among its loyal miners and miners that infiltrated it. At stable state this is
\begin{equation}
\begin{aligned}
r_1 = \frac{R_1 + \xij{1}{2} r_2}{m_1 + \xij{2}{1}} && \\
r_2 = \frac{R_2 + \xij{2}{1} r_1}{m_2 + \xij{1}{2}} && \,\,\, .
\end{aligned}
\end{equation}
Solving for $r_1$ and $r_2$ we obtain the following closed expressions for each. We express the revenues as functions of $\xij{1}{2}$ and $\xij{2}{1}$.
\begin{equation} \label{eqn:twoAttackers:rs}
\begin{aligned}
r_1(\xij{1}{2}, \xij{2}{1})
=
\frac{m_2 R_1+\xij{1}{2} (R_1+R_2)
}{
m_1 m_2 + m_1 \xij{1}{2} + m_2 \xij{2}{1}
} &&
\\
r_2(\xij{2}{1}, \xij{1}{2})
=
\frac{
m_1 R_2+\xij{2}{1} (R_1+R_2)
}{
m_1 m_2 + m_1 \xij{1}{2} + m_2 \xij{2}{1}
} &&
\,\,\, .
\end{aligned}
\end{equation}
Each pool controls only its own infiltration rate. In each round of the pool game, each pool will optimize its infiltration rate of the other. If pool~1 acts at step~$t$, it optimizes its revenue with
\begin{equation} \label{eqn:twoAttackers:x12}
\xij{1}{2}(t) \gets \arg\max_{x'} r_1(x', \xij{2}{1}(t-1)) \,\,\, ,
\end{equation}
and if pool~2 acts at step~$t$, it optimizes its revenue with
\begin{equation} \label{eqn:twoAttackers:x21}
\xij{2}{1}(t) \gets \arg\max_{x'} r_2(x', \xij{1}{2}(t-1)) \,\,\, .
\end{equation}
An equilibrium exists where neither pool~1 nor pool~2 can improve its revenue by changing its infiltration rate. That is, any pair of values $x_1', x_2'$ such that
\begin{equation} \label{eqn:twoPoolsProblem}
\left\{
\begin{array}{l}
\arg\max_{\xij{1}{2}} r_1(\xij{1}{2}, \xij{2}{1}') = \xij{1}{2}' \\
\arg\max_{\xij{2}{1}} r_2(\xij{1}{2}', \xij{2}{1}) = \xij{2}{1}'
\end{array}
\right.
\end{equation}
under the constraints
\begin{equation} \label{eqn:twoPoolsConstraints}
\begin{aligned}
0 < x_1 < m_1 && \\
0 < x_2 < m_2 && \,\,\, .
\end{aligned}
\end{equation}
The feasible region for the pool sizes is $m_1 > 0, m_2 > 0$, and $m_1 + m_2 \le m$.
The revenue function for $r_i$ is concave in $x_i$ ($\partial^2 r_i / \partial x_i^2 < 0$) for all feasible values of the variables. Therefore the solutions for equations~\ref{eqn:twoAttackers:x12} and~\ref{eqn:twoAttackers:x21} are unique and are either at the borders of the feasible region or where $\partial r_i / \partial \xij{i}{j} = 0$.
From Section~\ref{sec:twoPoolsOneAttacker} we know that no-attack is not an equilibrium point, since each pool can increase its revenue by choosing a strictly positive infiltration rate, that is, $\xij{1}{2} = \xij{2}{1} = 0$ is not a solution to Equations~\ref{eqn:twoPoolsProblem}--\ref{eqn:twoPoolsConstraints}.
Nash equilibrium therefore exists with \xij{1}{2}, \xij{2}{1} values where
\begin{equation} \label{eqn:twoPoolsNE}
\left\{
\begin{array}{l}
\displaystyle
\frac{\partial r_1(\xij{1}{2}, \xij{2}{1})}{\partial{\xij{1}{2}}} = 0 \\
\displaystyle
\frac{\partial r_2(\xij{2}{1}, \xij{1}{2})}{\partial{\xij{2}{1}}} = 0
\end{array}
\right.
\,\,\, .
\end{equation}
Using symbolic computation tools, we see that there is a single pair of values for which Equation~\ref{eqn:twoPoolsNE} holds for a choice of $m_1$ and $m_2$.
\subsection{Numerical Analysis}
A numerical analysis confirms these observations. We simulate the pool game for a range of pool sizes.
For each choice of pool sizes, we start the simulation when both pools do not infiltrate each other, $\xij{1}{2} = \xij{2}{1} = 0$, and the revenue densities are $r_1 = r_2 = 1$.
At each round one pool chooses its optimal infiltration rate based on the pool sizes and the rate with which it is infiltrated, and we calculate the revenue after convergence with Equation~\ref{eqn:twoAttackers:rs}. Recall the players in the pool game are chosen with the Round Robin policy, so the pools take turns, and we let the game run until convergence. The results are illustrated in Figure~\ref{fig:twoPools}.
Each run with a some $m_1, m_2$ values results in a single point in each graph in Figure~\ref{fig:twoPools}. We depict the infiltration rates of both pools $\xij{1}{2}, \xij{2}{1}$ in Figures~\ref{fig:twoPools:x1}--\ref{fig:twoPools:x2} and the pools' revenue densities~$r_1, r_2$ in Figures~\ref{fig:twoPools:r1}--\ref{fig:twoPools:r2}.
So for each choice of $m_1$ and $m_2$, the values of $\xij{1}{2}$, $\xij{2}{1}$, $m_1$ and $m_2$ are the points in each of the graphs with the respective coordinates.
As before, for the $\xij{i}{j}$ graphs we draw a border around the region where there is no-attack by $i$ in equilibrium. For the $r_i$ graphs we draw a line around the region where the revenue is the same as in the no-attack scenario, namely~1.
We first observe that only in extreme cases a pool does not attack its counterpart.
Specifically, at equilibrium a pool will refrain from attacking only if the other pool is larger than about $80\%$ of the total mining power.
But, more importantly, we observe that a pool improves its revenue compared to the no-pool-attacks scenario only when it controls a strict majority of the total mining power.
These are the small triangular regions in Figures~\ref{fig:twoPools:r1} and~\ref{fig:twoPools:r2}.
\subsection{The Prisoner's Dilemma}
\begin{figure*}[t]
\begin{tabular}{|l|c|c|}
\hline
\backslashbox{Pool 2\kern-0.5em}{Pool 1} & no attack & attack \\
\hline
no attack & $(r_1 = 1, r_2 = 1)$ & $(r_1 > 1, r_2 = \tilde{r}_2 < 1)$ \\
\hline
attack & $(r_1 = \tilde{r}_1 < 1, r_2 > 1)$ & $(\tilde{r}_1 < r_1 <1 , \tilde{r}_2 < r_2 < 1)$ \\
\hline
\end{tabular}
\caption{Prisoner's Dilemma for two pools. The revenue density of each pool is determined by the decision of both pools whether to attack or not.
The dominant strategy of each player is to attack, however the payoff of both would be larger if they both refrain from attacking.}
\label{tbl:prisoners}
\end{figure*}
In a stable Bitcoin environment with two pools, where neither controls a strict majority of the mining power, both pools will earn less at equilibrium than if both pools ran without attacking.
We can analyze in this case a game where each pool chooses either to attack, or not to attack. If it attacks, the pool optimizes its revenue.
Consider pool~1 without loss of generality.
As we have seen in Section~\ref{sec:twoPoolsOneAttacker}, if pool~2 does not attack, pool~1 can increase its revenue above~1 by attacking, setting~$\xij{1}{2}$ to the optimum.
If pool~2 does attack but pool~1 does not, we denote the revenue of pool~1 by $\tilde{r}_1$. The exact value of $\tilde{r}_1$ depends on the values of~$m_1$ and~$m_2$, but it is always smaller than one. As we have seen above, if pool~1 does choose to attack, its revenue increases, but does not surpass one. The game is summarized in Figure~\ref{tbl:prisoners}.
When played once, this is the classical prisoner's dilemma. Attack is the dominant strategy: Whether pool~2 chooses to attack or not, the revenue of pool~1 is larger when attacking than when refraining from attack, and the same for pool~2.
At equilibrium of this attack-or-don't game, when both pools attack, the revenue of each pool is smaller than its revenue if neither pool attacked.
However, the game is not played once, but rather continuously, forming a super-game, where each pool can change its strategy between attack and no-attack.
If the pools agree (even implicitly) to coordinate, in each round a pool can detect whether it is being attacked and deduce that the other pool is violating the agreement.
In this super-game, cooperation where neither pool attacks is a possible stable state~\cite{friedman1971nonCooperative,aumann1994longTerm} despite the fact that the single Nash equilibrium in every round is to attack.
\section{\texorpdfstring{$p$}{p} Identical Pools} \label{sec:pPools}
Let there be any number of pools of identical size that engage in block withholding against one another.
In this case there exists a symmetric equilibrium.
Consider, without loss of generality, a step of pool~1.
It controls its attack rates each of the other pools, and due to symmetry they are all the same.
Denote by $\xij{1}{\lnot 1}$ the attack rate of pool~1 against any other pool.
Each of the other pools cab attack its peers as well.
Due to symmetry, all attack rates by all attackers are identical.
Denote by $\xij{\lnot 1}{*}$ the attack rate of any pool other than~1 against any other pool, including pool~1.
Denote by $R_1$ the direct revenue (from mining) of pool~1 and by $R_{\lnot 1}$ the direct revenue of each of the other pools. Similarly denote by $r_1$ and $r_{\lnot 1}$ the revenue densities of pool~1 and other pools, respectively.
The generic equations~\ref{eqn:RiFull} and~\ref{eqn:riFull} are instantiated to
\begin{equation} \label{eqn:RsSymm}
\begin{aligned}
&R_1
=
\frac{
m_i - (p-1) \xij{1}{\lnot 1}
}{
m - (p - 1) (p - 1) \xij{\lnot 1}{*} - (p - 1) \xij{1}{\lnot 1}
}
\\
&R_{\lnot 1}
=
\frac{
m_i - (p - 1) \xij{\lnot 1}{*}
}{
m - (p - 1) (p - 1) \xij{\lnot 1}{*} - (p - 1) \xij{1}{\lnot 1}
}
\end{aligned}
\end{equation}
and
\begin{equation} \label{eqn:rsSymm}
\begin{aligned}
&r_1
=
\frac{
R_1 + (p - 1) \xij{1}{\lnot 1} r_{\lnot 1}
}{
m_i + (p - 1) \xij{\lnot 1}{1}
}
\\
&r_{\lnot 1}
=
\frac{
R_{\lnot 1} + (p - 2) \xij{\lnot 1}{*} r_{\lnot 1} + \xij{\lnot 1}{*} r_1
}{
m_i + (p - 2) \xij{\lnot 1}{*} + \xij{1}{\lnot 1}
}
\end{aligned}
\,\,\, .
\end{equation}
Substituting Equations~\ref{eqn:RsSymm} in Equation~\ref{eqn:rsSymm} and solving we obtain a single expression for any $r_i$, since in the symmetric case we have $r_1 = r_{\lnot 1}$. The expression is shown in Equation~\ref{eqn:rsSymm} (Figure~\ref{fig:r1Symm}).
\begin{figure*}
\begin{equation} \label{eqn:r1Symm}
r_i
=
-\frac{m_i^2+m_i \xij{1}{\lnot 1}-(p-1) \xij{1}{\lnot 1} ((p-1) \xij{\lnot 1}{*}+\xij{1}{\lnot 1})}{\left((p-1) \xij{1}{\lnot 1}+(p-1)^2 \xij{\lnot 1}{*}-1\right) ((m_i+\xij{1}{\lnot 1}) (m_i+(p-1) \xij{\lnot 1}{1})-(p-1) \xij{1}{\lnot 1} \xij{\lnot 1}{*})}
\end{equation}
\caption{
Expression for $r_i$ in a system with pools of equal size.
}
\label{fig:r1Symm}
\end{figure*}
Given any value of $p$ and $m_i$ (where $p m_i < 1$), the feasible range of the infiltration rates is $0 \le \xij{i}{j} \le m_i / p$. Within this range $r_i$ is continuous, differentiable, and concave in $\xij{1}{\lnot 1}$.
Therefore, the optimal point for pool~1 is where $\partial r_1 / \partial \xij{1}{\lnot 1} = \nobreak 0$.
Since the function is concave the equation yields a single feasible solution, which is a function of the attack rates of the other pools, namely $\xij{\lnot 1}{1}$ and $\xij{\lnot 1}{*}$.
To find a symmetric equilibrium, we equate $\xij{1}{\lnot 1} = \xij{\lnot 1}{1} = \xij{\lnot 1}{*}$ and obtain a single feasible solution. The equilibrium infiltration rate and the matching revenues are shown in Equation~\ref{eqn:ppoolsStableX} (Figure~\ref{fig:ppoolsStableX}).
\begin{figure*}
\begin{equation} \label{eqn:ppoolsStableX}
\begin{aligned}
&\xijbar{1}{\lnot 1}
=
\xijbar{\lnot 1}{1}
=
\xijbar{\lnot 1}{*}
=
\frac{
p - m_i - \sqrt{(m_i - p)^2 - 4 (m_i)^2 (p - 1)^2 p}
}{
2 (p - 1)^2 p)
}
\\
&\bar{r}_1
=
\bar{r}_{\lnot 1}
=
\frac{
2 p
}{
p - m_i + 2 m_i p + \sqrt{(m_i - p)^2 - 4 (m_i)^2 (p - 1)^2 p}
}
\end{aligned}
\end{equation}
\caption{
Symmetric equilibrium values for a system of $p$~pools of equal sizes.
}
\label{fig:ppoolsStableX}
\end{figure*}
As in the two-pool scenario, the revenue at the symmetric equilibrium is inferior to the no-one-attacks non-equilibrium strategy.
\section{Discussion} \label{sec:discussion}
\subsection{Bitcoin's Health} \label{sec:implications}
Large pools hinder Bitcoin's distributed nature as they put a lot of mining power in the hands of a few pool managers. This has been mostly addressed by community pressure on miners to avoid forming large pools~\cite{andresen2014centralized}. However such recommendations had only had limited success, and mining is still dominated by a small number of large pools.
As a characteristic example, in the period of November 2--8, 2014, three pools generated over $50\%$ of the proofs of work~\cite{organofcorti2014poolStats}.
Long term block withholding attacks are difficult to hide, since miners using an attacked pool would notice the reduced revenue density.
Nevertheless, such attacks are rarely reported\footnote{A recent example is an attack that was partially subverted due to limited efforts of the attacker to hide itself and an alert pool manager~\cite{wizkid2013eligius}. It is unknown whether this was a classical block withholding attack or a more elaborate scheme.}, and we can therefore conclude that they are indeed rare.
The fact that such attacks do not persist may indicate that the active pools have reached an implicit or explicit agreement not to attack one another.
However, an attacked pool cannot detect which of its miners are attacking it, let alone which pool controls the miners.
At some point a pool might miscalculate and decide to try and increase its revenue.
One pool might be enough to break the agreement, possibly leading to a constant rate of attacks among pools and a reduced revenue.
If open pools reach a state where their revenue density is reduced due to attacks, miners will leave them in favor of other available options: Miners of sufficient size can mine solo; smaller miners can form private pools with closed access, limited to trusted participants.
Such a change in the mining forces may be in favor of Bitcoin as a whole. Since they require such intimate trust, we believe private pools are likely to be smaller, and lead to a fine grained distribution of mining power with many small pools and solo miners.
\subsection{Miners and Pools}
\subsubsection{Direct Pool Competition}
Since miners evidently prefer to work with public pools rather than solo~\cite{organofcorti2014poolStats}, a pool may engage in an attack against another pool not to increase its absolute revenue, but rather to attract miners by temporarily increasing its revenue compared to a competing pool.
Our analysis addressed the eventual revenue of pools under block withholding attacks, after the Bitcoin system has normalized the revenues by adjusting difficulty.
Before this normalization, the revenue of an attacking pool is reduced due to the reduction in revenue of both the attacking and attacked pools.
Nevertheless, the attacker's revenue density compared to the victim's revenue density is immediately improved.
This is an an enhanced version of the classical sabotage block withholding with a lower overhead for the attacker. Eventually, once difficulty is adjusted, the attacker may see an absolute benefit in attacking.
The pool game model does not cover the dynamic interplay of pools and miners, which we leave for future work.
\subsubsection{Pool Fees}
We assumed in our analysis that pools do not charge fees from their members since such fees are typically nominal ($0$ -- $3\%$ of a pool's revenue~\cite{btcWiki2014pools}).
The model can be extended to include pools fees.
Fees would add a friction element to the flow of revenue among infiltrated and infiltrating pools.
Specifically, Equation~\ref{eqn:riFull} would change to take into account a pool fee of $f$
\begin{equation} \label{eqn:riFullWithFee}
r_i(t)
=
\frac{
R_i(t) + \sum_{j = 1}^{p} \xij{i}{j}(t) (1 - f) r_j(t)
}{
m_i + \sum_{j = 1}^{p} \xij{j}{i}(t)
}
\,\,\, .
\end{equation}
A pool with a fee of $f$ is a less attractive target for block withholding, since the attacker's revenue is reduced by $f$.
However it is also less attractive for miners in general.
Trading off the two for best protection is left for future work, as part of the treatment of the miner-pool interplay.
\section{Related Work} \label{sec:related}
\subsection{The Block Withholding Attack}
The danger of a block withholding attack is as old as Bitcoin pools.
The attack was described by Rosenfeld~\cite{rosenfeld2011analysis} as early as~2011, as pools were becoming a dominant player in the Bitcoin world.
The paper described the standard attack, used by a miner to sabotage a pool at the cost of reducing its own revenue. Early work did not address the possibility of pools infiltrating other pools for block withholding.
Courtois and Bahack~\cite{courtois2014subversive} have recently noted that a pool can increase its overall revenue with block withholding if all other mining is performed by honest pools.
We consider the general case where not all mining is performed through public pools, and analyze situations where pools can attack one another.
The discrepancy between the calculations of~\cite{courtois2014subversive} and our results for the special case analyzed there can be explained by the strong approximations in that work.
For example, we calculate exactly how infiltrating miners reduce the revenue density of the infiltrated pool.
\subsection{Temporary Block Withholding}
In the Block withholding attack discussed in this work the withheld blocks are never published.
However, blocks can be withheld temporarily, not following the Bitcoin protocol, to improve an attacker's revenue.
An attacker can perform a double spending attack as follows~\cite{rosenfeld2011analysis}. He intentionally generates two conflicting transactions, places one in a block it withholds, and publishes the other transaction.
After the recipient sees the published transaction, the attacker publishes the withheld block to revoke the former transaction. This attack is performed by miners or pools against service providers that accept Bitcoin, and it unrelated to this work.
A miner or a pool can perform a selfish mining attack. Here, the attacker increases its revenue by temporarily withholding its blocks and publishing them in response to block publication by other pools and miners~\cite{eyal2013majority}. This attack is independent of the block withholding attack we discuss here and the two can be performed concurrently.
\subsection{Block Withholding Defense}
Most crypto-currencies use a proof-of-work architecture similar to Bitcoin, where finding proof of work is the result of solution guessing and checking. All of the algorithms we are aware of are susceptible to the block withholding attack, as in all of them the miner can check whether she found a full solution or a partial proof of work.
Prominent examples are Litecoin~\cite{litecoin2013site}, Dogecoin~\cite{dogecoin2013site} and Permacoin~\cite{miller2014permacoin}.
Rosenfeld~\cite{rosenfeld2011analysis} suggested a change of the block structure that would allow a pool to probe for block withholding with a honey-pot technique. A pool could generate miner tasks that it knows would lead to a (useless) block solution. An attacker would withhold the solution and expose itself.
This fix, a different proof of work algorithm, or another solution, could reduce or remove the danger of block withholding. However, this may not be in the interest of the community: Pool block withholding, or even its potential, could lead to a reduction of pool sizes, as explained in Section~\ref{sec:implications}.
\subsection{Decentralized Pools}
Although most pools use a centralized manager, a prominent exception is P2Pool~-- a distributed pool architecture with no central manager~\cite{forrsetv2011p2pool}.
But the question of whether a pool is run by a centralized manager or with a decentralized architecture is almost immaterial for the attack we describe.
An open P2Pool group can be infiltrated and attacked, and the P2Pool code can be changed to support attacks against other pools.
On the other hand, P2Pool can be used by groups of miners to easily form closed pools. These do not accept untrusted miners, and are therefore protected against block withholding.
\section{Conclusion} \label{sec:conclusion}
We explored a block withholding attack among Bitcoin mining pools~--- an attack that is possible in any similar system that rewards for proof of work. Such systems are gaining popularity, running most digital currencies and related services.
We observe that no-pool-attacks is not a Nash equilibrium: If none of the other pools attack, a pool can increase its revenue by attacking the others.
When two pools can attack each other, they face a version of the Prisoner's Dilemma. If one pool chooses to attack, the victim's revenue is reduced, and it can retaliate by attacking and increase its revenue. However, when both attack, at Nash equilibrium both earn less than they would have if neither attacked. With multiple pools of equal size a similar situation arises with a symmetric equilibrium.
The fact that block withholding is not common may be explained by modeling the attack decisions as an iterative prisoner's dilemma. However, we argue that since the attack can be done anonymously by any of the pools, this situation is unstable. Eventually one pool may decide to increase its revenue and drag the others to attack as well, ending with a reduced revenue for all. This would push miners to join private pools which can verify that their registered miners do not withhold blocks. This may lead to smaller pools, and so ultimately to a better environment for Bitcoin as a whole.
\vspace{1ex}
\paragraph*{Acknowledgements}
The author is grateful to
Ken Birman,
Fred B.\ Schneider, and
Eva Tardos for their valuable advice.
\vfill\eject
\bibliographystyle{plain}
|
\section{Introduction}
\label{sec:intro}
\input{intro_journal}
\section{Notation and preliminaries}
\label{sec:prelim}
\input{preliminaries}
\section{Problems of Type 2}
\label{sec:type2}
\input{type2}
\section{Problems of Type 3}
\label{sec:type3}
\input{type3}
\section{Lower bound for Planar Disjoint Paths }
\label{sec:pdp}
\input{disjointPaths}
{\small
\bibliographystyle{abbrv}
\section{Proof of Theorem~\ref{th:lb3c}}
\label{sec:3coloring}
We start with defining some planar gadgets.
The first one is depicted in Fig. \ref{fig:C} and called \emph{color gadget}, C-\emph{gadget} for short.
This gadget ensures that two vertices $u$ and $u'$ are in the same color class.
Note that we can extend the C-gadget for three vertices $u$, $u'$, and $u''$ and ensure the three vertices to be in the same color class by fixing a C-gadget between $u$ and $u'$ and another C-gadget between $u'$ and $u''$.
The second gadget is depicted in Fig. \ref{fig:CC} and called \emph{cross-color} gadget, CC-\emph{gadget} for short.
In this gadget, originally introduced in~\cite{GJE76}, one can check that if $u$, $v$, $u'$, and $v'$ are in the same face before being connected by the gadget, and oriented in this order around the face, then $u$ and $u'$ are in the same color class and $v$ and $v'$ are in the same color class.
\input{type1gadget}
We reduce from \textsc{3-Colorability}. Let $G=(V,E)$ be an input general graph with $n = |V|$ and $V = \{v_1, \dots, v_n\}$, and we define the planar graph $H$, illustrated in Fig.~\ref{fig:Hx} for $n=4$, as follows:
\begin{itemize}
\item[$\bullet$] For each $i \in [n]$, $u_{H,i}, v_{H,i}, w_{H,i} \in V(H)$;
\item[$\bullet$] For each $i,j \in [n]$, $i < j$, $\alpha_{H,i,j} \in V(H)$ and $\beta_{H,i,j} \in V(H)$;
\item[$\bullet$] For each $i \in \{1, \dots, n-1\}$, there is a C-gadget between $u_{H,i}$ and $\alpha_{H,i-1,i}$;
\item[$\bullet$] For each $i \in \{2, \dots, n\}$, there is a C-gadget between $u_{H,i}$ and $\beta_{H,i-1,i}$;
\item[$\bullet$] There is a C-gadget between $u_{H,n}$ and $w_{H,n}$;
\item[$\bullet$] There is a C-gadget between $u_{H,1}$ and $v_{H,1}$;
\item[$\bullet$] For each $i,j \in \{2, \dots, n-1\}$, $i<j$ there is a CC-gadget between $\alpha_{H,i,j}$, $\beta_{H,i,j}$, $\alpha_{H,i,j+1}$, and $\beta_{H,i-1,j}$;
\item[$\bullet$] For each $i \in \{2, \dots, n-1\}$, $i<j$ there is a CC-gadget between $\alpha_{H,i,n}$, $\beta_{H,i,n}$, $w_{H,i}$, and $\beta_{H,i-1,n}$;
\item[$\bullet$] For each $j \in \{2, \dots, n-1\}$, $i<j$ there is a CC-gadget between $\alpha_{H,1,j}$, $\beta_{H,1,j}$, $\alpha_{H,1,j+1}$, and $v_{H,j}$;
\item[$\bullet$] There is a CC-gadget between $\alpha_{H,1,n}$, $\beta_{H,1,n}$, $w_{H,1}$, and $v_{H,n}$;
\item[$\bullet$] For each $i,j \in [n]$, $i<j$, if $(v_{i},v_{j}) \in E$, then $(\alpha_{H,i,j}, \beta_{H,i,j}) \in E(H)$.
\end{itemize}
As the C-gadget and the CC-gadget are planar, $H$ is indeed planar (see Fig. \ref{fig:Hx}).
Because of the properties on the C-gadget and the CC-gadget, for each $i$ in $[n]$, $u_{H,i}$, $v_{H,i}$, and $w_{H,i}$ are in the same color class.
Because of the edges $(\alpha_{H,i,j}, \beta_{H,i,j})$, if there is an edge between $v_i$ and $v_j$ in $G$, then $u_{H,i}$ and $u_{H,j}$ should receive different colors.
If we have a 3-coloring of $G$, then by coloring $u_{H,i}$ with the color of $v_i$ for each $i \in [n]$, we find a 3-coloring of $H$. Conversely, if we have a coloring of $H$, for each $i \in [n]$ we color each vertex $v_i$ of $V$ with the color of $u_{H,i}$.
Let us now argue about the maximum degree of the graph $H$. With the previous construction described so far, $H$ has maximum degree 7. In order to restrict it to 5, we replace each vertex of degree 6 or 7 with the two color gadgets shown in Fig.~\ref{fig:7to5}.
Let us finally argue about the number of vertices of $H$. Note that $H$ can be seen as a spanning subgraph of a grid of size $n$, where each vertex either has been replaced by a C-gadget or a CC-gadget, or it has been removed. As these two gadgets have at most $13$ vertices, and in the worst case, all these new vertices have degree 7 and we need to replace them with two color gadgets, that have 7 vertices each, we have that $|V(H)| \leq 65\cdot n^2$. As {\sc 3-Colorability} cannot be solved in time $2^{o(n)}\cdot n^{O(1)}$ unless the ETH fails~\cite{IPZ01}, the theorem follows.
\subsection{Algorithm for Planar Cycle Packing}
\probls
{Cycle Packing}
{An $n$-vertex graph $G=(V,E)$ and an integer $\ell_0$.}
{Does $G$ contain $\ell_0$ pairwise vertex-disjoint cycles?}
{The treewidth $\text{tw}$ of $G$.}
It is proved in~\cite{CNP11} that \textsc{Cycle Packing} {\sl cannot} be solved in time $2^{o(\text{tw} \log \text{tw})} \cdot n^{O(1)}$ on general graphs unless the ETH fails. On the other hand, a dynamic programming algorithm for \textsc{Planar Cycle Packing} running in time $2^{O(\text{tw})} \cdot n^{O(1)}$ can be found in~\cite{KLL02}. Therefore, it follows that \textsc{Cycle Packing} is of Type~2. In Lemma~\ref{lem:CyclePacking} below we provide an alternative algorithm for \textsc{Planar Cycle Packing} running in time $2^{O(\text{tw})} \cdot n^{O(1)}$, which is a direct application of the techniques based on \emph{Catalan structures} introduced in~\cite{DPBF10}. We include its proof here for completeness, as it yields slightly better constants than the algorithm of~\cite{KLL02}, and because we will use similar terminology in the more involved algorithm of Lemma~\ref{lem:algoMonochromDisjointPaths} in Section~\ref{sec:type3}.
\begin{lemma}\label{lem:CyclePacking
\textsc{Planar Cycle Packing} can be solved in time $2^{O(\text{tw})} \cdot n^{O(1)}$.
\end{lemma}
\begin{proof} We prove the lemma for ${\mathbf{bw}}$, but as $\text{tw} \leq \lfloor\frac{3}{2} {\mathbf{bw}} \rfloor -1$, it will imply the same asymptotic upper bound for $\text{tw}$. Let $G$ be a graph, $X \subseteq V(G)$, and $M$ a matching on $V(G)\backslash X$.
Intuitively, $M$ represents the endpoints of the paths we are building and $X$ is the set of vertices that are already inside a path but they are not an endpoint of any path.
We define $G[(X,M,\ell)] = (\set{M},M)$.
We say that $G[(X_1, M_1,\ell_1), (X_2,M_2,\ell_2)]$ is \emph{defined} if $X_1 \cap (X_2 \cup \set{M_2}) = X_2 \cap (X_1 \cup \set{M_1}) = \emptyset$ and we define $G[(X_1, M_1,\ell_1), (X_2,M_2,\ell_2)] = G[(X_1,M_1,\ell_1)] \cup G[(X_2, M_2,\ell_2)]$. Otherwise, we say that $G[(X_1, M_1,\ell_1), (X_2,M_2, \ell_2)]$ is \emph{undefined}.
We say that $cp(G,X,M) \geq \ell$ if $G$ contains paths joining each pair of vertices given by $M$ and $\ell$ cycles, all pairwise vertex-disjoint.
We now consider $G=(V,E)$ to be our $\Sigma$-plane input graph and $\ell_0$ our integer. Let $(T,\mu,\pi)$ be a sc-decomposition of $G$ of width ${\mathbf{bw}}$. As in \cite{DPBF10}, we root $T$ by arbitrarily choosing an edge $e$ and we subdivide it by inserting a new node $s$. Let $e'$ and $e''$ be the new edges and set ${\bf mid}(e') = {\bf mid}(e'')={\bf mid}(e)$. We create a new node root $r$, we connect it to $s$ by an edge $e_r$, and set ${\bf mid}(e_r) = \emptyset$. The root $e_r$ is not considered as a leaf.
Let $e \in E(T)$ and $\mathcal{R}_e = \{(X,M,\ell) | X \subseteq {\bf mid} (e)$, $M$ is a matching of a subset of ${\bf mid} (e) \backslash X$, and $cp(G_e,X,M) \geq \ell\}$. We observe that there exist $\ell_0$ pairwise vertex-disjoint cycles in $G$ if and only if $(\emptyset, \emptyset, \ell_0) \in \mathcal{R}_{e_r}$.
We should now compute $\mathcal{R}_{e_r}$. If $e$ is a leaf then $G_e = (\{x,y\}, \{(x,y)\})$ and $\mathcal{R}_e = \{(\emptyset,\emptyset,0), (\emptyset,\{(x,y)\},0)\}$.
Otherwise, let $e_1$ and $e_2$ be the two children of $e$ in $E(T)$. $\mathcal{R}_e$ is the set of all triples $(X,M,\ell)$ such that there exist $(S_1,S_2) = ((X_1,M_1,\ell_1),(X_2,M_2,\ell_2)) \in \mathcal{R}_{e_1} \times \mathcal{R}_{e_2}$ such that
$M \subseteq ((\set{M_1}\cup \set{M_2})\cap ({\bf mid} (e)\backslash X))^2$, $G[S_1,S_2]$ is defined, all vertices in ${\bf mid} (e)$ of degree at least two in $G[S_1,S_2]$ are in X, and we can find in $G[S_1,S_2]$ $\ell_3$ cycles and a path $x \dots y$ for each $(x,y) \in M$
such that $\min(\ell_1 + \ell_2 + \ell_3, \ell_0) \geq \ell$.
Note that $G[S_1,S_2]$ is a minor of $G$ so $G[S_1,S_2]$ is also planar. As we have considered a sc-decomposition and all the paths we consider in $G[S_1,S_2]$ are pairwise vertex-disjoint, since each vertex has degree at most two, the maximum number of distinct matchings $M$ is bounded by the number of non-crossing matchings on $|{\bf mid}(e)|$ elements, which is at most $2^{|{\bf mid}(e)|}$. As we have at most $3^{|{\bf mid}(e)|}$ choices for $X$ and $\set{M}$, it follows that for each $e \in E(T)$, $|\mathcal{R}_e| \leq 6^{|{\bf mid}(e)|}\cdot \ell_0$.
As for each $e \in E(T)$ such that $e$ is not a leaf, we have to merge the tables of the two children $e_1$ and $e_2$ of $e$, this algorithm can check in time $O(36^{{\mathbf{bw}}}\cdot \ell_0^2 \cdot |V(G)|)$ whether $G$ contains at least $\ell_0$ vertex-disjoint cycles. We note that the constant can probably be optimized, for example by using fast matrix multiplication (see for instance~\cite{Wil12}), but this is outside of the scope of this paper.\end{proof}
\vspace{.25cm}
We now prove that the running time given by Lemma~\ref{lem:CyclePacking} is asymptotically tight.
\begin{theorem}
\label{th:lbcp}
\textsc{Planar Cycle Packing} cannot be solved in time $2^{o(\sqrt{n})} \cdot n^{O(1)}$ unless the ETH fails. Therefore, \textsc{Planar Cycle Packing} cannot be solved in time $2^{o(\text{tw})} \cdot n^{O(1)}$ unless the ETH fails.
\end{theorem}
\begin{proof}
To prove this theorem, we reduce from \textsc{Planar 3-Colorability} where the input graph has maximum degree at most 5. Let $G=(V,E)$ be a planar graph with maximum degree at most 5 with $V = \{v_1, \dots, v_n\}$. We proceed to construct a planar graph $H$ together with a planar embedding of it, where we will ask for an appropriate number $\ell_0$ of vertex-disjoint cycles.
In this proof, we abuse notation and say that we \emph{ask} for $x$ cycles in a gadget to say that the number of cycles we are looking for in the {\sc Planar Cycle Packing} problem is increased by $x$.
We will ask for a certain number of cycles in each of the introduced gadgets, which by construction will lead to a set of cycles of maximum cardinality in $H$.
We start by introducing some gadgets.
For each $i \in [n]$, corresponding to the vertices $v_1, \dots, v_n$ of $G$, we add to $H$ the SC$_i$-gadget depicted in Fig. \ref{fig:CPSCG}. More precisely, $SC_i = (\{a_i,b_i,c_i, u_{i,0}, u_{i,1}, u_{i,2}, u_{i,3}\},\{(u_{i,0},u_{i,1}),(u_{i,0},u_{i,2}),(u_{i,0},u_{i,3}), (a_i,u_{i,1}),(a_i,u_{i,2}),$ $(b_i,u_{i,2}), (b_i,u_{i,3}), (c_i,u_{i,1}), (c_i,u_{i,3}) \})$. We ask for a cycle inside this gadget. This cycle imposes that at least one of the vertices $\{a_i, b_i, c_i\}$, named a \emph{selected vertex} of the SC$_i$-gadget, is used by the inner cycle and leaves the possibility that the two others are free.
The intended meaning of each SC$_i$-gadget is as follows. The three vertices $a_i$, $b_i$, and $c_i$ correspond to the three colors in the 3-coloring of $G$, namely $a$, $b$, and $c$. If for instance $a_i$ is a selected vertex for index $i$, it will imply that vertex $v_i$ can be colored with color $a$. Therefore, each SC$_i$-gadget defines the available colors for vertex $v_i$, which we call the \emph{color output} of vertex $v_i$.
\input{cycle_packing_figure}
In order to construct a graph $H$ that defines a valid 3-coloring of $G$, we need to propagate the color output of $v_i$ as many times as the degree of $v_i$ in $G$. For this, we introduce a gadget called \emph{bifurcate} gadget.
Before proceeding to the description of the gadget, let us describe its intended functionality. The objective is, starting with the vertices $a_i$, $b_i$, and $c_i$ of the SC$_i$-gadget, to construct a set of triples $\{a_{i,k}, b_{i,k}, c_{i,k}\}$ for $1 \leq k \leq \degs{G}{v_i}$ such that in each triple there will be again at least one selected vertex, defined by the cycles that we will construct in the bifurcate gadgets. Note that in the SC$_i$-gadget the choice of a selected vertex in each triple $\{a_{i,k}, b_{i,k}, c_{i,k}\}$ naturally defines a color output for vertex $v_i$. The crucial property of the gadget is that the intersection of the color outputs given by all the triples is non-empty if and only if the graph $H$ contains enough vertex-disjoint cycles. In other words, the existence of the appropriate number of vertex-disjoint cycles in $H$ will define an available color for each vertex $v_i$ of $G$.
We now proceed to the construction of the bifurcate gadget. First we need to introduce three other auxiliary gadgets. The first two ones, called \emph{expel} and \emph{double-expel} gadgets, are depicted in Fig. \ref{fig:CPE}. Formally, for two vertices $u$ and $u'$, the expel gadget is defined as $EG_{u, u'} =(\{u,u',v,v'\}, \{(u,v), (u,v'), (u',v), (u',v'), (v,v')\})$, and we ask for a cycle inside each such expel gadget. This gadget ensures that if $u$ is in another cycle, then $u'$ is necessarily used by the internal cycle and vice-versa.
Similarly, the double-expel gadget for three vertices $u$, $u'$, and $u''$ is defined as $DEG_{u, u',u''} =(\{u,u', u'', v, v'\}, \{(u,v), (u,v'), (u',v), (u'',v'), (u',u''), (v,v')\})$, and we also ask for a cycle inside each such gadget. This gadget ensures that if $u$ is in another cycle, then $u'$ and $u''$ are necessarily used by the internal cycle and that if $u'$ or $u''$ are in an external cycle, then $u$ is necessarily used by the internal cycle.
\input{bifurcate}
As in our construction the edges of the expel gadgets will cross, we need a gadget that replaces each edge-crossing with a planar subgraph while preserving the existence of the original edges, in the sense that each of the crossing edges gets replaced by a path joining the endvertices of the original edge. This gadget is called \emph{path-crossing} gadget and is depicted in Fig. \ref{fig:CPPC}. Formally, the path-crossing gadget $PCG$ is such that
$ \{pc_{1}, pc_{2}, pc_{3}, pc_{4}, w_0, w_{ 1,1}, w_{ 1,2}, w_{ 2,1}, w_{ 2,2}, w_{ 3,1}, w_{ 3,2}, w_{ 4,1}, w_{ 4,2}\} \subseteq V(PCG)$, $E(PCG)$ contains two paths $pc_{1}, w_{ 1,1}, w_{ 1,2}, w_{0}, w_{ 3,2}, w_{ 3,1}, pc_{3}$ and $pc_{2}, w_{ 2,1}, w_{ 2,2}, w_{0}, w_{ 4,2}, w_{ 4,1}, pc_{4}$, and we add 4 expel gadgets $EG_{w_{1,1},w_{2,2}}$, $EG_{w_{2,1},w_{3,2}}$, $EG_{w_{3,1},w_{4,2}}$, $EG_{w_{4,1},w_{1,2}}$ to $PCG$.
We ask in this gadget only the 4 cycles asked in the expel gadgets. This gadget ensures that, in order to have enough vertex-disjoint cycles, an external cycle that contains an edge from a path-crossing gadget should go \emph{straight}, i.e., for all $\alpha \in [4]$, if the cycle arrives at a vertex $pc_{\alpha}$ it should leave by $pc_{(\alpha+1 \pmod{4})+1}$. If a cycle does not respect this property, we say that the cycle \emph{turns} inside the path-crossing gadget.
That is, the gadget preserves the existence of the original crossing edges whenever there are no cycles that turn inside it. Note that the two paths corresponding to the two original crossing edges cannot be used simultaneously by a set of cycles in the planar graph $H$. We can now define the bifurcate gadget, which is depicted in Fig.~\ref{fig:CPEC}(a), and where each of the 12 edge-crossings should be replaced by a path-crossing gadget. Note that each bifurcate gadgets contains 6 expel and 3 double-expel gadgets. We ask in this gadget the 48 cycles of the path-crossing gadgets, the 3 cycles of the double expel gadgets, and the 6 cycles of the expel gadgets. Note that, indeed, given a triple $\{a_i, b_i, c_i\}$ defining a color output for a vertex $v_i$, the cycles asked in the bifurcate gadget define two triples $\{a_{i,1}, b_{i,1}, c_{i,1}\}$ and $\{a_{i,2}, b_{i,2}, c_{i,2}\}$, which in turn define two color outputs compatible with the one defined by $\{a_i, b_i, c_i\}$, in the sense that there is a common available color for $v_i$. For example, in Fig.~\ref{fig:CPEC}(b) vertex $a_i$ is the only selected vertex of $\{a_i, b_i, c_i\}$ (given by the corresponding SC$_i$-gadget, which is not shown in the figure for the sake of visibility), and the bold cycles define the selected vertices for the triples $\{a_{i,1}, b_{i,1}, c_{i,1}\}$ and $\{a_{i,2}, b_{i,2}, c_{i,2}\}$. Note that color $a$ is simultaneously available for the three triples. We would like to stress that there are other choices of a maximum-cardinality set of cycles in the bifurcate gadget of Fig.~\ref{fig:CPEC}(b), but all of them yield color $a$ available.
For each vertex $v_i$, we need as many triples $\{a_{i,k}, b_{i,k}, c_{i,k}\}$ as $\degs{G}{v_i}$. For that, we concatenate the bifurcate gadgets $deg_G(v_i) -1$ times in the following way. Inductively, we consider the triple $\{a_{i,2}, b_{i,2}, c_{i,2}\}$ of Fig. \ref{fig:CPEC}(a) as the original triple $\{a_i, b_i, c_i\}$, and plug another bifurcate gadget starting from this triple.
With the gadgets defined so far, we have a representation of the colored vertices of $G$ in $H$. We now proceed to capture the edges of $G$ in $H$. For this, we introduce for each $\{v_i, v_j\} \in E$, $i,j \in [n]$, an \emph{edge} gadget depicted in Fig. \ref{fig:CPEG}, where all the 12 edge-crossings should be replaced by a path-crossing gadget. We ask in this gadget 51 new cycles (3 for the expel gadgets and 48 for the path-crossing gadgets). We plug one side of this gadget to a triple $\{a_{i,k}, b_{i,k}, c_{i,k}\}$ defining a color output of $v_i$ and the other side to a triple $\{a_{j,k'}, b_{j,k'}, c_{j,k'}\}$ defining a color output of $v_j$. The edge gadget ensures that the intersection of the two color outputs is empty. This completes the construction of $H$, which is clearly a planar graph, and we set $\ell_0$ to be the sum of the number of cycles asked in each of the introduced gadgets.
\input{edge-gadget}
\begin{claimN}\label{claim1
In any solution of \textsc{Cycle Packing} in $H$, each expel gadget, double-expel gadget, and SC$_i$-gadget contains a cycle, and each cycle is contained inside such a gadget.
\end{claimN}
\begin{proof} In this proof, we say that a cycle $C$ \emph{kills} an another cycle $C'$ if, for any set $S$ of vertex-disjoint cycles containing $C$, $(S \backslash \{C\}) \cup \{C'\}$ is also a set of vertex-disjoint cycles. When dealing with a gadget $F$, we say that a cycle intersecting $F$ is \emph{internal} if it contains only vertices in $F$, and \emph{external} otherwise.
First note that any internal cycle in an expel or a double-expel gadget should use both vertices $v$ and $v'$. Also note that if some external cycle in an expel or a double-expel gadget uses the vertex $v$ or $v'$ of an expel or a double-expel gadget, then it also uses the vertex $u$ (or $u$ and $u''$), and then we are not able to find an internal cycle anymore. Therefore, any external cycle containing $v$ or $v'$ kills the cycle on the set of vertices $\{u, v, v'\}$ or $\{ u', u'', v', v''\}$.
Note that if an external cycle of a path-crossing gadget turns inside it, then without loss of generality it uses a path of the form $pc_1, w_{1,1}, w_{1,2}, w_0, w_{2,2}, w_{2,1}, pc_2$ inside the path-crossing gadget. This external cycle kills the cycle inside the expel gadget between $w_{1,1}$ and $w_{2,2}$. Moreover, note that another disjoint external cycle turning in the same path-crossing gadget kills another internal cycle in the path-crossing gadget, namely the one inside the expel gadget between $w_{3,1}$ and $w_{4,2}$.
Let $C$ be a cycle in $H$ that is not entirely contained in only one expel, double-expel, or SC$_i$-gadget. Because of the previous remarks, we have that $C$ cannot turn in two different path-crossing gadgets, and that if it does {\sl not} turn in any path-crossing gadget, then by construction it uses at least two expel or double-expel gadgets and kills their internal cycles. In both configurations, adding $C$ to the solution decreases the number of vertex-disjoint cycles that we can find in $H$.
The only remaining choice for $C$ is to turn exactly once in one path-crossing gadget. If it happens inside a bifurcate gadget, then $C$ uses vertices of two expel gadgets, namely $expel_1$ and $expel_2$, corresponding to two different colors.
The only way to connect vertices corresponding to different colors outside the path-crossing gadget is by using an SC$_i$-gadget.
So either $C$ kills the cycles of $expel_1$ and $expel_2$, or it may also use a path leading to an edge gadget. If $C$ turns in a path-crossing gadget inside an edge gadget, then the analysis is similar, but there is an extra case where the edge gadget representing the edge between $v_i$ and $v_j$ is directly plugged into the SC$_j$-gadget. In this case, note that none of the vertices $a_i, b_i, c_i$ can be a selected vertex with the set of cycles we currently ask for, and therefore in order to allow it we need to decrease the number of cycles in the solution. \end{proof}
\vspace{.2cm}
If we are given a solution of \textsc{Planar Cycle Packing} in $H$, then for each $i \in [n]$, by Claim~\ref{claim1} the selection of a cycle in the SC$_i$-gadget selects a color for $v_i$, that can be any color that belongs simultaneously to all color outputs of $v_i$, and the edge gadgets ensure that two adjacent vertices are in two different color classes. So in this way we obtain a solution of \textsc{Planar 3-Colorability} in $G$.
Conversely, given a solution of \textsc{Planar 3-Colorability} in $G$, we construct a solution of \textsc{Planar Cycle Packing} in $H$ as follows. For each $i \in [n]$ we choose in the SC$_i$-gadget the cycle of length 4 that contains $u_{i,0}$ and the vertex in $\{a_i, b_i, c_i\}$ that corresponds to the color of $v_i$.
We also choose in the bifurcates gadgets the cycles selecting vertices in $\{
|
a_{i,1}, b_{i,1}, c_{i,1}, a_{i,2}, b_{i,2}, c_{i,2}\}$ that lead to two identical color outputs coinciding with the color output of $\{a_i, b_i, c_i\}$. This choice has the property that the color output of $\{a_i, b_i, v_i\}$ is a subset of the color output of $\{a_{i,1}, b_{i,1}, c_{i,1}\}$ and the color output of $\{a_{i,2}, b_{i,2}, c_{i,2}\}$, and leaves as many free vertices as possible for other cycles in other gadgets. Inside the edge gadget representing $\{v_i, v_j\} \in E$, we select the three cycles that are allowed by the free vertices. We complete our cycle selection by selecting a cycle in each expel gadget contained in a path-crossing gadget. By Claim~\ref{claim1}, this choice leads to a solution of {\sc Planar Cycle Packing} in $H$.
As the degree of each vertex in $G$ is bounded by 5, the number of gadgets we introduce for each $v_i \in V(G)$ to construct $H$ is also bounded by a constant, so
the total number of vertices of $H$ is linear in the number of vertices of $G$.
Therefore if we could solve {\sc Planar Cycle Packing} in time $2^{o(\sqrt{n})}\cdot n^{O(1)}$ then we could also solve {\sc Planar 3-coloring} in time $2^{o(\sqrt{n})} \cdot n^{O(1)}$, which is impossible by Theorem \ref{th:lb3c} unless the ETH fails.\end{proof}
\vspace{.15cm}
\paragraph{\textbf{\emph{Other problems of Type~2.}}} We can provide other examples of problems of Type~2. This is the case, for instance, of \textsc{Cycle Cover}, for which the lower bound has been proved in~\cite{CNP11}, and the upper bound can be proved similarly to Lemma~\ref{lem:CyclePacking}.
Other problems of Type~2 are those where one wants to {\sl maximize} the number of connected components induced by the vertices in a solution. It has been proved in~\cite{CNP11} that \textsc{Maximally Disconnected Dominating Set} cannot be solved in time $2^{o(\text{tw} \log \text{tw})} \cdot n^{O(1)}$ unless the ETH fails. Again, the upper bound can be proved similarly to Lemma~\ref{lem:CyclePacking}. We can define more problems of this flavor, such as the following one.
\probls
{{\sc Maximally Disconnected Feedback Vertex Set}}
{A graph $G=(V,E)$ and two integers $\ell$ and $r$.}
{Does $G$ contain a feedback vertex set of size at most $\ell$ that induces at least $r$ connected components?}
{The treewidth $\text{tw}$ of $G$.}
The following lemma can be proved by using the reduction given in~\cite{CNP11} for {\sc Maximally Disconnected Dominating Set}, just by appropriately redefining the so-called \emph{force} and \emph{one-in-many} gadgets.
\begin{lemma}
\textsc{Maximally Disconnected Feedback Vertex Set} cannot be solved in time $2^{o(\text{tw} \log \text{tw})} \cdot n^{O(1)}$ unless the ETH fails.
\end{lemma}
And again, the following lemma can be proved using standard dynamic programming techniques.
\begin{lemma}\label{lem:algoDisconnectedFVS}
\textsc{Maximally Disconnected Feedback Vertex Set} can be solved in time $2^{O(\text{tw} \log \text{tw})} \cdot n^{O(1)}$, and \textsc{Planar Maximally Disconnected Feedback Vertex Set} can be solved in time $2^{O(\text{tw})} \cdot n^{O(1)}$.
\end{lemma}
\subsection{Algorithm for Monochromatic Disjoint Paths}
The proof of the following lemma is inspired from the algorithm given in~\cite{Sch94} for the {\sc Disjoint Paths} problem on general graphs.
\begin{lemma}\label{lem:algoMonochromDisjointPaths
\textsc{Monochromatic Disjoint Paths} can be solved in time $2^{O(\text{tw} \log \text{tw})} \cdot n^{O(1)}$.
\end{lemma}
\begin{proof} Again, we prove the lemma using branch-decomposition, which will lead the same asymptotic upper bounds in terms of the treewidth. Let $G$ be a colored graph and
let $\gamma: V(G) \rightarrow \{0,\dots, \text{tw}\}$ be a coloring of $V(G)$.
Let $\{\mathcal{N}_i=\{s_i,t_i\}\}_{i\in [m]}$ be the endvertices of the $m$ paths we are looking for, and
let $(T,\mu)$ a branch-decomposition of $G$ of width ${\mathbf{bw}} = {\mathbf{bw}} (G)$.
As in \cite{DPBF10}, we root $T$ by arbitrarily choosing an edge $e$ and subdivide it by inserting a new node $s$.
Let $e'$ and $e''$ be the new edges and set ${\bf mid} (e') = {\bf mid} (e'')= {\bf mid} (e)$.
We create a new root node $r$, connect it to $s$ by an edge $e_r$, and set ${\bf mid}(e_r) = \emptyset$. The root $e_r$ is not considered as a leaf.
Let now $e$ be an edge of $T$, let $X, P \subseteq {\bf mid} (e)$ with $X \cap P = \emptyset$, and let $M, L$ be two disjoint matchings of ${\bf mid} (e) \backslash (X \cup P)$. Let $\gamma_0: P \cup \set{M} \cup \set{L} \rightarrow \{0,\dots, \text{tw}\}$ be a color function, and let $\varphi : P \rightarrow [m]$ be an injective function.
Intuitively, we want to keep track of the (partial) paths inside $G_e$, and to this end $P$ will correspond to the virtual sources of terminals, $M$ to the pairs of virtual sources to be linked by a path, $L$ to pairs of vertices $\{x,y\}$ such that there is a path in $G_e$ linking $x$ and $y$, and $X$ to vertices that are already inside a path or that are both an endpoint and a terminal.
We say that $mdp(G_e, {\bf mid} (e), X, P, M, L, \gamma_0,\varphi) = true$ if the following conditions are fulfilled:
\begin{itemize}
\item[$\circ$] For all $\{s_i,t_i\}$ in $\mathcal{N} \cap V(G_e)^2$,
\begin{itemize}
\item There exists a monochromatic path $s_i\dots t_i$ in $G_e$, or
\item There exist $\{s'_i, t'_i\} \in M$ and two monochromatic paths in $G_e$ $s_i\dots s'_i$ colored $\gamma_0(s'_i)$ and $t_i\dots t'_i$ colored $\gamma_0(t'_i)$ with $\gamma_0(s'_i) \equiv \gamma_0(t'_i)$.
\end{itemize}
\item[$\circ$] For all $\{s_i,t_i\}$ in $\mathcal{N}$, such that $s_i \in V(G_e)$ and $t_i \not\in V(G_e)$ or vice-versa,
\begin{itemize}
\item There exist $s'_i \in P$ such that $\varphi(s'_i) = i$ and a monochromatic path $s_i=v_0\dots v_k=s'_i$ colored $\gamma_0(s'_i)$.
\end{itemize}
\item[$\circ$] For all $\{x_i,y_i\}$ in $L$,
\begin{itemize}
\item There exists in $G_e$ a monochromatic path $x_i\dots y_i$ colored $\max (\gamma_0(x_i),\gamma_0(y_i) )$.
\end{itemize}
\item[$\circ$] All these paths are vertex-disjoint and all vertices in ${\bf mid} (e)$ with degree at least 2 are in $X$.
\end{itemize}
Let $S_1 = (X_1, P_1, M_1, L_1, \gamma_1, \varphi_1)$ and $S_2 = (X_2, P_2, M_2, L_2, \gamma_2, \varphi_2)$ with $X_1, X_2, P_1,P_2, \dots $ defined as above. We define $G[S_1] = (P_1\cup \set{M_1} \cup \set{L_1}, \{\{x,y\} \in L_1\})$ and colored by $\gamma_1$, and we define $G[S_2]$ analogously.
We say that $G[S_1, S_2]$ is \emph{defined} if for all $x\in V(G[S_1]) \cap V(G[S_2])$, $\gamma_1(x) \equiv \gamma_2(x)$, $X_1 \cap V(G[S_2]) = X_2 \cap V(G[S_1]) = X_1 \cap X_2 = \emptyset$, and we define $G[S_1, S_2] = G[S_1] \cup G[S_2]$ and colored by $\gamma_{12}$ such that for all $x \in V(G[S_1,S_2])$, $\gamma_{12} = \max (\gamma_1(x), \gamma_2 (x))$.
Otherwise, we say that $G[S_1,S_2]$ is \emph{undefined}.
For each $e \in E(T)$, we define $\mathcal{R}_e = \{(X, P, M, L, \gamma, \varphi) | X \subseteq {\bf mid} (e), P \subseteq {\bf mid} (e)$, $X \cap P = \emptyset$, $M$ and $L$ are disjoint matchings on $ {\bf mid} (e) \backslash (X \cup P)$, $\set{M} \cap \set{L} = \emptyset$ \ and\ $mdp(G_e, {\bf mid} (e), X,P, M, L, \gamma, \varphi) = true $. We want to know whether $(\emptyset, \emptyset, \emptyset,\emptyset, \emptyset,\emptyset) \in \mathcal{R}_{e_r}$. For each $e \in E(T)$, we can compute $\mathcal{R}_e$ as follows:
\begin{itemize}
\item[$\circ$] if $e$ is a leaf, then $G_e = (\{x,y\}, \{(x,y)\}$, and
\begin{itemize}
\item if $\{x,y\} \in \mathcal{N}$, then \\
$\mathcal{R}_e = \{(\{x,y\}, \emptyset, \emptyset, \emptyset, \emptyset, \emptyset)\}$.
\item if $x \in \mathcal{N}_i, y \in \mathcal{N}_j$, $i \not = j$, then \\
$\mathcal{R}_e = \{(\emptyset, \{x,y\}, \emptyset, \emptyset, \{(x,\gamma(x)),(y,\gamma(y))\}, \{(x,i), (y,j)\})\}$.
\item if $x \in \mathcal{N}_i$ and $\forall j \in [m], y \not\in \mathcal{N}_j$ and $\gamma(x) \not\equiv \gamma(y)$, then \\
$\mathcal{R}_e = \{ (\emptyset,\{x\}, \emptyset, \emptyset, \{(x,\gamma(x)\}, \{(x,i)\})\}$.
\item if $x \in \mathcal{N}_i$ and $\forall j \in [m], y \not\in \mathcal{N}_j$ and $\gamma(x) \equiv \gamma(y)$, then
$\mathcal{R}_e = \{ (\emptyset,\{x\}, \emptyset, \emptyset, \{(x,\gamma(x)\}, \{(x,i)\}),(\{x\}, \{y\}, \emptyset, \emptyset, \{(y,\max (\gamma(x), \gamma(y)))\}, \{(y,i)\})\}$.
\end{itemize}
\item[$\circ$] if $e$ is not a leaf, let $e_1$ and $e_2$ be the two children of $e$ in $E(T)$.
We construct $\mathcal{R}_e$ as the set of all 6-tuples $(X, P, M, L, \gamma_0, \varphi)$ such that there exist $S_1 = (X_1, P_1, M_1, L_1, \gamma_1, \varphi_1) \in \mathcal{R}_{e_1}$ and $S_2 = (X_2, P_2, M_2, L_2, \gamma_2, \varphi_2) \in \mathcal{R}_{e_2}$ fulfilling the following properties:
\begin{itemize}
\item $H = G[S_1, S_2]$ is defined;
\item For all $\{x_i, y_i\} \in L$, there exists a monochromatic path $x_i \dots y_i$ in $H$ and we have $\gamma_0 (x_i) = \gamma_0 (y_i) = \gamma_{12} (x_i \dots y_i)$;
\item All vertices in ${\bf mid} (e)$ of degree at least 2 in $G[S_1,S_2]$ are in $X$;
\item For all $\{v,w\} \in M_i$, $i \in \{1,2\}$, there is a monochromatic color-compatible path from $v$ to $w$ in $G[S_1,S_2]$ or two vertices $\{v', w'\} \in M$, and two monochromatic color-compatible paths $v \dots v'$ and $w \dots w'$ with $\gamma_0 (v') = \gamma_0 (w') = \max (\gamma_{12} (v \dots v'), \gamma_{12} (w \dots w'))$;
\item For all $i \in \{1,2\}$ an for all $v$ in $P_i$, there exist $w \in P$ and a monochromatic color-compatible path $v \dots w$, or there exist $w \in P_{3-i}$ such that $\varphi_i(v) = \varphi_{3-i}(w)$ and a monochromatic path $v \dots w$ such that $\gamma_0(w) = \gamma_{12} (v\dots w)$;
\item All these paths are pairwise vertex-disjoint.
\end{itemize}
\end{itemize}
As in the graph $G[S_1,S_2]$ by construction all vertices have degree at most two, we can easily check all the previous properties in polynomial time, as we just have to compare two sets or traverse a path in $G[S_1,S_2]$ to verify each property. Therefore, we can compute each element of $\mathcal{R}_e$ in time $\mbox{poly}({\bf mid} (e))$. As $(X,P,\set {M}, \set {L})$ forms a partition of a subset of ${\bf mid} (e)$, there are at most $5^{{\bf mid}(e)}$ such 4-tuples. There are at most $\text{tw}+1$ colors and at most $(\text{tw} +1)^{{\bf mid} (e)}$ choices for $\gamma_0$.
As $|\{\varphi(x) | x\in P\}| \leq |P| \leq {\bf mid} (e)$, there are at most ${\bf mid} (e) ^ {{\bf mid} (e)}$ possible different color functions $\varphi$.
As ${\mathbf{bw}} -1 \leq \text{tw}$ we have that for all $e$ in $E(T)$, $|{\bf mid} (e)| \leq \text{tw} +1$, hence
for all $e$ in $E(T)$, $ |\mathcal{R}_e| \leq 5^{\text{tw}+1} \cdot (\text{tw} +1)^{2 (\text{tw}+1)}$.
As for each $e \in E(T)$ such that $e$ is not a leaf, we have to merge the tables of the two children $e_1$ and $e_2$ of $e$, the above dynamic programming algorithm can solve \textsc{Monochromatic Disjoint Paths} in time $O(25^{\text{tw}+1}\cdot (\text{tw} +1)^{4 (\text{tw}+1)} \cdot |V(G)|)$. Again, we note that the constant can probably be optimized by using fast matrix multiplication~\cite{Wil12}.\end{proof}
\vspace{.3cm}
We need to define the \textsc{$k \times k$-Hitting Set} problem, first introduced in~\cite{LMS11a}.
\probls
{\textsc{$k \times k$-Hitting Set}}
{A family of sets $S_1,S_2, \dots, S_m \subseteq [k]\times[k]$, such that each set contains at most one element from each row of $[k]\times[k]$.}
{Is there a set $S$ containing exactly one element from each row such that $S\cap S_i \not = \emptyset$ for any $1 \leq i \leq m$?}
{$k$.}
\begin{theorem}[Lokshtanov \emph{et al}.~\cite{LMS11a}]
\label{th:hittingSet}
{\textsc{$k \times k$-Hitting Set}} cannot be solved in time $2^{o(k \log k)} \cdot m^{O(1)}$ unless the ETH fails.
\end{theorem}
We state the following theorem in terms of the pathwidth of the input graph, and as any graph $G$ satisfies $\text{tw}(G) \leq \text{pw}(G)$, it implies the same lower bound in the treewidth.
\begin{theorem}
\label{th:lbmdp}
\textsc{Planar Monochromatic Disjoint Paths} cannot be solved in time $2^{o(\text{pw} \log \text{pw})} \cdot n^{O(1)}$ unless the ETH fails
\end{theorem}
\begin{proof}
We reduce from \textsc{$k \times k$-Hitting Set}. Let $k$ be an integer and $S_1, S_2 , \dots, S_m \subseteq [k]\times[k]$ such that each set contains at most one element from each row of $[k]\times[k]$.
We will first present an overview of the reduction with all the involved gadgets, and then we will provide a formal definition of the constructed planar graph $G$.
We construct a gadget for each row $\{r\} \times [k]$, $r \in [k]$, which selects the unique pair $p$ of $S$ in this row.
First, for each $r \in [k]$, we introduce two new vertices $s_r$ and $t_r$, a request $\{s_r,t_r\}$, $m+1$ vertices $v_{r,i}$, $i \in \{0, \dots, m\}$, and $m+2$ edges $\{e_{r,0} = (s_r,v_{r,0})\} \cup \{e_{r,i} = (v_{r,i-1},v_{r,i}) | i \in [m] \} \cup \{e_{r,m+1} = (v_{r,m},t_r)\}$. That is, we have a path with $m+2$ edges between $s_r$ and $t_r$.
Each edge of these paths, except the last one, will be replaced with an appropriate gadget.
Namely, for each $r \in [k]$, we replace the edge $e_{r,0}$ with the gadget depicted in Fig.~\ref{fig:CS}, which we call \emph{color-selection} gadget.
In this figure, vertex $u_{r,i}$ is colored $i$.
The color used by the path from $s_r$ to $t_r$ in the color-selection gadget will define the pair of the solution of $S$ in the row $\{r\} \times [k]$.
\input{type3gadget}
Now that we have described the gadgets that allow to define $S$, we need to ensure that $S\cap S_i \not = \emptyset$ for any $i \in [m]$.
For this, we need the gadget depicted in Fig.~\ref{fig:EE}, which we call \emph{expel} gadget. Each time we introduce this gadget, we add to $\mathcal{N}$ the request $\{s,t\}$.
This new requested path uses either vertex $u$ or vertex $v$, so only one of these vertices can be used by other paths.
For each $i \in [m]$, we replace all the edges $\{e_{r,i} | r \in [k]\}$ with the gadget depicted in Fig. \ref{fig:S}, which we call \emph{set} gadget.
In this figure, $a_{r,i}$ is such that if $(\{r\} \times [k]) \cap S_i = \{\{r,c_{r,i}\}\}$ then $a_{r,i}$ is colored $c_{r,i}$, and if $(\{r\} \times [k]) \cap S_i = \emptyset$ then vertex $a_{r,i}$ is removed from the gadget.
\input{FigSetGadget}
This completes the construction of the graph $G$, which is illustrated in Fig.~\ref{fig:FG}. Note that $G$ is indeed planar. Formally, the graph we obtain is $G = (V,E)$, where $V = \{s_r | r \in [k] \} \cup \{ t_r | r \in [k] \} \cup \{ v_{r,i} | r \in [k], i \in \{0,m\} \} \cup \{ u_{r,c} | r \in [k], c \in [k]\} \cup (\{ w_{r,i,b} | r \in [k], i \in [m], i \in \{1,2\} \} \backslash \{w_{r,i,b} | i \in [m], (r,b) \in \{(1,1), (k,2)\}\} )\cup \{s_{r,i} | r \in [k-1], i \in [m]\} \cup \{ t_{r,i} | r \in [k-1], i \in [m]\} \cup \{ a_{r,i} | \exists c \in [k], (r,c) \in S_i$ and
$E = \{\{s_r, u_{r,c}\} \in V^2 | r \in [k], c \in [k] \} \cup \{\{u_{r,c}, v_{r,0}\} \in V^2 | r \in [k], c \in [k]\} \cup \{\{v_{r,i-1},w_{r,i,b}\} \in V^2 | r \in [k], i \in [m], b \in \{1,2\}\} \cup \{\{w_{r,i,b},v_{r,i}\} \in V^2 | r \in [k], i \in [m], b \in \{1,2\}\} \cup \{\{v_{r,i-1},a_{r,i}\} \in V^2 | r \in [k], i \in [m] \} \cup \{\{a_{r,i},v_{r,i}\} \in V^2 | r \in [k], i \in [m]\} \cup \{\{v_{r,m}, t_r\} \in V^2 | r \in [k]\} \cup \{\{s_{r,i}, w_{r,i,2} \} \in V^2 | r \in [k-1], i \in [m] \} \cup \{\{s_{r,i},w_{r+1,i,1}\} \in V^2 | r \in [k-1], i \in [m] \} \cup \{\{ t_{r,i}, w_{r,i,2}\} \in V^2 | r \in [k-1], i \in [m]\} \cup \{\{ t_{r,i},w_{r+1,i,1}\} \in V^2 | r \in [k-1], i \in [m]\}$.
\input{type3final}
The color function $\gamma$ of $G$ is defined such that for each $r \in [k]$ and $c \in [k]$, $\gamma(u_{r,c}) = c$, and for each $i \in [m]$ and $(r,c) \in S_i$, $\gamma(a_{r,i}) = c$. For any other vertex $v \in V(G)$, we set $\gamma (v) = 0$. Finally, the input of \textsc{Planar Monochromatic Disjoint Paths} is the planar graph $G$, the color function $\gamma$, and the $k+(k-1)\cdot m$ requests $\mathcal{N} = \{\{s_r,t_r\} | r \in [k]\} \cup \{\{s_{r,i},t_{r,i}\}| r \in [k-1], i \in [m]\}$, the second set of requests corresponding to the ones introduced by the expel gadgets.
Note that because of the expel gadgets, the request $\{s_r, t_r\}$ imposes a path between $v_{r,i-1}$ and $v_{r,i}$ for each $r \in [k]$.
Note also that because of the expel gadgets, at least one of the paths between $v_{r,i-1}$ and $v_{r,i}$ should use an $a_{r,i}$ vertex, as otherwise at least two paths would intersect.
Conversely, if one path uses a vertex $a_{r,i}$, then we can find all the desired paths in the corresponding set gadgets by using the vertices $w_{r,i,b}$.
Given a solution of \textsc{Planar Monochromatic Disjoint Paths} in $G$, we can construct a solution of \textsc{$k \times k$-Hitting Set} by letting $S = \{(r, c) | r \in [k]$ such that the path from $s_r$ to $t_r$ is colored with color $c \}$.
We have that $S$ contains exactly one element of each row, so we just have to check if $S \cap S_i \not = \emptyset$ for each $i \in [m]$.
Because of the property of the set gadgets mentioned above, for each $i \in [m]$, the set gadget labeled $i$ ensures that $S \cap S_i \not = \emptyset$.
Conversely, given a solution $S$ of \textsc{$k \times k$-Hitting Set}, for each $\{r,c\} \in S$ we color the path from $s_r$ to $t_r$ with color $c$. We assign an arbitrary coloring to the other paths.
For each $i \in [m]$, we take $\{r,c \} \in S \cap S_i$ and in the set gadget labeled $i$, we impose that the path from $v_{r, i-1}$ to $v_{r,i}$ uses vertex $a_{r,i}$. By using the vertices $w_{r,i,b}$ for the other paths, we find the desired $k + (k-1)\cdot m$ monochromatic paths.
Let us now argue about the pathwidth of $G$. We define for each $r,c \in [k]$ the bag $B_{0,r,c} = \{s_{r'} | r' \in [k]\} \cup \{ v_{r',0} | r' \in [k]\} \cup \{ u_{r,c} \}$, for each $i \in [m]$, the bag $B_i = \{v_{r,i-1} | r \in [k]\} \cup \{v_{r,i} | r \in [k]\} \cup \{a_{r,i}\in V(G) | r \in [k]\} \cup \{w_{r,i,b}\in V(G) | r \in [k], b \in [2]\} \cup \{ s_{r,i} | r \in [m-1]\} \cup \{ t_{r,i} | r \in [m-1]\}$, and the bag $B_{m+1} = \{v_{r,m} | r \in [k] \} \cup \{ t_r | r\in [k]\}$. We note that the size of each bag is at most $ 2\cdot (k-1) + 5\cdot k-2 = O(k)$. A path decomposition of $G$ consists of all bags $B_{0,r,c}$, $r,c \in [k]$ and $B_i$, $i \in [m+1]$ and edges $\{B_{i}, B_{i+1}\}$ for each $i \in [m]$, $\{B_{0,r,c},B_{0,r,c+1}\}$ for $r \in [k]$, $c\in [k-1]$, $\{B_{0,r,k}, B_{0,r+1,1}\}$ for $r \in [k]$, and $\{B_{0,k,k},B_1\}$. Therefore, as we have that $\text{pw} (G) = O(k)$, if one could solve \textsc{Planar Monochromatic Disjoint Paths} in time $2^{o(\text{pw} \log \text{pw})} \cdot n^{O(1)}$, then one could also solve {\textsc{$k \times k$-Hitting Set}} in time $2^{o(k \log k)} \cdot m^{O(1)}$, which is impossible by Theorem~\ref{th:hittingSet} unless the ETH fails.\end{proof}
|
\section{Introduction}
In quantum information theory \cite{Nielsen} a quantum channel is
represented by a completely positive trace preserving map (CPT)
between states of two quantum systems living in ${\cal H}_A$ and
${\cal H}_B$. Consider ${\cal H}_A = {\cal H}_B = \Cd$. Then the
states of both systems are defined by semi-positive elements from
$M_d \cong \mathbb{C}^d \otimes \mathbb{C}^d$. Due to the
Kraus-Choi representation theorem \cite{Kraus} any CPT map
\begin{equation}\label{CPT}
\Phi\ :\ M_d \ \longrightarrow\ M_d \
,
\end{equation}
may be represented by
\begin{equation}\label{KK}
\Phi(\rho) = \sum_\alpha\, K_\alpha\, \rho\, K^*_\alpha\ ,
\end{equation}
where the Kraus operators $K_\alpha \in M_d$ satisfies
trace-preserving condition $\sum_\alpha\, K_\alpha^*\, K_\alpha =
I_d$. It is, therefore, clear that all the properties of $\Phi$
are encoded into the family $K_\alpha$. In the present paper we
show how the structure of $\Phi$ depends upon the rank of Kraus
operators. In particular it is well known
\cite{Shor-Horodecki,Ruskai} that if all $K_\alpha$ are rank one
then $\Phi$ defines so called entanglement breaking channel (EBT),
that is, for any state $\rho$ from $M_d {\,\otimes\,} M_d$, $ (\mbox{id}_d \otimes
\Phi)\rho\ $
is separable in $M_d {\,\otimes\,} M_d$.
\begin{definition} We call a channel (\ref{CPT}) an
$r$--partially entanglement breaking channel ($r$--PEBT) iff for an arbitrary
$\rho$
\begin{equation}\label{}
\mbox{SN}[(\mbox{id}_d \otimes \Phi)\rho] \leq r \ ,
\end{equation}
where $\mbox{SN}(\sigma)$ denotes the Schmidt number of $\sigma$.
\end{definition}
Clearly, EBT channels are 1--PEBT. Let us recall
\cite{Terhal-Horodecki} that
\begin{equation}\label{SN-rho}
\mbox{SN}(\sigma) = \min_{p_k,\psi_k}\, \left\{ \,
\max_{k}\, \mbox{SR}(\psi_k)\, \right\} \ ,
\end{equation}
where the minimum is taken over all possible pure states
decompositions
\begin{equation}\label{}
\sigma = \sum_k \, p_k\, |\psi_k\>\<\psi_k|\ , \nonumber
\end{equation}
with $p_k\geq 0$, $\sum_k\, p_k =1$ and $\psi_k$ are normalized
vectors in $\mathbb{C}^d \otimes \mathbb{C}^d$. The Schmidt rank
SR$(\psi)$ denotes the number of non-vanishing Schmidt
coefficients in the Schmidt decomposition of $\psi$. This number
characterizes the minimum Schmidt rank of the pure states that are
needed to construct such density matrix. It is evident that $ 1
\leq \mbox{SN}(\rho) \leq d $ and $\rho$ is separable iff
$\mbox{SN}(\rho) =1 $. Moreover, it was proved
\cite{Terhal-Horodecki} that the Schmidt number is non-increasing
under local operations and classical communication.
Let us denote by $S_k$ the set of density matrices on
$\mathbb{C}^d \otimes \mathbb{C}^d$ that have Schmidt number at
most $k$. One has ${\cal S} = S_1 \subset S_2 \subset \ldots
\subset S_d = {\cal P}$ with ${\cal S}$ and ${\cal P}$ being the
sets of separable and all density matrices, respectively. Recall,
that a positive map $ \Lambda : M_d \longrightarrow M_d$ is
$k$-positive, if $(\mbox{id}_k \otimes \Lambda)$ is positive on
$M_k\otimes M_d$. Due to Choi \cite{Choi} $\Lambda$ is completely
positive iff it is $d$-positive. Now, $\Lambda$ is $k$-positive
iff $(\mbox{id}_d \otimes \Lambda)$ is positive on $S_k$. The set of
$k$-positive maps which are not $(k+1)$-positive may be used to
construct a Schmidt number witness operator $W$ which is
non-negative on all states in $S_{k-1}$, but detects at least one
state $\rho$ belonging to $S_k$ \cite{Bruss1,Bruss2} (see also
\cite{Eisert}), i.e.
\begin{equation}\label{}
\mbox{Tr}\, (W\sigma) \geq 0 \ , \ \ \ \ \sigma \in S_{k-1} \ ,
\end{equation}
and there is a $\rho \in S_k$ such that $\mbox{Tr}\, (W\rho) < 0$.
In the next section we investigate basic properties of PEBT
channels. Then in section \ref{Multi} we generalize the
discussion to multipartite entangled states.
\section{Properties of PEBT channels}
Using well know duality between quantum CPT maps (\ref{CPT}) and
states of the composite quantum system living in $\mathbb{C}^d
\otimes \mathbb{C}^d$ \cite{Zyczkowski,Kossakowski} we may assign
a Schmidt number to any CPT map. Take any CPT map $\Phi$ and
define a state \cite{Jam}
\begin{equation}\label{J}
\rho_\Phi = (\mbox{id}_d \otimes \Phi)\, P^+_d \ ,
\end{equation}
where $P^+_d = |\psi^+_d\>\<\psi^+_d|$ with $\psi^+_d =
d^{-1/2}\sum_k\, e_k \otimes e_k$ being a maximally entangled
state in $\mathbb{C}^d \otimes \mathbb{C}^d$ ($e_k\, ; \
k=1,2,\ldots,d$ denote the orthonormal base in $\mathbb{C}^d$).
\begin{definition}
A Schmidt number of $\Phi$ is defined by
\begin{equation}\label{}
\mbox{SN}(\Phi) = \mbox{SN}(\rho_\Phi)\ ,
\end{equation}
where $\rho_\Phi$ stands for the `dual' state defined in
(\ref{J}).
\end{definition}
Actually, in \cite{Kossakowski} a CPT map $\Phi : M_d
\longrightarrow M_d$ was called an $r$--CPT iff SN$(\Phi) \leq r$.
We show that $r$--PEBT channels are represented by $r$--CPT maps.
Note, that using Kraus decomposition (\ref{KK}) we may express the
Schmidt number of $\Phi$ in analogy to (\ref{SN-rho}) as follows:
\begin{equation}\label{SN-Phi}
\mbox{SN}(\Phi) = \min_{K_\alpha}\, \left\{ \,
\max_{\alpha}\, \mbox{rank}\, K_\alpha\, \right\} \ .
\end{equation}
The analogy between
(\ref{SN-rho}) and (\ref{SN-Phi}) is even more visible if we make
the following observation: any vector $\psi \in \mathbb{C}^d
\otimes \mathbb{C}^d$ may be written as $\psi = \sum_{i,j=1}^d
x_{ij} e_i \otimes e_j$ and hence, introducing a $\psi$-dependent
operator $F \in M_d$ such that $x_{ij}= \<j|F|i\> $, one has
\begin{equation}\label{psi-F}
\psi = \sum_{i=1}^d\, e_i \otimes F e_i \ .
\end{equation}
Using the maximally entangled state $\psi^+_d$ it may be rewritten
in perfect analogy to (\ref{J}):
\begin{equation}\label{J-F}
\psi = \sqrt{d}\, (\mbox{id}_d {\,\otimes\,} F)\psi^+_d\ .
\end{equation}
Clearly, the above formula
realizes an isomorphism between $\mathbb{C}^d \otimes
\mathbb{C}^d$ and $M_d$. Note, that the normalization condition
$\<\psi|\psi\> = 1$ implies $\mbox{Tr}(F^* F)=1$. Moreover, two
vectors $\psi_1$ and $\psi_2$ are orthogonal iff the corresponding
operators $F_1$ and $F_2$ are trace-orthogonal, i.e.
$\mbox{Tr}(F_1^\dag F_2)=0$. It is evident that $\mbox{SR}(\psi) =
\mbox{rank}\, F$. Moreover, the singular values of $F$ are nothing
but the Schmidt coefficients of $\psi$. Hence, the separable pure
states from $\mathbb{C}^d \otimes \mathbb{C}^d$ correspond to rank
one operators from $M_d$.
Consider now the corresponding one-dimensional projector
$|\psi\>\<\psi|$. It may be written as
\begin{equation}\label{psi-FF}
|\psi\>\<\psi| = \sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F e_{ij} F^* \
,
\end{equation}
with $\mbox{Tr}(F^\dag F)=1$. In (\ref{psi-FF}) a rank one
operator $e_{ij} \in M_d$ equals to $|i\>\<j|$ in Dirac notation.
Hence the Schmidt class $S_k$ may be defined as follows: $\rho \in
S_k$ iff
\begin{equation}\label{}
\rho = \sum_\alpha\, p_\alpha P_\alpha\ ,
\end{equation}
with $p_\alpha\geq 0$, $\sum_\alpha\, p_\alpha=1$ and
\begin{equation}\label{P-alpha}
P_\alpha = \sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F_\alpha e_{ij}
F_\alpha^*\ ,
\end{equation}
with $\mbox{rank}\, F_\alpha \leq k$, and $\mbox{Tr}(F_\alpha
F^*_\alpha)=1$. That is, $S_k$ is a convex combination of one
dimensional projectors corresponding to $F$'s of rank at most $k$.
\begin{theorem} A quantum channel $\Phi \in $ $r$--PEBT iff
$\ SN(\Phi)\leq r$.
\end{theorem}
{\it Proof.} Note, that $\mbox{SN}(\Phi) \leq r$ iff there exists
a Kraus decomposition such that all Kraus operators $K_\alpha$
satisfy $\mbox{rank}\,K_\alpha \leq r$. Indeed, using (\ref{KK})
and (\ref{P-alpha}) one has
\begin{eqnarray}\label{}
(\mbox{id}_d {\,\otimes\,} \Phi) \, P^+_d = \sum_{i,j=1}^d \, e_{ij} {\,\otimes\,}
\Phi(e_{ij}) = \sum_\alpha\, p_\alpha P_\alpha\ , \nonumber
\end{eqnarray}
with
\[ p_\alpha = \frac 1d \mbox{Tr}(K^\dag_\alpha K_\alpha)\ , \ \ \
\ \ F_\alpha = \frac{1}{\sqrt{dp_\alpha}}\,K_\alpha \ . \] The
above relations simply translate the isomorphism between states
and CPT maps in terms of operators $K_\alpha$ and $F_\alpha$.
Suppose now that $\Phi$ is $r$-PEBT and let $\rho$ be an arbitrary
state in $M_d$
\[ \rho = \sum_\beta p_\beta\, \sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F_\beta\, e_{ij}\, F_\beta^*\ , \]
with arbitrary $F_\alpha \in M_d$ such that $\mbox{Tr}(F_\alpha
F^*_\alpha)=1$. One has
\begin{eqnarray}\label{}
(\mbox{id}_d \otimes \Phi)\rho \ = \ \sum_{\alpha,\beta}\,p_{\alpha\beta}\,
\sum_{i,j=1}^d\, e_{ij} \otimes \widetilde{F}_{\alpha\beta}
e_{ij} \widetilde{F}_{\alpha\beta}^*\ ,
\end{eqnarray}
with
\[ p_{\alpha\beta} \ =\ \frac 1d\,
\mbox{Tr}(K_\alpha K^*_\alpha)\, p_\beta\ , \ \ \ \
\widetilde{F}_{\alpha\beta} \ = \
\sqrt{\frac{dp_\beta}{p_{\alpha\beta}}}\, K_\alpha F_\beta\ ,
\]
where $K_\alpha$ are Kraus operators representing an $r$--CPT map
$\Phi$ satisfying rank$K_\alpha\leq r$. Now,
\begin{equation}\label{}
\mbox{rank}\, (K_\alpha F_\beta) \leq \min \{\mbox{rank}\,
K_\alpha\, ,
\mbox{rank}\,F_\beta\} \leq r \ , \nonumber
\end{equation}
and hence $(\mbox{id}_d {\,\otimes\,} \Phi)\, \rho \in S_r$. The converse follows
immediately. \hfill $\Box$
\noindent As a corollary note that since $\mbox{rank}\, (K_\alpha
F_\beta) \leq \mbox{rank}\, F_\beta$ one finds
\begin{equation}\label{}
\mbox{SN}( (\mbox{id}_d {\,\otimes\,} \Phi)\, \rho ) \leq \mbox{SN}(\rho) \
,
\end{equation}
which shows that indeed SN does not increase under a local
operation defined by $\mbox{id}_d {\,\otimes\,} \Phi$.
\begin{theorem}A map $\Phi$ is $r$-CPT iff $\Lambda \circ \Phi$
is CPT for any $r$-positive map $\Lambda$.
\end{theorem}
{\it Proof.} Suppose that $\Phi$ is $r$-CPT and take an
arbitrary $k$-positive $\Lambda$:
\begin{equation}\label{Proof}
(\mbox{id}_d {\,\otimes\,} \Lambda \circ \Phi) \, P^+_d = (\mbox{id}_d {\,\otimes\,} \Lambda)\left[ (\mbox{id}_d {\,\otimes\,} \Phi) \,
P^+_d\right] \geq 0 \ , \nonumber
\end{equation}
since $(\mbox{id}_d {\,\otimes\,} \Phi) \, P^+_d \in S_r$. Conversely, let
$\Lambda \circ \Phi$ be CPT for any $r$-positive $\Lambda$, then
$ (\mbox{id}_d {\,\otimes\,} \Lambda \circ \Phi) \, P^+_d \geq 0 $ implies that
$(\mbox{id}_d {\,\otimes\,} \Phi) \,P^+_d \in S_r$ and hence $\Phi$ is $r$-CPT.
Actually, the same is true for $\Phi \circ \Lambda$. \hfill $\Box$
To introduce another class of quantum operations let us recall the
notion of co-positivity: a map $\Lambda$ is $r$--co-positive iff
$\tau \circ \Lambda$ is $r$-positive, where $\tau$ denotes
transposition in $M_d$. In the same way $\Phi$ is completely
co-positive (CcP) iff $\tau \circ \Phi$ is CP. Let us define the
following convex subsets in $M_d \otimes M_d$: $S^r = (\mbox{id}_d {\,\otimes\,}
\tau)\, S_r$. One obviously has: $S^1 \subset S^2 \subset \ldots
\subset S^n$. Note, that $S^1=S_1 ={\cal S}$ and $S_n \cap S^n$ is
a set of all PPT states.
Now, following \cite{Kossakowski} we call a CcPT map $\Phi$ an
$(r,s)$-CPT if
\begin{equation}\label{r-CPT}
(\mbox{id}_d {\,\otimes\,} \Phi) \, P^+_d \in S_r \cap S^s\ ,
\end{equation}
that is
\[ \rho_\Phi \in S_r \ \ \ \ \mbox{and}\ \ \ \
(\mbox{id}_d {\,\otimes\,} \tau)\rho_\Phi \in S_s \ . \]
Hence, if $\rho_\phi$ is
a PPT state, then $\Phi$ is $(r,s)$-CPT for some $r$ and $s$. In
general there is no relation between $(r,s)$-CPT and $(k,l)$-CPT
for arbitrary $r,s$ and $k,l$. However, one has
\[ (1,1)\mbox{-CPT} \subset (2,2)\mbox{-CPT} \subset \ldots \subset
(n,n)\mbox{-CPT} \ , \] and $(n,n)\mbox{-CPT} \equiv \mbox{CPT}
\cap \mbox{CcPT}$.
\noindent {\bf Theorem 3:} A map $\Phi$ is $(r,s)$-CPT iff for any
$r$-
|
positive map $\Lambda_1$ and $s$--co-positive map $\Lambda_2$
the composite map $\Lambda_1 \circ \Lambda_2 \circ\Phi$ is CPT.
\section{Examples}
\noindent {\bf Example 1:} Let us consider so called isotropic
state in $d$ dimensions
\begin{equation}\label{}
{\cal I}_\lambda = \frac{1-\lambda}{d^2} I_d \otimes I_d + \lambda
P^+_d \ ,
\end{equation}
with $-1/(d^2-1) \leq \lambda \leq 1$. It is well known
\cite{Horodecki} that ${\cal I}_\lambda$ is separable iff $\lambda
\leq 1/(d+1)$. Now, let $\Psi : M_d \longrightarrow M_d$ be an
arbitrary positive trace preserving map and define a CPT map
$\Phi_\lambda$ by
\begin{equation}\label{}
(\mbox{id}_d {\,\otimes\,} \Phi_\lambda) P^+_d = (\mbox{id}_d {\,\otimes\,} \Psi){\cal I}_\lambda\ .
\end{equation}
One easily finds
\begin{equation}\label{}
\Phi_\lambda(\rho) = \frac{1-\lambda}{d}\,\mbox{Tr}\rho\, I_d +
\lambda\Psi(\rho)\ .
\end{equation}
Clearly, for $\lambda \leq 1/(d+1)$ (i.e. when ${\cal I}_\lambda$
is separable) $\Phi_\lambda$ is $(1,1)$-CPT, i.e. both
$\Phi_\lambda$ and $\tau \circ \Phi_\lambda$ are EBT.
\noindent {\bf Example 2:} Let us rewrite an isotropic state
${\cal I}_\lambda$ in terms of fidelity $f=\mbox{Tr}({\cal
I}_\lambda\, P^+_d)$:
\begin{equation}\label{}
I_f = \frac{1-f}{d^2-1} (I_d {\,\otimes\,} I_d - P^+_d) + fP^+_d \ .
\end{equation}
It was shown in \cite{Terhal-Horodecki} that SN$({\cal I}_f) = k$
iff
\begin{equation}\label{f}
\frac{k-1}{d} < f \leq \frac{k}{d} \ .
\end{equation}
Defining a CPT map $\Phi_f$
\begin{equation}\label{}
(\mbox{id}_d {\,\otimes\,} \Phi_f) P^+_d = {\cal I}_f \ ,
\end{equation}
one finds
\begin{equation}\label{}
\Phi_f(\rho) = \frac{1-f}{d^2-1}\, \mbox{Tr}\rho\, I_d +
\frac{d^2f-1}{d^2-1}\, \rho\ .
\end{equation}
This map is $k$--CPT iff $f$ satisfies (\ref{f}) and hence it
represents an $r$--PEBT channel.
\noindent {\bf Example 3:} Consider
\begin{equation}\label{FU}
\rho = \sum_{\alpha=1}^{d^2}\, p_\alpha\,
\sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F_\alpha\, e_{ij}\, F^*_\alpha\ ,
\end{equation}
where
\begin{equation}\label{}
p_\alpha\geq 0\ ,\ \ \ \ \ \ \
\sum_{\alpha=1}^{d^2}\, p_\alpha=1\ ,\ \ \ \ \ \ \
F_\alpha = \frac{U_\alpha}{\sqrt{d}}\ ,
\end{equation}
and $U_\alpha$ defines a family of unitary operators from $U(d)$
such that
\begin{equation}\label{}
\mbox{Tr}(U_\alpha\, U^*_\beta) = \delta_{\alpha\beta} \ , \ \
\ \ \ \ \alpha,\beta = 1,2,\ldots,d^2\ .
\end{equation}
The corresponding `dual' quantum channel $\Phi$ is given by
\begin{equation}\label{Phi-d}
\Phi(\sigma) = \sum_{\alpha=1}^{d^2}\, K_\alpha\, \sigma\,
K^*_\alpha\ ,
\end{equation}
with $K_\alpha= \sqrt{p_\alpha}\, U_\alpha$. Note, that for
$p_\alpha = 1/d^2$ one obtains a completely depolarizing channel,
i.e.
\begin{equation}\label{depolar}
\frac{1}{d^2}\, \sum_{\alpha=1}^{d^2}\, U_\alpha\, e_{ij}\,
U^*_\alpha\ = \delta_{ij}\ .
\end{equation}
Now, following \cite{Tomiyama} consider a map
\begin{equation}\label{}
\Lambda_\mu(\sigma) = I_d\, \mbox{Tr}\, \sigma - \mu \sigma\ ,
\end{equation}
which is $k$ (but not $(k+1)$)--positive for
\begin{equation}\label{k-mu}
\frac{1}{k+1} \leq \mu \leq \frac 1k\ .
\end{equation}
One has
\begin{eqnarray}\label{}
(\mbox{id}_d {\,\otimes\,} \Lambda_\mu) \rho &=& \sum_{\alpha=1}^{d^2}\, p_\alpha\,
\sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} \left[ I_d\, \mbox{Tr}(F_\alpha\, e_{ij}\,
F^*_\alpha) - \mu\, F_\alpha\, e_{ij}\,
F^*_\alpha \right] \nonumber \\
&=& \frac 1d\, I_d {\,\otimes\,} I_d - \sum_{\alpha=1}^{d^2}\, \mu p_\alpha\,
\sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F_\alpha\, e_{ij}\, F^*_\alpha
\nonumber \\
&=& \frac 1d\, \sum_{\alpha=1}^{d^2}\, (1 - d\mu p_\alpha)\,
\sum_{i,j=1}^d\, e_{ij} {\,\otimes\,} F_\alpha\, e_{ij}\, F^*_\alpha\ ,
\end{eqnarray}
where we have used (\ref{depolar}). It is therefore clear that if
for some $1\leq \alpha \leq d^2$, $\, p_\alpha > 1/(d\mu)$ and
$\mu$ satisfies (\ref{k-mu}), then $\mbox{SN}(\rho) \geq k+1$.
Equivalently, a `dual' quantum channel (\ref{Phi-d}) belongs to
$\{\, d$--PEBT $-$ $k$--PEBT$\}$.
\section{PEBT channels and multipartite entanglement}
\label{Multi}
Consider now a multipartite entangled state living in ${\cal H} =
( \mathbb{C}^d)^{{\,\otimes\,} N}$ for some $N\geq 2$. Any $\psi \in {\cal
H}$ may be written as follows:
\begin{equation}\label{multi-F}
\psi = \sum_{i_1,\ldots,i_K=1}^d \, e_{i_1} {\,\otimes\,} \ldots {\,\otimes\,} e_{i_K}
{\,\otimes\,} F ( e_{i_1} {\,\otimes\,} \ldots {\,\otimes\,} e_{i_K}) \ ,
\end{equation}
where $F$ is an operator
\[ F \ :\ (\mathbb{C}^d)^{{\,\otimes\,} K}\ \longrightarrow\ (\mathbb{C}^d)^{{\,\otimes\,}
N-K} \ , \] and $1\leq K \leq N-1$. Again, normalization of
$\psi$ implies $\mbox{Tr}(F^*F)=1$. Clearly, such representation
of $\psi$ is highly non-unique. One may freely choose $K$ and take
$K$ copies of $\Cd$ out of $(\Cd)^{{\,\otimes\,} N}$. Any specific choice of
representation depends merely on a specific question we would like
to ask. For example (\ref{multi-F}) gives rise to the following
reduced density matrices:
\begin{equation}\label{}
\rho_B = \mbox{Tr}_A \, |\psi\>\<\psi| = \mbox{Tr}_{12\ldots
K}\, |\psi\>\<\psi| \, = \, FF^* \, \in \, M_d^{{\,\otimes\,} N-K} \ ,
\end{equation}
and
\begin{equation}\label{}
\rho_A = \mbox{Tr}_B \, |\psi\>\<\psi| = \mbox{Tr}_{K+1\ldots
N}\, |\psi\>\<\psi| \, = \, F^*F \, \in \, M_d^{{\,\otimes\,} K} \ .
\end{equation}
A slightly different way to represent $\psi$ reads as follows
\begin{equation}\label{}
\psi = \sum_{i_1,\ldots,i_{N-1}=1}^d \, e_{i_1} {\,\otimes\,} \ldots {\,\otimes\,}
e_{i_{N-2}} {\,\otimes\,} e_{i_{N-1}}
{\,\otimes\,} F_{i_1\ldots i_{N-2}} e_{i_{N-1}} \ ,
\end{equation}
where
\[ F_{i_1\ldots i_{N-2}} \ :\ \mathbb{C}^d\ \longrightarrow\
\mathbb{C}^d\ , \] for any $i_1,\ldots,i_{N-2}=1,2,\ldots,d$.
Now, normalization of $\psi$ implies
\begin{equation}\label{}
\sum_{i_1,\ldots,i_{N-2}=1}^d\, \mbox{Tr}\left( F_{i_1\ldots
i_{N-2}}^*F_{i_1\ldots i_{N-2}}\right) =1 \ .
\end{equation}
One has the following relation between different
representations:
\begin{equation}\label{}
\< e_{i_N} |F_{i_1\ldots i_{N-2}}|e_{i_{N-1}} \>\, =\,
\< e_{i_1} {\,\otimes\,} \ldots {\,\otimes\,} e_{i_{N-1}}|F|e_{i_N}\> \ .
\end{equation}
\noindent {\bf Example 4.}
For $N=3$ we have basically three representations:
\begin{equation}\label{3-I}
\psi = \sum_{i=1}^d\, e_i {\,\otimes\,} F e_i \ ,
\end{equation}
\begin{equation}\label{3-II}
\psi = \sum_{i,j=1}^d \, e_i {\,\otimes\,} e_j {\,\otimes\,} F'(e_i {\,\otimes\,} e_j) \ ,
\end{equation}
and
\begin{equation}\label{3-III}
\psi = \sum_{i,j=1}^d \, e_i {\,\otimes\,} e_j {\,\otimes\,} F_{i}\, e_j \ ,
\end{equation}
with
\[ F \ :\ \mathbb{C}^d\ \longrightarrow\
(\mathbb{C}^d)^{{\,\otimes\,} 2}\ , \ \ \ \ F' = F^T \ :\
(\mathbb{C}^d)^{{\,\otimes\,} 2}\ \longrightarrow\ \mathbb{C}^d \ , \ \ \ \
F_{i} \ :\ \mathbb{C}^d\ \longrightarrow\ \mathbb{C}^d \ . \] As
an example take $d=2$ and let us consider two well known 3-qubit
states \cite{GHZ}:
\begin{equation}\label{GHZ}
|\mbox{GHZ}\> = \frac{1}{\sqrt{2}}\, \left( |000\> + |111\> \right)\ ,
\end{equation}
and
\begin{equation}\label{W}
|W\> = \frac{1}{\sqrt{3}}\, \left( |100\> + |010\> + |001\> \right)\
.
\end{equation}
One finds for GHZ--state:
\begin{equation}\label{}
F' = (F_1,F_2) = \frac{1}{\sqrt{2}}\, \left( \begin{array}{cccc}
1&0&0&0\\ 0&0&0&1
\end{array} \right) = F^T \ ,
\end{equation}
and for W--state:
\begin{equation}\label{}
\widetilde{F}' = (\widetilde{F}_1,\widetilde{F}_2) = \frac{1}{\sqrt{3}}\,
\left( \begin{array}{cccc} 0&1&1&0\\ 1&0&0&0
\end{array} \right) = \widetilde{F}^T \ .
\end{equation}
Note, that for both states $\mbox{rank}(F) =
\mbox{rank}(\widetilde{F}) =2$. There is, however, crucial
difference between $F_i$ and $ \widetilde{F}_i$:
$\mbox{rank}(F_i)=1$, whereas $\mbox{rank}( \widetilde{F}_1)=2$.
Both states possess genuine 3--qubit entanglement. The difference
consists in the fact that GHZ--state is 2--qubit separable whereas
W--state is 2--qubit entangled \cite{Maciek}:
\begin{equation}\label{}
\rho^{\mbox{\scriptsize GHZ}}_{\, 23} = \mbox{Tr}_1
|\mbox{GHZ}\>\<\mbox{GHZ}| = \sum_{k=0}^1\, \sum_{i,j=0}^1\,
e_{ij} {\,\otimes\,} F_k\, e_{ij}\, F^*_k \ ,
\end{equation}
with $ \mbox{SN}( \, \rho^{\mbox{\scriptsize GHZ}}_{\, 23}\, ) =
1\, $ ,
and
\begin{equation}\label{}
\rho^{\mbox{\scriptsize W}}_{\, 23} = \mbox{Tr}_1
|\mbox{W}\>\<\mbox{W}| = \sum_{k=0}^1\, \sum_{i,j=0}^1\, e_{ij}
{\,\otimes\,} \widetilde{F}_k\, e_{ij}\, \widetilde{F}^*_k \ ,
\end{equation}
with $\mbox{SN}( \, \rho^{\mbox{\scriptsize W}}_{\, 23}\, ) = 2\, $.
If $N=2K$ any state vector $\psi \in (\Cd)^{{\,\otimes\,} N} = (\Cd)^{{\,\otimes\,} K}
{\,\otimes\,} (\Cd)^{{\,\otimes\,} K}$ may be represented by (\ref{multi-F}) with
\begin{equation}\label{}
F \ :\ (\mathbb{C}^d)^{{\,\otimes\,} K}\ \longrightarrow\ (\mathbb{C}^d)^{{\,\otimes\,}
K} \ .
\end{equation}
Hence, an arbitrary state $\rho$ from $M_d^{{\,\otimes\,} K} {\,\otimes\,} M_d^{{\,\otimes\,}
K}$ reads as follows
\begin{equation}\label{2K-F}
\rho = \sum_\alpha\, p_\alpha \sum_{i_1,\ldots,i_K=1}^d\,
\sum_{j_1,\ldots,j_K=1}^d \, e_{i_1j_1} {\,\otimes\,} \ldots {\,\otimes\,} e_{i_Kj_K}
{\,\otimes\,} F_\alpha ( e_{i_1j_1} {\,\otimes\,} \ldots {\,\otimes\,} e_{i_Kj_K}) F_\alpha^* \
.
\end{equation}
Clearly, $\mbox{SN}(\rho) \leq r$ iff $\mbox{rank}(F_\alpha)\leq
r$ for all $F_\alpha$ appearing in (\ref{2K-F}). Then the
corresponding quantum channel
\begin{equation}\label{}
\Phi \ :\ M_d^{{\,\otimes\,} K} \ \longrightarrow\ M_d^{{\,\otimes\,} K} \ ,
\end{equation}
possesses Kraus decomposition with $K_\alpha =
\sqrt{d^Kp_\alpha}\, F_\alpha$ and hence is $r$--PEBT. For other
aspects of multipartite entanglement se e.g. \cite{multi}.
\section*{Acknowledgments}
This work was partially supported by the Polish State Committee
for Scientific Research Grant {\em Informatyka i in\.zynieria
kwantowa} No PBZ-Min-008/P03/03.
|
\section{Introduction}
Since the 1990s, there had been a significant renewal of interest in the possibility
that the seemingly absolute Lorentz and CPT symmetries of the standard model and
gravity might actually be very weakly violated. At this time, there is no
no compelling evidence for such symmetry breaking. However, if violations of
isotropy, Lorentz boost invariance, or CPT were ever observed experimentally, the
discovery would obviously be of fundamental importance. It would change our
understanding of how physics works at the very deepest levels. Even if there is
no current evidence for Lorentz or CPT violation, these symmetries are so basic
(as fundamental building blocks of both quantum field theories and the general
theory of relativity), that they are worthy of careful study.
There are also other reasons to be interested in Lorentz and CPT tests. Attempts
to develop a quantum theory of gravity have shown that many of the
speculative frameworks that have been suggested to describe quantum gravity
seem to allow for the existence of Lorentz violation, at least within certain regimes.
Moreover, with the explication of a comprehensive effective field theory (EFT)
capable of describing Lorentz-violating phenomena, it came to be realized that
the symmetry violations could come in a much wider variety of forms than previous
unsystematic analyses had considered. Large regions of the EFT parameter space
were scarcely constrained by earlier generations of experiments. CPT symmetry is
also closely tied to Lorentz symmetry, so that even with non-local interactions,
CPT violation in a quantum field theory (QFT) automatically entails Lorentz
violation~\cite{ref-greenberg}, as long as the theory has a well-defined
$S$-matrix.
The general EFT describing Lorentz violation in particle physics therefore includes
the most general CPT violation as well. This EFT is known as the standard model
extension (SME)~\cite{ref-kost1,ref-kost2}. The action for the SME contains
operators that can be constructed out of the standard model fields.
The usual standard model action is formed by writing down all the local, renormalizable,
$SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$
gauge-invariant, Lorentz-invariant operators that can be constructed from those fields.
The SME is constructed in much the same way, but with the Lorentz invariance requirement
dropped; specifically, the minimal SME (mSME) keeps all of the other requirements---locality,
renormalizability, and gauge invariance. The mSME is now the usual framework
used for parametrizing the results of experimental Lorentz and CPT
tests. However, since the mSME action contains quite a large number of parameters,
many different types of experiments have turned out to be useful for establishing
bounds on the mSME parameters. An up-to-date summary of the results of these
experiments is given in~\cite{ref-tables}.
Studies of possible Lorentz and CPT violation have also been fruitful theoretically,
providing new insights, especially into the structure of QFTs.
The radiative corrections to the Chern-Simons term in Lorentz-violating
quantum electrodynamics (QED) have been one of the most studied topics
related to the SME---and almost certainly the most controversial.
There is a similarly-structured gravitational Chern-Simons term in the gravitational version
of the SME. Although the radiative corrections in the gravitational sector have been
examined to a limited extent, a profound and significant puzzle exists in that
sector---which has previously been overlooked. This paper will both introduce
this puzzle and proceed to solve it.
\section{The Puzzle of Gravitational Radiative Corrections}
The basic outline of the puzzle is the following. We must begin with a discussion of the
simpler electromagnetic Lorentz violation term. The Chern-Simons term in the (3+1)-dimensional
Abelian gauge sector takes the form ${\cal L}_{AF}=\frac{1}{2}k_{AF}^{\mu}\epsilon_{\mu\alpha\beta\gamma}
F^{\alpha\beta}A^{\gamma}$~\cite{ref-carroll1}. (When $k_{AF}^{\mu}$ is purely timelike, this term
in the Lagrange density is proportional to $\vec{A}\cdot\vec{B}$, which breaks P and CPT symmetries.)
Coming from the charged fermion sector,
there is a radiative correction to ${\cal L}_{AF}$ that is necessarily finite, but whose coefficient
is undetermined. At the quantum level, there is an infinite family of
different theories that correspond to the same classical Lagrangian. The
differences between these theories are in how they are regulated, but there is
nothing that singles out one regulator as being unambiguously correct. Different high-momentum regulators
lead to radiatively-generated terms with different finite
coefficients~\cite{ref-coleman,ref-jackiw1,ref-victoria1,ref-chung1,ref-chung2,ref-chen,ref-chung3,
ref-volovik,ref-chan,ref-bonneau1,ref-chaichian,
ref-victoria2,ref-battistel,ref-andrianov,ref-altschul1,ref-altschul2}.
Various schemes have been suggested for identifying a single
correct result; some authors have argued
that only their certain specific regulators were appropriate for the calculation---and thus
that there was one true correct answer.
However, all such unambiguous answers appear to suffer from one
of two deficiencies. The most naive schemes fix the term by demanding that
the Lagrange density be gauge invariant; since the Chern-Simons term is not
gauge invariant (it changes by a total derivative under a gauge transformation),
the term is automatically ruled out. However, this is not a legitimate
result, because it excludes the term of interest {\em a priori}. Gauge
invariance of the Lagrange density is an unnecessarily strong condition; if
we instead only demand that the integrated action be invariant, the
Chern-Simons term is fully allowed. Alternatively, nonperturbative schemes for
fixing the radiative correction have also been suggested. However, for a
nonperturbative framework to make sense, it must provide a way of
determining higher-order radiative corrections as well as first-order ones; and unfortunately,
all the proposed nonperturbative methodologies that lead to particular nonzero values of
the induced Chern-Simons coefficient appear to fail at higher order.
Calculations appear to show that the gravitational sector has the same
kind of ambiguity~\cite{ref-mariz2,ref-mariz1,ref-gomes1,ref-filipe,ref-assuncao}.
The Lorentz-violating Chern-Simons term for (3+1)-dimension gravity is
\begin{equation}
{\cal L}_{\Gamma}=-\frac{1}{4}v_{\mu}\epsilon^{\mu\alpha\beta\gamma}\left(
\Gamma_{\alpha\tau}^{\sigma}\partial_{\beta}\Gamma_{\gamma\sigma}^{\tau}
+\frac{2}{3}\Gamma_{\alpha\tau}^{\sigma}\Gamma_{\beta\eta}^{\tau}\Gamma_{\gamma\sigma}^{\eta}
\right),
\end{equation}
in terms of the Christoffel symbols $\Gamma_{\alpha\beta}^{\gamma}$
[and with $\kappa=(8\pi G)^{-1}=1]$. In the linearized
gravity limit, this may be expressed more conveniently directly in terms of the metric fluctuations.
${\cal L}_{\Gamma}$ is proportional to $v^{\mu}\epsilon_{\mu\alpha\beta\gamma}h^{\beta\nu}
\partial^{\gamma}(\partial_{\sigma}\partial^{\sigma}h^{\alpha}_{\nu}-\partial_{\nu}
\partial_{\sigma}h^{\alpha\sigma})$, where $g^{\mu\nu}=\eta^{\mu\nu}+h^{\mu\nu}$.
Not surprisingly, the gravitational Chern-Simons term contains two more derivatives than
the electromagnetic term (to match the number
of free $h^{\mu\nu}$ indices); but otherwise the two types of terms appear (in the weak field limit) to be
quite similar in structure. However, there is actually a fundamental difference between the
electromagnetic Chern-Simons term that may be radiatively generated in
Lorentz-violating QED and the corresponding gravitational Chern-Simons term.
The difference is that the gravitational Chern-Simons term is, in spite of appearances,
actually Lorentz invariant. The profound puzzle that faces us is that it appears to be
possible for a Lorentz-violating $b^{\mu}$ term in the fermion sector to generate
a radiative correction that is proportional to $b^{\mu}$ and fully
P violating~\cite{ref-alexander1,ref-smith,ref-alexander2}, yet which is invariant
under all rotations and Lorentz boosts.
The Lorentz invariance of a pure gravity theory that includes a Chern-Simons
term is rather subtle. In fact, this was itself a bit of a puzzle when the term was
first introduced~\cite{ref-jackiw5}; it appeared that there were no physical
distinctions between versions of the theory with explicit (externally imposed)
symmetry breaking and certain types of dynamical symmetry breaking. However, this was ultimately explained, and
the Lorentz symmetry of the gravitational Chern-Simons theory was demonstrated
by constructing the conserved gravitational energy-momentum (pseudo-)tensor
$\Theta^{\mu\nu}$~\cite{ref-guarrera}. This $\Theta^{\mu\nu}=\Theta^{\nu\mu}$
has a symmetric form, and symmetry of the energy-momentum tensor is equivalent to Lorentz
invariance of the $S$-matrix (because the rotation and boost generators can be expressed as
integrals of moments of $\Theta^{\mu\nu}$).
Evidently, the dependence of the theory on the preferred vector
$v^{\mu}$ is illusory. It is not possible to write down such a Chern-Simons
term without introducing such a vector, but the particular spacetime direction of $v^{\mu}$
turns out to have no bearing on the physics. This is directly related to
the gauge invariance of the theory; the semblance of Lorentz violation is essentially a
gauge artifact.
Nonetheless, the gravitational Chern-Simons term really does break the
discrete symmetries of general relativity. For a timelike $v^{\mu}$, the boost
violation that is seemingly apparent in the form of the term is unphysical, but
the parity violation is quite real. Boost invariance manifests itself in the fact
that all gravitational waves in the theory propagate at the speed of light. However,
the P breaking means that right- and left-polarized waves are coupled
to their sources with different strengths. Note that the lack of Lorentz
violation in the CPT-violating gravitational Chern-Simons theory demonstrates
that CPT violation in the gravitational sector does not automatically need to be accompanied by
Lorentz violation. This is a somewhat surprising result, although it is clear upon
careful reinspection
that the formal derivation~\cite{ref-greenberg} of the result that CPT violation
requires Lorentz violation does not technically apply in the context of a metric
theory of gravity.
Purely gravitational theories are not formulated using QFT to begin with, and
stability of the quantum vacuum (required for the definition of the $S$-matrix)
is thus not a condition that can formally be applied.
Physically, this corresponds to the fact that there is generally
no reason to expect that matter in a universe cannot progressively coalesce into heavier and
heavier black holes, there being no ultimately stable lowest-energy configuration.
So while $b^{\mu}$ and $v^{\mu}$ terms have the same discrete symmetries, $b^{\mu}$ breaks
Lorentz invariance, while $v^{\mu}$ does not. Returning to the main problematical
observation, it appears that if a fermion species with a Lorentz-violating $b^{\mu}$
term is coupled to gravity, radiative corrections may produce a ${\cal L}_{\Gamma}$ term with
coefficient $v^{\mu}$ proportional to $b^{\mu}$. The radiative correction would thus
possess a much greater degree of symmetry than the novel term that generated
it. It is not immediately clear whether this is possible or whether it should
be ruled out by some general principles of field theory. Whichever option is
correct, there is evidently quite a bit more to be understood about how
radiative corrections work in these kinds of theories.
Having established the existence of this open question,
we shall show that the resolution of this enigma is rather subtle. The
key pieces of information necessary to construct the solution are embedded in the
theory, but they need to be pieced together, in conjunction with what is
already known about the general structure of Lorentz-violating field theories.
The ultimate answer will be tied to the fact that gravitational theories
are fundamentally different from other field theories when it comes to
Lorentz violation. In particular, Lorentz violation in a metric theory must arise
spontaneously.
\section{Structure of Ambiguous Corrections}
The fact that Lorentz symmetry in a metric theory of gravity can only be broken
spontaneously will have profound consequences for the radiative corrections to the
gravitational Chern-Simons term. To understand these consequences, we must look very
carefully at the structures of both the electromagnetic and gravitational Chern-Simons
terms. This will reveal a close connection to chiral anomalies.
\subsection{Abelian Theory}
\begin{figure}
\centering
\includegraphics{gravcs-fig-1a.jpg}
\includegraphics{gravcs-fig-1b.jpg}
\caption{One-loop diagrams that can contribute to the radiatively-generated
Lorentz-violating Chern-Simons terms.
The dots represent the $b^{\mu}$ insertions appearing in the fermion propagator
$S_{b}$. (a) The two triangle
diagrams that exist in the radiative calculation of the Abelian Chern-Simons term.
(b) The additional contributing diagram that appears in the gravitational theory.
\label{fig-contribute}}
\end{figure}
The QED Lagrange density, including the only mSME term that has the
right structure to make a radiative contribution to ${\cal L}_{AF}$, is
\begin{equation}
{\cal L}_{{\rm QED}}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\psi}(i\!\!\not\!\partial-m-
q\!\!\not\!\!A\,+\!\not\!b\gamma_{5})\psi.
\end{equation}
The pair of diagrams that contribute to the undetermined radiative correction in an
Abelian gauge theory is shown in Figure~\ref{fig-contribute}a. These are
essentially just the same dia\-grams---with fermion triangles and two external photons---that
are responsible for the chiral anomaly.
The two diagrams in Figure~\ref{fig-contribute}a differ in the direction of the fermion number flow around the
triangular loop. Alternatively (taking the viewpoint suggested by the
nonperturbative treatment in~\cite{ref-jackiw1}), there is just a single
one-loop diagram---the usual vacuum polarization diagram, but with the modified
fermion propagator
\begin{equation}
S_{b}(l)=\frac{i}{\!\not l-m\,+\!\not\!b\gamma_{5}}
\approx\frac{i}{\!\not l-m}+\frac{i}{\!\not l-m}\left(i
\!\not\!b\gamma_{5}\right)\frac{i}{\!\not l-m}.
\end{equation}
(Since the Lagrange density involves no nonstandard time derivatives, the fermion sector
may be quantized without any changes to the spinor representation~\cite{ref-kost3,ref-kost5},
and the $b^{\mu}$-exact propagator may simply be read off from ${\cal L}_{{\rm QED}}$.)
The two triangles then arise from the fact that, at first order in the Lorentz violation, there
may be a $b^{\mu}$ insertion on exactly one of the two internal fermion lines.
The two triangle diagrams are very similar to those that arise in the
calculation of the QED chiral anomaly---for example, in its original context of
$\pi^{0}$ decay~\cite{ref-bell1}. Each triangle has two vertices attached to outgoing gauge
boson propagators, and a third axial vector vertex with a $\gamma_{5}$. The presence of the
$\gamma_{5}$ is what ensures that, when the loop momentum is very large, the
contributions from the two different diagrams cancel out, since the fields
passing through the $\gamma_{5}$ vertex have opposite chiralities. A great deal
is known about the structure of these kinds of diagrams, from analyses of the
chiral anomaly.
However, there is a subtle but fundamental difference between
how the sum of the two triangle diagrams should be evaluated, in the
contexts of Lorentz violation versus $\pi^{0}$ decay. The issue is that, when
a meson vertex is involved, there is an additional leg attached there, which
represents the incoming decaying particle. That particle carries momentum,
so the two fermion propagators attached to that vertex will have different momenta.
In contrast, in a theory with explicit Lorentz violation, the $b^{\mu}$ vertex cannot
carry any momentum whatsoever, because $b^{\mu}$ is constant across all spacetime.
Surprisingly, this changes the way the calculations can proceed in a significant way.
It was initially argued~\cite{ref-coleman}---incorrectly---that
the triangle diagrams could not generate
a Chern-Simons term, because of the theory's gauge invariance properties.
In each diagram, there are external photons attached to two of the triangle's corners, carrying
momenta $p_{1}$ and $p_{2}$. Ward identities then imply that the amplitude ${\cal M}_{\mu\nu}$
corresponding to the sum of the two fermion loops must be transverse to both $p_{1}$ and
to $p_{2}$. That is,
\begin{equation}
\label{eq-MWard}
{\mathcal M}_{\mu\nu}p_{1}^{\mu}={\mathcal M}_{\mu\nu}p_{2}^{\nu}=0.
\end{equation}
The implies that the amplitude must be ${\cal O}(p_{1})$ and separately ${\cal O}(p_{2})$.
If $p_{1}$ and $p_{2}$ are allowed to be different (that is, if the axial vector vertex can
carry a nonzero momentum), this implies that ${\mathcal M}_{\mu\nu}$ is ${\mathcal O}(p_{1}p_{2})$.
When we set $p_{1}=-p_{2}$, corresponding to the physical situation, the amplitude
must be ${\mathcal O}(p_{1}^{2})$. Since the Chern-Simons term is only ${\mathcal O}(p)$,
it would appear that the Abelian Chern-Simons term cannot be generated by radiative corrections.
However, this simple argument fails when there is no momentum input at the fermion triangles' third
vertices. If $p_{1}$ is always identically equal to $-p_{2}$, then the two transversality
conditions in (\ref{eq-MWard}) are redundant, and the matrix element only needs to be ${\cal O}(p_{1})$,
meaning a Chern-Simons term is actually allowed.
Without two independent Ward identities to be satisfied, the sum of the two triangle diagrams is
actually undetermined, because the diagrams are each naively linear divergent, and there is no unique
way to regulate them. However they are regulated, the divergent parts of the two diagrams will cancel,
producing a finite result. A specific regulator is often most conveniently expressed
in terms of a relationship between the loop momenta $k$ and $k'$ in the two diagrams.
If the amplitude really had needed to be transverse to two different photon momenta $p_{1}$
and $p_{2}$, it would have been necessary to choose $k'=k+3p_{1}$ and then (after Wick rotation) to perform
a spherically symmetric integration over $k$. This is why when the axial vector vertex represents a physical
$\pi^{0}$---which carries a nonvanishing momentum---the chiral anomaly gives a unique result
for the meson decay rate. However, when only a single transversality condition is imposed, it is possible
to have $k'=k+(3+\xi)p_{1}$ for any real value of $\xi$. While the induced Chern-Simons term vanishes for
$\xi=0$, with nonzero values of $\xi$ there is a $k_{AF}^{\mu}=-\xi q^{2}b^{\mu}/16\pi^{2}$ proportional to
$\xi$~\cite{ref-jackiw1}.
Each value of $\xi$ essentially defines a different quantum theory, all based on the same classical
Lagrangian. Lorentz- and CPT-violating QED is thus an example of a QFT with finite
but undetermined radiative corrections; some general characteristics of such theories are discussed
in~\cite{ref-jackiw3}.
With a momentum cutoff regulator, the shift in the integration by $\xi p_{1}$ produces a surface term,
which is allowed to be nonzero because the full diagram is divergent. This kind of surface term is
well known to create problems with gauge invariance. However, because of the presence of the $\gamma_{5}$
in the fermion loop, there must be a Levi-Civita tensor $\epsilon_{\mu\alpha\beta\gamma}$ in the
resulting radiative correction to the photon two-point function; and because of the total antisymmetry of
$\epsilon_{\mu\alpha\beta\gamma}$, the radiative correction
|
(i.e.\ the induced
Chern-Simons term ${\cal L}_{AF}$) still obeys the Ward identity. This is
what ensures that the integrated action remains gauge invariant, even though gauge invariance is lost at
the level of the Lagrange density.
Since surface terms are involved, it seems like it might be possible to
avoid the Chern-Simons ambiguity by using a better regulator. However,
both Pauli-Villars and dimensional regularization---normally the best
choices when there are potential problems with maintaining gauge
invariance---reintroduce the ambiguity in other ways. The Pauli-Villars
method entails introducing additional families of fictitious heavy fermions,
whose contributions to the photon self-energy are subtractive. However, the
new fermions will posses their own $b^{\mu}$ terms, whose sizes are not
determined by the classical Lagrangian. In dimensional regularization,
there is no unique extension of $\gamma_{5}$ to $4-\epsilon$ dimensions,
and different extensions will produce different Chern-Simons terms. With
other regulation methods, the source of the radiative ambiguity may sometimes
be disguised, but the ambiguity always appears to be present somewhere.
Several specific nonzero values for the induced $k_{AF}^{\mu}$ were suggested in
the literature. These were typically based on various nonperturbative
arguments for how the momentum integrations in the two triangle diagrams
should be performed. However, any nonperturbative method should determine
the structure of the radiative corrections not just at ${\cal O}(b)$,but
also at ${\cal O}(b^{2})$. It turns out that any choice of regulator that
gives a specific nonzero coefficient for the Chern-Simons term at first
order also breaks gauge invariance by producing
a Lorentz-violating photon mass~\cite{ref-altschul8} term at second order
in $b^{\mu}$ (e.g~\cite{ref-altschul1,ref-altschul2,ref-filipe}). The one natural
exception is a regulator that produces a
vanishing $k_{AF}^{\mu}$; it is always possible to enforce the maximal degree of gauge
invariance at first order without spoiling gauge symmetry at higher
orders.
\subsection{Gravitational Theory}
Having pointed out all the key properties of the ambiguity in the Abelian
Chern-Simons term in Lorentz-violating QED, we now turn our attention to
the even trickier case of the gravitational Chern-Simons term.
The gravitational SME action includes the $b^{\mu}$ term in the form
\begin{equation}
\label{eq-Spsi}
S_{\psi}=\int d^{4}x\, ee^{\mu}\,_{a}\bar{\psi}\left(\frac{i}{2}\gamma^{a}
\overset{\text{\tiny$\leftrightarrow$}}{D}_{\mu}+
b_{\mu}\gamma^{a}\gamma_{5}\right)\psi.
\end{equation}
where the fermions are taken to be massless (purely for simplicity). The vierbein (tetrad) is $e^{\mu}\,_{a}$,
and its determinant is $e$. The coupling to gravitation occurs through $e^{\mu}\,_{a}$ and through
the gravitational covariant derivative, which is
\begin{equation}
D_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{2}\omega_{\mu cd}\sigma^{cd}\psi,
\end{equation}
with the usual spin connection $\omega_{\mu}\,^{cd}$ including derivatives of
the vierbein.
Because of the required vierbein factors, (\ref{eq-Spsi}) is written as an integrated action $S$, although
for linearized gravity it would actually be sufficient to work with just a Lagrange density.
In a the linearized theory and in harmonic gauge, chosen for its simplicity and convenience, especially
when dealing with gravitational
anomalies~\cite{ref-alvarez}, the vierbein has a very simple representation in terms of the metric:
$e_{\mu a}=\eta_{\mu a}+\frac{1}{2}h_{\mu a}$ and $e^{\mu}\,_{a}=\eta^{\mu}\,_{a}-\frac{1}{2}h^{\mu}\,_{a}$.
\begin{figure}
\centering
\includegraphics{gravcs-fig-2.jpg}
\caption{Feynman rules for the fermion-graviton vertices in the presence
of $b^{\mu}$.
\label{fig-rules}}
\end{figure}
Neglecting $h^{\mu}\,_{\mu}$ interactions, which cannot contribute to the gravitational Chern-Simons
term, the linearized Lagrange density for the fermions coupled to gravity is
\begin{equation}
{\cal L}_{\psi}=\bar{\psi}\left\{\frac{i}{2}\left(\gamma^{\mu}-\frac{1}{2}h^{\mu\nu}\gamma_{\nu}\right)
\overset{\text{\tiny$\leftrightarrow$}}{\partial}_{\mu}
-h_{\mu\nu}\left[\frac{1}{2}b^{\mu}\gamma^{\nu}\gamma_{5}+
\frac{i}{96}(\partial_{\rho}h_{\alpha\beta})\eta^{\beta\nu}\gamma^{(\nu\beta\rho)}\right]
+\!\not\!b\gamma_{5}\right\}\psi.
\end{equation}
This expression involves the
antisymmetrized product of three distinct $\gamma$-matrices, $\gamma^{(\nu\beta\rho)}=
\gamma^{\nu}\gamma^{\beta}\gamma^{\rho}\pm({\rm all\, permutations})$.
The corresponding Feynman rules for the perturbative interactions of gravitons with
Lorentz-violating fermions are given in Figure~\ref{fig-rules}. There are the
usual vertices for single or paired gravitational excitations $h^{\mu\nu}$ interacting
with a fermion line, and there is also a new vertex in which $b^{\mu}$ appears.
The new vertex exists because of the presence of $b^{\mu}$ in the energy-momentum
tensor for the fermionic sector. However, it turns out that the new vertex does not
actually make any contribution to the radiatively-induced gravitational Chern-Simons term.
On the other hand, both the usual three-particle vertex and the four-particle vertex involving
$\gamma^{(\nu\beta\rho)}$ play potentially important roles.
All the two-point graviton diagrams derived from ${\cal L}_{\psi}$ that have a single fermion
loop are shown in Figures~\ref{fig-contribute}
and~\ref{fig-notcontrib}. However, only those in
Figure~\ref{fig-contribute} can actually contribute to the Chern-Simons term (see below for details),
and we shall therefore concentrate our attention on those three diagrams.
Besides the presence of an additional diagram with a two-graviton vertex, there is another
way in which the gravitational radiative corrections are more complicated than those in the
Abelian theory.
Because the metric modes couple to the fermions' energy-momentum, there are additional
factors of the loop momentum appearing at the fermion-boson vertices. This gives the two triangle
diagrams in Figure~\ref{fig-contribute}a a naive cubic degree of divergence. The new diagram with
the two-graviton vertex would also have a cubic divergence if the axial vector vertex could carry
a nonzero momentum. However, with only a strictly constant $b^{\mu}$ inserted into the fermion
propagator, the degree of divergence is reduced. In order to obtain
a finite final results for the induced $v^{\mu}$, both the cubic and linear divergences
in the sum of the diagrams must be canceled.
\begin{figure}
\centering
\includegraphics[align=c]{gravcs-fig-3a.jpg}
\includegraphics[align=c]{gravcs-fig-3b.jpg}
\caption{Four diagrams that do not contribute to the radiatively induced gravitational
Chern-Simons term. (a) Diagrams with a $b^{\mu}$-modified fermion-photon vertex, which leads
to a term in which $b^{\mu}$ is contracted directly with one of the indices of the external
graviton $h^{\mu\nu}$. (b) Pure tadpole diagrams.
\label{fig-notcontrib}}
\end{figure}
The more elaborate cancelation is generally possible (and necessary in order to preserve
gravitational gauge invariance) because there are now three contributing diagrams---as
opposed to the just two that were present in the Abelian gauge theory. Note
that with a symmetric $k''$ integration, the diagram in Figure~\ref{fig-contribute}b actually gives
no net contribution to the Chern-Simons term; however, shifting the integration momentum by a
multiple of $p_{1}$ does yield a surface term with a linear divergence. The
calculation of the sum of the three diagrams
proceeds along much the same lines as in the original papers on the gravitational
contribution to the partially conserved axial current (PCAC)~\cite{ref-kimura,ref-delbourgo1,ref-delbourgo2}
(describing, for example, the potential rate of decay of a $\pi^{0}$ into two gravitons). The cubic
divergences in the triangle diagrams automatically cancel, because they have opposite momentum
routings and $\gamma_{5}$ is present. However, the cubic cancelation still leaves a residual linear
divergence. In the Abelian case, the residual finite term in the sum of the two triangle
diagrams was set by a $p_{1}$-dependent difference in the integration momenta $k$ and $k'$; here in
the gravitational calculation, the remaining linear divergence is set by the difference in $k$ and
$k'$. However, with a shift in the integration momentum $k''$ in the tadpole diagram, this linear
divergence may also be canceled.
These results, for both kinds of gauge theories,
can also be expressed in terms of the number of Ward identities that need to be satisfied.
When the axial current vertex carries an external momentum (as it does in $\pi^{0}$ decay), there
there are two independent Ward identities, because there are two different external momenta to
which the fermion loop matrix element must be transverse. Violations of gauge invariance must come
from divergent loop integrals. Imposing the first Ward identity always forces the strongest
naive divergence to vanish. However, there are still possible gauge symmetry violations coming
from surface terms associated with the Feynman diagram divergences. In the electromagnetic case (with
a naive linear divergence), there is one possible surface term, which may be adjusted to zero by
choosing $k-k'$. This enforces the second independent Ward identity.
In the gravitational version, there are two possible surface terms (a
linearly divergent one and a finite one), because the initial degree of divergence is cubic.
Again however, by suitable choices of both $k-k'$ and $k-k''$, both the surface terms may be
eliminated, again making the matrix element transverse to both boson momenta.
The situation is somewhat different when the axial vector vertex originates from
a constant Lorentz-violating background $b^{\mu}$. In that case, as we have noted, there is
only one Ward identity to enforce, because the momenta $p_{1}$ and $p_{2}$ of the external
gauge bosons are redundant. In the Abelian case, that means that $k-k'$ may be chosen to
be any multiple of $p_{1}$, yielding an undetermined Chern-Simons term. In the gravitational
theory, there is still one nontrivial affine condition relating $k$, $k'$, and $k''$ that
is needed to ensure the linear divergence cancels. However, this means once again that
there remains one undetermined parameter, and different choices of this parameter will produce
different values for the induced $v^{\mu}$.
Other regulators for the naively divergent diagrams introduce the ambiguity in other ways,
just as in the Abelian theory. It may seem natural, therefore, to conclude that the
coefficient of the induced gravitational Chern-Simons term should be entirely undetermined,
just as in the Abelian case.
Before continuing, we shall pause briefly to point out why the only possible contributions to the induced
Chern-Simons term actually come from the Feynman diagrams in Figure~\ref{fig-contribute}, even though
there are several other diagrams that can be constructed formally as contributions to the
graviton two-point function. The non-contributing diagrams are shown in Figure~\ref{fig-notcontrib}.
The two diagrams appearing in Figure~\ref{fig-notcontrib}a have the Lorentz violation
entering through the $b^{\mu}$-modified
interaction vertex from Figure~\ref{fig-rules}. However, any diagram with
a $b^{\mu}$ vertex could only give a contribution to the effective action with
the Lorentz index of $b^{\mu}$ directly contracted with one of the external $h^{\mu\nu}$
metric modes---and such a term is not of the Chern-Simons form.
Moreover, the other two diagrams in Figure~\ref{fig-notcontrib}b are not one-particle irreducible.
They have pure tadpole forms, with the intermediate graviton propagator necessarily carrying
vanishing momentum, so no information about the momentum of the external gravitons can reach
the divergent fermion loop. In any reasonable renormalization scheme, the sum of all the one-point
graviton tadpole diagrams will be set to vanish (so that the nonfluctuating part of the metric
$g^{\mu\nu}$ takes on its proper background value).
More details of the integrals, covering both the Abelian and QED cases, are given in the appendix.
\section{Resolving the Puzzle}
The results of the previous section
bring us back to the extremely puzzling point that while a $b^{\mu}$ term in the fermion
sector is Lorentz violating, any proportional $v^{\mu}$ that is induced in the linearized
gravity sector is not. It seems very counterintuitive that the quantum corrections
at first order in $b^{\mu}$ should somehow ``restore'' the broken Lorentz symmetry, at least in the
gravitational sector. There is no analogous symmetry restoration for the electromagnetic radiative
corrections; the $k_{AF}^{\mu}$ violates the same spacetime symmetries as a $b^{\mu}$ term. Doubly puzzling
is that while the gravitational Chern-Simons term avoids the Lorentz violation associated with $b^{\mu}$,
it still has the same discrete symmetries (and hence the same CPT violation) as the fermion sector
term.
However, this paradoxical---or at the least, extremely curious---behavior turns out to be an artifact
of having used an oversimplified description of Lorentz violation in conjunction with gravity.
In fact, the direct generation of a Lorentz-invariant radiative correction by a
Lorentz-violating term in the fermion sector does not actually occur.
We have repeatedly noted that when the axial vector vertex insertion in the fermion propagator can
carry a nonzero momentum, the number of constraints on a bosonic two-point function changes.
If momentum can be exchanged with the axial vector background, then the momenta of the
two attached bosons are not identical, and then there are two independent transverse Ward identities
that must hold. Then the argument that the radiative corrections to the boson propagator
must be at least ${\cal O}(p_{1}^{2})$ is entirely correct, and a Chern-Simons term is excluded. Notice
that insisting that the two separate Ward identities both hold when there is a net external momentum being
inserted into the loop diagrams is equivalent to demanding that the Fourier transform of the effective
Lagrange density be gauge invariant at the value of the external momentum in question. In fact, the only Fourier
component of the (electromagnetic or gravitational) Chern-Simons term that is invariant under gauge
transformations is the zero-momentum component---which is just the integrated action. If a stronger
form of gauge invariance---invariance of the density ${\cal L}$, rather than just $S$---is required,
then the coefficient $v^{\mu}$ in the gravitational sector must vanish.
It happens to be the case that
Lorentz violation in a metric theory of gravity must arise
spontaneously. Spontaneous Lorentz symmetry breaking is analogous to
other, more familiar, types of spontaneous symmetry breaking. A bosonic field
acquires a vacuum expectation value (vev), so that the vacuum state of the
theory does not respect all the symmetries of the underlying Lagrangian. If the
field with the vev has tensor indices, then the vev becomes a preferred
tensor in the vacuum. Couplings of other fields to the symmetry-breaking field
give rise to SME-type Lorentz-violating operators.
In a flat-space QFT like QED, Lorentz violation might arise
spontaneously, or it might be explicit. In the latter case, the fundamental
Lagrangian for the theory contains operators that violate Lorentz symmetry.
Either possibility is internally consistent, although what we know about
symmetry breaking in real physical systems may suggest that the spontaneous
symmetry breaking might be more elegant. In a gravitational theory, however,
matters are quite different. Only spontaneous symmetry breaking is possible;
gravity theories with explicit Lorentz breaking turn out to be mathematically
inconsistent. In a metric theory with explicit symmetry breaking, the Bianchi identities
cannot be satisfied, and the theory fails~\cite{ref-kost12}.
The qualitative reason for the inconsistency is actually rather simple. The basic premise of
a metric theory of gravitation is that test particles are moving along geodesics of
a background spacetime configuration. Two particles with equal mass, beginning
at the same point and moving with the same initial speeds, must follow exactly
the same trajectory. There is no room for the dynamics to depend on anything
else; there is no way to incorporate the spin and orientation dependences normally
associated with the motion of different species of particles in a theory with explicitly
broken rotation or boost symmetry. A preferred direction like $b^{\mu}$ cannot affect the motion of a
fermion if the fermion's motion is entirely determined by the spacetime geometry that
it is passing through. This argument holds even in a pure gravity theory,
because gravity actually provides its own test quanta. Pure gravity theories
have propagating gravitons, which in the linear theory are still effectively passing through a
background geometry.
This geometric obstruction has spurred some interest in studying Lorentz violation in more general Finsler
spacetimes. However, this work is still in its infancy; basic constructions,
such as of a scalar field action or a spinor bundle have not yet been demonstrated.
For the present purposes, we shall continue to
suppose that gravitation is described by a metric theory like general
relativity. The structure and consequences of a gravitational Chern-Simons-like term
in a Finsler geometry are matters far beyond the
current state of understanding.
The fact that Lorentz violation in a metric theory of gravity must be spontaneous
has definite phenomenalistic consequences. There must be additional fluctuating
modes in the theory, which affect the physical observables in both the purely
gravitational sector~\cite{ref-bailey2} as well as with matter-gravity
couplings~\cite{ref-tasson1}.
The observation that the preferred $b^{\mu}$ is formed from the vacuum expectation value of
vector-valued fields on the spacetime manifold leads to a number of interesting conclusions
and opens up new avenues of investigation. The vector field underlying $b^{\mu}$
may have a global structure related to the topology of the spacetime. Moreover, there will be
additional quantized excitations which are coupled to the theory's fermions
in the same way as $b^{\mu}$ itself.
However, what is important here is that the $b^{\mu}$
that appears in the Feynman rules in the gravitational theory cannot just be a
fixed background vector. Instead, it is accompanied by additional fluctuating degrees
of freedom. While the fluctuations themselves may be extremely small, the very fact
that they must be possible changes the conceptual nature of the $b^{\mu}$ vertices.
Because $b^{\mu}$ is the vev of a dynamical field, it has to be possible for there
to be momentum exchange between the $b^{\mu}$ vertex and the gravitational field.
When a nonzero momentum can enter the fermion loops at the $b^{\mu}$ insertion, there
are going to be two nontrivial Ward identities, and the effective Lagrange density
itself---not just the integrated effective action---must be gauge invariant. This
returns the theory to the original situation that was studied in the context of PCAC,
in which the form of the radiative corrections is completely fixed. For the
coefficient of the induced gravitational Chern-Simons term, the resulting unambiguous
value is zero.
Thus the highly peculiar behavior of the radiative corrections---that there could be a
Lorentz-invariant correction that is linear in the Lorentz-violating parameter
$b^{\mu}$---has thus been avoided. The reason for this is that
the arbitrariness of the Chern-Simons-type radiative corrections
only exists when the Lorentz violation in the theory is explicit---which itself
may be a rather unexpected result, although not a potentially paradoxical one.
In fact, we may take the argument one step further and note that the when
Lorentz-violating QED is studied in the (realistic) context of a background spacetime
governed by general relativity, the $b^{\mu}$ in the fermion sector still has to be
just one piece of a dynamical field. This means that the radiative-induced Chern-Simons
term for the Abelian theory is also zero!
We have reached this level of understanding by drawing together earlier conclusions
about the mathematical structures of different kinds of Lorentz-violating field
theories. This further reinforces the observation that when basic symmetries such as
Lorentz symmetry or CPT are broken there may be some fairly subtle effects,
qualitatively unlike those seen in more symmetric models---especially in regard
to quantum corrections.
|
\section{Introduction}
In today's globalized just-in-time economy, production systems are facing greater challenges. Customers demand more specialized products which leads to increasing products varieties and demand variations. To deal with these challenges, companies need sophisticated decision-support systems. In addition, constraints to consider change significantly from one industry to another. This paper tackles the simultaneous lot sizing and scheduling problem for the tire industry.
Lot sizing problems have been widely studied in the literature over the last decades. The expected output of lot sizing is to give a complete picture of how many pieces to produce and how many pieces to carry in inventory at each period over a planning horizon. It takes its origin in the well-known Economic Order Quantity (EOQ) model \cite{Harris} under the assumption of single item, constant demand and infinite planning horizon. Since then, numerous researchers have built more realistic models to cope with real-world problems. The production capacity limitation is a significant constraint that production managers have to cope with. The Capacitated Lot Sizing Problem (CLSP) has been proven to be NP-hard by Bitran \cite{bitran1982computational}. Since then, various extensions of the lot sizing problem have been studied extensively and can be classified based on several criteria. An exhaustive review on the CLSP can be found in \cite{drexl1997lot} and \cite{pinedo2005planning} and a classification of criteria is presented in \cite{brahimi2017production}. We limit our study to the multi-item version of the CLSP with a focus on the single-level simultaneous lot sizing and scheduling problem (see \cite{Worbelauer2019Simultaneous} and \cite{copil2017simultaneous} for a more exhaustive classification). Please refer to \cite{brahimi2017single} for single-item LSP and to \cite{buschkuhl2010dynamic} for multi-level LSP.
Our contribution is threefold. First, we tackle an industrial case and demonstrate the efficiency of our method to solve it. Second, twenty specific constraints such as the number of setup per period, upstream resources saturation and customer prioritization are presented. To the best of our knowledge, these constraints have never been modeled and considered together in the literature. Finally, we propose a MIP formulation and a problem-based matheuristic to solve the problem efficiently. Several objectives are taken in consideration and objective function parameters are calibrated thanks to the Taguchi procedure. A sensitivity analysis of one particular parameter is also conducted.
The remainder of this paper is organized as follows. Section \ref{Literature_review)} provides a literature review of the considered problem. The problem description and the assumptions of this study are described in Section \ref{Problem description}. Section \ref{MIP formulation} describes the mathematical formulation proposed to model the planning problem. In Section \ref{Matheuristic}, a matheuristic method based on an original hybrid sequential approach is introduced to solve the problem. A model size comparison is also presented. Section \ref{Computational Results} details the real case study results based on large performance testing campaigns and sensitivity analysis. Finally, Section \ref{conclusion} summarizes the study and gives future research directions.
\section{Literature Review}
\label{Literature_review)}
As mentioned before, the underlying constraints to consider vary significantly from one industry to another. Recent papers dealing with simultaneous lot sizing and scheduling (LSS) problems put more focus on particular features of these industries \cite{copil2017simultaneous}: the beverage industry \cite{toledo2015synchronized}, \cite{toledo2015relax} and \cite{baldo2017alternative}, steel manufacturing \cite{li2017fix}, \cite{de2008lot} and \cite{de2007joint}, the automotive industry \cite{deeratanasrikul2017multiple} and \cite{gnoni2003production}, glass-container manufacturing \cite{toledo2016mathematical}, the tile industry \cite{ramezanian2017simultaneous}, the chemical industry \cite{cunha2018integrated}, and the paper industry \cite{figueira2013hybrid}. Additionally some research has been conducted in the tire industry \cite{lasdon1971efficient} and \cite{jans2004industrial} and more specifically in the off-the-road tire industry \cite{degraeve1997tire} and \cite{koch2020dedicated}.
Some researchers used meta-heuristic to deal with the studied NP-hard problem. One of the most popular meta-heuristic is the genetic algorithm (GA). Babaei et al. \cite{babaei2014genetic} applied it on the LSS problem in capacitated flow shop environment with consideration of backlogging and sequence-dependent setups. They used a heuristic proposed by Mohammadi et al. \cite{mohammadi2011genetic} to set the initial population and crossover operators from Ruis and Maroto \cite{ruiz2005solving}. Vincent et al. \cite{vincent2020population} recently presented a population-based heuristic to solve a multi-item capacitated lot sizing problem with setup time and unrelated parallel machines. A dynamic constructive heuristic was proposed to generate a set of initial solutions. A diversification procedure was realised using a path-relinking strategy and finally the population was intensified with a local search method. Another well known meta-heuristic is the Simulated Annealing (SA) algorithm. Ceschia et al. \cite{ceschia2017solving} introduced a SA approach to cope with the multi-item single-machine LSS problem. The search space was a vector $V$ (size: number of periods) with values $v_t$ representing the items to produce. The neighbourhood relation considered was the composite Swap $\bigcup$ Insert firstly proposed by Della Croce \cite{della1995generalized}. They also provided a hybrid method combining a MILP formulation and SA. The SA was run for a short time period and then the solution was injected as initial solution for the MILP using \textit{warm start} functionality of the CPLEX solver.
Beyond general meta-heuristic approaches, an increasing number of researchers bring forward decomposition heuristics. They usually start from a new MIP formulation from a real-world case study and propose approaches that can be product-, machine- or time based- decomposition heuristics. For instance, Meyr and Mann \cite{meyr2013decomposition} described a heuristic for the general LSS problem for parallel production lines with consideration of backlogging. The multi-line problem was divided into a series of single-line problems easier to solve. They discussed priority rules for product assignment to avoid setups and save production and inventory costs. Time-based decomposition approaches are much more widespread, mostly with Relax-and-Fix (R\&F) and Fix-and-optimize (F\&O) heuristics. De Araujo et al. \cite{de2008lot} applied a R\&F approach to solve an integrated two-level lot sizing and furnace scheduling problem in small foundries. They presented a MIP formulation for the case study and solved each macro-period divided in 10 micro periods one after another. A descent heuristic procedure was also proposed and improved using diminished neighbourhood search and simulating annealing. Rodoplu et al. \cite{rodoplu2020fix} used a R\&F heuristic to address a single-item lot sizing problem with a flow shop system and energy constraints. A worst-case analysis of R\&F algorithm for lot sizing problems have recently been conducted by Absi and Van den Heuvel \cite{absi2019worst}. They analysed the impact of time window length, effect of overlapping time windows and capacity constraints and showed that even for simple instances with time-invariant parameters, the worst-case ratio may be unbounded. R\&F procedures are often coupled with F\&O heuristics as in \cite{deeratanasrikul2017multiple} where a multi-level multi-machine CLSP with setup times was dealt with. The R\&F heuristic proposed was a two-level partition of the problem. Each sub-problem related to a time-window of $\lambda$ consecutive periods in one stage. The R\&F solution was used as an initial solution for the F\&O procedure which was based on machine-decomposition strategy. The first to introduce F\&O were Helbert and Sahling \cite{helber2010fix} and they also applied it to the multi-level CLSP. Toscano et al. \cite{toscano2020formulation} have recently developed a F\&O algorithm to solve a synchronized two-stage CLSP with mandatory temporal cleaning and sequence-dependent changeover in a soft drink company. F\&O heuristics have also been combined with local search such variable neighborhood search as in \cite{chen2015fix} and \cite{li2017fix}. Li et al. \cite{li2017fix} dealt with a multi-item lot sizing problem in the steel industry with demand class formulation and stochastic demand where backlogging and overtime costs were also incurred. An original F\&O approach based on ``k-degree decomposition'' that proposes several decomposition among products, resources and time horizon was presented. Then they proposed an integrative F\&O and VNS procedure. The demand class formulation is an original contribution of this paper. The work of Gruson et al. \cite{gruson2018impact} is also to be noted on this topic. They provided a service level analysis for lot sizing problems with backlogging. They introduced two service-level definitions and modelling based on backorder fix and variable costs. Moreover, Gören and Tunali \cite{goren2015solving} used a sequential hybrid approach with F\&O and GA for the multi-level CLSP with setup carryover. They used a time window of five consecutive periods and the sub-problems were then solved with a GA. Besides, Toledo et al. \cite{toledo2015relax} and Baldo et al. \cite{baldo2017alternative} both brought forward for the soft drink and brewery industry MIP-based heuristics with binary variables relaxation for two-level multi-item and multi-machine CLSP with sequence-dependent setup at each level. In \cite{toledo2015relax} they suggested two relaxation of binary variables into continuous variables to minimize setup, production and inventory costs. In \cite{baldo2017alternative}, they added backorder consideration and coped with two-level problem with tanks providing bottling lines. The MIP-based sequential approach focused on the decomposition of stages to minimize setup, production, inventory and backorder costs. Finally, an interesting fuzzy mathematical programming and self-adaptating artificial fish swarm algorithm for just-in-time energy aware flow shop scheduling problem with outsourcing option was presented by Tirkolaee et al. \cite{tirkolaee2020fuzzy}. They proposed a novel bi-objective mixed-integer linear programming model (MILP). They treated it as a single objective MILP using a multi-objective fuzzy mathematical programming technique and implemented a sensitivity analysis to study the behavior of the objectives with real-world conditions.
\section{Problem Description}
\label{Problem description}
\subsection{Industrial Process}
\answer{
This work tackles a planning problem in tire manufacturing. A tire is a complex structure composed of many layers. The main components can be summarized as follow (this list is non-exhaustive, see Figure 1): (i) The inner liner plays the role of an inner tube; (ii) One to three casing plies. It is also called the carcass and refers to the main body of the tire; (iii) The crown ply. This belt is placed on top of the casing and consists of layers of fabric to avoid tire deformation; (iv) The tire tread. It is the part of the tire that comes into direct contact with the road; (v) The sidewalls are found at the sides of the tires. It protects the carcass and prevents tears or punctures on the side of the tire; (vi) The tire bead is composed of steel wire wrapped into a thin layer of rubber. These large steel cords are wound together to form a cable to compose the bead cores.}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.7]{architecture_leg1.png}
\caption{Tire architecture}
\end{figure}
\answer{
The whole production process can be divided in five major sub-processes (see Figure 2\footnote{Tire architecture description provided by the company}) \cite{boudha_kumar_srimannarayana_koyan_2011}. The first subprocess is the production of a homogeneous rubber material based on four main compounds: elastomers, renforcing fillers, platicizers and other chemicals elements. These components are mixed in a banbury mixer to obtain a homogeneous rubber material in the form of thin layers. In addition to the rubber compounds, two more raw materials are necessary to build a tire: textile reinforcements and steel wires, ensuring its rigidity and geometry. The second sub-process is the production of semi-finite products using two different technologies: first the extruding and calendaring process; and second the profiling and cutting process. Then the task of building the tire begins. This third sub-process is called the assembling sub-process. The tire building machine also needs a resource called a drum (a rotative cylinder) on which the different parts of the tire are assembled. The inner liner, the casing plies,the bead cores and the sidewalls are put together. After that the building machine operates the conformation operation. The edges of the drum are brought together and the center is inflated: this operation gives the final toroidal shape of the tire. Finally the crown ply and the tire tread are incorporated in that structure to obtain what is called a ``green'' or uncured tire. Fourth is the curing and vulcanizing process. A tire-specific mold is placed into the curing press, which is also called a heater or curing press, and the green tire is put into that mold. A bladder filled with pressurized hot fluid in the center of the mold forces the still malleable substance of the green tire to flow into all the cavities of the tread pattern engraved inside the mold. The heat of the fluid starts the curing process. The increase of temperature causes the sulfur contained in the rubber compound to bound with the rubber molecules. This is what we call vulcanization. The rubber is then transformed from a plastic to an elastic state. When ejected from the mold and after cooling the tire has taken on his final shape and properties. Finally inspection and finishing operations remains before the tire is stored in the warehouse.
}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.8]{Manufacturing_process.jpg}
\caption{Tire Manufacturing Process from \cite{boudha_kumar_srimannarayana_koyan_2011}}
\end{figure}
\subsection{Curing Workshop Planning}
\answer{To deal with the planning problem of this plant
|
, the company should adress the balance between the assembling and curing subprocesses capacity. Indeed this production environnement can be described as a hybrid flow shop environnement with two stages and a different number of machines at each stage with tire-dependent processing times with high variability. The assembling workshop has twenty-two assembling machines for a mean processing time around fourty minutes while seventy curing presses are available for a mean processing time of two hours. The throughput analysis of the production process provided by the company allows to identify the curing workshop as the bottleneck. However, depending on the mix of tires to produce, the bottleneck can move from curing to assembling workshop from time to time. Therefore, the stability of the production process between assembling and curing workshop is balanced on a knife edge. Hence, the idea is to organise the production of the plant based on the curing workshop planning problem while considering at the same time the saturation of assembling workshop.}\\
\answer{
We therefore focus on the curing workshop simultaneous planning and scheduling problem. There is a wide portfolio of around 170 tires to produce on 70 curing presses over 42 periods. In addition, the portfolio is getting wider to match customer expectations and makes the considered problem more difficult. The production is based on a make-to-stock inventory policy, so that the inventory level stays between a minimum and a maximum level calculated to prevent shortage and keep Working Capital Requirement to a minimum. The number of campaign endings per week is limited to avoid too much raw material loss in semi-finite upstream workshop. Also, the number of setups per period is limited by human ressources. A new mold setup requires consequent setup times (20\% of the usual daily production yield) and the planning problem is highly restricted by the eligibility matrix between tires and curing presses.\\}
During the curing process, the green tire is put into a mold that provides a specific pattern for the tire. Each mold is tire-specific: it can be used for exactly one type of tire. For some tire references several molds are available, though for most tires there is only one mold. Every mold can be placed in several curing presss, respecting the eligibility matrix. Nonetheless, each press can contain at most one mold at a time. The curing time depends on the tire produced and the curing press used. The curing presses capacity therefore links together different tire references that compete for the same resource - available time of a given curing press where the molds can be placed in. Except for the first and the last period of the production campaign, tires are produced in a continuous run and production is always done at full capacity. This type of production is often referred to as ``all-or-nothing'' production. Also, only one type of tires can be cured in a curing press within one period. Thus, our problem is classified as a small-bucket lot sizing problem.
\section{MIP formulation}
\label{MIP formulation}
The production planning problems encountered in the industry may be intractable in numerous situations due to several practical constraints. The demand over the planning horizon is known in advance (deterministic). In order to deal with situations when demand cannot be met in time, the company allows backlogging. Specific constraints are also added such as the number of ``campaign endings'' within the planning horizon or the number of different tires produced at each period. In Figure 3 is presented in the red circles the constraints to be considered. Blue circles denote the different sources where the constraints comes from. The objective of the proposed approach is to find a good feasible solution for the single-level multi-item multi-machine with deterministic demand, backlogging, sequence independent setup times and specific constraints problem. The method also takes in consideration different classes of demand representing client prioritization.
\begin{figure}[!h]
\centering
\includegraphics[scale=1]{Constraints.JPG}
\caption{Constraints to be considered and variety of sources diagram}
\end{figure}
\textbf{Indexes and Sets}\\
$A$: Number of tires in the portfolio, $a$ = 1~..~$A$\\
$N$: Number of items to plan, $i$ = 1~..~$N$, $N\ge A$\\
$N_a$: Number of items $i$ for tire $a$, $i$ = 1$~$..$~N_a$\\
$P$: Number of curing presses, $p$ = 1~..~$P$\\
$T$: Number of micro-periods $t$ in one macro-period $h$, $t$ = 1~..~$T$\\
$H$: Number of macro-period, $h$ = 1~..~$H$\\
$W$: Number of workshops in assembling shop, $w$ = 1~..~$W$\\
$N_w$: Set of items $i$ doable on workshop $w$\\
$N_d$: Number of assembling machine resource types (drums), $d$ = 1~..~$N_d$\\
$C$: Set of demand classes for client prioritization for tire $a$ in 1..$A$, $c$= ${ C_1..C_\gamma}$\\
\textbf{Parameters}\\
Inventory and capacity parameters\\
$\overline{S_{at}}$: Overstock of tire $a$ at period $t$\\
$\underline{S_{at}}$: Understock of tire $a$ at period $t$\\
$D_{act}$: Demand of class $c$ for tire $a$ at period t\\
$\gamma_{c}$: Weight vector to balance demand class priorities\\
$K_a$: Number of curing press resources (molds) available for tire $a$\\
$(M_{ap})$: Eligibility matrix tire - curing press\\
$R_{it}$: Daily rate of a curing press for item $i$ during period $t$\\
Weights parameters \\
$V_{t}$: Targeted weight to produce within one micro-period $t$\\
\answer{
$\overline{v_t}$: Upper tolerance for targeted weight at micro-period $t$\\
$\underline{v_t}$: Lower tolerance for targeted weight at micro-period $t$\\
$\overline{v_h}$: Upper tolerance for targeted weight at macro-period $h$\\
$\underline{v_h}$: Lower tolerance for targeted weight at macro-period $h$\\
}
$\Omega_i$: Unit weight of item $i$\\
Upstream workshop saturation parameters\\
$\overline{S_w}$: Maximum saturation of workshop $w$ in assembling shop (time unit)\\
$N_{id}$: Set of item $i$ that can be produced with drum $d$ in assembling shop \\
$T_i$: Unit time of production of item $i$ in assembling shop \\
$K_d$: Number of assembling shop resources (drums) of type $d$ \\
$\varepsilon_{id}$: Number of molds for a given item $i$ that can be furnished by the production of one drum $d$ in upstream assembling shop\\
Flexibility parameters\\
$S$: Maximum number of items $i$ produced simultaneously per micro-period $t$\\
\answer{
$\overline{C_t}$: Maximum number of molds setup per micro-period $t$\\
$\overline{C_h}$: Maximum number of molds setup per macro-period $h$\\
$\overline{e_h}$: Maximum number of campaign ending per macro-period $h$\\
$\tau_m$: Minimum number of micro-periods $t$ of production after a setup \\
$\tau_s$: Duration of campaign suspension to count a mold setup\\
$\tau_e$: Duration of campaign suspension to count a campaign ending\\
}
Problem specific parameters\\
$(U_{pt})$: Matrix of unavailable curing press $p$ for maintenance in period $t$\\
$(P_{ipt})$: Matrix of production of item $i$ on curing press $p$ in micro-period $t$ to be enforced for industrial trial \\
$(F_{t})$: Vector of days off $t$ for each macro-period $h$\\
$\underline{Molds}$ : Minimum number of molds to plan at the same time\\
$Spec$: Set of specific items $i$ that needs to be produced with at least $\underline{Molds}$ molds at a time for quality reasons\\
$M = \max(R_{it})$: A large number that major the maximum yield $R_{it}$\\
\\
\textbf{Decision Variables}\\
$I_{at}$: Inventory level for tire $a$ at the end of micro-period $t$\\
$B_{act}$: Backorder level of class $c$ for tire $a$ at the end of micro-period $t$\\
$X_{ipt}$: Quantity of item $i$ to produce on curing press $p$ in micro-period $t$\\
\answer{$\Delta_{idt}$: Number of drums of type $d$ needed for item $i$ in micro-period $t$} \\
$Y_{ipt}$: Binary variable that equals 1 if there is a new production of an item $i$ on curing press $p$ at period $t$; 0 otherwise\\
\answer{$\sigma_{it}$: Binary variable that equals 1 if item $i$ is being cured at micro-period $t$; 0 otherwise\\
$s_{ipt}$: Binary variable that equals 1 if there is a mold setup for item $i$ on curing press $p$ in micro-period $t$; 0 otherwise\\
$e_{it}$: Binary variable that equals 1 if there is a campaign ending for item $i$ in micro-period $t$; 0 otherwise\\
$m_{ipt}$: Binary variable that equals 1 if in micro-period $t$ the minimum number of days of production after a setup have not been reached on curing press $p$ ; 0 otherwise\\
$\delta_{idt}$: Binary variable that equals 1 if in micro-period $t$ the drum $d$ is used to produce item $i$; 0 otherwise\\
}
\\
The objective of the production planner of the tire company is to optimize several criteria. The main objective is to prevent shortage. We consider $n$ demand classes ${ C_1}..{ C_\gamma}$. All demand of class ${ C_1}$ should be satisfied before demand of class ${ C_2}..{ C_\gamma}$, and so on. All demand of class ${ C_2}$ should be satisfied before demand of class ${ C_3}..{ C_\gamma}$. Once backordering has been minimized, the production planner tries to keep every tire between a minimum and a maximum inventory level set by the supply chain department. $\lambda_j$ coefficients allows the right balancing between chosen KPIs. The objective function is expressed as follows.\\\\
Minimize:
\begin{equation} \label{1}
Z = \sum_{c \in C}\lambda_{c}\sum_{a=1}^{A}\sum_{t=1}^{T}B_{act} ~+~ \lambda_{\overline{S}} \sum_{a=1}^{A}\sum_{t=1}^{T}\overline{S}_{at} ~+~ \lambda_{\underline{S}} \sum_{a=1}^{A}\sum_{t=1}^{T}\underline{S}_{at}
\end{equation}\\
To avoid scaling effects, another objective function with normalization coefficients $\mu_j$ is formulated below.\\
Minimize:
\begin{equation}\label{2}
Z'= \sum_{c \in C}\frac{\lambda_{c}}{\mu_c} \sum_{a=1}^{A}\sum_{t=1}^{T}B_{act}
~+~ \frac{\lambda_{\overline{S}}}{\mu_{\overline{S}}} \sum_{a=1}^{A}\sum_{t=1}^{T}\overline{S}_{at}
~+~ \frac{\lambda_{\underline{S}}}{\mu_{\underline{S}}} \sum_{a=1}^{A}\sum_{t=1}^{T}\underline{S}_{at}
\end{equation}\\
Where
\begin{equation}\label{3}
\mu_c=\sum_{a=1}^{A}B_{ac0}+\sum_{t=1}^{T}D_{act} , c \in C
\end{equation}
\begin{equation}\label{4}
\mu_{\overline{S}}=\sum_{a=1}^{A}(\overline{S}_{a0}+K_a R_{a} T)
\end{equation}
\begin{equation}\label{5}
\mu_{\underline{S}}=\sum_{a=1}^{A}\underline{S}_{a0} + \sum_{c \in C}\mu_{c}
\end{equation}
Subject to
\begin{equation}\label{6}
\begin{split}
I_{at-1} + \sum_{i=1}^{N_a}\sum_{p=1}^{P} X_{ipt} = I_{at} + \sum_{c \in C} (D_{act} - B_{act} + B_{act-1}) , a = 1..A, t = 1..T
\end{split}
\end{equation}
\begin{equation}\label{7}
I_{at-1} + \sum_{i=1}^{N_a}\sum_{p=1}^{P} X_{ipt} \ge I_{at} , a = 1..A, t = 1..T
\end{equation}
\begin{equation}\label{8}
I_{at-1} + \sum_{i=1}^{N_a}\sum_{p=1}^{P} X_{ipt} \ge D_{act} - B_{act} + B_{act-1} , a = 1..A, c = C_1 .. C_{n-1}, t = 1..T
\end{equation}
\begin{equation}\label{9}
B_{act} \le D_{act} + B_{act-1} , a = 1..A, c = C_2 .. C_{n}, t = 1..T
\end{equation}\\
Constraint (\ref{6}) is the inventory balance equation, with consideration of backorder. Thanks to Constraints (\ref{7}) to (\ref{9}) demand prioritization is correctly handled. Constraint (\ref{7}) avoid the creation of ``ghost'' inventory through inventory Equation (\ref{
|
,\dots,p}\norm{X_{j}}_\infty^2,
\end{equation}
\begin{equation}
\rho \le 2^3P^\frac{3}{2} \norm{O}_\infty\max_{j=1,\dots,p}\norm{X_{j}}_\infty^3.
\end{equation}
\end{theorem}
\begin{proof}
Since the cost function is a combination of sin and cosin functions, its derivatives exist and are bounded, and from this it follows that the cost function is strongly smooth and its Hessian is Lipschitz.
However, it is worth explicitly calculate $\beta$ and $\rho$ and bound them to verify, for example, the scaling with the number of qubits.
We have the $\beta-$smooth constant defined as
\begin{align}
\norm{\partial \mathcal{L}({\theta})-\partial \mathcal{L}( {\theta^\prime})}\le \beta\norm{{\theta}- {\theta^\prime}}
\end{align}
which means we need to consider the Lipschitz constant for the $p$-dimensional function $\partial \mathcal{L}({\theta})$.
Using Lemma $\ref{LemmaRPtoRMgeneral}$ where $g({\theta})=\partial \mathcal{L}$ and $M=p$, we have
\begin{equation}
\beta\le p\max_{i,j}\left(\sup_{{\theta}}\left|\frac{\partial^2 \mathcal{L}({\theta})}{\partial \theta_i\partial \theta_j}\right|\right).
\end{equation}
Applying Lemma \ref{SupDeriv},
we find
\begin{equation}
\beta\le 2^2p\norm{O}_\infty\max_{i}(\norm{X_i}_\infty)^2,
\end{equation}
where we have used the matrix spectral (operator) norm.
The $\rho-$Hessian constant
is defined as
\begin{align}
\norm{\partial^2 \mathcal{L}({\theta})-\partial^2 \mathcal{L}( {\theta^\prime})}_{\operatorname{H.S.}}\le \rho\norm{{\theta}- {\theta^\prime}}_2
\end{align}
where $\partial^2 \mathcal{L}$ is the Hessian matrix and we have used the Hilbert-Schmidt matrix norm.
Observing that the
Hilbert Schmidt norm of a matrix is the 2-norm of the matrix \emph{vectorization} $\operatorname{vec}(\cdot)$ we have $\norm{\partial^2 \mathcal{L}({\theta})-\partial^2 \mathcal{L}( {\theta^\prime})}_{\operatorname{H.S.}}=\norm{\operatorname{vec}\left(\partial^2 \mathcal{L}({\theta})\right)-\operatorname{vec}\left(\partial^2 \mathcal{L}( {\theta^\prime})\right)}_{2}$ and applying Lemma $\ref{LemmaRPtoRMgeneral}$ with $M=p^2$ and $g({\theta})=\operatorname{vec}\left(\partial^2 \mathcal{L}({\theta})\right)$, we have
\begin{equation}
\rho\le p^\frac{3}{2}\max_{i,j,k}\left(\sup_{{\theta}}\left|\frac{\partial^3 \mathcal{L}({\theta})}{\partial \theta_k \partial \theta_i\partial \theta_j}\right|\right) .
\end{equation}
Thus, applying Lemma~\ref{SupDeriv}, we have
\begin{equation}
\rho \le 2^3p^\frac{3}{2} \norm{O}_\infty\max_{i}(\norm{X_i}_\infty)^3.
\end{equation}
\end{proof}
It is important to observe that for typical VQAs cost function $O$ and $X_i$ have an operator norm that grows at most polynomially with the number of qubits, so also $\beta$ and $\rho$ will grow at most polynomially.
The previous calculations can be easily generalized for the case of differentiable and bounded cost functions which are functions of expectation values, i.e. ,
\begin{align}
\mathcal{L}(\theta )=f\left( \left\langle 0\left| {{U}^{\dagger }}(\theta )OU(\theta ) \right|0 \right\rangle\right)~.
\end{align}
In fact we observe that if $\mathcal{L}$ and $g$ are Lipschitz functions, then
\begin{align}
\norm{\mathcal{L}(g({\theta}))-\mathcal{L}(g( {\theta^\prime}))}\le L_\mathcal{L} \norm{g({\theta})-g( {\theta^\prime})}\le L_\mathcal{L} L_g \norm{{\theta}- {\theta^\prime}}.
\end{align}
In addition, if $\mathcal{L}$ is a differentiable function with bounded derivatives on a convex set, then (because of the mean value theorem) $\mathcal{L}$ is Lipschitz on this set. From this follows that if $\mathcal{L}$ is a differentiable function with bounded derivatives of a quantum expectation value (whose image defines a bounded $\mathbb{R}$ interval), then it is Lipschitz. Moreover the sum of Lipschitz functions is a Lipschitz function.
\section{Analytic heuristics}
In this section, we provide a set of analytic heuristics about predicting the noisy convergence and the critical noise with significant improvements in performance. Our derivation is physical and heuristic, but we expect that they will be helpful to understand the nature of the noisy dynamics during gradient descent in the quantum devices.
\subsection{Brownian motion and the Polya's constant}
One of the simplest heuristics about noisy gradient descent is the theory of Brownian motion.
Let $p(d)$ be the probability that a random walk on a $d$-dimensional lattice that could return to the origin. It has been proven that \cite{polya1921aufgabe},
\begin{align}
p(1)=p(2)=1
\end{align}
but
\begin{align}
p(d\ge 3) <1 .
\end{align}
It is shown that $p(d)$, in fact, could have a closed formula \cite{montroll1956random},
\begin{align}
p(d)=1-{{\left( \int_{0}^{\infty }{{{\left[ {{I}_{0}}\left( \frac{t}{d} \right) \right]}^{d}}}{{e}^{-t}}dt \right)}^{-1}},
\end{align}
where $d>3$ is the number of training parameter in our case, and $I$ is the modified Bessel function of the first kind. One could compute numerical values of the probability $p(d)$ for increasing $d$. From $d=4$ to $d=8$, it changes monotonically from 0.19 to 0.07. It is hard to compute the integral accurately because of damping, but it is clear that it is decaying and will vanish for large $d$.
In our problem, we could regard the process of noisy gradient descent as random walks in the space of variational angles. One could regard the returning probability roughly as the probability of coming back to the saddle point from the minimum. Thus, the statement about lattice random walk gives us intuition that it is less likely to return back when we have a large number of variational angles.
\subsection{Guessing $1/\epsilon^2$ by dimensional analysis}
One of the primary progress of the technical result \cite{ExpTimeSaddle} is the $1/\epsilon^2$ dependence on the convergence time $T$ with the size of the noise $\epsilon$. Here we show that one could guess such a result in the small $\eta$ limit (where $\eta$ is the learning rate) simply by dimensional analysis.
Starting from the definition of the gradient descent algorithm,
\begin{align}
\delta {\theta _i } = {\theta _i }(t + 1) - {\theta _i }(t) = - \eta \frac{{\partial L}}{{\partial {\theta _i }}}
\end{align}
we could instead study the variation of the loss function,
\begin{align}
&\delta L = L(t + 1) - L(t) \approx \sum\limits_i {\frac{{\partial L}}{{\partial {\theta _i }}}\delta {\theta _i }} = - \eta \sum\limits_i {\frac{{\partial L}}{{\partial {\theta _i }}}\frac{{\partial L}}{{\partial {\theta _i }}}} \nonumber\\
&= - 4\eta \sum\limits_i {\frac{{\partial \sqrt L }}{{\partial {\theta _i }}}\frac{{\partial \sqrt L }}{{\partial {\theta _i }}}} L .
\end{align}
Here, we use the assumption where $\eta$ is small, such that we could expand the loss function change $\delta L$ by the first order Taylor expansion. Now, we define
\begin{align}
{K_L} = 4\sum\limits_i {\frac{{\partial \sqrt L }}{{\partial {\theta _i }}}\frac{{\partial \sqrt L }}{{\partial {\theta _i }}}}
\end{align}
and we have
\begin{align}
\delta L = - \eta {K_L}L .
\end{align}
If $K_L$ is a constant (and we could assume it is true since we are doing dimensional analysis), we get
\begin{align}
L(t) = {(1 - \eta {K_L})^t} \approx {e^{ - \eta {K_L}t}}.
\end{align}
In general, we could assume a time-dependent solution,
\begin{align}
L(t) = {(1 - \eta {K_L (t)})^t} \approx {e^{ - \eta {K_L (t)}t}} .
\end{align}
Now let us think about how the scaling of convergence time will be with noise. First, in the $\eta \to 0$ limit, for small $\eta$ the convergence time would get smaller, not larger (since it is immediately dominated by noise). So it is not possible that $T\sim 1/\eta $ to some powers in $\eta$. So the only possibility is
\begin{align}
T = \mathcal{O}(1) + \mathcal{O}(\eta ) + \mathcal{O}({\eta ^2}) \ldots
\end{align}
in the scaling of $\eta$. We will thus focus on the first $\mathcal{O}(1)$ term in the small $\eta$ limit. Furthermore, from the form $e^{ - \eta {K_L}t}$ we know that $T\sim 1/K_L$.
Now let us count the dimension, assuming $\theta_i$ has the $\theta$-dimension 1 and $L$ has the $\theta$-dimension 0. From the gradient descent formula, $\eta$ has the $\theta$-dimension 2, $K_L$ has the $\theta$-dimension -2, and $\epsilon$ has the $\theta$-dimension 1. The time $T$ is dimensionless since $\eta {K_L}T$ is dimensionless and appeares in the exponent. Thus, since we know that $T\sim 1/K_L$, there must be an extra factor balancing the $\theta$-dimension of $K_L$. The only choice is $\epsilon^2$, and we cannot use $\eta$ because we are studying the term with the $\eta$-scaling $\mathcal{O}(1)$. Thus, we immediately get, $T\sim
{1}/{(K_L \epsilon^2)}$. That is how we get the dependence $T\sim 1/\epsilon^2$ by dimensional analysis. Note that the estimation only works in the small $\eta$ limit. More generally, we have
\begin{align}
T=\sum\limits_{m,n>2m}^{{}}{\mathcal{O}(\frac{{{\eta }^{n-2m}}}{K_{L}^{m}{{\epsilon }^{n}}})}
\end{align}
if we assume that the expression of $T$ is analytic.
\subsection{Large-width limit}
The dependence $T\sim 1/\epsilon^2$ can also be derived using the \emph{quantum neural tangent kernel} (QNTK) theory. The QNTK theory has been established \cite{Liu:2021wqr,Liu:2022eqa,Liu:2022rhw} in the limit where we have a large number of trainable angles $d$ and a small learning rate $\eta$, with the quadratic loss function. According to Ref.~\cite{Liu:2022rhw},
we use the loss function,
\begin{align}
\mathcal{L}(\theta)=\frac{1}{2}\left(\left\langle\Psi_{0}\left|U^{\dagger}(\theta) O U(\theta)\right| \Psi_{0}\right\rangle-O_{0}\right)^{2} \equiv \frac{1}{2} \varepsilon^{2} .
\end{align}
Here, we make predictions on the eigenvalue of the operator $O$ towards $O_0$. And we use $U(\theta) $ as the variational ansatz. The gradient descent algorithm is,
\begin{align}
{{\theta }_{i}}(t+1)-{{\theta }_{i}}(t)\equiv \delta {{\theta }_{i}}=-\eta \frac{\partial \mathcal{L}}{\partial {{\theta }_{i}}}
\end{align}
when there is no noise. Furthermore, we hereby model the noise by adding Gaussian random variables in each step of the update. Those random fluctuations are independently distributed through $\Delta \theta_i\sim \mathcal{N} (0,\epsilon^2)$.
Now, in the limit where $d$ is large, we have an analytic solution of the convergence time,
given by
\begin{align}
T\approx \frac{\log \left( \frac{\epsilon }{\sqrt{2{{\varepsilon }^{2}}(0)\eta -{{\varepsilon }^{2}}(0){{\eta }^{2}}K+{{\epsilon }^{2}}}} \right)}{\log (1-\eta K)}
\end{align}
where $K=K_L/2$. In the small $\eta$ limit, we have
\begin{align}
T\approx \frac{{{\varepsilon }^{2}}(0)}{\epsilon^2 K} .
\end{align}
This exactly verifies the result in
the dimensional analysis.
\subsection{Critical noise}
Moreover, using the result from Ref.~\cite{Liu:2022rhw}, we could also estimate the critical noise, namely, the critical value of phase transition of the noise size that leads to better performance and avoiding the saddle points.
According to Ref.~\cite{Liu:2022rhw}, we have
\begin{align}
\overline{{{\varepsilon }^{2}}}(t)={{(1-\eta K)}^{2t}}\left( {{\varepsilon }^{2}}(0)-\frac{\epsilon^{2}}{\eta (2-\eta K)} \right)+\frac{\epsilon^2}{\eta (2-\eta K)}
.
\end{align}
Here $\overline{\varepsilon^2}$ is the variance of the residual training error $\varepsilon$ after averaging over the realizations of the noise. Imagine that now the gradient descent process is running from the saddle point to the exact local minimum, we have
\begin{align}
\frac{1}{2}\left( {{\left| {{\varepsilon }_{\text{saddle}}} \right|}^{2}}-{{\left| {{\varepsilon }_{\text{min}}} \right|}^{2}} \right)=\Delta_{\mathcal{L}}\sim \frac{\epsilon ^{2}}{2\eta (2-\eta K)}
\end{align}
where $\Delta_{\mathcal{L}}$ is the distance of the loss function from the saddle point to the local minimum (defined also in the main text), $\Delta_{\mathcal{L}} = L_{\text{saddle}}-L_{\text{mininum}}=\frac{1}{2}\left( {{\left| {{\varepsilon }_{\text{saddle}}} \right|}^{2}}-{{\left| {{\varepsilon }_{\text{min}}} \right|}^{2}} \right)$. So we get
\begin{align}
\epsilon^2\sim\Delta_{\mathcal{L}}\left( 2\eta (2-\eta K) \right)\sim 4\eta \Delta_{\mathcal{L}}
\end{align}
Here in the most right hand side of the formula we use the approximation where $\eta$ is small enough. This formula might be more generic beyond QNTK, since one could regard it as an analog of Einstein's formula of Brownian motion,
\begin{align}
\overline{x^2}(t)= 2 D t
\end{align}
with the averaging moving distance square $\overline{x^2}$, mass diffusivity $D$ and time $t$ in the Brownian motion.
\bibliographystyle{apsrev}
\section{Introduction}\label{sec:introduction}
Quantum computing has for many years been a hugely inspiring
theoretical idea. Already in the 1980ies it was suggested that
quantum devices could possibly have superior computational
capabilities over computers operating based on classical laws
\cite{Feynman-1986,QuantumTuring}.
It is a relatively recent development that devices have been devised
that may indeed have computational capabilities beyond
classical means \cite{GoogleSupremacy,Boixo,SupremacyReview,jurcevic_demonstration_2021}.
These devices are going substantially beyond what was possible not
long ago. And still, they are unavoidably
noisy and imperfect for many years to come. The quantum devices that are available today and presumably will be in the near future are commonly conceived
as hybrid quantum devices running
variational quantum algorithms \cite{Cerezo_2021},
where a quantum circuit is addressed
by a substantially larger surrounding classical circuit. This classical
circuit takes measurements from the quantum device and appropriately
varies variational parameters of the quantum device in an update.
Large classes of \emph{variational quantum eigensolvers} (VQE), the \emph{quantum
approximate optimization algorithm} (QAOA) and models for quantum-assisted machine learning are thought to operate along those lines,
based on suitable \emph{loss functions} to be minimized
\cite{Peruzzo,Kandala,McClean_2016,QAOA,Lukin,bharti_2021_noisy,VariationalReview}.
In fact, with few exceptions,
near-term quantum algorithms in the era of
\emph{noisy intermediate-scale quantum} (NISQ) computing
\cite{preskill_quantum_2018}
are commonly seen as being
variational quantum algorithms.
While this is an exciting development, it puts a lot of burden to
understanding how reasonable and practical classical control can be conceived.
Generally, updates in quantum schemes are seen as deriving from
\emph{gradient
evaluations} \cite{PhysRevA.99.032331,bergholm2018pennylane,pennylane,Gradients},
in one way or the other. This makes a lot of sense, as one
may think that going downhill in a variational quantum algorithms is a
good idea. That said, the concomitant classical optimization
problems are generally not convex optimization problems and
the variational landscapes are marred by
\je{globally suboptimal}
local optima and
saddle points. In fact, it is known that the problems of
optimizing variational parameters of quantum circuits
\je{are}
computationally hard in worst case complexity \cite{Bittel}.
While this is not of too much concern in practical considerations
and resembles an analogous situation in classical machine learning,
it does point to the fact that one should expect
a rugged optimization landscape, featuring
different local minima
as well as saddle points. Such saddle points can indeed
be a burden to feasible and practical classical optimization
of variational quantum algorithms.
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/SaddlePoint.png}
\caption{Stochasticity in variational quantum algorithms can help in avoiding (strict) saddle points.}
\label{fig:sampling}
\end{figure}
In this work, we establish the notion that
in such situations,
\emph{small noise levels} can actually be of substantial help.
More precisely, we show that
some levels of \emph{statistical noise} (\je{specifically
the kind of}
noise that \je{naturally} arises from a finite number of measurements to estimate quantum expectation values) can even be
beneficial. We get inspiration from and build on a powerful
mathematical theory in \emph{classical machine learning}:
there, theorems have been established
that say that ``noise'' can help gradient-descent optimization not to get stuck at saddle points \cite{JordanSaddlePoints,jain2017non}.
Building on such ideas we show that they can be adapted and developed to be applicable to
the variational quantum algorithms setting.
Then we argue that the ``natural'' statistical noise of a quantum experiment can play the role of the artificial noise that is inserted by hand in classical machine learning algorithms to avoid saddle points. We maintain the
\je{precise and} rigorous mindset of Ref.~\cite{JordanSaddlePoints}, but show that the findings
have practical importance and can be made concrete use of
when running variational quantum algorithms on near-term quantum devices.
It is important to stress that for noise we refer to \je{in our theorems is} the type of noise that adds stochasticity to the gradient estimations, such as the use of a finite number of measurements or the zero-average fluctuations that are involved in real experiments. \je{Also, instances of depolarizing noise
are covered.}
Thus, in this case, noise does not mean the generic quantum noise that results from the interaction with the environment characterized by
\je{\emph{completely positive and trace-preserving}} (CPTP)
maps, which can be substantially detrimental to the
\je{performance of the} algorithm \cite{DePalmaVQAs, Stilck_Fran_a_2021},
\je{while at the same time we expect that for sufficiently
small noise levels, much of the intuition developed here
for the setting we can deliver precise theorems
carries over to that case as well}. In addition, it has been shown that noisy CPTP maps in the circuit \je{may}
significantly worsen
the problem of \emph{barren plateaus} \cite{NoisyBP_Wang_2021,McClean_2018}, i.e., that phenomenon that VQAs can suffer from whereby the landscape can become essentially flat.
We perform numerical experiments, and we show examples where optimizations with gradient descent without noise get stuck at saddle points, whereas if we add some noise, we can escape this problem and get to the minimum---convincingly demonstrating the functioning of the approach. We verify the latter not only in a numerical simulation, but also making use of data of
a real IBM quantum machine.
\section{Preliminaries}
\label{sec:pre}
In our work, we will show how a class of saddle points, the so-called strict saddle points, can be avoided in noisy gradient descent. In developing our machinery, we build strongly on the
rigorous results laid out in \je{Ref.}~\cite{JordanSaddlePoints}
\je{and uplift them to the quantum setting at hand}.
First, we introduce some useful definitions and theorems
(\je{see Ref.}~\cite{JordanSaddlePoints} for a more
in-depth discussion).
\subsection{Notation}
Throughout this work, we assume that to minimize a function $\mathcal{L}: \mathbb{R}^p \to \mathbb{R}$
as a function of
$\theta$. We indicate its gradient
at $\theta$ as $\partial \mathcal{L} (\theta)$ and its Hessian matrix at point $\theta$
as $\partial^2 \mathcal{L} (\theta)$.
We denote as $\abs{\cdot}_2$ the $l_2$ norm of a vector. $\abs{\cdot}_{HS}$ and $\abs{\cdot}_\infty$ denote respectively the Hilbert-Schmidt norm and the largest eigenvalue norm of a matrix. We denote as $\lambda_{\min}(\cdot)$ the minimum eigenvalue of a matrix.
\begin{comment}
Moreover, we
say that
\begin{itemize}
\item $f(x)=\mathcal{O}(g(x))$ if $\exists\,\, x_0,\,C\,\in \mathbb{R}$ such that for $x>x_0$, $0\le f(x)<C g(x)$.
\item $f(x)=\Omega(g(x))$ if $\exists\,\, x_0,\,C\,\in \mathbb{R}$ such that for $x>x_0$, $0\le g(x)<C f(x)$.
\item $f(x)=\Theta(g(x))$ if $f(x)=\mathcal{O}(g(x))$ and $f(x)=\Omega(g(x))$.
\end{itemize}
We use a tilde $\tilde{\mathcal{O}}(\cdot)$, $\tilde{\Omega}(\cdot)$, $\tilde{\Theta}(\cdot)$ to denote that we hide poly-logarithmic factors in the problem parameters.
\end{comment}
\subsection{Definitions and theorems}
\begin{definition}[$L$-Lipschitz function]
A function
$g: \mathbb{R}^p \to \mathbb{R}^d$ is $L$-Lipschitz if and only if
\begin{align}
\abs{g (\theta)- g (\phi)}_2\le L\abs{\theta - \phi}_2
\end{align}
for every $\theta$ and $\phi$.
\end{definition}
\begin{definition}[$\beta$-strong smoothness]
A differentiable function $\mathcal{L}: \mathbb{R}^p \to \mathbb{R}$ is called $\beta$-strongly smooth if and only if its gradient is a $\beta$-Lipschitz function, i.e.,
\begin{align}
\abs{\partial \mathcal{L} (\theta)-\partial \mathcal{L} (\phi)}_2\le \beta\abs{\theta - \phi}_2~,
\end{align}
for every $\theta$ and $\phi$.
\end{definition}
\begin{definition}[Stationary point]
If $\mathcal{L}$ is differentiable, ${\theta^*}$ is defined a stationary point
if
\begin{equation}
\abs{\partial \mathcal{L}({\theta^*})}_2=0.
\end{equation}
\end{definition}
\begin{definition}[$\epsilon$-approximate stationary point]
If $\mathcal{L}$ is
differentiable, ${\theta^*}$ is defined an $\epsilon$-approximate stationary point if
\begin{equation}
\abs{\partial \mathcal{L}({\theta^*})}_2\le \epsilon.
\end{equation}
\end{definition}
\begin{definition}[Local minimum, local maxima and saddle point]
If $\mathcal{L}$ is differentiable, a stationary point ${\theta^*}$ is a
\begin{itemize}
\item \emph{local minimum}, if there
exists $\delta>0$ such that $\mathcal{L}({\theta^*}) \leq \mathcal{L}(\theta)$ for any $\theta$ with $\abs{\theta-{\theta^*}}_2 \leq \delta$.
\item \emph{Local maximum},
if there exists $\delta>0$ such that $\mathcal{L}({\theta^*}) \geq \mathcal{L}(\theta)$ for any $\theta$ with $\abs{\theta-{\theta^*}}_2 \leq \delta$.
\item \emph{Saddle point}, otherwise.
\end{itemize}
\end{definition}
\begin{definition}[$\rho$-Lipschitz Hessian]
A twice differentiable function $\mathcal{L}$ has $\rho$-Lipschitz Hessian matrix $\partial^2 \mathcal{L}$ if and only if
\begin{align}
\abs{\partial^2 \mathcal{L} (\theta)-\partial^2 \mathcal{L} (\phi)}_{\operatorname{HS}}\le \rho\abs{\theta - \phi}_2
\end{align}
for every $\theta$ and $\phi$ (where $\abs{\cdot}_{\operatorname{HS}}$ is the Hilbert-Schmidt norm).
\end{definition}
\begin{definition}[Gradient descent]
Given a differentiable function $\mathcal{L}$, the gradient descent algorithm is defined by the update rule
\begin{align}
\theta_{i}^{t+1}=\theta_{i}^{t}-\eta {{\
|
partial }_{i}}\mathcal{L} (\theta^t)\je{,}
\end{align}
where $\eta>0$ is called \emph{learning rate}.
\end{definition}
The convergence time of the gradient descent algorithm is given by the following theorem \cite{JordanSaddlePoints}.
\begin{theorem}[Gradient descent complexity]
Given a $\beta$-strongly smooth function $\mathcal{L}(\cdot)$, for any $\epsilon>0$, if we set the learning rate as $\eta=1 / \beta$, then the number of iterations required by the gradient descent algorithm such that it will visit an $\epsilon$-approximate stationary point is
$$
\mathcal{O}\left({\frac{\beta\left(\mathcal{L}\left(\mathbf{\theta}_{0}\right)-\mathcal{L}^{\star}\right)}{\epsilon^{2}}}\right),
$$
where $\mathbf{\theta}_{0}$ is the initial point and $\mathcal{L}^\star$ is the value of $\mathcal{L}$ computed in the global minimum.
\end{theorem}
It is important to note that this result does not depend on the number of free parameters. Also, the stationary point at which the algorithm will converge is not necessarily a local minimum, but can also be a saddle point.
Note that a generic saddle point satisfies $\lambda_{\min}(\partial^2 \mathcal{L} (\theta_s))\le0$ where $\lambda_{\min}(\cdot)$ is the minimum eigenvalue. Now we define a subclass of saddle points.
\begin{definition}[Strict saddle point]
$\theta_s$ is a
\emph{strict saddle} point for a twice differentiable function $\mathcal{L}$ if and only if $\theta_s$ is a stationary point and if the minimum eigenvalue of the Hessian is $\lambda_{\min}(\partial^2 \mathcal{L} (\theta_s))<0$.
\end{definition}
Adding the \emph{strict} condition, we remove the case in which a saddle point satisfies $\lambda_{\min}(\partial^2 \mathcal{L} (\theta_s))=0$.
Analogously to Ref.~\cite{JordanSaddlePoints}, we focus on finding stationary points that are not strict saddle points (i.e. avoiding strict saddle points). Hence, it is useful to introduce the following definition.
\begin{definition}[Second-order stationary point]
Given a twice differentiable function $\mathcal{L}(\cdot), \mathbf{\theta^*}$ is a second-order stationary point if and only if
$$
\partial \mathcal{L}(\mathbf{\theta^*})=\mathbf{0}, \quad \text { and } \quad \lambda_{\min}(\partial^{2} \mathcal{L}(\mathbf{\theta^*})) \ge 0.
$$
\end{definition}
\begin{definition}[$\epsilon$-second-order stationary point]
For a $\rho$-Hessian Lipschitz function $\mathcal{L}(\cdot), \mathbf{\theta^*}$ is an $\epsilon$-second-order stationary point if
$$
\|\partial \mathcal{L}(\mathbf{\theta^*})\| \leq \epsilon \quad \text { and } \quad \lambda_{\min}(\partial^{2} \mathcal{L}(\mathbf{\theta^*})) \ge -\sqrt{\rho \epsilon} .
$$
\end{definition}
\begin{comment}
Even the function $\mathcal{L}$ is non-convex, we could have local convergence guarantees (under smoothness assumption)
\begin{theorem}
Say that $\eta \in [\frac{1}{2\beta},\frac{1}{\beta}]$. If we perform the gradient descent algorithm,
\begin{align}
\theta_{i}^{t+1}=\theta_{i}^{t}-\eta {{\partial }_{i}}\mathcal{L} (\theta^t)~,
\end{align}
then within $T=\mathcal{O}(1/\epsilon^2)$ steps, the algorithm will get to a stationary point. Namely, for
some $t\ge T$, we must have
\begin{align}
\left\|\partial \mathcal{L}\left(\theta^{t}\right)\right\|_2 \leq \epsilon~.
\end{align}
\end{theorem}
Moreover, we would like the function $\mathcal{L} (\theta)$ to have some convexity condition around the local minima. So we have
\begin{definition}[$\alpha$-strong convexity]
A function $\mathcal{L}: \mathbb{R}^p \to \mathbb{R}$ is called $\alpha$-strong convex if and only if
\begin{align}
\mathcal{L} (\phi)-\mathcal{L} (\theta)-\sum\limits_{i}{{{\partial }_{i}}\mathcal{L} (\theta)\left( {{\phi}_{i}}-{{\theta}_{i}} \right)}\ge \frac{\alpha }{2}{{\left\| \theta - \phi \right\|}^{2}}
\end{align}
for every $\theta$ and $\phi$.
\end{definition}
\end{comment}
\begin{comment}
Moreover, it is useful to define the following:
\begin{definition}[Strict saddle property (SSa)]
A function $\mathcal{L}: \mathbb{R}^p \to \mathbb{R}$ satisfies \emph{strict saddle property} (SSa) if the following statement is true: First, for every local minimum $\theta^*$ of the function, the function is $\alpha$-strongly convex in the region $\mathcal{B}(x,2\xi) \equiv \left\{ x:\left\| x-{{x}^{*}} \right\|<2\xi \right\}$. Second, for every point $x_0\in \mathbb{R}^p$, the function $\mathcal{L} (\theta)$ satisfies at least one of the following properties
\begin{itemize}
\item $x_0$ is not stationary: $\left\| {{\partial }_{i}}\mathcal{L}\left( {{x}_{0}} \right) \right\|\ge \kappa $.
\item $x_0$ is a strict saddle point. Namely, we have ${{\lambda }_{\min }}\left( {{\partial }_{i}}{{\partial }_{j}}f\left( {{x}_{0}} \right) \right)\le -\gamma $.
\item $x_0$ is closed to a local minimum $\theta^*$. Namely, we have $\left\| {{x}_{0}}-{{x}^{*}} \right\|\le \xi $.
\end{itemize}
\end{definition}
SSa is general enough, and we expect that for variational algorithms, we can prove some generic claims.
\subsection{How avoiding saddle points in non-convex optimization}
We introduce the noisy gradient descent algorithm.
\begin{center}
\begin{algorithmic}
\State Algorithm: Noisy Gradient Descent (NGD)
\\
\State Input: function $\mathcal{L}$, maximal learning rate $\eta_{\max}$, and precision $\epsilon$. \Comment{Input}
\\
\State Output: the local optimal $\theta^*$.
\Comment{Output}
\\
\State Initialize an arbitrary $x^1$. \Comment{Initialization}
\State Set $T=1/\eta^2$ with $ \eta =\min \left\{ {{\epsilon }^{2}}/{{\log }^{2}}(1/\epsilon ),{{\eta }_{\max }} \right\}$
\For{$t=1, t\le T, t=t+1$}
\State Sample a perturbation $\zeta^t$ uniformly on $S^{p-1}$, the $(p-1)$-sphere. \Comment{Sampling the noise}
\State $\theta_{i}^{t+1}=\theta_{i}^{t}-\eta {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+{{\zeta }^{t}}$. \Comment{Updating rule}
\EndFor
\State Return $x^T$.
\label{NGDalgorithm}
\end{algorithmic}
\end{center}
Now, we could establish the following theorem.
\begin{theorem}\label{mainthm}
For any $\epsilon,\delta>0$, suppose that NGD is executed on a function that is $\beta$-strongly
smooth, has $\rho$-Lipschitz Hessians, and satisfies the $(\alpha, \gamma, \kappa, \xi)$-SSa property, with a step length
\begin{align}
\eta<\eta_{\max }=\min \left\{\frac{\epsilon^{2}}{\log (1 / \epsilon \delta)}, \frac{\alpha}{\beta^{2}}, \frac{\xi^{2}}{\log (1 / \xi \delta)}, \frac{1}{(\beta+\rho)^{2}}, \frac{\kappa^{2}}{\beta}\right\}
\end{align}
Then, with the probably $1-\delta$, after $T\ge \log p/\eta^2 \times \log (2/\delta)$ iterations, NGD will produce $x^T$ satisfying
\begin{align}
\left\| {{x}^{T}}-{{x}^{*}} \right\|\le \epsilon \,
\end{align}
where $\theta^*$ is a local minimum.
\end{theorem}
Now, we need to use Theorem \ref{mainthm} to claim the usage about noise in variational quantum algorithms. Since in the NGD algorithm, the perturbation is set to be the unit size. We could parametrize the noise by $\sigma$. The rescaling will affect all other quantities, including $x$, $\mathcal{L}$, and $\eta$. One need to figure out how the rescaling affects Theorem \ref{mainthm}, and it should be straightforward. In fact, we could consider the following map,
\begin{align}
& \theta_{i}^{t}\mapsto {{\sigma }^{-1}}\theta_{i}^{t}~, \nonumber\\
& f\mapsto {{\sigma }^{-2}}f~, \nonumber\\
& {{\partial }_{i}}\mathcal{L}\mapsto {{\sigma }^{-1}}{{\partial }_{i}}\mathcal{L}~, \nonumber\\
& {{\partial }_{i}}{{\partial }_{j}}f\mapsto {{\partial }_{i}}{{\partial }_{j}}f~.
\end{align}
This will transform the gradient descent equation towards,
\begin{align}
& {{\sigma }^{-1}}\theta_{i}^{t+1}={{\sigma }^{-1}}\theta_{i}^{t}-{{\sigma }^{-1}}\eta {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+{{\zeta }^{t}} \nonumber\\
& \Rightarrow \theta_{i}^{t+1}=\theta_{i}^{t}-\eta {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+\sigma {{\zeta }^{t}}~.
\end{align}
Note that,
\begin{align}
& \eta <{{\eta }_{\max }}=\min \left\{ \frac{{{\epsilon }^{2}}}{\log (1/\epsilon \delta )},\frac{\alpha }{{{\beta }^{2}}},\frac{{{\xi }^{2}}}{\log (1/\xi \delta )},\frac{1}{{{(\beta +\rho )}^{2}}},\frac{{{\kappa }^{2}}}{\beta } \right\} \nonumber \\
& \mapsto \approx \min \left\{ \frac{{{\epsilon }^{2}}}{{{\sigma }^{2}}\log (\sigma )},\frac{{{\xi }^{2}}}{{{\sigma }^{2}}\log (\sigma )},\frac{{{\kappa }^{2}}}{{{\sigma }^{2}}\beta } \right\}~.
\end{align}
The last $\approx$ is estimating the effect for sufficiently large $\sigma$. So effectively, introducing more and more noise will make the convergence time to be longer.
\end{comment}
Gradient descent makes a non-zero step only when the gradient is non-zero, and thus in the non-convex setting it will be stuck at saddle points.
A simple variant of GD is \emph{perturbed gradient descent} (PGD) \cite{JordanSaddlePoints} which adds randomness to the iterates at each step.
\begin{definition}[Perturbed gradient descent]
Given a differentiable function $\mathcal{L} : \mathbb{R}^p \to \mathbb{R}$, the perturbed gradient descent algorithm is defined by the update rule
\begin{align}
\theta_{i}^{t+1}=\theta_{i}^{t}-\eta \left({{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+{{\zeta }^{t}}\right)~\je{,}
\end{align}
where $\eta>0$ is the learning rate and ${{\zeta }^{t}}$ is a normally distributed random variable with mean $\mu=0$ and variance $\sigma^{2}={r^{2}}/{p}$ with $r \in \mathbb{R}$.
\label{PerturbGF}
\end{definition}
In Ref.~\cite{JordanSaddlePoints},
\je{the authors} show that if we pick $r=\tilde{\Theta}(\epsilon)$, PGD will find an $\epsilon$-second-order stationary point in a number of iterations that has only a poly-logarithmic dependence on the number of free parameters, i.e.\je{,}
it has the same complexity of (standard) gradient descent up to poly-logarithmic dependence.
\begin{theorem}[\cite{JordanSaddlePoints}]
Let the function $\mathcal{L}: \mathbb{R}^d \to \mathbb{R}$ be $\beta$ strongly smooth and such that it has a $\rho$ Lipschitz-Hessian. Then, for any $\epsilon, \delta>0$, the $P G D$ algorithm starting by the point $\theta_{0}$, with parameters $\eta=\tilde{\Theta}(1 / \beta)$ and $r=\tilde{\Theta}(\epsilon)$, will visit an $\epsilon-$second-order stationary point at least once in the following number of iterations, with probability at least $1-\delta$
$$
\tilde{\mathcal{O}}\left(\frac{\beta\left(\mathcal{L}\left(\theta_{0}\right)-\mathcal{L}^{\star}\right)}{\epsilon^{2}}\right)
$$
where $\tilde{\mathcal{O}}$ and $\tilde{\Theta}$ hide poly-logarithmic factors in $p, \beta, \rho, 1 / \epsilon, 1 / \delta$ and $\Delta_{\mathcal{L}}:=\mathcal{L}\left(\theta_{0}\right)-\mathcal{L}^{\star}$. $\mathbf{\theta}_{0}$ is the initial point and $\mathcal{L}^\star$ is the value of $\mathcal{L}$ computed in the global minimum.
\label{GaussianPGD}
\end{theorem}
This theorem \je{has been} proven in \je{Ref.~}\cite{JordanSaddlePoints} for Gaussian distributions, but the authors have pointed out that this is not strictly necessary and that it can be generalized to other types of probability distributions in which appropriate concentration inequalities and bounds on the variance by below can be applied.
In Ref.~\cite{ExpTimeSaddle},
it has
been shown
that although the standard GD (without perturbations) almost always escapes the addle points asymptotically \cite{Lee2017}, there are (non-pathological) cases in which the optimization requires exponential time to escape.
This highlights the importance of using gradient descent with perturbations.
\section{Statistical noise in variational quantum algorithms}
Our analysis focuses on variational quantum algorithms in which the loss function to be minimized has the following form
$\mathcal{L}\left(\theta\right):=\bra{0} U^{\dagger}(\theta)OU(\theta )\ket{0}$ where $O$ is a Hermitian operator and $U(\theta)$ is a parameterized unitary of the form $U({\theta}):=\prod_{\ell=1}^{p} W_{\ell} \exp \left(i \theta_{\ell} X_{\ell}\right)$.
The above Theorem \ref{GaussianPGD} assumes that the loss function to minimize is $\beta$ strongly smooth and has a $\rho$ Lipschitz-Hessian. We show that $\mathcal{L}\left(\theta\right)$ satisfies such \je{a} condition with
\begin{align}
&\beta\le2^2p\abs{O}_\infty\max_{j=1,\dots,p}\abs{X_{j}}_\infty^2,\\
&\rho \le 2^3p^\frac{3}{2} \abs{O}_\infty\max_{j=1,\dots,p}\abs{X_{j}}_\infty^3.
\end{align}
We note that for typical VQA problems\je{,} the previous upper bounds will grow polynomially at most \je{in the system size}, so as not to affect the efficiency of the gradient descent algorithm.
Moreover\je{,} Theorem \ref{GaussianPGD} assumes that at each step of the gradient descent a normally distributed random variable is added to the gradient, namely $\theta_{i}^{t+1}=\theta_{i}^{t}-\eta\left( {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+{{\zeta }^{t}}\right)$.
In VQAs the partial derivatives are \je{commonly} estimated using a finite number of measurements, such as by the so-called \emph{parameter shift rule}~\cite{GradientsQcomp}.
Here, the update rule for the gradient descent is
\begin{equation}
\theta_{i}^{t+1}=\theta_{i}^{t}-\eta \hat{g}_i\left( { {\theta}^{t}} \right),
\end{equation}
where $\hat{g}_i\left( { {\theta}^{t}} \right)$ is an estimator of the partial derivative ${{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)$ obtained by a finite number of measurements $N_{\rm shots}$ from the quantum device. Moreover, we
define
\begin{equation}
\hat{\zeta}^t_{N_{\rm shots}} := {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)-\hat{g}_i\left( { {\theta}^{t}} \right).
\end{equation}
Note that $\hat{\zeta}^t_{N_{\rm shots}}$ is a random variable with zero expectation value.
Therefore\je{,} we have
\begin{equation}
\theta_{i}^{t+1}=\theta_{i}^{t}-\eta\left( {{\partial }_{i}}\mathcal{L}\left( { {\theta}^{t}} \right)+ \hat{\zeta}^t_{N_{\rm shots}}\right) \je{.}
\end{equation}
The ``noise'' $\hat{\zeta}^t_{N_{\rm shots}}$ will play the role of the noise that is added by hand in the perturbed-gradient descent algorithm \ref{PerturbGF}.
However, we cannot control exactly the distribution of such random variable, nor the variance.
\section{Numerical and quantum experiments}
In this section\je{,} we discuss the results of numerical and quantum experiments we \je{have} performed to show that stochasticity can help escape saddle points.
Our results suggest that statistical noise allows a probability of not getting stuck in a saddle point and escaping a lower value of the loss function.
\je{These numerical experiments also complement the
rigorous results that are proven to be valid
under very precisely defined conditions, while
the intuition developed here is expected to
be more broadly applicable, so that the
rigorous results can be seen as proxies for a
more general mindset.}
We \je{have} also observed this phenomena in a real IBM quantum device. \je{We have done so to convincingly stress the significance of our results
in practice.}
\subsection{Numerical simulations}
Let us first consider the Hamiltonian ${H}=\sum_{i=1}^{N=4}Z_i$. The loss function we consider is defined as the expectation value of such Hamiltonian over the parametrized quantum circuit $\texttt{qml.StronglyEntanglingLayers}$
in \emph{Pennylane}~\cite{bergholm2018pennylane}, where two layers of the circuit are used
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/NoiseFig2New.png}
\caption{Comparison of the loss evolution with or without noise. The noise
levels are manually-added Gaussian distributions, and we keep the same initial conditions. Bottom: Noiseless case and the noisy case with the standard deviation of the noise $r=0.1$. Top: Five different values of the noise norms.}
\label{fig:fake}
\end{figure}
In all our experiments we first select initial parameters (found through extensive search) so that the gradient descent in the noiseless case converges to a strict saddle point. We then use that initial condition to run noisy versions of the same experiment.
As a proof of principle, we first show the results of an exact simulation (i.e., the expectation values are not estimated using a finite number of shots, but are calculated exactly) in which noise is added manually at each step of the gradient descent. The probability distribution associated to the noise is chosen to be a Gaussian distribution with mean $\mu=0$ and variance $\sigma^2=r^2$.
Figure \ref{fig:fake} shows the difference between the noiseless, and noisy calculations with the same initial conditions of the gradient descent, when the noise is from random Gaussian perturbations added manually.
Figure \ref{fig:performanceGaussian} shows the performance of the experiment, defined as \je{${1}/({\mathcal{L}-\mathcal{L}_{opt}})$}
as a function of the noise parameter. Here, we can find a critical value of noise, leading to saddle-point avoidance.
Figure \ref{fig:sampling} specifically addresses quantum noise levels, with simulated results about purely statistical noise levels (shot noise) and device noise (simulated
\je{by making use of the actual quantum hardware} \texttt{IBM Qiskit}).
\je{It} should be noted that including device noise generally also means dealing with \emph{completely positive
trace preserving} maps that can lead to a different loss function, with new local minima, new saddle points and a flatter landscape \cite{NoisyBP_Wang_2021}. However, even in this case we observe an improvement in performance using the same initial parameters leading to the saddle point in the noiseless case. \je{This is perfectly in line with the intuition developed here, as long as the effective noise emerging can be seen as a small perturbation of the reference circuit featuring a given loss landscape that is then in effect perturbed by stochastic noise.}
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/NoiseFig3New.png}
\caption{We quantify the performance against the size of the noise $r$ (classical Gaussian noise) by ${1}/({\mathcal{L}-\mathcal{L}_{opt}})$.} %
\label{fig:performanceGaussian}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/NoiseFig4NewAlternative.png}
\caption{Saddle-point avoidance from quantum noise. We prepare 30 instances starting from the same initial condition. When noise levels are small (left up, with purely measurement noise, left down, including device noise, and shot number is 1000), most trajectories cannot jump out of the saddle points. When noise levels are larger (right up, with purely measurement noise, right down, including device noise, and shot number is 70), we have a probability to jump towards the global minimum.}
\label{fig:sampling}
\end{figure}
Aside from the quantum machine learning example, we also provide another instance in \emph{Variational Quantum Eigensolvers} (VQEs). Here, we use the Hamiltonian associated to the Hydrogen molecule $\text{H}_2$, which is a 4-qubit Hamiltonian obtained by the fermionic one performing a Jordan-Wigner transformation. We specifically use the same circuit from \texttt{h2.xyz}, the \emph{Hydrogen VQE} example in \emph{Pennylane} \cite{pennylane}.
Also here, given initial parameters that led to saddle points in the noiseless case, we find that starting by the same parameters and adding noise can lead to saddle-point avoidance. Results are shown in Fig.~\ref{fig:fake_vqe} where we compare the noiseless and noisy simulation.
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/NoiseFig5New}
\caption{Comparison of the loss evolution with or without noise with Hydrogen VQE. The noise is manually drawn from Gaussian distributions with the standard deviation $0.2$, and we keep the same initial conditions. We compare the noiseless case, noisy case and the exact solution.}
\label{fig:fake_vqe}
\end{figure}
\subsection{Real hardware experiments}
\je{To further provide evidence of the functioning of our
approach suggested and the rigorous insights established,
we put the findings into contact of the results of a}
real experiment example in the \texttt{IBM Qiskit} environment. We use the Hamiltonian ${H}=\sum_{i=1}^{N=4}Z_i$ that we used in our first numerical simulation with
\je{four} qubits and
\je{two}
layers. We run the experiment using as initial condition the one that lead to a saddle point in the noiseless case. We use \texttt{IBMQ\_Jakarta} device with 10000 shots. The result in \je{Fig.}~\ref{fig:converge_experiment} shows that it is possible to obtain a lower value of the cost function than that of the simulation without noise that
\je{has been stuck} in a saddle point.
\begin{figure}[h]
\centering
\includegraphics[width=0.48\textwidth]{figures/NoiseFig6New}
\caption{A real quantum experiment. We use \texttt{IBMQ\_Jakarta} device with 10000 shots. }
\label{fig:converge_experiment}
\end{figure}
\section{Conclusion and outlook}
In this work, we have proposed small \je{stochastic noise} levels as an instrument to facilitate variational quantum algorithms. \je{This noise can be substantial, but should not be too large:} The way a noise level can strike the balance in overcoming getting stuck in saddle points and being detrimental is in some ways reminiscent of
the phenomenon of \emph{stochastic resonance} in statistical
physics \cite{StochasticResonance}, where fine
tuned noise levels can facilitate resonance behaviour
and avoid getting trapped. On a higher level, this
work invites to think more deeply about
\emph{Metropolis sampling}
inspired classical algorithms, to avoid getting
stuck in rugged energy landscapes. It is the hope that the present work puts the role of stochasticity in variational quantum computing into a new perspective, and contributes to a line of thought exploring the use of suitable noise
and sampling for enhancing quantum computing schemes.
\begin{acknowledgments}
JL is
supported in part by International Business Machines (IBM) Quantum through the Chicago Quantum Exchange, and the Pritzker School of Molecular Engineering at the University of Chicago through AFOSR MURI (FA9550-21-1-0209). FW, AAM and JE thank the BMBF (Hybrid, MuniQC-Atoms, DAQC),
the BMWK (EniQmA, PlanQK),
the MATH+ Cluster of Excellence,
the Einstein Foundation (Einstein Unit on
Quantum Devices),
the QuantERA (HQCC), the Munich Quantum
Valley (K8), and the DFG (CRC 183)
for support. LJ acknowledges support from the ARO (W911NF-18-1-0020, W911NF-18-1-0212), ARO MURI (W911NF-16-1-0349), AFOSR MURI (FA9550-19-1-0399, FA9550-21-1-0209), DoE Q-NEXT, NSF (EFMA-1640959, OMA-1936118, EEC-1941583), NTT Research, and the Packard Foundation (2013-39273).
\end{acknowledgments}
\bibliographystyle{alpha}
\newcommand{\etalchar}[1]{$^{#1}$}
|
\section{Introduction}
Bessel operators appear in the setting of harmonic analysis related with Hankel transformations. In the one dimensional case, Bessel operators appear when we consider the Laplacian operator in polar coordinates. In this work we study the fractional Bessel operator and Liouville theorems of the $n$-dimensional versions in $\R^{n}_{+}=\ceroinf$ given by
\begin{equation}\label{nOBZ}
S_{\mu}=
\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}^{2}}+
\frac{4\mu_{i}^{2}-1}{4x_{i}^{2}}
\end{equation}
and
\begin{equation}\label{nOBH}
\Delta_{\mu}=
\sum_{i=1}^{n}
\frac{\partial^{2}}{\partial x_{i}^{2}}+
(2\mu_{i}+1)(x_{i}^{-1}\frac{\partial}{\partial x_{i}})
\end{equation}
where $\mu\in\R^n$, $\mu=(\mu_{1},\dots,\mu_{n})$ and $\mu_{i}>-\frac{1}{2}$.
In \cite{Mo18}, were studied the fractional powers of the one dimensional case of \eqref{nOBZ} and \eqref{nOBH} in the sense of the classical theory of fractional powers developed by Balakrishnan in \cite{Ba60} and using similarity of both operators. Let $X$ and $Y$ Banach spaces. Two linear operators $A$ and $B$, $A:D(A)\subset X\to X$ and $B:D(B)\subset Y\to Y$ are {\it similar} if there exists an isomorphism $T:X\to Y$ with inverse $T^{-1}:Y\to X$ such that $D(B)=\{x\in Y: T^{-1}x\in D(A)\}$ given by
\begin{equation}\label{OpSimilares}
B=TAT^{-1}.
\end{equation}
Similar operators have the same spectral properties and also that of being non-negative if one of them has this property. Thus, their powers are similar operators and verifies the same similarity relation, so $$ B^{\alpha}=TA^{\alpha}T^{-1}.$$
In this work, we generalize the results obtained in \cite{Mo18} to the $n$-dimensional case obtaining the fractional powers of Bessel operator \eqref{nOBZ} and \eqref{nOBH} in weighted Lebesgue spaces and in distributional spaces. As in \cite{Mo18} we first study the non-negativity of Bessel operator \eqref{nOBZ} in suitable weighted Lebesgue spaces and by similarity we obtain the non-negativity of \eqref{nOBH} in the corresponding Lebesgue space.
Analogously to the one-dimensional case, we construct a locally convex space $\mathcal{B}$ in which $-S_{\mu}$ is continuous and non-negative. Next, we can consider the dual space $\mathcal{B'}$ with the strong topology and thus obtaining non-negativity of $-S_{\mu}$ in this distributional space. $\mathcal{B'}$ is contained in the distributional Zemanian space and contain the weighted Lebesgue spaces in which non-negativity was studied. Consequently, if we denote with $(S_{\mu})_{\mathcal{B'}}$ the Bessel operator with domain $\mathcal{B'}$, we can consider the powers $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}$ with $\Re(\alpha)>0$ and it is verified the following relation inherited from the selfadjuncture of $S_{\mu}$
$$((-S_{\mu})^{\alpha}u,\phi)=(u,(-S_{\mu})^{\alpha}\phi),$$
for $\phi\in\mathcal{B}$ and $u\in\mathcal{B'}$.
In \cite{GMQ18}, a Liouville-type theorem was studied for a certain general class of Bessel-type operators. This class of operators contain as a particular case the Bessel operator \eqref{nOBZ}, and the Liouville theorem applied to this operator states that if $u$ is a Zemanian distribution that verifies that $S_{\mu}u=0$ then $u$ is a polynomial. This result is analogous to that existing for the Laplacian operator and temperate distributions that states that any harmonic tempered distribution is a polynomial. In \cite{BKN02}, \cite{Li06}, \cite{ZCCY14} and \cite{CDAL14} different version of Liouville theorem for the fractional Laplacian are studied. In this paper we prove the following Liouville theorems for the distributional fractional Bessel operators:
\begin{theorem}\label{Teorema5.1}
Let $u\in\mathcal{B'}$ and $\alpha\in{\mathbb C}$ with $\Re\alpha>0$. If $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}u=0$ then there exists a polynomial $p$ such that $u=x^{\mu+\frac{1}{2}}p(\|x\|^{2})$.
\end{theorem}
For the study of the powers of Bessel operator given by \eqref{nOBH} we introduce a locally convex space $\mathcal{F}$. This space verifies that its dual space $\mathcal{F'}$ with the strong topology is a suitable distributional space for the study of fractional powers $(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}$ and from similarity we conclude the following result
\begin{theorem} \label{teorema 8.2}
Let $u\in\mathcal{F'}$ and $\alpha\in{\mathbb C}$ with $\Re\alpha>0$. If $(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}u=0$ then there exists a polynomial $p$ such that $u=x^{2\mu+1}p(\|x\|^{2})$.
\end{theorem}
This paper is organized as follow. In section 2 we summarize basic results related with harmonic analysis in the Hankel setting. Section 3 contain a brief review of non-negative operators in Banach and locally convex spaces and properties of fractional powers of similar operators. In sections 4, 5 and 6 we study the non-negativity of Bessel operator \eqref{nOBZ} and \eqref{nOBH} . Finally, sections 6 and 7 contains Liouville's theorems for the two fractional Bessel operators.
\section{Preliminaries}
In this section we introduce the Lebesgue and distributional spaces necessary four our purposes.
We now present some notational conventions that will allow us to simplify the presentation of our results. Let $\R^n$ be the $n$-dimensional euclidean space, $\R^{n}_{+}=\ceroinf$ the $n$-tuples of real positive numbers. We denote by $x=(x_1,\ldots,x_n)$ and $y=(y_1,\ldots,y_n)$ to the elements of $\ceroinf$ or $\mathbb{R}^{n}$ and let $\mathbb{N}$ be the set $\{1,2,3,\ldots\}$ and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$. For $x\in\R^n$, the norm is given by $\|x\|=(x_{1}^{2}+\ldots + x_{n}^{2})^{\frac{1}{2}}$. The notations $x<y$ and $x\leq y$ mean, respectively $x_i<y_i$ and $x_i\leq y_i$, for $i=1,\ldots,n$. Moreover if $x\in\mathbb{R}^{n}$ and $a\in\mathbb{R}$, $x=a$ means $x_1=x_2=\ldots=x_n=a$. We denote $e_j$, for $j=1,\ldots,n$, the elements of the canonical basis $\mathbb{R}^{n}$.
An element $k=(k_1,\ldots,k_n)\in\mathbb{N}_{0}^{n}=\mathbb{N}_{0}\times\mathbb{N}_{0}\times\ldots\times\mathbb{N}_{0}$ is called multi-index. For $k,m$ multi-index we set $|k|=k_1+\ldots+k_n$.
Also we will note
\[
k!=k_1!\ldots k_n!
\qquad \text{and}
\qquad \binom{k}{m}=\binom{k_1}{m_1}\ldots
\binom{k_n}{m_n}
\qquad \text{for} \: k,m\in\mathbb{N}_{0}^{n}.
\]
If $x\in\R^n$ and $\beta\in\R^n$, we define
\begin{equation}\label{x^b}
x^{\beta}=x_{1}^{\beta_1}\ldots x_{n}^{\beta_n}.
\end{equation}
In particular if $a\in\mathbb{R}$, $a^{\beta}$ means
\begin{equation}\label{constante^b}
a^{\beta}=a^{\beta_1}\ldots a^{\beta_n},
\end{equation}
and if $\beta$ is a multi-index, $\beta\in\mathbb{N}_{0}^{n}$
\[a^{\beta}=a^{|\beta|}.\]
For $\alpha\in\mathbb{R}$ let $\bm{\alpha}=(\alpha,\ldots,\alpha)$,
then for $a\in\mathbb{R}\:\text{and}\:x\in\R^n$
\begin{equation}\label{x^{constante}}
a^{\bm{\alpha}}=(a^{n})^{\alpha}
\quad \text{y}\quad
x^{\bm{\alpha}}=x_{1}^{\alpha}\ldots x_{n}^{\alpha}=x^{\alpha}.
\end{equation}
\noindent If $\beta$ is a multi-index, $\beta=(\beta_{1},\ldots,\beta_{n})$ y $\alpha\in\mathbb{R}$ let
\begin{equation}\label{multi-real}
\beta-\alpha=
(\beta_{1}-\alpha,\ldots,\beta_{n}-\alpha)=
\beta-\bm{\alpha}
\end{equation}
\bigskip
\noindent If $D_j=\frac{\partial}{\partial x_j}$, $j=1,\ldots,n$ then the partial derivatives respect to $x$ is denoted by
\[D^{k}=D_{1}^{k_1}\ldots D_{n}^{k_n}.\]
where $k$ is a multi-index. We define the operators \[T_{j}=x_{j}^{-1}D_{j}\]
for $j=1,\ldots,n$. For a multi-index $k$ we shall write
\[T^{k}=T_{n}^{k_n}\circ T_{n-1}^{k_{n-1}}\circ\ldots\circ T_{1}^{k_1}.\]
\begin{remark}
Let $k$ be a multi-index and $\theta,\varphi$ differentiable functions up to order $|k|$, the following equality is valid
\begin{equation}\label{Leibniz}
T^{k}\{\theta\cdot\varphi\}=\sum_{j=0}^{k}\binom{k}{j}T^{k-j}\theta. T^{j}\varphi,
\end{equation}
\noindent where $\:"\cdot\:"$ denote the usual product of functions.
\end{remark}
\bigskip
Hankel transformation appears in mathematical literature in various forms, two classical versions correspond to the versions studied by A. H. Zemanian \cite{Ze87,Ze66}
\begin{equation}\label{1HZ}
(h_{\alpha}f)(t)=
\int_{0}^{\infty}
f(x)\sqrt{xt}J_{\alpha}(xt)\:dx,
\qquad t\in(0,\infty)
\end{equation}
and I. I. Hirschman \cite{Hi60}
\begin{equation}\label{1HH}
(H_\alpha f)(t)=
\int_{0}^{\infty} f(x)(xt)^{-\alpha}J_{\alpha}(xt)\:x^{2\alpha+1}\:dx,
\qquad t\in(0,\infty)
\end{equation}
where $\alpha>-\frac{1}{2}$ and $J_{\alpha}$ is the well known Bessel function of first kind and order $\alpha$.
\medskip
S. Molina y S. Trione studied in \cite{MT07,MT08} a $n$-dimensional generalization of \eqref{1HZ}, given by $h}%{\mathbbl{h}_{\mu}$ and defined by
\begin{equation}\label{nHZ
(h}%{\mathbbl{h}_{\mu}\phi)(y)=
\int_{\R^{n}_{+}}\phi(x_{1},\ldots,x_{n})\prod_{i=1}^{n}
\{\sqrt{x_{i}y_{i}}J_{\mu_{i}}(x_{i}y_{i})\}
\;dx_{1}\ldots dx_{n}.
\end{equation}
Analogously it is possible to define a $n$-dimensional generalization for
\eqref{1HH}, given by $H}%{\mathbbl{H}_{\mu}$ and defined by
\begin{equation}\label{nHH}
(H}%{\mathbbl{H}_{\mu}\phi)(y)=
\int_{\R^{n}_{+}}\phi(x_1,\ldots,x_n)\left\{\prod\limits_{i=1}^{n}
(x_iy_i)^{-\mu_i}J_{\mu_i}(x_iy_i)\,x_i^{2\mu_i+1}\right\}
\: dx_1\ldots dx_n.
\end{equation}
\noindent In both, \eqref{nHZ} and \eqref{nHH}, $\mu=(\mu_{1},\ldots,\mu_{n})$, $\mu_{i}>-\frac{1}{2}$ and $J_{\mu_i}$ represents the Bessel function of first kind and order $\mu_i$ for $i=1,\ldots n$.
\bigskip
Next we define certain weighted $L^{p}$-spaces for $1\leq p\leq\infty$. Let
\begin{equation}\label{peso-s}
s(x)=\frac{x^{2\mu+1}}{C_\mu}
\end{equation}
\begin{equation}\label{peso-r}
r(x)=x^{-\mu-\frac{1}{2}}
\end{equation}
where $\mu=(\mu_{1},\ldots,\mu_{n})$, $x\in\R^{n}_{+}$, $C_\mu=2^\mu\,\Gamma(\mu_1+1)\ldots \Gamma(\mu_n+1)$ and $dx$ is the usual $n$-dimensional Lebesgue and the powers $x^{2\mu+1}$ and $x^{-\mu-1/2}$ are given by \eqref{x^b} and $2^{\mu}$ is given by \eqref{constante^b}. Let $L^{p}(\R^{n}_{+},sr^p)$, $1\leq p<\infty$ the space of measurable functions $f$ defined over $\R^{n}_{+}$ with norm
\[\|f\|_{L^{p}(\R^{n}_{+},sr^p)}=
\left(\int_{\R^{n}_{+}} |f(x)|^{p}\,s(x)r^p(x)\:dx\right)^{1/p} \quad 1\leq p<\infty.\]
Moreover, $L^{\infty}(\R^{n}_{+},r)$, is the space of measurable functions over $\R^{n}_{+}$ such that
\[\|f\|_{L^{\infty}(\R^{n}_{+},r)}= \mathop{\mathrm{ess\, sup \;}}_{x\in\R^{n}_{+}}\:|r(x)f(x)|<\infty.\]
In particular, if $p=2$, $L^{2}(\R^{n}_{+},sr^2)=L^{2}(\R^{n}_{+})$.
\bigskip
For simplicity sometimes we write $L^{p}(sr^p)$ and $L^{\infty}(r)$ instead of $L^{p}(\R^{n}_{+},sr^p)$ and $L^{\infty}(\R^{n}_{+},r)$.
\medskip
By $\mathcal{D}(\R^{n}_{+})$ we denote the space of functions in $C^{\infty}(\R^{n}_{+})$ with compact support in $\R^{n}_{+}$ with the usual topology, and by $\mathcal{D'}(\R^{n}_{+})$ the space of classical distributions in $\R^{n}_{+}$.
\bigskip
We consider the Zemanian space $\mathcal{H}_{\mu}$ of the functions $\phi\in C^{\infty}(\R^{n}_{+})$ such that
\[
\sup_{x\in\R^{n}_{+}}
|(1+\|x\|^{2})^{m}T^{k}\{x^{-\mu-1/2}\phi(x)\}|<\infty:
m\in\mathbb{N}_{0},\;k\in\mathbb{N}_{0}^{n}
\]
\noindent endowed with the topology generated by the family of seminorms $\{\nu_{m,k}^{\mu}\}$, given by
\begin{equation}\label{Topologia H_mu}
\nu_{m,k}^{\mu}(\phi)=
\sup_{x\in\R^{n}_{+}}
|(1+\|x\|^{2})^{m}T^{k}\{x^{-\mu-1/2}\phi(x)\}|
\end{equation}
\noindent where $-\mu-1/2=(-\mu_1-1/2,\ldots,-\mu_n-1/2)$ and the operators $T^{k}$ are given by
\[
T^{k}=T_{n}^{k_n}\circ T_{n-1}^{k_{n-1}}\circ\ldots\circ T_{1}^{k_1},
\]
\noindent where $T_{i}=x_{i}^{-1}\frac{\partial}{\partial x_{i}}$ and $k=(k_1,\ldots,k_n)$. $\mathcal{H}_{\mu}$ is a Fréchet space (see \cite{MT07}). The dual space of $\mathcal{H}_\mu$ is denoted by $\mathcal{H'}_\mu$.
\begin{remark}
Sometimes we will consider the family of seminorms
\begin{equation}\label{Topologia H_mu-2}
\gamma_{m,k}^{\mu}(\phi)=
\sup_{x\in\R^{n}_{+}}
|x^{m}T^{k}\{x^{-\mu-1/2}\phi(x)\}|
\end{equation}
with $m,k\in\mathbb{N}_{0}^{n}$, which are equivalent to $\nu_{m,k}^{\mu}$.
\end{remark}
\begin{lemma}\label{lema3.1}
The following inclusions hold
\begin{equation}\label{lema3.1-eq1}
\mathcal{H}_{\mu}\subset
L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)\subset
L^{p}(\R^{n}_{+},sr^{p}),
\quad 1\leq p<\infty
\end{equation}
where $s$ and $r$ are given by \eqref{peso-s} and \eqref{peso-r}
respectively.
\end{lemma}
\begin{proof}
Let $\phi\in\mathcal{H}_{\mu}$,
\begin{equation}\label{lema3.1-eq2}
\|\phi\|_{L^{\infty}(\R^{n}_{+},r)}=
\sup\limits_{x\in\R^{n}_{+}}|x^{-\mu-1/2}\phi(x)|=
\gamma_{0,0}^{\mu}(\phi),
\end{equation}
then $\phi\in L^{\infty}(\R^{n}_{+},r)$.
\bigskip
\noindent To show that $\mathcal{H}_{\mu}\subset
L^{1}(\R^{n}_{+},sr)$, let $\phi\in\mathcal{H}_{\mu}$, $m\in\mathbb{N}$ such that $m>2\mu_{i}+2$, for $i=1,\ldots,n$, then
\begin{align*}
\int_{\R^{n}_{+}}|\phi(x)|s(x)r(x)\;dx
&=
\int_{(0,1]^{n}}
|x^{-\mu-1/2}\phi(x)|\;\frac{x^{2\mu+1}}{C_{\mu}} dx +
\int_{\R^{n}_{+}-(0,1]^{n}}
x^{m} |x^{-\mu-1/2}\phi(x)|\;\frac{x^{2\mu+1-m}}{C_{\mu}} dx\\
& \leq
\gamma_{0,0}^{\mu}(\phi)\:C_{\mu}^{-1}
\int_{(0,1]^{n}}
x^{2\mu+1} dx +
\gamma_{m,0}^{\mu}(\phi)\:C_{\mu}^{-1}
\int_{\R^{n}_{+}-(0,1]^{n}}
x^{2\mu+1-m} dx
<\infty.
\end{align*}
\noindent Thus
\begin{equation}\label{lema3.1-eq3}
\|\phi\|_{L^{1}(\R^{n}_{+},sr)}\leq
C\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\quad \phi\in\mathcal{H}_{\mu}.
\end{equation}
\bigskip
\noindent Now let us see that $L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)\subset L^{p}(\R^{n}_{+},sr^{p})$. Let $\phi\in L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)$
\begin{align*}
\int_{\R^{n}_{+}} |\phi(x)|^{p}& s(x)r^{p}(x)\;dx
=
\int_{\R^{n}_{+}} |\phi(x)|^{p-1}r(x)^{p-1}\;|\phi(x)|s(x)r(x)\;dx\\
&=
\int_{\R^{n}_{+}} |r(x)\phi(x)|^{p-1}\;|\phi(x)|s(x)r(x)\;dx\\
&\leq
\|\phi\|_{L^{\infty}(\R^{n}_{+},r)}^{p-1}
\|\phi\|_{L^{1}(\R^{n}_{+},sr)},
\end{align*}
\noindent from where
\begin{equation}\label{lema3.1-eq4}
\|\phi\|_{L^{p}(\R^{n}_{+},sr^{p})}\leq
\|\phi\|_{L^{\infty}(\R^{n}_{+},r)}^{\frac{p-1}{p}}
\|\phi\|_{L^{1}(\R^{n}_{+},sr)}^{\frac{1}{p}}
\end{equation}
From \eqref{lema3.1-eq2} and \eqref{lema3.1-eq3} we can consider that there exist constants $C_{1}$ y $C_{2}$ such that
\begin{equation}\label{lema3.1-eq5}
\|\phi\|_{L^{\infty}(\R^{n}_{+},r)}\leq
C_{1}\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\quad \phi\in\mathcal{H}_{\mu}.
\end{equation}
\begin{equation}\label{lema3.1-eq6}
\|\phi\|_{L^{1}(\R^{n}_{+},sr)}\leq
C_{2}\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\quad \phi\in\mathcal{H}_{\mu}.
\end{equation}
Then from \eqref{lema3.1-eq4}, \eqref{lema3.1-eq5} y \eqref{lema3.1-eq6} we can consider a constant $C_{3}$ such that
\begin{equation}\label{lema3.1-eq7}
\|\phi\|_{L^{p}(\R^{n}_{+},sr^{p})}\leq
C_{3}\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\quad \phi\in\mathcal{H}_{\mu}.
\end{equation}
\end{proof}
\bigskip
\begin{remark}
If $\phi\in L^{1}(\R^{n}_{+},sr)$, then Hankel transform $h}%{\mathbbl{h}_{\mu}\phi$ is well defined because the kernel $(x_iy_i)^{-\mu_i}J_{\mu_i}(x_iy_i)$ is bounded for $\mu_{i}>-\frac{1}{2}$, $i=1,\ldots,n$ (see \cite[(1), pp.49]{Wa}),
\begin{align*}
&\int_{\R^{n}_{+}}|\phi(x)|
\prod_{i=1}^{n}\{(x_iy_i)^{\mu_{i}+1/2}
|(x_iy_i)^{-\mu_{i}} J_{\mu_i}(x_iy_i)|\}\;dx\\
&\leq
y^{\mu+1/2} M^{n}\int_{\R^{n}_{+}} |\phi(x)|\;x^{\mu+1/2}\;dx
= C y^{\mu+1/2} \|\phi\|_{L^{1}(\R^{n}_{+},sr)}<\infty
\end{align*}
\bigskip
By Lemma \ref{lema3.1}, $h}%{\mathbbl{h}_{\mu}\phi$ is well defined for all $\phi\in\mathcal{H}_{\mu}$ and is an automorphism of $\mathcal{H}_{\mu}$ (see \cite{Ze87} for the $1$-dimensional case and \cite{MT07} for the $n$-dimensional case.)
\end{remark}
\bigskip
The space of continuous linear functions $T:\mathcal{H}_{\mu}\to {\mathbb C}$ is denoted by $\mathcal{H'}_{\mu}$. We call a function $f\in L^{1}_{loc}(\R^{n}_{+})$ a regular element of $\mathcal{H'}_{\mu}$ if the application $T_{f}\in \mathcal{H'}_{\mu}$ where $T_{f}(\phi)=\int_{\R^{n}_{+}}f\phi$, with $\phi\in\mathcal{H}_{\mu}$.
\begin{lemma}\label{lema3.2}
Let $1\leq p <\infty$. A function in $L^{p}(\R^{n}_{+},sr^{p})$ or in $L^{\infty}(\R^{n}_{+},r)$ is a regular element of $\mathcal{H'}_{\mu}$. In particular, the functions in $\mathcal{H}_{\mu}$ can be considered as regular elements of $\mathcal{H'}_{\mu}$.
\end{lemma}
\begin{proof}
Let $f\in L^{\infty}(\R^{n}_{+},r)$ and $\phi\in\mathcal{H}_{\mu}$. Since $\mathcal{H}_{\mu}\subset L^{1}(\R^{n}_{+},sr)$, then $\phi\in L^{1}(\R^{n}_{+},r^{-1})$ and $(T_{f},\phi)=\int_{\R^{n}_{+}}f\phi$ is well defined. So, by \eqref{lema3.1-eq3}
\begin{align*}
|(T_{f},\phi)|
&\leq
\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\|\phi\|_{L^{1}(\R^{n}_{+},r^{-1})}=
C_{\mu}
\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\|\phi\|_{L^{1}(\R^{n}_{+},sr)}\\
&\leq C\;C_{\mu}\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\end{align*}
\noindent consequently, $f$ is a regular element of $\mathcal{H'}_{\mu}$.
\medskip
Now, let $f\in L^{p}(\R^{n}_{+},sr^{p})$ with $1\leq p<\infty$ and $\phi\in\mathcal{H}_{\mu}$, then
\begin{equation}\label{lema3.2-eq1}
\begin{split}
|(T_{f},\phi)|
&\leq
\int_{\R^{n}_{+}}|f(x)\phi(x)|\;dx=
\int_{\R^{n}_{+}} |r(x)f(x)|\;|s^{-1}(x)r^{-1}(x)\phi(x)|\;s(x)\;dx\\
&=
\int_{\R^{n}_{+}} |r(x)f(x)|\;C_{\mu}|r(x)\phi(x)|\;s(x)\;dx.
\end{split}
\end{equation}
Since $r|f|\in L^{p}(\R^{n}_{+},s)$ and $r|\phi|\in L^{q}(\R^{n}_{+},s)$, being $q$ such that $\frac{1}{p} + \frac{1}{q} = 1$, then due to H\"{o}lder's inequality and \eqref{lema3.1-eq7} we obtain that
\[
|(T_{f},\phi)|
\leq
C_{\mu} \|f\|_{L^{p}(\R^{n}_{+},sr^{p})}
\|\phi\|_{L^{q}(\R^{n}_{+},sr^{q})}
\leq
C\;C_{\mu} \|f\|_{L^{p}(\R^{n}_{+},sr^{p})}
\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\}
\]
with $m>2\mu_{i}+2$, for $i=1,\ldots,n$. Therefore $f$ is a regular element of $\mathcal{H'}_{\mu}$.
\end{proof}
\begin{remark}\label{L^2 subset H'mu}
In particular if $p=2$, $L^{p}(\R^{n}_{+},sr^{p})=L^{2}(\R^{n}_{+})$ and from the previous Lemma we have that the functions in $L^{2}(\R^{n}_{+})$ can be considered as regular elements of $\mathcal{H'}_{\mu}$.
\end{remark}
\bigskip
Given $f,g$ defined on $\R^{n}_{+}$, the Hankel convolution associated to the transformation $h}%{\mathbbl{h}_{\mu}$ is defined formally by
\begin{equation}\label{nconv_Z}
(f\sharp g)(x)=
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
\textbf{D}_{\mu}(x,y,z)f(y)g(z)\;dy\;dz
\end{equation}
where for every $x,y,z\in\R^{n}_{+}$,
\begin{equation}\label{nDZ}
\textbf{D}_{\mu}(x,y,z)=
\prod_{i=1}^{n} D_{\mu_i}(x_i,y_i,z_i)
\end{equation}
\noindent where $D_{\alpha}$ is the Delsarte kernel defined in \cite{Delsarte}, given by
\begin{equation}\label{DZ}
D_{\alpha}(u,v,w)=
\frac{2^{\alpha-1}\;(uvw)^{-\alpha+1/2}}
{\Gamma(\alpha+1/2)\sqrt{\pi}}
A(u,v,w)^{2\alpha-1}
\end{equation}
\noindent and $A(u,v,w)$ is the area of the triangle with sides $u,v,w\in\R_+$ y $\alpha\in\mathbb{R}$, $\alpha>-\frac{1}{2}$.
\medskip
Note that $|u-v|<w<u+v$ is the condition for such triangle to exist, and in this case
\begin{equation}\label{Área-lados}
A(u,v,w)=
\begin{cases}
\frac{1}{4}\sqrt{[(u+v)^{2}-w^{2}][w^{2}-(u-v)^{2}]}
&|u-v|<w<u+v\\
\hfil 0
& 0<w<|u-v|\quad\text{or}\quad w>u+v,
\end{cases}
\end{equation}
\begin{remark}
If $u,v$ and $w$ are the sides of a triangle and $\theta$ is the angle opposite the side $w$, then
\[A(u,v,w)= \frac{uv\sin\theta}{2}\]
\end{remark}
\begin{proposition}\label{prop-nDZ}
\begin{itemize}
\item[]
\item[(i)] $\textbf{D}_{\mu}(x,y,z)\geq 0,\quad x,y,z\in\R^{n}_{+}$.
\item[(ii)]$\int_{\R^{n}_{+}}\textbf{D}_{\mu}(x,y,z)
\prod\limits_{i=1}^{n}
\{\sqrt{z_it_i}J_{\mu_i}(z_it_i)\}\;dz=
t^{-\mu-1/2}
\prod\limits_{i=1}^{n}
\{\sqrt{x_it_i}J_{\mu_i}(x_it_i)\}
\prod\limits_{i=1}^{n}
\{\sqrt{y_it_i}J_{\mu_i}(y_it_i)\}$
\item[(iii)] $\int_{\R^{n}_{+}}z^{\mu+1/2}\textbf{D}_{\mu}(x,y,z)\;dz=
C_{\mu}^{-1} x^{\mu+1/2}y^{\mu+1/2}$
\end{itemize}
\end{proposition}
\begin{proof}
For the proof of this result, we refer the reader to the Appendix, page \pageref{Proof-prop-nDZ}.
\end{proof}
\begin{lemma}\label{lema3.3}
Let $f\in L^{1}(\R^{n}_{+},sr)$.
\begin{itemize}
\item[(i)] If $g\in L^{\infty}(\R^{n}_{+},r)$, then the convolution $f\sharp g(x)$ exists for every $x\in\R^{n}_{+}$, $f\sharp g(x)\in L^{\infty}(\R^{n}_{+},r)$ and
\begin{equation}\label{lema3.3-eq1}
\|f\sharp g\|_{L^{\infty}(\R^{n}_{+},r)}
\leq
\|f\|_{L^{1}(\R^{n}_{+},sr)}\|g\|_{L^{\infty}(\R^{n}_{+},r)}.
\end{equation}
\item[(ii)] If $g\in L^{p}(\R^{n}_{+},sr^{p})$, $1\leq p<\infty$, then the convolution $f\sharp g(x)$ exists for almost every $x\in\R^{n}_{+}$, $f\sharp g(x)\in L^{p}(\R^{n}_{+},sr^{p})$ and
\begin{equation}\label{lema3.3-eq2}
\|f\sharp g\|_{L^{p}(\R^{n}_{+},sr^{p})}
\leq
\|f\|_{L^{1}(\R^{n}_{+},sr)}\|g\|_{L^{p}(\R^{n}_{+},sr^{p})}.
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
For the proof of Lemma \ref{lema3.3}, refer to the Appendix, page \pageref{Proof-lema3.3}.
\end{proof}
\vspace{.2in}
The proof of the following results uses standard arguments and it will be omitted.
\begin{lemma}\label{lema3.4}
Let $f,g\in L^{1}(\R^{n}_{+},sr)$, then
\begin{equation}\label{lema3.4-eq1}
h}%{\mathbbl{h}_{\mu}(f\sharp g) = rh}%{\mathbbl{h}_{\mu}(f) h}%{\mathbbl{h}_{\mu}(g).
\end{equation}
\end{lemma}
\begin{lemma}
Let $f\in L^{1}(sr)$, then the Hankel transform $h}%{\mathbbl{h}_{\mu}f\in L^{\infty}(r)$ and \[\|h}%{\mathbbl{h}_{\mu}f\|_{L^{\infty}(r)}\leq \|f\|_{L^{1}(sr)}.\]
\end{lemma}
\begin{remark}\label{remark7.4 [Mo18]}
Given $f\in L^{1}(\R^{n}_{+})$ we have that $h}%{\mathbbl{h}_{\mu}f$ is continuous, is in $L^{\infty}(\R^{n}_{+})$ and
\[
\|h}%{\mathbbl{h}_{\mu}f\|_{\infty}\leq C\|f\|_{1}.
\]
\end{remark}
\begin{proposition}\label{prop7.5 [Mo18]}
$h}%{\mathbbl{h}_{\mu}(L^{1}(\R^{n}_{+}))\subset C_{0}(\R^{n}_{+})$
\end{proposition}
\begin{proof}
First, we observe that
\begin{equation}\label{prop7.5 [Mo18]-eq1}
L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)
\subset
L^{1}(\R^{n}_{+}).
\end{equation}
Let $Q$ the cube $Q=[0,1]^{n}$, then
\begin{align*}
\int_{\R^{n}_{+}}|f(x)|\:dx
&=
\int_{\R^{n}_{+}}|f(x)|\:r(x)\:r^{-1}(x)\:dx\\
&=
\int_{Q\cap\R^{n}_{+}}|f(x)|\:r(x)\:r^{-1}(x)\:dx
+
\int_{Q^{c}\cap\R^{n}_{+}}|f(x)|\:r(x)\:r^{-1}(x)\:dx\\
&\leq
\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\int_{Q\cap\R^{n}_{+}} r^{-1}(x)\:dx
+
\int_{Q^{c}\cap\R^{n}_{+}}|f(x)|\:r^{-1}(x)\:dx\\
&\leq
C\|f\|_{L^{\infty}(\R^{n}_{+},r)}
+
C_{\mu}\|f\|_{L^{1}(\R^{n}_{+},sr)},
\end{align*}
because $r(x)<1$ for $\|x\|>1$, $\mu_{i}>-\frac{1}{2}$, $i=1,\ldots,n$ and $r(x)s(x)=C_{\mu}^{-1}r^{-1}(x)$.
\bigskip
By \eqref{lema3.1-eq1} and \eqref{prop7.5 [Mo18]-eq1} we deduce that $\mathcal{H}_{\mu}\subset L^{1}(\R^{n}_{+})$. Since $\mathcal{D}(\R^{n}_{+})\subset\mathcal{H}_{\mu}$ then $\mathcal{H}_{\mu}$ is dense in $L^{1}(\R^{n}_{+})$. Given $f\in L^{1}(\R^{n}_{+})$ and $\{\phi_{m}\}\in \mathcal{H}_{\mu}$ such that $\phi_{m}\to f$ in $L^{1}(\R^{n}_{+})$, then by Remark \ref{remark7.4 [Mo18]} $h}%{\mathbbl{h}_{\mu}(\phi_{m})\toh}%{\mathbbl{h}_{\mu}(f)$ uniformly. Since $h}%{\mathbbl{h}_{\mu}(\phi_{m})\in C_{0}(\R^{n}_{+})$ then $h}%{\mathbbl{h}_{\mu}(f)\in C_{0}(\R^{n}_{+})$.
\smallskip
\end{proof}
\clearpage
\begin{lemma}\label{lema3.5}
Let $\{\phi_{m}\}\subset L^{1}(\R^{n}_{+},sr)$ such that
\begin{enumerate}
\item[(1)] $\phi_{m}\geq 0$ in $\R^{n}_{+}$,
\item[(2)] $\int_{\R^{n}_{+}} \phi_{m}(x)\;s(x)r(x)\;dx=1$
for all $m\in\mathbb{N}$,
\item[(3)] For all $\eta>0$,
$\lim\limits_{m\to\infty}
\int_{\|x\|>\eta} \phi_{m}(x)\;r(x)s(x)\;dx=0$.
\end{enumerate}
\noindent Let $f\in L^{\infty}(\R^{n}_{+},r)$ and continuous in $x_{0}\in\R^{n}_{+}$, then $\lim\limits_{m\to\infty} f\sharp \phi_{m}(x_{0}) = f(x_{0})$. Moreover, if $rf$ is uniformly continuous in $\R^{n}_{+}$ then \[\lim\limits_{m\to\infty} \|f\sharp\phi_{m}(x)-f(x)\|_{L^{\infty}(\R^{n}_{+},r)} = 0.\]
\end{lemma}
\begin{proof}
First let us observe that
\begin{align}\label{lema3.5-eq1}
\int_{\R^{n}_{+}}&
\int_{\R^{n}_{+}}
x_{0}^{-\mu-1/2} y^{\mu+1/2}
\textbf{D}_{\mu}(x_{0},y,z)\:\phi_{m}(z)\:dy
dz\\
&=
\int_{\R^{n}_{+}}
x_{0}^{-\mu-1/2} \phi_{m}(z)
\left\{\int_{\R^{n}_{+}}
y^{\mu+1/2}
\textbf{D}_{\mu}(x_{0},y,z)\:dy
\right\}dz\nonumber\\
&=
\int_{\R^{n}_{+}}
x_{0}^{-\mu-1/2} \phi_{m}(z)
C_{\mu}^{-1} x_{0}^{\mu+1/2} z^{\mu+1/2}\:dz \nonumber\\
&=
\int_{\R^{n}_{+}}
\phi_{m}(z)\:s(z)r(z)\:dz = 1.\nonumber
\end{align}
\bigskip
\noindent Let $\varepsilon>0$, then since $f$ is a continuous function in $x_{0}$, there exists $\delta>0$ such that if $\|y-x_{0}\|<\delta$, then $|y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})|<\frac{\varepsilon}{2x_{0}^{\mu+1/2}}$.
\begin{align*}
& f\sharp \phi_{m}(x_{0})-f(x_{0})
=
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
[y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})]\:dy\:dz\\
& =
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
[y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})]\:dy\:dz\\
& +
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
[y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})]\:dy\:dz.
\end{align*}
\noindent Calling
\begin{align*}
I_{1} &=
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
[y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})]\:dy\:dz\\
I_{2} &=
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
[y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})]\:dy\:dz,\\
\end{align*}
\noindent from where
\[
|f\sharp \phi_{m}(x_{0})-f(x_{0})| \leq |I_{1}|+|I_{2}|.
\]
\noindent Since $f\in L^{\infty}(r)$,
\[|y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})|\leq 2\|f\|_{L^{\infty}(r)}.\]
\noindent Moreover, since $\lim\limits_{m\to\infty}\int_{\|z\|>\frac{\delta}{\sqrt{n}}} \phi_{m}(z)\:s(z)r(z)\:dz =0$, there exists $N_{0}\in\mathbb{N}$ such that
\[
\int_{\|z\|>\frac{\delta}{\sqrt{n}}} \phi_{m}(z)\:s(z)r(z)\:dz <
\frac{\varepsilon}{4\|f\|_{L^{\infty}(r)} x_{0}^{\mu+1/2}},
\quad \forall\:m>N_{0}.
\]
\begin{align*}
|I_{1}| &\leq
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
|y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})|\:dy\:dz\\
& =
2\|f\|_{L^{\infty}(r)}
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)
\left\{\int_{\R^{n}_{+}}
y^{\mu+1/2}\:\textbf{D}_{\mu}(x_{0},y,z)
\:dy\right\}dz\\
& =
2\|f\|_{L^{\infty}(r)}
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)C_{\mu}^{-1} x_{0}^{\mu+1/2} z^{\mu+1/2}
dz\\
& =
2\|f\|_{L^{\infty}(r)} x_{0}^{\mu+1/2}
\int_{\|z\|>\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)\:s(z)r(z)\:dz\\
& <
2\|f\|_{L^{\infty}(r)} x_{0}^{\mu+1/2}
\frac{\varepsilon}{4\|f\|_{L^{\infty}(r)} x_{0}^{\mu+1/2}}=\frac{\varepsilon}{2}
\end{align*}
\noindent On the other hand, if $\|z\|<\frac{\delta}{\sqrt{n}}$, then $z_{i}\in(0,\delta/\sqrt{n})$ for all $i=1,\cdots,n$. Moreover, if we consider $\textbf{D}_{\mu}(x_0,y,z)$ as a function depending on $y$ then
\begin{align*}
\mathop{\mathrm{supp}}\textbf{D}_{\mu}(x_0,y,z)
&\subset
(|x_{1}^{0}-z_{1}|, x_{1}^{0}+z_{1})\times\cdots\times (|x_{n}^{0}-z_{n}|, x_{n}^{0}+z_{n}) \\
&\subset
(x_{1}^{0}-\delta/\sqrt{n}, x_{1}^{0}+\delta/\sqrt{n})\times\cdots\times (x_{n}^{0}-\delta/\sqrt{n}, x_{n}^{0}+\delta/\sqrt{n}).
\end{align*}
So, if $y\in\mathop{\mathrm{supp}}\textbf{D}_{\mu}(x_0,y,z) $ then $|y_{i}-x_{i}^{0}|< \delta/\sqrt{n}$, for all $i=1,\cdots,n$. Thus
\[\|y-x_{0}\|_{\infty} = \max\limits_{1\leq i\leq n}\{|y_{i}-x_{i}^{0}|\}<\delta/\sqrt{n}.\]
\noindent Since the euclidean norm is equivalent to the uniform norm and $\|\cdot\|_{\infty}\leq\|\cdot\|\leq\sqrt{n} \|\cdot\|_{\infty}$, then we obtain that $\|z\|<\frac{\delta}{\sqrt{n}}$ implies $\|y-x_{0}\|<\delta$, then
\begin{align*}
|I_{2}| &\leq
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
|y^{-\mu-1/2}f(y)-x_{0}^{-\mu-1/2}f(x_{0})|\:dy\:dz\\
&\leq
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}\int_{\R^{n}_{+}}
y^{\mu+1/2}\phi_{m}(z)\:\textbf{D}_{\mu}(x_{0},y,z)
\frac{\varepsilon}{2x_{0}^{\mu+1/2}}\:dy\:dz\\
& =
\frac{\varepsilon}{2x_{0}^{\mu+1/2}}
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)
\left\{\int_{\R^{n}_{+}}
y^{\mu+1/2}\:\textbf{D}_{\mu}(x_{0},y,z)
\:dy\right\}dz\\
& =
\frac{\varepsilon}{2x_{0}^{\mu+1/2}}
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)\:C_{\mu}^{-1}x_{0}^{\mu+1/2}z^{\mu+1/2}
dz\\
& =
\frac{\varepsilon}{2}
\int_{\|z\|<\frac{\delta}{\sqrt{n}}}
\phi_{m}(z)\:s(z)r(z)\:dz
\leq
\frac{\varepsilon}{2}
\int_{\R^{n}_{+}}
\phi_{m}(z)\:s(z)r(z)\:dz = \frac{\varepsilon}{2}.
\end{align*}
\noindent From where we have proved that given $\varepsilon>0$, there exists $N_{0}\in\mathbb{N}$ such that $|f\sharp \phi_{m}(x_{0})-f(x_{0})|<\varepsilon$, for all $n>N_{0}$. The uniform convergence is obtained analogously to the uniform continuity of $rf$.
\end{proof}
\bigskip
We are going to consider Bessel operators in $\R^{n}_{+}$ given by \eqref{nOBZ} and \eqref{nOBH} which are related through
\begin{equation}\label{similaridad Op de Bessel}
S_{\mu}=x^{\mu+1/2}\Delta_{\mu} x^{-\mu-1/2},
\end{equation}
see remark \ref{proof similaridad Op de Bessel} for a proof.
\bigskip
Bessel operator \eqref{nOBZ} and Hankel transform \eqref{nHZ} were studied in the distributional setting over the Zemanian spaces $\mathcal{H}_{\mu}$ and $\mathcal{H'}_{\mu}$ in \cite{Ze66} ($1$-dimensional case), \cite{Mo03} and \cite{MT07} ($n$-dimensional case).
\bigskip
$S_{\mu}$ is a continuous operator in $\mathcal{H}_{\mu}$ and selfadjoint, so the generalized Bessel operator $S_{\mu}$ can be extended to $\mathcal{H'}_{\mu}$ by transposition
\begin{equation*}
(S_{\mu} f, \phi)=(f, S_{\mu}\phi),
\quad f\in\mathcal{H'}_{\mu}
\quad \phi\in\mathcal{H}_{\mu}.\\
\end{equation*}
\medskip
\noindent Analogously, generalized Hankel transform $h_{\mu}f$ can be extended to $\mathcal{H'}_\mu$ by
\begin{equation*}
(h_{\mu}f, \phi)=(f, h_{\mu}\phi),
\quad f\in\mathcal{H'}_{\mu},
\quad \phi\in\mathcal{H}_\mu
\end{equation*}
\noindent for $\mu=(\mu_1,\ldots,\mu_n)$, $\mu_{i}>-\frac{1}{2}$, $i=1,\ldots,n$. Then $h_{\mu}$ is an automorfism over $\mathcal{H}_{\mu}$ and $\mathcal{H'}_{\mu}$.
\clearpage
There exist different proofs for the inversion theorem of the Hankel transform for the $1$-dimensional case. In this work we present a proof for the inversion theorem for the $n$-dimensional case, in the same way of the classic versions of the results known for the inversion of the Fourier transform in Lebesgue spaces.
\begin{theorem}\label{TeoInv-nHZ}
Let $f\in L^{1}(\R^{n}_{+},x^{\mu+1/2})$ and $h}%{\mathbbl{h}_{\mu}f\in L^{1}(\R^{n}_{+},x^{\mu+1/2})$ where $x^{\mu+1/2}$ is given by \eqref{x^b}. Then $f(x)$ may be redefined on a set of measure zero so that it is continuous on $\R^{n}_{+}$ and
\begin{equation}\label{FormulaInv-nHZ}
f(x) = h}%{\mathbbl{h}_{\mu}(h}%{\mathbbl{h}_{\mu}f)(x),
\end{equation}
\noindent for almost every $x\in\R^{n}_{+}$.
\end{theorem}
\begin{proof}
For the proof of this result we refer the reader to the Appendix. Details can be found in page \pageref{Proof-TeoInv-nHZ}.
\end{proof}
\medskip
\begin{remark}\label{TeoInv-nHZ en Hmu}
From Theorem \ref{TeoInv-nHZ} we deduce immediately the validity of equality \eqref{FormulaInv-nHZ} in $\mathcal{H}_{\mu}$ and $\mathcal{H'}_{\mu}$.
\end{remark}
\vspace{.2in}
\noindent For the proof of the next results we refer the reader to \cite{MT07}.
\begin{lemma}\label{lema3.8}
Let $\phi\in\mathcal{H}_{\mu}$, then
\begin{itemize}
\item[(i)] $h}%{\mathbbl{h}_{\mu}S_{\mu}\phi=-\|y\|^{2}h}%{\mathbbl{h}_{\mu}\phi$.
\item[(ii)]$\nBZh}%{\mathbbl{h}_{\mu}\phi=h}%{\mathbbl{h}_{\mu}(-\|x\|^{2}\phi)$.
\end{itemize}
\end{lemma}
\begin{lemma}\label{lema3.10}
If $u\in\mathcal{H'}_{\mu}$, then
\begin{itemize}
\item[(i)] $h}%{\mathbbl{h}_{\mu}S_{\mu} u = -\|x\|^{2}h}%{\mathbbl{h}_{\mu} u$.
\item[(ii)]$\nBZh}%{\mathbbl{h}_{\mu} u = h}%{\mathbbl{h}_{\mu}(-\|y\|^{2} u)$.
\end{itemize}
\end{lemma}
\begin{remark}\label{lema3.9}
According to Lemma 3.2 in \cite{Mo03} the functions $(\lambda+\|x\|^{2})$ for $\lambda\geq 0$ and $(\lambda+\|x\|^{2})^{-1}$ for $\lambda> 0$ belong to the space of multipliers of $\mathcal{H}_{\mu}$ and $\mathcal{H'}_{\mu}$.
\end{remark}
So, the next result holds.
\begin{lemma}\label{lema3.11}
The following equalities are valid in $\mathcal{H}_{\mu}$ and in $\mathcal{H'}_{\mu}$ for $m\in\mathbb{N}$ and $\lambda\in{\mathbb C}$.
\begin{itemize}
\item[(i)] $(-S_{\mu}+\lambda)^{m}h}%{\mathbbl{h}_{\mu} = h}%{\mathbbl{h}_{\mu}(\lambda+\|y\|^{2})^{m}$.
\item[Si] $\lambda>0$,
\item[(ii)] $h}%{\mathbbl{h}_{\mu} (-S_{\mu}+\lambda)^{-m} = (\lambda+\|y\|^{2})^{-m}h}%{\mathbbl{h}_{\mu}$
\item[(iii)] $h}%{\mathbbl{h}_{\mu}(-S_{\mu}(-S_{\mu}+\lambda)^{-1})^{m}=
\|y\|^{2m}(\lambda+\|y\|^{2})^{-m}h}%{\mathbbl{h}_{\mu}$
\end{itemize}
\end{lemma}
\begin{proof}
We refer the reader to page \pageref{proof-lema3.11} in the Appendix for details.
\end{proof}
\section{Non-negativity and fractional powers of similar operators}
In this section we include a brief review of non-negative operators in Banach spaces and locally convex spaces.
\medskip
Let $X$ be a Banach space (real or complex). Let $A$ be a closed linear operator $A:D(A)\subset X\to X$ and $\rho(A)$ the resolvent set of $A$. We say that $A$ is non-negative if $(-\infty,0)\subset \rho(A)$ and
\[
\sup_{\lambda>0}
\{\|\lambda(\lambda+A)^{-1}\|\}<\infty.
\]
\vspace{.2in}
Now, let $X$ is a locally convex space with a Hausdorff topology generated by a directed family of seminorms $\{\|\:\:\|_{\alpha}\}_{\alpha\in\Lambda}$. A family of linear operators $\{A_{\lambda}\}_{\lambda\in\Gamma}$, $A_{\lambda}:D(A_{\lambda})\subset X\to X$, is equicontinuous if for each $\alpha\in\Lambda$ there are $\beta=\beta(\alpha)\in\Lambda$ and a constant $C=C_{\alpha}\geq 0$ such that for all $\lambda\in\Gamma$
\[
\|A_{\lambda}\phi\|_{\alpha}
\leq C\|\phi\|_{\beta},
\quad \phi\in X.
\]
Under the above conditions, we say that a closed linear operator $A:D(A)\subset X\to X$ is non-negative if $(-\infty,0)\subset\rho(A)$ and the family of operators
\[
\{\lambda(\lambda+A)^{-1}\}_{\lambda>0}
\]
is equicontinuous.
\vspace{.2in}
Now, we will briefly describe the theory of fractional powers of operators. According to \cite[Proposition 3.1.3]{MS01}, we can define the Balakrishnan operator $J^{\alpha}$ in the following way.
\bigskip
Let $A$ be a non-negative operator in a Banach space or in a locally convex and sequentially complete space. Let
$\alpha\in{\mathbb C}_{+}$ and $n>\Re\alpha$, $n\in\mathbb{N}$. If $\phi\in D(A^{n})$ and $m\geq n$ is a positive integer, then
\begin{equation}\label{prop4.1-eq1}
J^{\alpha}\phi=
\frac{\Gamma(m)}{\Gamma(\alpha)\Gamma(m-\alpha)}
\int_{0}^{\infty}
\lambda^{\alpha-1}\big[A(\lambda+A)^{-1}\big]^{m}\phi\:d\lambda.
\end{equation}
\bigskip
If $A$ is bounded, $J_{A}^{\alpha}$ can be considered as the fractional power of $A$. In other cases we can consider the following representation for the fractional power stated in \cite[Theorem 5.2.1]{MS01}
\begin{theorem}
Let $A$ be a non-negative operator, $\alpha\in{\mathbb C}_{+}$, $\lambda\in\rho(-A)$ and $n\in\mathbb{N}$. Then
\begin{equation}\label{Theorem 5.2.1-MS01}
A^{\alpha}=(A+\lambda)^{n}J_{A}^{\alpha}(A+\lambda)^{-n}.
\end{equation}
(If $n>\Re\alpha$, the operator $\overline{J_{A}^{\alpha}}$ can be replaced by $J_{A}^{\alpha}$ in the preceding formula.)
\end{theorem}
\bigskip
Similar operators has been described in the introduction. Let $A$ and $B$ similar operators and $T$ the isomorphism that verifies \eqref{OpSimilares} then
\[(zId-B)^{-1}=T(zId-A)^{-1}T^{-1},\]
\noindent for $z$ a complex number, from where we deduce immediately that $A$ is non-negative operator if and only if so is $B$.
\bigskip
When two operators are similar, the fractional powers also meet this property. Thus we have the following result which holds in Banach spaces and in locally convex and sequentially complete spaces.
\begin{proposition}\label{prop-similares Banach}
Let $A$ and $B$ be similar non-negative operators. If $\alpha\in{\mathbb C}_{+}$ then
\begin{equation}\label{prop4.1-1}
J_{B}^{\alpha}=T J_{A}^{\alpha}T^{-1},
\end{equation}
and
\begin{equation}\label{prop4.1-2}
B^{\alpha}=TA^{\alpha}T^{-1},
\end{equation}
where $T$ is the isometric isomorphism that verifies $B=TAT^{-1}$.
\end{proposition}
\section{Fractional powers of \texorpdfstring{$S_{\mu}$}{Smu} in Lebesgue spaces}
Let $s$ and $r$ as in Section 2 and let $1\leq p<\infty$. We will denote by $S_{\mu,p}$ the part of $S_{\mu}$ in $L^{p}(\R^{n}_{+},sr^{p})$, that is to say, the operator $S_{\mu}$ with domain
\[
D(S_{\mu,p})=
\{f\in L^{p}(\R^{n}_{+},sr^{p}):
S_{\mu} f\in L^{p}(\R^{n}_{+},sr^{p})\}
\]
and given by $S_{\mu,p}f=S_{\mu} f$.
\medskip
Analogously, with $S_{\mu,\infty}$ we will denote the part of $S_{\mu}$ in $L^{\infty}(\R^{n}_{+},r)$, $\Delta_{\mu,p}$ and $\Delta_{\mu,\infty}$ the part of $\Delta_{\mu}$ in $L^{p}(\R^{n}_{+},s)$ and $L^{\infty}(\R^{n}_{+})$ respectively.
\bigskip
\noindent Let $L_{r}$ the isometric isomorphism
\[
L_{r}: L^{p}(\R^{n}_{+},sr^{p})\to L^{p}(\R^{n}_{+},s),
\quad\text{with}\quad
1\leq p<\infty
\]
(or $L_{r}: L^{\infty}(\R^{n}_{+},r)\to L^{\infty}(\R^{n}_{+})$) given by \[L_{r}(f)=rf.\]
Let
then
\[
S_{\mu,p}=
L_{r}^{-1}\circ
\Delta_{\mu,p}\circ L_{r}.
\]
\vspace{.2in}
Consequently it is enough to study the operator $S_{\mu}$ in the spaces $L^{p}(\R^{n}_{+},sr^{p})$ (or $L^{\infty}(\R^{n}_{+},r)$). In order to study the non-negativity of operators $-S_{\mu,p}$ and $-S_{\mu,\infty}$ we consider the following function given by
\begin{equation}
\mathcal{N}_{\nu}(w)=
\int_{0}^{\infty}
e^{-t-\frac{w^{2}}{4t}}\:\frac{dt}{t^{\nu+1}}
\end{equation}
which is defined for all $\nu\in\mathbb{R}$ and $w\in\R_+$.
\bigskip
\noindent Let $t\in\R_+$, if $\mu=(\mu_{1},\ldots,\mu_{n})$, then $t^{\mu+1}$ means
\[
t^{\mu+1}=
t^{\mu_{1}+1}\ldots t^{\mu_{n}+1}=
t^{\mu_{1}+\ldots+\mu_{n}+n},
\]
from where
\begin{equation}
\mathcal{N}_{\mu_{1}+\ldots+\mu_{n}+n-1}(\|x\|)
=\int_{0}^{\infty}
e^{-t-\frac{\|x\|^{2}}{4t}}\:\frac{dt}{t^{\mu_{1}+\ldots+\mu_{n}+n-1+1}}
=\int_{0}^{\infty}
e^{-t-\frac{\|x\|^{2}}{4t}}\:\frac{dt}{t^{\mu+1}}.
\end{equation}
\vspace{.2in}
\noindent Given $\lambda>0$, let us consider the function
\smallskip
\begin{equation}\label{nucleo}
N_{\lambda}(x)=
2^{-\mu-1}x^{\mu+1/2}\lambda^{\mu}\lambda^{n-1}
\mathcal{N}_{\mu_{1}+\ldots+\mu_{n}+n-1}(\|\sqrt{\lambda}x\|),
\quad x\in\R^{n}_{+}.
\end{equation}
\medskip
\begin{lemma}\label{Lema4.3}
Given $\mu=(\mu_{1},\ldots,\mu_{n})$, $\mu_{i}>-\frac{1}{2}$ and $\lambda>0$ then
\begin{enumerate}
\item[(a)] $N_{\lambda}\in L^{1}(\R^{n}_{+},sr)$ and
\[\|N_{\lambda}\|_{L^{1}(\R^{n}_{+},sr)}=\frac{1}{\lambda}\]
\item[(b)]
\[h}%{\mathbbl{h}_{\mu}N_{\lambda}(y)=\frac{y^{\mu+1/2}}{\lambda+\|y\|^{2}}\]
\end{enumerate}
\end{lemma}
\begin{proof}[Proof (a)]\phantom{\qedhere}
\begin{align*}
\|N_{\lambda}\|_{L^{1}(\R^{n}_{+},sr)}
&=
\int_{\R^{n}_{+}}|N_{\lambda}(x)|\:\frac{x^{\mu+1/2}}{C_{\mu}} dx\\
&=
\int_{\R^{n}_{+}} 2^{-\mu-1} x^{\mu+1/2}\lambda^{\mu}\lambda^{n-1}
\mathcal{N}_{\mu_{1}+\ldots+\mu_{n}+n-1}(\|\sqrt{\lambda}x\|)
\:\frac{x^{\mu+1/2}}{C_{\mu}} dx\\
&=
2^{-\mu-1}\lambda^{\mu}\lambda^{n-1}\frac{1}{C_{\mu}}
\int_{\R^{n}_{+}}\left\{\int_{0}^{\infty}
e^{-t-\frac{\lambda\|x\|^{2}}{4t}}\frac{dt}{t^{\mu+1}}
\right\} x^{2\mu+1} dx\\
&=
2^{-\mu-1}\lambda^{\mu}\lambda^{n-1}\frac{1}{C_{\mu}}
\int_{0}^{\infty}\left\{\int_{\R^{n}_{+}}
e^{-\frac{\lambda\|x\|^{2}}{4t}}x^{2\mu+1} dx
\right\} e^{-t}\frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}\lambda^{\mu}\lambda^{n-1}\frac{1}{C_{\mu}}
\int_{0}^{\infty} \prod_{i=1}^{n}\left\{\int_{0}^{\infty}
e^{-\frac{\lambda x_{i}^{2}}{4t}} x_{i}^{2\mu_{i}+1} dx_{i}
\right\} e^{-t}\frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}\lambda^{\mu}\lambda^{n-1}\frac{1}{C_{\mu}}
\int_{0}^{\infty} \prod_{i=1}^{n}\left\{
2^{\mu_{i}}\Gamma(\mu_{i}+1) \left(\frac{2t}{\lambda}\right)^{\mu_{i}+1}
\right\} e^{-t}\frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}\lambda^{\mu}\lambda^{n-1}\frac{1}{C_{\mu}}
2^{\mu+1}\lambda^{-\mu}\lambda^{-n}C_{\mu} \int_{0}^{\infty}
t^{\mu+1} e^{-t}\frac{dt}{t^{\mu+1}}=\frac{1}{\lambda}
\end{align*}
where we have used the formula \eqref{ApA-eq2}.
\end{proof}
\begin{proof}[Proof (b)]
\begin{align*}
h}%{\mathbbl{h}_{\mu} & N_{\lambda}(y)=
\int_{\R^{n}_{+}} N_{\lambda}(x)
\prod_{i=1}^{n}\{\sqrt{x_{i}y_{i}} J_{\mu_{i}}(x_{i}y_{i})\} dx\\
&=
\int_{\R^{n}_{+}}
2^{-\mu-1}x^{\mu+1/2}\lambda^{\mu}\lambda^{n-1}
\left\{\int_{0}^{\infty}
e^{-t-\frac{\lambda\|x\|^{2}}{4t}}
\frac{dt}{t^{\mu+1}}
\right\}
\prod_{i=1}^{n}
\{\sqrt{x_{i}y_{i}}J_{\mu_{i}}(x_{i}y_{i})\} dx\\
&=
2^{-\mu-1}y^{1/2}\lambda^{\mu}\lambda^{n-1}
\int_{0}^{\infty} \left\{
\int_{\R^{n}_{+}}
x^{\mu+1}e^{-\frac{\lambda\|x\|^{2}}{4t}}
\prod_{i=1}^{n}
\{J_{\mu_{i}}(x_{i}y_{i})\}\:dx
\right\} e^{-t} \frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}y^{1/2}\lambda^{\mu}\lambda^{n-1}
\int_{0}^{\infty} \prod_{i=1}^{n} \left\{
\int_{0}^{\infty}
x^{\mu_{i}+1} e^{-\frac{\lambda x_{i}^{2}}{4t}}
J_{\mu_{i}}(x_{i}y_{i})\:dx_{i}
\right\} e^{-t} \frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}y^{1/2}\lambda^{\mu}\lambda^{n-1}
\int_{0}^{\infty} \prod_{i=1}^{n} \left\{
\left(\frac{\lambda}{2t}\right)^{-\mu_{i}-1} y_{i}^{\mu_{i}}
e^{-\frac{ty_{i}^{2}}{\lambda}}
\right\} e^{-t} \frac{dt}{t^{\mu+1}}\\
&=
2^{-\mu-1}y^{1/2}\lambda^{\mu}\lambda^{n-1}
2^{\mu+1}\lambda^{-\mu-1}y^{\mu}
\int_{0}^{\infty}
t^{\mu+1}
e^{-\frac{t\|y\|^{2}}{\lambda}} e^{-t}
\frac{dt}{t^{\mu+1}}\\
&=
y^{\mu+1/2}\lambda^{\mu}\lambda^{n-1}
\lambda^{-\mu}\lambda^{-n}
\int_{0}^{\infty}
e^{-t(1+\frac{\|y\|^{2}}{\lambda})}
\:dt\\
&=
y^{\mu+1/2}\lambda^{-1} \frac{\lambda}{\lambda+\|y\|^{2}}
\int_{0}^{\infty} e^{-s}\:ds\\
&=
\frac{y^{\mu+1/2}}{\lambda+\|y\|^{2}}
\end{align*}
where we have used \eqref{Prop3.14-eq1}.
\end{proof}
\begin{lemma}
Let $1\leq p<\infty$. If $f\in L^{p}(\R^{n}_{+},sr^{p})$ or $f\in L^{\infty}(\R^{n}_{+},r)$ then the following equality holds on $\mathcal{H'}_{\mu}$
\begin{equation}
h}%{\mathbbl{h}_{\mu}(N_{\lambda}\sharp f) =
\frac{1}{\lambda+\|y\|^{2}} h}%{\mathbbl{h}_{\mu}f
\end{equation}
\end{lemma}
\begin{proof}
Suppose that $f\in L^{p}(\R^{n}_{+},sr^{p})$ and $\psi\in\mathcal{H}_{\mu}$, we claim that
\begin{equation}\label{Lema4.4-eq1}
\int_{\R^{n}_{+}}
(N_{\lambda}\sharp f)(x)\psi(x)\;dx
=
\int_{\R^{n}_{+}}
f(z)(N_{\lambda}\sharp \psi)(z)\;dz
\end{equation}
\begin{equation}\label{Lema4.4-eq2}
\int_{\R^{n}_{+}}
f(z)(N_{\lambda}\sharp \psi)(z)\;dz=
\int_{\R^{n}_{+}} f(z)
\left\{
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
N_{\lambda}(y)\psi(x)\;\textbf{D}_{\mu}(x,y,z)\;dy\;dx
\right\}dz.
\end{equation}
\noindent Let us see that $ \int_{\R^{n}_{+}} |f(z)|
\left\{
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
|N_{\lambda}(y)|\;|\psi(x)|\;\textbf{D}_{\mu}(x,y,z)\;dy\;dx
\right\}dz$ is finite.
\noindent Let
\[G(z)=
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
|N_{\lambda}(y)|\;|\psi(x)|\;\textbf{D}_{\mu}(x,y,z)\;dy\;dx\]
\noindent and let $q$ such that $\frac{1}{p}+\frac{1}{q}=1$. The function $G$ is the convolution of $|N_{\lambda}|$ and $|\psi|$. From Lemma \ref{lema3.3}, since $|N_{\lambda}|\in L^{1}(\R^{n}_{+},sr)$ and $|\psi|\in L^{q}(\R^{n}_{+},sr^{q})$ we have that $G\in L^{q}(\R^{n}_{+},sr^{q})$, then
\begin{align*}
\int_{\R^{n}_{+}}
|f(z)|\;|G(z)|\;dz&=
\int_{\R^{n}_{+}}
(r|f(z)|)\;(r^{-1}s^{-1}|G(z)|)\;s\;dz\\
&=
\int_{\R^{n}_{+}}
(r|f(z)|)\;(C_{\mu}r|G(z)|)\;s\;dz\\
&=
C_{\mu}\int_{\R^{n}_{+}}
|rf(z)|\;|rG(z)|\;s\;dz\\
&\leq
C_{\mu}\|rf\|_{L^{p}(\R^{n}_{+},s)}\|rG\|_{L^{q}(\R^{n}_{+},s)}\\
&=
C_{\mu}\|f\|_{L^{p}(\R^{n}_{+},sr^{p})}\|G\|_{L^{q}(\R^{n}_{+},sr^{q})}
\end{align*}
then it is possible to change the order of integration in \eqref{Lema4.4-eq2}.
\begin{align*}
\int_{\R^{n}_{+}}f(z)\;(N_{\lambda}\sharp\psi)(z)\;dz
&=
\int_{\R^{n}_{+}}
\left\{
\int_{\R^{n}_{+}}\int_{\R^{n}_{+}}
N_{\lambda}(y)\;f(z)\;\textbf{D}_{\mu}(x,y,z)\;dy\;dz
\right\}\;\psi(x)\;dx\\
&=
\int_{\R^{n}_{+}}
(N_{\lambda}\sharp f)(x)
\;\psi(x)\;dx\\
\end{align*}
So, we have proved \eqref{Lema4.4-eq2}.
\medskip
\noindent Now let $f\in L^{\infty}(\R^{n}_{+},r)$. To see that \eqref{Lema4.4-eq1} holds, it will be enough to see that
\begin{align*}
\int_{\R^{n}_{+}}&\left\{
\int_{\R^{n}_{+}}\left\{
\int_{\R^{n}_{+}}
|f(z)|\;|N_{\lambda}(y)|\;|\psi(x)|\;\textbf{D}_{\mu}(x,y,z)\;dz
\right\}\;dy\right\}\;dx\\
&\leq
\|rf\|_{L^{\infty}(\R^{n}_{+})}
\int_{\R^{n}_{+}}\left\{
\int_{\R^{n}_{+}}
|N_{\lambda}(y)|\;|\psi(x)|\;
\left\{
\int_{\R^{n}_{+}}
z^{\mu+1/2}\textbf{D}_{\mu}(x,y,z)\;dz
\right\}\;dy\right\}\;dx\\
&= C_{\mu}\;\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\;\|N_{\lambda}\|_{L^{1}(\R^{n}_{+},sr)}
\;\|\psi\|_{L^{1}(\R^{n}_{+},sr)}<\infty.
\end{align*}
\medskip
\noindent Let $\phi\in\mathcal{H}_{\mu}$ and $f\in L^{p}(\R^{n}_{+},sr)$ or $f\in L^{\infty}(\R^{n}_{+},r)$, from \eqref{Lema4.4-eq1} we have that
\begin{equation}\label{Lemma4.4-eq2}
(h}%{\mathbbl{h}_{\mu}(N_{\lambda}\sharp f),\phi) =
((N_{\lambda}\sharp f),h}%{\mathbbl{h}_{\mu}\phi) =
\int_{\R^{n}_{+}} (N_{\lambda}\sharp f)(x)\;(h}%{\mathbbl{h}_{\mu}\phi)(x)\;dx=
\int_{\R^{n}_{+}} f(z)\;(N_{\lambda}\convZh}%{\mathbbl{h}_{\mu}\phi)(z)\;dz.
\end{equation}
\noindent From Lemma \ref{lema3.4}, Theorem \ref{TeoInv-nHZ} and item (b) of Lemma \ref{Lema4.3} we obtain that
\[
h}%{\mathbbl{h}_{\mu}(N_{\lambda}\sharp h}%{\mathbbl{h}_{\mu}\phi)(y)=
r(h}%{\mathbbl{h}_{\mu}N_{\lambda})(h}%{\mathbbl{h}_{\mu}(h}%{\mathbbl{h}_{\mu}\phi))(y)=
y^{-\mu-1/2}\frac{y^{\mu+1/2}}{\lambda+\|y\|^{2}}\phi(y)=
\frac{\phi(y)}{\lambda+\|y\|^{2}}.
\]
\noindent Then
\begin{equation}\label{Lemma4.4-eq3}
N_{\lambda}\sharp h}%{\mathbbl{h}_{\mu}\phi=
h}%{\mathbbl{h}_{\mu}\left(\frac{\phi}{\lambda+\|y\|^{2}}\right).
\end{equation}
\noindent Finally, from \eqref{Lemma4.4-eq2} and \eqref{Lemma4.4-eq3} we obtain that for $\phi\in\mathcal{H}_{\mu}$ that
\begin{align*}
(h}%{\mathbbl{h}_{\mu}(N_{\lambda}\sharp f),\phi)
&=
\int_{\R^{n}_{+}}
f(x)(N_{\lambda}\convZh}%{\mathbbl{h}_{\mu}\phi)(x)\;dx\\
&=
\int_{\R^{n}_{+}}
f(x)\;h}%{\mathbbl{h}_{\mu}\left(\frac{\phi}{\lambda+\|y\|^{2}}\right)(x)\;dx\\
&=
\int_{\R^{n}_{+}}
\frac{1}{\lambda+\|x\|^{2}}
h}%{\mathbbl{h}_{\mu}f(x)\;\phi(x)\;dx\\
&=
\left(\frac{h}%{\mathbbl{h}_{\mu}f}{\lambda+\|x\|^{2}},\phi\right)
\end{align*}
\end{proof}
\begin{theorem}\label{Teorema4.7}
Given $\mu=(\mu_{1},\ldots,\mu_{n})$, $\mu_{i}>-\frac{1}{2}$, then $S_{\mu,p}$ and $S_{\mu,\infty}$ are closed and non-negative operators.
\end{theorem}
\begin{proof} Since convergence in $L^{\infty}(\R^{n}_{+},r)$ and $L^{p}(\R^{n}_{+},sr^p)$ implies convergence in $\mathcal{D'}(\R^{n}_{+})$, then $S_{\mu,\infty}$ and $S_{\mu,p}$ are closed.
\medskip
Now let $\lambda>0$ and $f\in D(S_{\mu,\infty})$ such that $(\lambda-S_{\mu,\infty})f=0$. So,
\[h}%{\mathbbl{h}_{\mu}(\lambda-S_{\mu,\infty})f=0\]
in $\mathcal{H'}_{\mu}$. By Lemma \ref{lema3.10} we obtain that
\[(\lambda+\|y\|^{2})h}%{\mathbbl{h}_{\mu}f=0\]
in $\mathcal{H'}_{\mu}$ and hence by Lemma \ref{lema3.9}
\[h}%{\mathbbl{h}_{\mu}f=(\lambda+\|y\|^{2})^{-1}(\lambda+\|y\|^{2})h}%{\mathbbl{h}_{\mu}f=0.\]
Then, $f=0$ as element of $\mathcal{H'}_{\mu}$ and we conclude that $f=0$ a.e. in $x\in\R^{n}_{+}$ and $\lambda-S_{\mu,\infty}$ is injective.
\medskip
Let $f\in L^{\infty}(\R^{n}_{+},r)$ and $g=N_{\lambda}\sharp f$. Then, by Lemma \ref{lema3.3} $g\in L^{\infty}(\R^{n}_{+},r)$ and
\[
h}%{\mathbbl{h}((\lambda-S_{\mu,\infty})g)=
(\lambda+\|y\|^{2})h}%{\mathbbl{h}_{\mu}g=
(\lambda+\|y\|^{2})h}%{\mathbbl{h}_{\mu}(N_{\lambda}\sharp f)=
h}%{\mathbbl{h}_{\mu}f.
\]
\noindent By injectivity of Hankel transform in $\mathcal{H'}_{\mu}$ we obtain that
\[(\lambda-S_{\mu,\infty})g=f,\]
so, $\lambda-S_{\mu,\infty}$ is onto. Also
\begin{align*}
\|(\lambda-S_{\mu,\infty})^{-1}f\|_{L^{\infty}(\R^{n}_{+},r)}
&=
\|g\|_{L^{\infty}(\R^{n}_{+},r)}=
\|N_{\lambda}\sharp f\|_{L^{\infty}(\R^{n}_{+},r)}\\
&\leq
\|N_{\lambda}\|_{L^{1}(\R^{n}_{+},sr)}\|f\|_{L^{\infty}(\R^{n}_{+},r)}\\
&=
\frac{1}{\lambda}\|f\|_{L^{\infty}(\R^{n}_{+},r)},
\end{align*}
hence
\[
\|\lambda(\lambda-S_{\mu,\infty})^{-1}f\|_{L^{\infty}(\R^{n}_{+},r)}
\leq \|f\|_{L^{\infty}(\R^{n}_{+},r)}\]
and $-S_{\mu,\infty}$ is non-negative.
\medskip
\noindent The proof of the non-negativity of $S_{\mu,p}$ is similar.
\end{proof}
\vspace{.2in}
Since we have proved that both $-S_{\mu,p}$ and $-S_{\mu,\infty}$ are non-negative we can consider the fractional powers of them. If $\alpha\in{\mathbb C}$, $\Re(\alpha)>0$ and $n>\Re(\alpha)$ then the fractional power of $-S_{\mu,\infty}$ can be represented from \eqref{Theorem 5.2.1-MS01} by:
\[(-S_{\mu,\infty})^{\alpha}=
(-S_{\mu,\infty}+1)^{n}\mathcal{J}_{\infty}^{\alpha}(-S_{\mu,\infty}+1)^{-n},
\]
where with $\mathcal{J}_{\infty}^{\alpha}$ we denote the Balakrishnan operator associated to $-S_{\mu,\infty}$ given by:
\[
\mathcal{J}_{\infty}^{\alpha}\phi=
\frac{\Gamma(n)}{\Gamma(\alpha)\Gamma(n-\alpha)}
\int_{0}^{\infty} \lambda^{\alpha-1}
[-S_{\mu,\infty}(\lambda-S_{\mu,\infty})^{-1}]^{n}\phi\;d\lambda,
\]
for $\alpha\in{\mathbb C}$, $0<\Re(\alpha)<n$ and $\phi\in D[(-S_{\mu,\infty})^{n}]$.
\medskip
Analogously for the representation of fractional powers of $-S_{\mu,p}$.
\section{Non-negativity of Bessel operator \texorpdfstring{$S_{\mu}$}{Smu} in the space \texorpdfstring{$\mathcal{B}$}{B}}
\begin{remark}
The operator $-S_{\mu}$ is not non-negative in $\mathcal{H}_{\mu}$.
\end{remark}
\noindent If $-S_{\mu}$ were non-negative in $\mathcal{H}_{\mu}$, since $-S_{\mu}$ is continuous in $\mathcal{H}_{\mu}$, given $\alpha\in{\mathbb C}$, $0<\alpha<1$ and according to \eqref{prop4.1-eq1} and \eqref{complementosEuler}, we have that fractional power $(-S_{\mu})^{\alpha}$ would be given by
\begin{equation}\label{eq1-obs4.7}
(-S_{\mu})^{\alpha}\phi=
\frac{\sin\alpha\pi}{\pi}
\int_{0}^{\infty}
\lambda^{\alpha-1}
(-S_{\mu})(\lambda-S_{\mu})^{-1}\phi\:d\lambda
\end{equation}
and $D[(-S_{\mu})^{\alpha}]=D(-S_{\mu})=\mathcal{H}_{\mu}$. Applying the Hankel transform in \eqref{eq1-obs4.7} we obtain
\begin{align*}
h}%{\mathbbl{h}_{\mu}(-S_{\mu})^{\alpha}\phi
&=
\frac{\sin\alpha\pi}{\pi}
\int_{0}^{\infty}
\lambda^{\alpha-1}
h}%{\mathbbl{h}_{\mu}[(-S_{\mu})(\lambda-S_{\mu})^{-1}\phi]\:d\lambda\\
&=
\frac{\sin\alpha\pi}{\pi}
\int_{0}^{\infty}
\lambda^{\alpha-1} \|y\|^{2}(\lambda+\|y\|^{2})^{-1}
h}%{\mathbbl{h}_{\mu}\phi(y)\:d\lambda\\
&=
(\|y\|^{2})^{\alpha}h}%{\mathbbl{h}_{\mu}\phi(y),
\end{align*}
where we have interchanged the Bochner integral with the Hankel transform, and the we have applied item $(iii)$ of Lemma \ref{lema3.11} and \cite[Remark 3.1.1]{MS01}. This would imply that $(\|y\|^{2})^{\alpha}h}%{\mathbbl{h}_{\mu}\phi(y)\in\mathcal{H}_{\mu}$ which is not true in general.
\vspace{.5in}
Now we consider the Banach space $Y=L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)$, with norm
\[
\|f\|_{Y}=
\max\bigl\{
\|f\|_{L^{1}(\R^{n}_{+},sr)},
\|f\|_{L^{\infty}(\R^{n}_{+},r)}
\bigr\},
\]
and the part of the Bessel operator in $Y$, $(S_{\mu})_{Y}$, with domain given by \[
D[(S_{\mu})_{Y}]=
\{f\in Y:\: S_{\mu} f\in Y\}.
\]
\noindent From Theorem \ref{Teorema4.7} we have that $-(S_{\mu})_{Y}$ is closed and non-negative.
\clearpage
\begin{proposition}\label{Prop4.7}
If $k>\frac{n}{2}$ then $D[((S_{\mu})_{Y})^{k+1}]\subset C_{0}(\R^{n}_{+})$.
\end{proposition}
\begin{proof}
\[
D[((S_{\mu})_{Y})^{k+1}]=
\{
\phi\in D[((S_{\mu})_{Y})^{k}]:
\quad
((S_{\mu})_{Y})^{k}\phi\in D[(S_{\mu})_{Y}]
\}
\]
\noindent Let $f\in D[((S_{\mu})_{Y})^{k+1}]$, then $f$ and $((S_{\mu})_{Y})^{k}f$ are in $D[(S_{\mu})_{Y}]$.
\bigskip
\noindent From Lemma \ref{lema3.1} and \eqref{prop7.5 [Mo18]-eq1} we have that
\[
L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)
\subset
L^{1}(\R^{n}_{+})\cap L^{2}(\R^{n}_{+}).
\]
Then $f$ and $((S_{\mu})_{Y})^{k}f$ are in $L^{1}(\R^{n}_{+})$. From Remark \ref{remark7.4 [Mo18]} we obtain that
$h}%{\mathbbl{h}_{\mu}f$ and $h}%{\mathbbl{h}_{\mu}((S_{\mu})_{Y})^{k}f$ are in $L^{\infty}(\R^{n}_{+})$,
that is to say that there exist $M>0$ such that
\[
|(1+\|y\|^{2k})\:h}%{\mathbbl{h}_{\mu}f|
\leq M.
\]
\noindent Since for $k>\frac{n}{2}$, $(1+\|y\|^{2k})^{-1}$ is integrable in $\R^{n}_{+}$, then $h}%{\mathbbl{h}_{\mu}f\in L^{1}(\R^{n}_{+})$. Then, we have proved that $f\in D[((S_{\mu})_{Y})^{k+1}]$, $f$ and $h}%{\mathbbl{h}_{\mu}f\in L^{1}(\R^{n}_{+})\cap L^{2}(\R^{n}_{+})$.
\medskip
\noindent From Remark \ref{L^2 subset H'mu} we have that $L^{1}(\R^{n}_{+})\cap L^{2}(\R^{n}_{+})\subset \mathcal{H'}_{\mu}$ and from Remark \ref{TeoInv-nHZ en Hmu}
we have
\[
h}%{\mathbbl{h}_{\mu}(h}%{\mathbbl{h}_{\mu}f)(x)=f(x),
\quad\text{a.e}\quad x\in\R^{n}_{+},
\]
considering $f$ as a regular distribution in $\mathcal{H'}_{\mu}$. Since $h}%{\mathbbl{h}_{\mu}f\in L^{1}(\R^{n}_{+})$, then by Proposition \ref{prop7.5 [Mo18]} we have that $f=g\quad\text{a.e.}$
in $\R^{n}_{+}$ with $g\in C_{0}(\R^{n}_{+})$.
\end{proof}
\vspace{.2in}
We now consider the following space:
\begin{equation}\label{eqB}
\mathcal{B}=
\{f\in Y: \:(S_{\mu})^{k}f\in Y
\quad\text{for}\quad
k=0,1,2,\ldots\}
=\bigcap\limits_{k=0}^{\infty}
D[((S_{\mu})_{Y})^{k}],
\end{equation}
with seminorms
\[
\rho_{m}(f)=\max_{0\leq k\leq m}
\bigl\{
\|(S_{\mu})^{k}f\|_{Y},
\quad m=0,1,2,\ldots
\bigr\}.
\]
\begin{remark}\label{Remark4.8 bis}
From proposition \ref{Prop4.7} is evident that $\mathcal{B}\subset C_{0}(\R^{n}_{+})$. Moreover, from Lemma \ref{lema3.1} we obtain that $\mathcal{B}\subset L^{p}(\R^{n}_{+},sr^{p})$ for all $1\leq p<\infty$, and considering that $S_{\mu}$ is a continuous operator from $\mathcal{H}_{\mu}$ in itself then $\mathcal{H}_{\mu}\subset\mathcal{B}$ and the topology of $\mathcal{H}_{\mu}$ induced by $\mathcal{B}$ is weaker than the usual topology generated by the seminorms given by \eqref{Topologia H_mu}. In fact, from \eqref{lema3.1-eq5} and \eqref{lema3.1-eq6} we have that
\begin{equation}
\|\phi\|_{Y}\leq
C\{\gamma_{0,0}^{\mu}(\phi)+\gamma_{m,0}^{\mu}(\phi)\},
\quad \phi\in\mathcal{H}_{\mu}.
\end{equation}
for $m>2\mu_i+2$, $i=1,\ldots,n$ and by the continuity of $S_{\mu}$ in $\mathcal{H}_{\mu}$ we deduce that given a seminorm $\rho_{m}$, there exists a finite set of seminorms $\{\gamma_{m_i,k_i}^{\mu}\}_{i=1}^{r}$ and constants $c_1,\ldots,c_r$ such that
\[
\rho_{m}(\phi)\leq
\sum_{i=1}^{r} c_{i}\:
\gamma_{m_i,k_i}^{\mu}(\phi),
\quad \phi\in\mathcal{H}_{\mu}.
\]
\noindent From the density of $\mathcal{D}(\R^{n}_{+})$ in $\mathcal{B}$ we deduce the density of $\mathcal{H}_{\mu}$ in $\mathcal{B}$.
\end{remark}
\bigskip
We denote with $(S_{\mu})_{\mathcal{B}}$ the part of Bessel operator $S_{\mu}$ in $\mathcal{B}$, so the domain of the operator $(S_{\mu})_{\mathcal{B}}$ is $\mathcal{B}$ and the following result holds.
\begin{theorem}\label{Teorema4.8}
$\mathcal{B}$ is a Fréchet space and $-(S_{\mu})_{\mathcal{B}}$ is a continous and non-negative operator on $\mathcal{B}$.
\end{theorem}
\begin{proof}
Let $\{\phi_{k}\}$ a Cauchy sequence in $\mathcal{B}$, then the convergence of $\{\phi_{k}\}$ follows considering the seminorm $\rho_{0}$ and the completeness of $L^{1}(\R^{n}_{+},sr)$ and $L^{\infty}(\R^{n}_{+},r)$.
Since $\rho_{m}(S_{\mu}\phi)=\rho_{m+1}(\phi)$ then $(S_{\mu})_{\mathcal{B}}$ is continuous.
The non-negativity follows from Proposition 1.4.2 in \cite{MS01}.
\end{proof}
\section{Non-negativity of Bessel operator \texorpdfstring{ $S_{\mu}$}{Smu} in the distributional space \texorpdfstring{$\mathcal{B'}$}{B}}
We will study the non-negativity of Bessel operator in the topological dual space of $\mathcal{B}$ with the strong topology, that is to say, the space $\mathcal{B'}$ endowed with the topology generated by the family of seminorms $\{|\cdot|_{B}\}$, where the sets $B$ are bounded sets in $\mathcal{B}$, and the seminorms are given by
\[
|T|_{B}=
\sup\limits_{\phi\in B} |(T,\phi)|,
\quad T\in\mathcal{B'}.
\]
\begin{proposition}\label{B' secuencialmente completo}
$\mathcal{B'}$ is sequentially complete.
\end{proposition}
\begin{proof}
Let $\{T_{m}\}\subset\mathcal{B'}$ a Cauchy sequence, then for all bounded set $B\subset\mathcal{B}$,
\[
|T_{k}-T_{m}|_{B}\to 0,
\quad\text{for}\quad
k,m\to\infty,
\]
i.e, for all $\varepsilon>0$ there exists $N$ such that for all $k,m\geq N$ then
\[
|T_{k}-T_{m}|_{B}<\varepsilon.
\]
So, in particular, since the unit sets $\{\phi\}\subset\mathcal{B}$ are bounded,
\[
|T_{k}-T_{m}|_{\{\phi\}}
=
|(T_{k}-T_{m},\phi)|
<\varepsilon,
\quad\forall\:k,m\geq N,
\]
then $\{(T_{m},\phi)\}$ is a Cauchy sequence in ${\mathbb C}$, with which is convergent and there exists $T:\mathcal{B}\to{\mathbb C}$ such that
\[
(T,\phi)=
\lim\limits_{m\to\infty}(T_{m},\phi).
\]
\noindent Since $\mathcal{B}$ is barrelled, for being a Fréchet space, and from a generalization of the Banach–Steinhaus theorem (see \cite[Theorem 4.7, pp.86]{Schaefer}), it has to $T\in\mathcal{B'}$.
\end{proof}
\bigskip
\begin{remark}\label{Remark4.9} $\quad L^{p}(\R^{n}_{+},sr^{p})$ and $L^{\infty}(\R^{n}_{+},r)$ are included in $\mathcal{B'},\quad(1\leq p<\infty)$.
\medskip
\noindent Let $f\in L^{p}(\R^{n}_{+},sr^{p})$, $\phi\in\mathcal{B}$ and $q$ such that $\frac{1}{p} + \frac{1}{q} = 1$, then
\begin{align}
\left|\int_{\R^{n}_{+}}f(x)\phi(x)\:dx\right|
&=
\left|\int_{\R^{n}_{+}}
f(x)\phi(x)s^{-1}(x)r^{-p}(x)s(x)r^{p}(x)\:dx\right|\nonumber\\
&\leq
\|f\|_{L^{p}(\R^{n}_{+},sr^{p})}
\|\phi\:s^{-1}r^{-p}\|_{L^{q}(\R^{n}_{+},sr^{p})}\label{Remark4.9-eq1}
\end{align}
\noindent and
\begin{align}
\|\phi\:s^{-1}r^{-p}\|_{L^{q}(\R^{n}_{+},sr^{p})}
&=
\left\{
\int_{\R^{n}_{+}}
|\phi\:s^{-1}r^{-p}|^{q} sr^{p}
\right\}^{\frac{1}{q}}
=
\left\{
\int_{\R^{n}_{+}}
|\phi|^{q}\:(C_{\mu}r^{2}r^{-p})^{q} sr^{p}
\right\}^{\frac{1}{q}}\nonumber\\
&=
C_{\mu}
\left\{
\int_{\R^{n}_{+}}
|\phi|^{q}\:r^{2q-pq+p} s
\right\}^{\frac{1}{q}}
=
C_{\mu}
\left\{
\int_{\R^{n}_{+}}
|\phi|^{q}\: sr^{q}
\right\}^{\frac{1}{q}}\label{Remark4.9-eq2}.
\end{align}
\end{remark}
\noindent Furthermore, from \eqref{lema3.1-eq4}
\[
\|\phi\|_{L^{q}(\R^{n}_{+},sr^{q})}
\leq
\left\{\|\phi\|_{L^{\infty}(\R^{n}_{+},r)}\right\}^{\frac{q-1}{q}}
\left\{\|\phi\|_{L^{1}(\R^{n}_{+},sr)}\right\}^{\frac{1}{q}},
\]
\noindent from where
\begin{equation}\label{Remark4.9-eq3}
\|\phi\|_{L^{q}(\R^{n}_{+},sr^{q})}
\leq \rho_{0}(\phi).
\end{equation}
\noindent Then, from \eqref{Remark4.9-eq1}, \eqref{Remark4.9-eq2} and \eqref{Remark4.9-eq3} we obtain that $f\in\mathcal{B'}$.
\medskip
Now, let $B$ a bounded set in $\mathcal{B}$, then
\[
|f|_{B}
=\sup_{\phi\in B} \left|\int_{\R^{n}_{+}}f\phi\right|
\leq C_{\mu}\|f\|_{L^{p}(\R^{n}_{+},sr^{p})}
\sup_{\phi\in B} \|\phi
|
\|_{L^{q}(\R^{n}_{+},sr^{q})}
\leq C_{\mu}\|f\|_{L^{p}(\R^{n}_{+},sr^{p})}
\sup_{\phi\in B}\:\rho_{0}(\phi)
\]
Thus, the topology in $L^{p}(\R^{n}_{+},sr^{p})$ induced by $\mathcal{B'}$ with the strong topology is weaker than the usual topology.
\begin{remark}
Since $\mathcal{H}_{\mu}$ is dense in $\mathcal{B}$ and since the topology of $\mathcal{H}_{\mu}$ induced by $\mathcal{B}$ is weaker than the generated by the seminorms given by \eqref{Topologia H_mu} then $\mathcal{B'}\subset\mathcal{H'}_{\mu}$. Moreover, from the continuity of the Bessel operator in $\mathcal{B}$, we can consider $S_{\mu}$ in $\mathcal{B'}$ as the adjoint operators of $S_{\mu}$ in $\mathcal{B}$, that is to say
\[
(S_{\mu} T,\phi)=(T,S_{\mu}\phi),
\quad T\in\mathcal{B'},
\phi\in\mathcal{B},
\]
and we denote with $(S_{\mu})_{\mathcal{B'}}$ the part of Bessel operator in $\mathcal{B'}$.
\end{remark}
\begin{theorem}\label{Teorema4.9}
The operator $-(S_{\mu})_{\mathcal{B'}}$ is continuous and non-negative considering the strong topology in $\mathcal{B'}$.
\end{theorem}
\begin{proof}
Given a bounded set $B\subset\mathcal{B}$ and $T\in\mathcal{B'}$, then
\[
|(S_{\mu})_{\mathcal{B'}}T|_{B}=
\sup\limits_{\phi\in B}
|((S_{\mu})_{\mathcal{B'}}T,\phi)|=
\sup\limits_{\phi\in B}
|(T,(S_{\mu})_{\mathcal{B}}\phi)|=
|T|_{E}
\]
where the set $E=\{(S_{\mu})_{\mathcal{B}}\phi:\:\phi\in B\}$ is also bounded. Then it follows that $(S_{\mu})_{\mathcal{B'}}$ is continuous.
\bigskip
\noindent Let now $\lambda>0$ and $T\in\mathcal{B'}$. It is not difficult to see that the linear map $G:\psi\to (T,(\lambda-(S_{\mu})_{\mathcal{B}})^{-1}\psi)$ is continuous and $(\lambda-(S_{\mu})_{\mathcal{B'}})G=T$.
\noindent Therefore $(\lambda-(S_{\mu})_{\mathcal{B'}})$ is surjective.
\bigskip
\noindent To prove the injectivity, let $T\in\mathcal{B'}$ be such that $(\lambda-(S_{\mu})_{\mathcal{B'}})T=0$. Then, for all $\phi\in\mathcal{B}$,
\[
((\lambda-(S_{\mu})_{\mathcal{B'}})T,\phi)=
(T,(\lambda-(S_{\mu})_{\mathcal{B}})\phi)=0,
\]
and thus $T=0$ as $R(\lambda-(S_{\mu})_{\mathcal{B}})={\mathcal{B}}$, due to the fact that $-(S_{\mu})_{\mathcal{B}}$ is a non-negative operator.
\bigskip
\noindent To see that $(\lambda-(S_{\mu})_{\mathcal{B'}})^{-1}$ is continuous let $T\in\mathcal{B'}$, $B\subset\mathcal{B}$ a bounded set and let us consider the set $F=\{(\lambda-(S_{\mu})_{\mathcal{B}})^{-1}\phi:\:\phi\in B\}$, then
\[
|(\lambda-(S_{\mu})_{\mathcal{B'}})^{-1}T|_{B}
=
|G|_{B}
=
\sup\limits_{\psi\in B}
|(G,\psi)|
=
\sup\limits_{\psi\in B}
|(T,(\lambda-(S_{\mu})_{\mathcal{B}})^{-1}\psi)|_{B}
=
|T|_{F}.
\]
\noindent For every bounded set $B\subset\mathcal{B}$ and $T\in\mathcal{B'}$, since $-(S_{\mu})_{\mathcal{B}}$ is non-negative, the set $D=\{\eta(\eta-(S_{\mu})_{\mathcal{B}})^{-1}\phi:\:\phi\in B,\:\eta>0\}$ is also bounded and thus, for $\lambda>0$,
\begin{align*}
|\lambda(\lambda-(S_{\mu})_{\mathcal{B'}})^{-1}T|_{B}
&=
\sup\limits_{\phi\in B}
|(\lambda(\lambda-(S_{\mu})_{\mathcal{B'}})^{-1}T,\phi)|\\
&=
\sup\limits_{\phi\in B}
|(T,\lambda(\lambda-(S_{\mu})_{\mathcal{B}})^{-1}\phi)|\\
&\leq
|T|_{D}.
\end{align*}
We now conclude that the operator $-(S_{\mu})_{\mathcal{B'}}$ is non-negative.
\end{proof}
\begin{remark}
The operator $(S_{\mu})_{\mathcal{B'}}$ is not injective because the function $x^{\mu+\frac{1}{2}}$ is solution of $S_{\mu} u=0$ and belongs to $\mathcal{B'}$, in fact
\[
|(x^{\mu+\frac{1}{2}},\phi)|
\leq
C_{\mu}\|\phi\|_{L^{1}(\R^{n}_{+},sr)}
\leq
C_{\mu}\rho_{0}(\phi),
\quad \phi\in\mathcal{B}.
\]
\end{remark}
\vspace{.2in}
According to the representation of fractional powers of operators in locally convex spaces given in \cite{MS01}, it has to $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}$ is given by
\[
(-(S_{\mu})_{\mathcal{B'}})^{\alpha}T=
\frac{\Gamma(n)}{\Gamma(\alpha)\Gamma(n-\alpha)}
\int_{0}^{\infty}
\lambda^{\alpha-1}
[-(S_{\mu})_{\mathcal{B'}}
(\lambda-(S_{\mu})_{\mathcal{B'}})^{-1}
]^{n}\:T\:d\lambda.
\]
for $\Re\alpha>0$, $n>\Re\alpha$, $T\in\mathcal{B'}$.
\vspace{.3in}
From the general theory of fractional powers in sequentially complete locally convex spaces (see \cite[pp.134]{MS01}), we deduce some properties of powers such as multiplicativity and
\begin{itemize}
\item[(1)] If $\Re\alpha>0$ then
\begin{equation}\label{Obs-1}
\left((-(S_{\mu})_{\mathcal{B}})^{\alpha}\right)^{*}=
\left((-(S_{\mu})_{\mathcal{B}})^{*}\right)^{\alpha}
\end{equation}
\noindent Since $(-(S_{\mu})_{\mathcal{B}})^{*}=-(S_{\mu})_{\mathcal{B'}}$ then from \eqref{Obs-1} we obtain the following duality formula
\[
( (-(S_{\mu})_{\mathcal{B'}})^{\alpha}T , \phi)
=
( T, (-(S_{\mu})_{\mathcal{B}})^{\alpha} \phi),
\quad \phi\in\mathcal{B},
T\in\mathcal{B'}.
\]
\item[(2)] Since the usual topology in $L^{p}(\R^{n}_{+},sr^{p})$ is stronger than the topology induced by $\mathcal{B'}$ we can deduce that
\[
((-(S_{\mu})_{\mathcal{B'}})^{\alpha})_{L^{p}(\R^{n}_{+},sr^{p})}
=
(-(S_{\mu,p}))^{\alpha},
\]
for $\Re\alpha>0$, (see \cite[Theorem 12.1.6, pp.284]{MS01}).
\vspace{.2in}
This last property expresses a very desirable property in the theory of powers since it tells us that the restriction of the distributional power of $-S_{\mu}$ to $L^{p}(\R^{n}_{+},sr^{p})$ coincides with the power of $-S_{\mu}$ in $L^{p}(\R^{n}_{+},sr^{p})$.
\end{itemize}
\section{Distributional Liouville theorem for \texorpdfstring{$(-(S_{\mu}))^{\alpha}$}{Smualpha}.}
In this section we include the proof of Theorem \ref{Teorema5.1}. Before that, we will show the following Lemma.
\begin{lemma}\label{lemmamultpot}
Let $\psi\in\mathcal{H}_{\mu}$ such that $\mathop{\mathrm{supp}}\psi\subset \R^{n}_{+}\cap\{x:\|x\|\geq a\}$ with $a>0$ and $\alpha\in {\mathbb C}$ with $\Re\alpha>0$. Then $\|x\|^{-2\alpha}\psi(x)\in\mathcal{H}_{\mu}$.
\end{lemma}
\begin{proof} It is evident that $\|x\|^{-2\alpha}\psi(x)\in C^{\infty}(\R^{n}_{+})$. We are going to see that
\[
\sup_{x\in\R^{n}_{+}}
\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}
\Bigr|<\infty,
\]
with $k,m\in\mathbb{N}_{0}^{n}$.
Since $\mathop{\mathrm{supp}}\psi\subset \R^{n}_{+}\cap\{x:\|x\|\geq a\}$ with $a>0$, then
\begin{align*}
&\sup_{x\in\R^{n}_{+}}
\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}\Bigr|
=
\sup_{x\in\R^{n}_{+}: \|x\|\geq a}
\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}\Bigr|\\
&\quad\leq
\sup_{a\leq\|x\|\leq 1}\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}\Bigr|
+
\sup_{\|x\|\geq 1}\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}\Bigr|
\end{align*}
The first term in the last inequality is bounded because it is a continuous function over a compact set. On the other hand, since equality \eqref{Leibniz} holds then,
\[
\sup_{\|x\|\geq 1}
\Bigl|x^{m}T^{k}
\{x^{-\mu-\frac{1}{2}}\|x\|^{-2\alpha}\psi(x)\}\Bigr|\leq
\]
\[
\sup_{\|x\|\geq 1}
\Bigl|x^{m}\sum_{j=0}^{k}
\binom{k}{j}T^{k-j}
\{x^{-\mu-\frac{1}{2}}\psi(x)\}\cdot
T^{j}\|x\|^{-2\alpha}\Bigr|\leq
\]
\[
\sum_{j=0}^{k} \binom{k}{j} C(j,\alpha)\:\:\gamma_{m,k-j}^{\mu}(\psi),
\]
where $C(j,\alpha)$ are constants depending on $\alpha$ and $j$ such that $\sup\limits_{\|x\|\leq 1} |T^{j}\|x\|^{-2\alpha}|\leq C(j,\alpha)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{Teorema5.1}]
Let $u\in \mathcal{B'}$ such that $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}u=0$. Then for all $\phi\in \mathcal{B}$
\begin{equation}\label{eqequal0}
((-(S_{\mu})_{\mathcal{B'}})^{\alpha}u,\phi)=
(u,(-(S_{\mu})_{\mathcal{B}})^{\alpha}\phi)=0.
\end{equation}
Since $S_{\mu}$ is a continuous operator in $\mathcal{B}$ (see Theorem \ref{Teorema4.8}), then $(-(S_{\mu})_{\mathcal{B}})^{\alpha}\phi$ is given by the Balakrishnan operator as:
\begin{equation}\label{eqpot1}
(-(S_{\mu})_{\mathcal{B}})^{\alpha}\phi=
\frac{\Gamma(\alpha)\Gamma(m-\alpha)}{\Gamma(m)}
\int_{0}^{\infty}
\lambda^{\alpha-1}
[(-(S_{\mu})_{\mathcal{B}})
(\lambda-(S_{\mu})_{\mathcal{B}})^{-1}]^{m}
\phi\:\:d\lambda.
\end{equation}
By definition of $\mathcal{B}$ and the fact that $L^{1}(\R^{n}_{+},sr)\cap L^{\infty}(\R^{n}_{+},r)\subset L^{p}(\R^{n}_{+},sr^{p})$ for all $1\leq p\leq\infty$ then $\mathcal{B}\subset D(S_{\mu,p})$ for all $1\leq p\leq \infty$, in particular, $\mathcal{B}\subset D(S_{\mu,2})$. Then from Propositions 8.3 and 8.4 in \cite{MT07} we obtain that:
\begin{equation}\label{eqpot2}
(-S_{\mu,2})^{\alpha}\phi=
\frac{\Gamma(\alpha)\Gamma(m-\alpha)}{\Gamma(m)}
\int_{0}^{\infty}
\lambda^{\alpha-1}
[-S_{\mu,2}(\lambda-(S_{\mu,2})^{-1}]^{m}
\phi\:\:d\lambda.
\end{equation}
Since for $\phi\in\mathcal{B}$, the integrating into the expressions are equal and the fact that the convergence in $\mathcal{B}$ implies the convergence in $L^{2}(\R^{n}_{+})$ (see Lemma 2.1 and Remark 5.3 in \cite{Mo18}) we obtain the equality of (\ref{eqpot1}) and (\ref{eqpot2}) as functions.
\bigskip
We conclude that
\[
(-(S_{\mu})_{\mathcal{B}})^{\alpha}\phi=
h_{\mu }\|y\|^{2\alpha}h_{\mu}\phi,
\quad \phi\in\mathcal{B},
\]
(see \cite[Proposition 8.4 ]{Mo18}). From the last equality and \eqref{eqequal0}, we have that
\begin{equation} \label{eqpot3}
((-(S_{\mu})_{\mathcal{B'}})^{\alpha}u,\phi)=
(u,h_{\mu }\|y\|^{2\alpha}h_{\mu }\phi)=0,
\end{equation}
for all $\phi\in\mathcal{B}$.
Since $\mathcal{B'}\subset\mathcal{H'}_{\mu} $ (see \cite[Remark 6.2]{Mo18}), we can consider the Hankel transform in $\mathcal{B'}$. We are going to see that the following affirmation holds:
\medskip
\begin{center}
" \sl{If $u\in\mathcal{B'}$ is such that \eqref{eqpot3} is verified, then $(h_{\mu}u,\psi)=0$ for all $\psi\in \mathcal{H}_{\mu}$ such that $\mathop{\mathrm{supp}}\psi\subset \R^{n}_{+}\cap\{x:\|x\|\geq a\}$ with $a>0$.}"
\end{center}
\medskip
Let $u\in\mathcal{B'}$ such that \eqref{eqpot3} is valid and $\psi\in \mathcal{H}_{\mu}$ such that $\mathop{\mathrm{supp}}\psi\subset \R^{n}_{+}\cap\{x:\|x\|\geq a\}$ with $a>0$. Then, by Lemma \ref{lemmamultpot}, $\|x\|^{-2\alpha}\psi(x)\in\mathcal{H}_{\mu}$ and since the Hankel transform is an isomorphism in $\mathcal{H}_{\mu}$, there exists $\phi\in\mathcal{H}_{\mu}$ such that $h_{\mu}\phi=\|x\|^{-2\alpha}\psi(x)$. So,
\[(h_{\mu}u,\psi)=
(h_{\mu}u,\|x\|^{2\alpha}\|x\|^{-2\alpha}\psi)=
(h_{\mu}u,\|x\|^{2\alpha}h_{\mu}\phi)=
(u,h_{\mu}\|x\|^{2\alpha}h_{\mu}\phi).
\]
Consequently, from \eqref{eqpot3} we conclude that $(h_{\mu}u,\psi)=0$, then the assertion is valid. Thus by \cite[Theorem 4.1]{GMQ18}, there exist $N\in\mathbb{N}_{0}$ and scalars $c_{k}$ with $|k|<N$ such that $h_{\mu}u=\sum_{|k|<N}c_{k}S^{k}_{\mu}\delta_{\mu}$ where $\delta_{\mu}$ is given by \cite[equation (2.3)]{GMQ18} for $k=0$. Then,
\[
u=x^{\mu+\frac{1}{2}}
\sum_{|k|\leq N}c_{k}(-1)^{|k|}\|x\|^{2k}
\]
\end{proof}
\begin{remark}[Regular distributions in $\mathcal{B'}$]
\end{remark}
If $f\in L_{loc}^{1}(\R^{n}_{+})$ and $f=O(x^{\mu+\frac{1}{2}})$ then $f$ is a regular distribution in $\mathcal{B'}$ given by
\[
(f,\phi)=
\int_{\R^{n}_{+}} f(x)\phi(x)\:dx,
\quad \phi\in\mathcal{B},
\]
and
\begin{align*}
|(f,\phi)|&=
\Bigl|\int_{\R^{n}_{+}} f(x)\phi(x)\:dx\Bigr|\\
&\leq
\Bigl|\int_{\|x\|\leq M} f(x)\phi(x)dx\Bigr|+
\Bigl|\int_{\|x\|\geq M} f(x)\phi(x)dx\Bigr|\\
&\leq
\int_{\|x\|\leq M} |r^{-1}(x)f(x)|dx \:\|\phi\|_{L^{\infty}(r)}
+
\int_{\|x\|\geq M} c x^{\mu+\frac{1}{2}}|\phi(x)|dx\\
&=
C\|\phi\|_{L^{\infty}(r)}+c\:C_{\mu}\|\phi\|_{L^{1}(rs)}\leq C' \rho_{0}(\phi). \end{align*}
\medskip
\begin{corollary}
If $f\in L_{loc}^{1}(\R^{n}_{+})$, $f=O(x^{\mu+\frac{1}{2}})$ and $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}f=0$ then $f=C\;x^{\mu+\frac{1}{2}}$.
\end{corollary}
\section{Distributional Liouville theorem for \texorpdfstring{$(-(\Delta_{\mu}))^{\alpha}$}{Deltamualpha}.}
From theory of similar operators given in \cite{Mo18}, by the similarity of $S_{\mu}$ and $\Delta _{\mu}$, and by the non-negativity of the part of $-S_{\mu}$ in $L^{1}(\R^{n}_{+},sr)$ and $L^{\infty}(\R^{n}_{+},r)$ we deduce the non-negativity of the part of $-\Delta_{\mu}$ in $L^{1}(\R^{n}_{+},s)$ and $L^{\infty}(\R^{n}_{+})$. Consequently, we infer the non-negativity of the part of $-\Delta_{\mu}$ in the Banach space $Z=L^{1}(\R^{n}_{+},s)\cap L^{\infty}(\R^{n}_{+})$ with norm
\begin{equation*}
\left\| f\right\|_{Z}=
\max\left\{ \left\| f\right\|_{L^{1}(\R^{n}_{+},s)},
\left\| f\right\| _{L^{\infty }(\R^{n}_{+})}\right\}.
\end{equation*}
\bigskip
\noindent Thus, if we consider $Y$ as in Section 5 and $L_{r}:Y\to Z$ given by $L_{r}f=rf$ then
\begin{align*}
\left\| rf\right\|_{Z}
&=
\max \left\{ \left\| rf\right\|_{L^{1}(\R^{n}_{+},s)},
\left\| rf\right\|_{L^{\infty }(\R^{n}_{+})}\right\}\\
&=
\max \left\{ \left\| f\right\|_{L^{1}(\R^{n}_{+},rs)},
\left\| f\right\|_{L^{\infty }(\R^{n}_{+},r)}\right\}\\
&=
\left\| f\right\|_{Y},
\end{align*}
so, $L_{r}$ is an isometric isomorphism.
\vspace{.2in}
Moreover, we can consider the locally convex space $\mathcal{F}$ given by:
\[
\mathcal{F}=
\{f\in Z: (\Delta_{\mu})^{k}f\in Z
\quad\mbox {for}\quad k=0,1,2,\cdots\}
=
\bigcap_{k=0}^{\infty}D\bigl[((\Delta_{\mu})_{Z})^{k}\bigr],\]
\noindent where with $(\Delta_{\mu})_{Z}$ we denote the part of $\Delta_{\mu}$ in $Z$. The space $\mathcal{F}$ is endowed with the topology generated by the family of seminorms given by
\[
\gamma_{m}(f)=
\max_{0\leq k\leq m}
\{\| (\Delta_{\mu})^{k}f \|_{Z}\},
\quad m=0,1,2,\cdots
\]
Thus, the space $\mathcal{F}$ verifies that is a Fréchet space and from Remarks \ref{Remark4.8 bis} and \ref{Remark4.9} we deduce that $\mathcal{F}\subset C_{0}(\R^{n}_{+})$, $\mathcal{F}\subset L^{p}(\R^{n}_{+},s)$ for all $1\leq p<\infty$, $\mathcal{F}\subset L^{\infty}(\R^{n}_{+})$, $\mathcal{H}_{\mu}\subset \mathcal{F}$ and the topology of $\mathcal{H}_{\mu}$ induced by $\mathcal{F}$ is weaker than the usual topology in $\mathcal{H}_{\mu}$. Moreover, the operator $\Delta_{\mu}$ verifies that
\[
\gamma_{m}(\Delta_{\mu}f)=
\max_{0\leq k\leq m}
\{\| (\Delta_{\mu})^{k}\Delta_{\mu}f \|_{Z}\}
=\gamma_{m+1}(f),
\]
\noindent for all $f\in \mathcal{F}$. Then $(\Delta_{\mu})_{\mathcal{F}}$ the part of $\Delta_{\mu}$ in $\mathcal{F}$, is a continuous
\[
(\Delta_{\mu})_{\mathcal{F}}:\mathcal{F}\to\mathcal{F}.
\]
\noindent If $f\in\mathcal{B}$, (see \eqref{eqB}), then $rf\in\mathcal{F}$ and
\begin{align*}
\gamma_{m}(rf)&=
\max_{0\leq k\leq m}
\{\| (\Delta_{\mu})^{k}rf \|_{Z}\}
=
\max_{0\leq k\leq m}
\{\| r(S_{\mu})^{k}r^{-1}rf \|_{Z}\}\\
&=
\max_{0\leq k\leq m}
\{\| r(S_{\mu})^{k}f \|_{Z}\}
=\max_{0\leq k\leq m}
\{\| (S_{\mu})^{k}f \|_{Y}\}\\
&=\rho_{m}(f),
\end{align*}
\noindent where we have consider \eqref{similaridad Op de Bessel}. So, the application $L_{r}:\mathcal{B}\to \mathcal{F}$, given by $L_{r}f=rf$ is an isomorphism of locally convex spaces with inverse given by $L_{r^{-1}}:\mathcal{F}\to \mathcal{B}$.
\begin{remark} Since $\mathcal{B}$ and $\mathcal{F}$ are isomorphic then we can deduce that $\mathcal{F'}$ is sequentially complete as $\mathcal{B'}$ is also sequentially complete (see Proposition \ref{B' secuencialmente completo}).
\end{remark}
So, if we consider the continuous operator $(S_{\mu})_{\mathcal{B}}:\mathcal{B}\to \mathcal{B}$, then by \eqref{similaridad Op de Bessel} we obtain the similarity relation,
\begin{equation}\label{relopBesselHir}
(\Delta_{\mu})_{\mathcal{F}}=
L_{r}\:(S_{\mu})_{\mathcal{B}}\:L_{r^{-1}}.
\end{equation}
\noindent We deduce by \eqref{relopBesselHir} the non-negativity of $-(\Delta_{\mu})_{\mathcal{F}}$ and by \cite[Proposition 1.1]{Mo18}, for $\alpha\in{\mathbb C}$, $\Re\alpha>0$, we have that
\begin{equation}\label{relopBessHirF}
(-(\Delta_{\mu})_{\mathcal{F}})^{\alpha}=L_{r}\:(-(S_{\mu})_{\mathcal{B}})^{\alpha}\:L_{r^{-1}}.
\end{equation}
\noindent Consequently,
\[
(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}=
((-(\Delta_{\mu})_{\mathcal{F}})^{*})^{\alpha}=
((-(\Delta_{\mu})_{\mathcal{F}})^{\alpha})^{*},
\]
where we have considered in the second equality that $\mathcal{F}$ is a Fréchet space, (see \cite[pp.134]{MS01}). Thus,
\begin{equation}\label{eqpotBessHirdistr1}
(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}=
(L_{r^{-1}})^{*}\:((-(S_{\mu})_{\mathcal{B}})^{\alpha})^{*}\:(L_{r})^{*}=
(L_{r^{-1}})^{*}\:(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:(L_{r})^{*},
\end{equation}
and for $T\in\mathcal{F'}$, $\phi\in\mathcal{F}$
\begin{align}
((-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}T,\phi)
&=
((L_{r^{-1}})^{*}\:(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:(L_{r})^{*}T,\phi)\nonumber\\
&=(T,L_{r}(-(S_{\mu})_{\mathcal{B}})^{\alpha}L_{r^{-1}}\phi).\label{eqpotBessHirdistr2}
\end{align}
From now on, we will use the notation:
\[(\Delta_{\mu})_{\mathcal{F}}=
x^{-\mu-\frac{1}{2}}\:
(S_{\mu})_{\mathcal{B}}\:
x^{\mu+\frac{1}{2}},
\]
\[(-(\Delta_{\mu})_{\mathcal{F}})^{\alpha}=
x^{-\mu-\frac{1}{2}}\:
(-(S_{\mu})_{\mathcal{B}})^{\alpha}\:
x^{\mu+\frac{1}{2}}\]
and
\[(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}=
x^{\mu+\frac{1}{2}}\:
(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:
x^{-\mu-\frac{1}{2}}\]
to refer to \eqref{relopBesselHir}, \eqref{relopBessHirF} and \eqref{eqpotBessHirdistr1}. In the last equation, the operators $x^{\mu+\frac{1}{2}}$ and $x^{-\mu-\frac{1}{2}}$ represent $(L_{r^{-1}})^{*}$ and $(L_{r})^{*}$, so,
\[x^{\mu+\frac{1}{2}}:\mathcal{B'}\to \mathcal{F'},\]
\[x^{-\mu-\frac{1}{2}}:\mathcal{F'}\to\mathcal{B'},\]
\noindent are given by
\[(x^{\mu+\frac{1}{2}}T_{1},\phi)=
(T_{1},x^{\mu+\frac{1}{2}}\phi),
\qquad (T_{1}\in \mathcal{B'}), (\phi\in\mathcal{F})\]
\[(x^{-\mu-\frac{1}{2}}T_{2},\psi)=
(T_{2},x^{-\mu-\frac{1}{2}}\psi),
\qquad (T_{2}\in \mathcal{F'}), (\psi\in\mathcal{B})\]
\vspace{.2in}
Now we are able to establish the following theorem:
\begin{theorem}
Let $u\in\mathcal{F'}$ and $\alpha\in{\mathbb C}$ with $\Re\alpha>0$. If $(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}u=0$ then there exists a polynomial $p$ such that $u=x^{2\mu+1}p(\|x\|^{2})$.
\end{theorem}
\begin{proof}
Let $u\in \mathcal{F'}$ such that $(-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}u=0$. Then
\begin{equation}\label{eqequal1}
((-(\Delta_{\mu})_{\mathcal{F'}})^{\alpha}u,\phi)=
(x^{\mu+\frac{1}{2}}\:
(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:
x^{-\mu-\frac{1}{2}}u,\phi)=0
\end{equation}
for all $\phi\in\mathcal{F}$. Since $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:x^{-\mu-\frac{1}{2}}u \in \mathcal{B'}$, then given $\psi \in\mathcal{B}$ and considering \eqref{eqequal1}, we obtain that
\[((-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:x^{-\mu-\frac{1}{2}}u,\psi)=
((-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:x^{-\mu-\frac{1}{2}}u,x^{\mu+\frac{1}{2}}x^{-\mu-\frac{1}{2}}\psi)=\]
\[(x^{\mu+\frac{1}{2}}(-(S_{\mu})_{\mathcal{B'}})^{\alpha}\:x^{-\mu-\frac{1}{2}}u,x^{-\mu-\frac{1}{2}}\psi)=0.\]
\noindent By Theorem \ref{Teorema5.1} we deduce that there exists a polynomial $p$ such that $x^{-\mu-\frac{1}{2}}u=x^{\mu+\frac{1}{2}}p(\|x\|^{2})$ and consequently $u=x^{2\mu+1}p(\|x\|^{2})$.
\smallskip
\end{proof}
\begin{corollary}
If $f\in L_{loc}^{1}(\R^{n}_{+})$, $f=O(x^{2\mu+1})$ and $(-(S_{\mu})_{\mathcal{B'}})^{\alpha}f=0$ then $f=C\;x^{2\mu+1}$.
\end{corollary}
|
\section{Introduction}
Seminal results by M. Dekking \cite{Dekking} and B. Host \cite{Host} state that eigenvalues of primitive substitution dynamical systems are always associated to continuous eigenfunctions. Thus the topological and measure theoretical Kronecker factors coincide. It is natural to ask whether this phenomenon is still true for other classes of minimal Cantor systems. Most of the answers we have are negative.
Substitution dynamical systems correspond to expansive minimal Cantor systems having a periodic or stationary Bratteli-Vershik representation \cite{dhs}. A natural class to explore extending the former one are linearly recurrent minimal Cantor systems, which correspond to those systems having a Bratteli-Vershik representation with a bounded number of incidence matrices. In \cite{lr} and \cite{necesariasuficiente} necessary and sufficient conditions based only on the combinatorial structure of the Bratteli diagrams are given for this class of systems, allowing to differentiate continuous and measure theoretical but non continuous eigenvalues. The more general class of topological finite rank minimal Cantor systems is explored in \cite{rangofinito}, providing new examples and conditions to differentiate the topological and measure theoretical Kronecker factors.
It is known that any countable subgroup of the torus $\mathbb{S}^1=\{z \in {\mathbb C} \ ; |z|=1 \}$ containing infinitely many rationals can be the set of eigenvalues of a Toeplitz system \cite{iwanik,toeplitzpreset}. Nevertheless, in the class of finite rank systems, Toeplitz systems exhibit a completely different behavior. Indeed, if a Toeplitz system is linearly recurrent then all its eigenvalues are associated to continuous eigenfunctions and if it has finite topological rank just a few extra non continuous eigenvalues can appear and they are rational \cite{rangofinito}.
So the assumption of finite topological rank restricts the possibilities of non continuous eigenvalues to some particular ones. The purpose of this work is to study the nature of these particular non continuous eigenvalues of finite rank Toeplitz systems.
\medskip
Our main result (Theorem \ref{theo:principal}) states a necessary and sufficient condition for $\lambda=\exp{(2i\pi a/b)}$,
where $a,b$ are integers with $(a,b)=1$, to be a non continuous eigenvalue of a finite topological rank Toeplitz system. This condition shows that non continuous eigenvalues are very rare and impose particular local orders to the associated Bratteli-Vershik representations. In addition, even if this condition looks abstract, it is easily computable and allows to produce concrete examples, showing particular behaviors of the group of eigenvalues in relation to the set of ergodic measures.
\medskip
The article is organized as follows. Section 2 contains the main definitions concerning eigenvalues of dynamical systems and Bratteli-Vershik representations, in particular the concept of Toeplitz minimal Cantor system of finite topological rank. In Section 3 we give the main result of the article and its corollaries. In particular, we exhibit a relation between the number of ergodic measures and the number of non continuous eigenvalues in the class of Toeplitz minimal Cantor systems of finite topological rank. Main technical lemmas used in the proofs are given in Section 4 and the proofs of the main result and its corollaries in Section 5. Finally, in Section 6 we provide several examples to illustrate the main result, its consequences and the fact that our condition is computable.
\section{Basic definitions}
\subsection{Dynamical systems and eigenvalues}
A \emph{topological dynamical system}, or just dynamical system, is a compact Hausdorff space $X$ together with a homeomorphism
$T:X\rightarrow X$. We use the notation $\left( X,T\right)$. If $X$ is a Cantor set
(i.e., $X$ has a countable basis of closed and open sets and it has no isolated points) we say that the system is Cantor. A dynamical system is \emph{minimal} if all orbits are dense in $X$,
or equivalently the only non empty closed invariant set is $X$.
A complex number $\lambda$ is a {\it continuous eigenvalue} of $(X,T)$ if there exists a continuous function $f : X\to {\mathbb C}$, $f\not = 0$, such
that $f\circ T = \lambda f$; $f$ is called a {\it continuous eigenfunction} (associated to $\lambda$). Let $\mu$ be a $T$-invariant probability measure, i.e., $T\mu = \mu$, defined on the Borel $\sigma$-algebra of $X$. A complex number $\lambda$ is an {\it eigenvalue} of the
dynamical system $(X,T)$ with respect to $\mu$ if there exists $f\in L^2 (X,\mu)$, $f\not = 0$, such that $f\circ T = \lambda f$; $f$ is
called an {\it eigenfunction} (associated to $\lambda$). If $\mu$ is ergodic, then every eigenvalue has modulus 1 and every eigenfunction has a constant modulus $\mu$-almost surely. Of course, continuous eigenvalues are eigenvalues.
\subsection{Bratteli-Vershik representations}
Let $(X,T)$ be a minimal Cantor system. It can be represented by an ordered Bratteli diagram together with the Vershik transformation acting on it. For details on this theory see \cite{hps} or \cite{review}. This couple is called a Bratteli-Vershik representation of the system. We give a brief outline of this construction emphasizing the notations in this paper.
\subsubsection{Bratteli diagrams}
A Bratteli diagram is an infinite graph $\left( V,E\right)$ which consists of a vertex set $V$
and an edge set $E$, both of
which are divided into levels $V=V_{0}\cup V_{1}\cup \ldots$, $E=E_{1}\cup E_{2}\cup \ldots$
and all levels are pairwise disjoint.
The set $V_{0}$ is a singleton $\{v_{0}\}$ and for all $n\geq 1$ edges in $E_{n}$ join vertices in $V_{n-1}$ to
vertices in $V_{n}$. It is also required that every vertex in $V_{n}$ is the ``end-point'' of some edge in $E_{n}$ for $n\geq 1$ and an ``initial-point'' of some edge in $E_{n+1}$ for $n\geq 0$. We set $\#V_{n}=d_{n}$ for all $n\geq 1$.
\smallskip
Fix $n\geq 1$. We call \emph{level} $n$ of the diagram to the subgraph consisting of the vertices in $V_{n-1}\cup V_{n}$
and the edges $E_{n}$ between these vertices. Level $1$ is called the \emph{hat} of the Bratteli diagram.
We describe the edge set $E_n$ using a $V_{n-1}\times V_{n}$ incidence matrix $M_{n}$ for which
its $(t_{1},t_{2})$ entry is the number of edges in $E_{n}$ joining vertex $t_{1}\in V_{n-1}$
with vertex $t_{2} \in V_{n}$. We also set $P_{n}=M_{2}\cdots M_{n}$ with the convention that $P_{1}=I$, where $I$ denotes the identity matrix. The number of paths joining $v_{0} \in V_{0}$ and a vertex $t\in V_{n}$ is given by coordinate $t$ of the \emph{height row vector} $h_{n}=\left(h_n(t) ; t \in V_{n} \right) \in {\mathbb N} ^{d_{n}}$. Notice that $h_{1}=M_{1}$ and $h_{n}=h_{1}P_{n}$.
\smallskip
We also consider several levels at the same time.
For integers $0 \leq m< n$ we denote by $E_{m,n}$ the set of all paths in the graph joining vertices of
$V_{m}$ with vertices of $V_{n}$. We define matrices $P_{m,n}=M_{m+1} \cdots M_{n}$ with the convention that $P_{n,n}=I$
for $1\leq m \leq n$. Clearly, coordinate $\matrizp{m}{n}{t_{1}}{t_{2}}$ of matrix $P_{m,n}$ is the number of paths in $E_{m,n}$ from vertex $t_{1} \in V_{m}$ to vertex $t_{2} \in V_{n}$.
It can be verified that $h_{n}=h_{m}P_{m,n}$.
\smallskip
We need to notice that the incidence matrices defined above correspond to the transpose of the matrices defined at the classical reference in this theory \cite{hps}. This choice is done to simplify the understanding and reading of the article.
\subsubsection{Ordered Bratteli diagrams and Bratteli-Vershik representations}
An \emph{ordered} Bratteli diagram is a triple \( B=\left( V,E,\preceq \right) \), where \( \left( V,E\right) \) is a Bratteli diagram and \( \preceq \) is a partial ordering on \( E \) such that: edges
\( e \) and \( e' \) are comparable if and only if they have the same end-point.
This partial ordering naturally defines maximal and minimal edges and paths. Also,
the partial ordering of $E$ induces another one on paths of $E_{m,n}$, where $0 \leq m < n$:
$\left(e_{m+1},\ldots,e_{n}\right) \preceq \left(f_{m+1},\ldots ,f_{n}\right)$ if and only if
there is $m+1\leq i\leq n$ such that $e_{j}=f_{j}$ for $i<j\leq n$ and $e_{i}\preceq f_{i}$.
Given a strictly increasing sequence of integers
$\left(n_{k}\right)_{k\geq 0}$ with $n_{0}=0$ one defines the \emph{contraction} or \emph{telescoping} of
$B=\left(V,E,\preceq \right)$ with respect to $\left(n_{k} \right)_{k\geq 0}$ as
$$\left(\left(V_{n_{k}}\right)_{k\geq
0},\left( E_{n_{k},n_{k+1}}\right)_{k\geq 0},\preceq \right), $$ where $\preceq$ is the order induced in each set of edges
$E_{n_{k},n_{k+1}}$. The converse operation is called {\it microscoping} (see \cite{hps} for more details).
\smallskip
Given an ordered Bratteli diagram \( B=\left( V,E,\preceq \right) \) one defines \( X_{B} \) as the set of infinite paths \( \left(
x_{1},x_{2},\ldots \right) \) starting in \( v_{0} \) such that for all \( n\geq 1 \) the end-point of \( x_{n}\in E_{n} \) is the
initial-point of \( x_{n+1}\in E_{n+1} \). We topologize \( X_{B} \) by postulating a basis of open sets, namely the family of \emph{cylinder sets}
$$
\left[ e_{1},e_{2},\ldots ,e_{n}\right]
=\left\{ \left( x_{1},x_{2},\ldots \right) \in X_{B} \textrm{ } ; \textrm{ }
x_{i}=e_{i},\textrm{ for }1\leq i\leq n\textrm{ }
\right\} .$$
Each \( \left[ e_{1},e_{2},\ldots ,e_{n}\right] \) is also closed, as is
easily seen, and so \( X_{B} \) is a compact, totally disconnected metrizable space.
When there is a unique \( \left( x_{1},x_{2},\ldots \right) \in X_{B} \) such that \( x_{n} \) is (locally) maximal for any \( n\geq 1 \) and a unique
\( \left( y_{1},y_{2},\ldots \right) \in X_{B} \) such that \( y_{n} \) is (locally) minimal for any \(n \geq 1 \), one says that \( B=\left(
V,E,\preceq \right) \) is a \emph{properly ordered} Bratteli diagram. Call these particular points \( x_{\mathrm{max}} \) and \(
x_{\mathrm{min}} \) respectively. In this case one defines the dynamic \( V_{B} \) over \( X_{B} \) called the \emph{Vershik map}. Let \( x=\left( x_{1},x_{2},\ldots \right) \in X_{B}\setminus \left\{ x_{\mathrm{max}}\right\} \) and let \(
n\geq 1 \) be the smallest integer so that \( x_{n} \) is not a maximal edge. Let \( y_{n} \) be the successor of \( x_{n} \) for the local order and \( \left(
y_{1},\ldots ,y_{n-1}\right) \) be the unique minimal path in \( E_{0,n-1} \) connecting \( v_{0} \) with the initial vertex of \( y_{n} \). One
sets \( V_{B}\left( x\right) =\left( y_{1},\ldots ,y_{n-1},y_{n},x_{n+1},\ldots \right) \) and \( V_{B}\left( x_{\mathrm{max}}\right)
=x_{\mathrm{min}} \).
The dynamical system \( \left( X_{B},V_{B}\right) \) is minimal. It is called the \emph{Bratteli-Vershik system} generated by \( B=\left( V,E,\preceq \right)
\). The dynamical system induced by any telescoping of \( B \) is topologically conjugate to \( \left( X_{B},V_{B}\right) \). In \cite{hps} it
is proved that any minimal Cantor system \( \left( X,T\right) \) is topologically conjugate to a Bratteli-Vershik system \( \left(
X_{B},V_{B}\right) \). One says that \( \left( X_{B},V_{B}\right) \) is a \emph{Bratteli-Vershik representation} of \( \left( X,T\right) \).
In what follows we identify $(X,T)$ with any of its Bratteli-Vershik representations.
\subsubsection{Minimal Cantor systems of finite topological rank}
A minimal Cantor system is of finite (topological) rank if it admits a Bratteli-Vershik representation such that the number of vertices per level is uniformly bounded by some integer $d$. The minimum possible value of $d$ is called the \emph{topological rank} of the system. We observe that topological and measure theoretical finite rank notions are completely different. For instance, systems of topological rank one correspond to odometers, whereas in the measure theoretical sense there are rank one systems that are expansive as classical Chacon's example.
To have a better understanding of the dynamics of a minimal Cantor system, and in particular
to understand its group of eigenvalues, one needs to work with a ``good'' Bratteli-Vershik representation.
In the context of minimal Cantor systems of finite rank $d$ we will consider representations verifying:
\smallskip
(H1) The entries of $h_{1}$ are all equal to $1$.
(H2) For every $n\geq 2$, $M_{n}>0$.
(H3) For every $n\geq 2$, $d_{n}$ is equal to $d$.
(H4) For every $n\geq 2$, all maximal edges of $E_n$ start in the same vertex of $V_{n-1}$.
\medskip
A Bratteli-Vershik representation of a minimal Cantor system $(X,T)$ verifying (H1), (H2), (H3) and (H4) will be called \emph{proper}. In this case, to simplify notations and avoid the excessive use of indexes, we will identify $V_{n}$ with $\{1,\ldots, d\}$ for all $n\geq 1$. The level $n$ will be clear from the context.
It is not difficult to prove that a minimal Cantor system of finite rank $d$ has a proper representation. We give a brief outline for completeness. We start from a given Bratteli-Vershik representation that we transform by telescoping. Condition (H1) follows by splitting the first level to separate all arrows in the hat and then duplicating accordingly the arrows of the second level. By minimality there is a telescoping of the diagram such that (H2) holds \cite{hps}. Another telescoping to the levels where $\#V_{n}=d$ produces (H3). Property (H4) follows from a compactness argument and a series of telescopings: if this is not possible, then we can construct two disjoint maximal points and we get a contradiction.
\medskip
A minimal Cantor system is \emph{linearly recurrent} if it admits a proper Bratteli-Vershik representation such that the set $\{M_{n};n\geq 1\}$ is
finite. Clearly, linearly recurrent minimal Cantor systems are of finite rank (see \cite{dhs}, \cite{du1}, \cite{du2} and \cite{lr} for more details on this class of systems).
\subsubsection{Associated Kakutani-Rohlin partitions}
Let $B=\left( V,E,\preceq \right)$ be a properly ordered Bratteli diagram and $(X,T)$ the associated minimal Cantor system. This diagram defines for each $n\geq 0$ a clopen {\it Kakutani-Rohlin} partition of $X$: for $n=0$,
${\mathcal P}_{0}=\{B_{0}(v_{0})\}$, where $B_{0}(v_{0})=X$, and for $n\geq 1$
$$
{\mathcal P}_{n}=\{T^{-j}B_{n}(t); t \in V_n, \ 0 \leq j < h_n(t) \} \ ,
$$
where $ B_n(t) = [e_1 , \dots ,e_n]$ and $(e_1 , \dots ,e_n)$ is the unique maximal path from $v_0$ to vertex $t \in V_{n}$. For each $t \in V_n$ the set $\{T^{-j}B_n(t); 0 \leq j < h_n(t) \}$ is called the {\it tower} $t$ of ${\mathcal P}_{n}$.
It corresponds to the set of all paths from $v_0$ to $t\in V_n$ (there are exactly $h_n(t)$ of such paths).
Denote by ${\mathcal T}_{n}$ the $\sigma$-algebra generated by the partition ${\mathcal P}_{n}$.
The map $\tau_n: X \to V_n$ is
given by $\tau_n(x)=t$ if $x$ belongs to tower $t$ of ${\mathcal P}_{n}$. The entrance time of
$x$ to $B_n({\tau_n(x)})$ is given by $r_n(x)=\min\{ j\geq 0; T^jx \in B_n({\tau_n(x)}) \}$.
\smallskip
For each $x=(x_1,x_{2},\ldots) \in X$ and $n\geq 0$ define the row vector
$s_n(x)\in {\mathbb N}^{d_{n}}$, called the {\it suffix vector of order $n$} of $x$, by
$$
s_n(x,t)=\# \{e \in E_{n+1}; x_{n+1} \preceq e, x_{n+1}\not = e, t \hbox{ is the initial vertex of } e\}
$$
at each coordinate $t\in V_n$.
A classical computation gives for all $n\geq 1$ (see for example \cite{necesariasuficiente})
\begin{align}
\label{eq:formulareturn} r_n(x)= s_0(x)+\sum_{i=1}^{n-1} \la s_i(x),h_{1}P_{i} \ra=s_0(x)+\sum_{i=1}^{n-1} \la s_i(x),h_{i} \ra \ ,
\end{align}
where $\la \cdot, \cdot \ra$ is the euclidean inner product. Observe that under the hypothesis (H1), i.e., $h_{1}=(1,\ldots,1)$, we have $s_0(x)=0$.
\subsubsection{Invariant measures}
Let $\mu$ be an invariant probability measure of the system $(X,T)$ associated to a properly ordered Bratteli diagram
$B$, like in the previous subsection.
It is determined by the values assigned to $B_n(t)$ for all $n\geq 0$ and $t \in V_n$. Define the column vector
$\mu_{n}=(\mu_n(t)\ ; \ t \in V_{n})$ with $\mu_n(t)=\mu(B_n(t))$.
A simple computation allows to prove the following useful relation:
\begin{equation}\label{eq:measure}
\mu_{m}=P_{m,n}\mu_{n}
\end{equation}
for integers $0\leq m < n$. Also, $\mu(\tau_{n}=t)=h_{n}(t) \mu_{n}(t)$ for all $n\geq 1$ and $t\in V_{n}$.
\subsubsection{Clean Bratteli-Vershik representations}
Let $B$ be a proper ordered Bratteli diagram of finite rank $d$ and $(X,T)$ the corresponding minimal Cantor system.
Recall that in this case we identify $V_{n}$ with $\{1,\ldots, d\}$ for all $n\geq 1$.
Then, by Theorem 3.3 in \cite{bkms}, there exist a telescoping of the diagram (which keeps the diagram proper)
and $\delta >0$ such that:
\begin{enumerate}
\item For any ergodic measure $\mu$ there exists $I_{\mu} \subseteq \{1,\ldots,d\}$ verifying:
\begin{enumerate}
\item $\mu(\tau_{n}=t) \geq \delta$ for every $t \in I_{\mu}$ and $n \geq 1$, and
\item $\lim_{n\to \infty} \mu(\tau_{n}=t)=0$ for every $t \not \in I_{\mu}$.
\end{enumerate}
\item If $\mu$ and $\nu$ are different ergodic measures then $I_{\mu}\cap I_{\nu}=\emptyset$.
\end{enumerate}
When an ordered Bratteli diagram verifies the previous properties we say it is \emph{clean}.
We remark that this is a modified version of the notion of \emph{clean} Bratteli diagram given in
\cite{rangofinito} that is inspired by the results of \cite{bkms}. This property will be very
relevant for formulating our main result. In \cite{bkms}, systems such that $I_{\mu}=\{1,\ldots,d\}$
for some ergodic measure
$\mu$ are called of \emph{exact finite rank}. Those systems are uniquely ergodic.
\smallskip
Let $\lambda\in \mathbb{S}^{1}$ be an eigenvalue of the system $(X,T)$ associated to $B$ for an ergodic measure $\mu$. Let $f\in L^{2}(X,\mu)$ be an associated eigenfunction with $|f|=1$.
For $n\geq 1$ define $c_n:V_n\to {\mathbb R}_0^{+}$ and $\rho_n:V_n\to [0,1)$ by the relation
\begin{equation}\label{eq:def_c_y_ro}
\frac{1}{\mu_n(t)}\int_{B_n(t)}f \, d\mu=c_n(t)\lambda^{-\rho_n(t)}, \textrm{\ \; for\ \; } t\in V_n.
\end{equation}
Notice that $0\leq c_{n}(t) \leq 1$.
The sequence $(f_n;n\geq 1)$ of conditional expectations of $f$ with respect to the sigma algebras $({\mathcal T}_{n}; n\geq 1)$ generated by the Kakutani-Rohlin partitions satisfies
$$f_n(x)=\econd{f}{\mathcal{T}_{n}}(x)=c_n(\tau_n(x))\lambda^{-r_n(x)-\rho_n(\tau_n(x))}.$$
It can be proved that $\lambda^{-(r_n+\rho_n\circ\torref{n})}$ converges $\mu$--a.e. (for a slightly deeper discussion we refer the reader to \cite{necesariasuficiente}).
Also, rephrasing a known result from \cite{rangofinito} we have
\begin{lemm}\label{lemm:c_tiende_a_uno}
If $B$ is a clean Bratteli diagram and $\mu$ an ergodic measure for the associated minimal Cantor system,
then
\begin{enumerate}
\item for any $t \in \{1,\ldots , d\}$, $\lim_{n\to \infty}\mu(\torref{n}=t)(c_n(t)-1)\to 0$,
\item for $t\in I_{\mu}$, $\lim_{n\to \infty} c_n(t)\to 1$.
\end{enumerate}
\end{lemm}
\subsection{Bratteli-Vershik systems of Toeplitz type}
A properly ordered Bratteli diagram $B=(V,E,\preceq)$ is of \emph{Toeplitz type} if for all $n\geq 1$
the number of edges in $E_{n}$ finishing at a fixed vertex of $V_{n}$ is constant independently of the vertex.
Denote this number by $q_n$ and set $p_n = q_1 q_{2} \cdots q_n$. Observe that $p_{n}$ is the number of paths from $v_{0}$ to any vertex of $V_{n}$.
Thus $h_{n}(t)=p_{n}$ for any $t \in V_{n}$. We say that $(q_n;n\geq 1)$ is the \emph{characteristic sequence} of the diagram.
This class was obtained in \cite{gjtoeplitz} when characterizing Toeplitz
subshifts.
The main object in this study are eigenvalues of minimal Cantor systems of finite rank $d$, having a proper Bratteli-Vershik representation of Toeplitz type. It is known that finite rank minimal Cantor systems are either odometers or subshifts \cite{DM}, so in our study we will be dealing only with Toeplitz subshifts or odometers.
To state our main results we will need some extra notations. Fix a minimal Cantor system $(X,T)$ with a Toeplitz type proper Bratteli-Vershik representation of rank $d$ and characteristic sequence $(q_n;n\geq 1)$.
For $0 \leq m < n$ define $q_{m,n}=q_{m+1}\cdots q_{n}$,
the number of paths in $E_{m,n}$ finishing
in any fixed vertex $t\in V_n$. Clearly $q_{\ell,n}=q_{\ell,m}q_{m,n}$ if $0\leq \ell < m <n$.
Also, for $x=(x_{1}, x_{2}, \ldots) \in X$ define the integer $\sufijofbb{m}{n}(x)$ as the number of paths
in $E_{m,n}$ which end at $\torref{n}(x)$ that are strictly bigger than
$(x_{m+1},\ldots, x_{n})$ with respect to the induced partial order in $E_{m,n}$.
Finally, define the set $\sufijocbb{m}{n}{t_{1}}{t_{2}}$ for $t_{1}\in V_m$ and $t_{2}\in
V_n$ by $$\sufijocbb{m}{n}{t_{1}}{t_{2}}=\set{\sufijofbb{m}{n}(x); \tau_m(x)=t_{1} \; \textrm{and} \; \tau_n(x)=t_{2}}.$$
It is not difficult to prove that the cardinality of $\sufijocbb{m}{n}{t_{1}}{t_{2}}$ is equal to
$\matrizp{m}{n}{t_{1}}{t_2}$, that is, the number of paths from $t_{1} \in V_{m}$ to $t_{2} \in V_{n}$.
If necessary, to simplify notations we will denote $\sufijocbb{n}{n+1}{{t_{1}}}{t_{2}}$ by
$\sufijocb{n}{t_{1}}{t_{2}}$ and $\sufijofbb{n}{n+1}$ by $\sufijofb{n}$.
Notice that $\sufijofb{n}(x)=\la s_{n}(x), (1,\ldots,1) \ra=\sum_{t\in V_{n}} s_{n}(x,t)$ for any $x\in X$.
\medskip
We will need the following simple relations.
For $0\leq \ell < m < n$, $ t_{1}\in V_{\ell}$ and $x\in X$ the following equalities hold:
\begin{eqnarray}
\label{eq:retorno_toeplitz} r_\ell(x) &=& \sufijofb{0}(x) + \sum_{i=1}^{\ell-1}p_i\sufijofb{i}(x), \\
\label{eq:sufijo_dos_niveles}\sufijofbb{\ell}{m}(x) &=& \sufijofb{\ell}(x)+\sum_{i=\ell+1}^{m-1}q_{\ell+1}q_{\ell+2}\cdots q_{i}\sufijofb{i}(x) \nonumber \\
&=& \frac{r_m(x)-r_\ell(x)}{p_\ell}, \\
\label{eq:interpolacion_sufijos} \sufijofbb{\ell}{n}(x) &=& \sufijofbb{\ell}{m}(x)+q_{\ell,m}\sufijofbb{m}{n}(x),
\end{eqnarray}
\begin{eqnarray}
\label{eq:base_en_bases_superiores} B_{\ell}(t_{1})&=&\bigcup_{t_{2}\in V_{m}}\bigcup_{s\in\sufijocbb{\ell}{m}{t_{1}}{t_{2}}}T^{-p_\ell s}B_{m}(t_{2}),
\end{eqnarray}
where the union in the right hand side is disjoint.
\section{Eigenvalues of Toeplitz systems of finite rank}
As was mentioned in the introduction, any countable subgroup of $\mathbb{S}^1=\{z \in {\mathbb C} \ ; |z|=1 \}$ containing infinitely many rationals can be the set of eigenvalues of a Toeplitz subshift for a given invariant measure \cite{iwanik,toeplitzpreset}.
Also, $\exp(2 i \pi \, \alpha) \in \mathbb{S}^{1}$ is a continuous eigenvalue of a minimal Cantor system with a Toeplitz type proper Bratteli-Vershik representation
if and only if $\alpha = a/p_m$ for some $a \in \mathbb Z$ and $m\geq 1$ \cite{Williams,Keane}.
A direct proof can be given using the particular combinatorial structure of the Brattelli-Vershik representation
of a minimal Cantor system of Toeplitz type. We sketch it here.
Using \eqref{eq:retorno_toeplitz} and the fact that $p_{m}$ divides $p_{n}$ when $m\leq n$, one gets that
$r_{n}(x) /p_{m} = ({\bar s}_{0}(x) +\sum_{i=1}^{m-1} p_{i}{\bar s_{i}}(x))/p_{m} \mod \mathbb Z$, for all $n\geq m$.
Hence, $\exp(2i\pi r_{n}(x) /p_{m})$ converges uniformly when $n\to \infty$, which is a necessary and sufficient condition for
$\exp(2i\pi /p_{m})$, and thus $\exp(2 i \pi \ a/p_{m})$ for every $a \in \mathbb Z$, to be continuous eigenvalues in this context (see Proposition 12 in \cite{necesariasuficiente}).
In the opposite direction, using the same criterion, if
$\exp(2i\pi /b)$ with $b \in \mathbb Z$ is a continuous eigenvalue, then
$(r_{n+1}(x)-r_{n}(x))/b=p_{n}{\bar s}_{n}(x)/b \mod \mathbb Z$ is close to $0$
for any large enough $n \geq 1$ and uniformly in $x$. Taking a point $x$ such that ${\bar s}_{n}(x)=1$ allows to conclude that $1/b=a/p_{n}$ for some large $n\geq 1$ and $a\in \mathbb Z$. More details about continuous eigenvalues of Toeplitz type Bratteli-Vershik systems can be found in \cite{rangofinito}.
In the class of minimal Cantor systems with a Toeplitz type representation, the assumption of finite topological rank restricts the possibilities for non continuous eigenvalues. But, importantly, all are rational. In addition, if the characteristic sequence of a proper representation is bounded (or equivalently, a proper representation gives a linearly recurrent system), then all the eigenvalues are continuous. The following theorem gives a very restrictive condition verified by non continuous eigenvalues of Toeplitz systems in the finite rank case that are not linearly recurrent.
\begin{theo}{\cite{rangofinito}}
\label{theo:vpsonracionales}
Let $(X,T)$ be a minimal Cantor system with a Toeplitz type proper Bratteli-Vershik representation of rank $d$ and characteristic sequence $(q_n;n\geq 1)$. Let $\mu$ be an ergodic probability measure. If $\exp(2i \pi \, a/b)$, with $(a,b)=1$, is a non continuous
rational eigenvalue of $(X,T)$ for $\mu$, then $b/(b,p_n) \leq d$ for all $n$ large enough.
\end{theo}
Let $\lambda=\exp(2 i\pi \, a/b)$, with $a,b$ integers such that $(a,b)=1$, be a non continuous rational eigenvalue as in the previous theorem. We notice that $b/(b,p_n) >1$ for all $n$ large enough. Indeed, if $b/(b,p_n)=1$ for some $n\geq 1$, then
$1/b=a'/p_{n}$ for some $a' \in \mathbb Z$, which by the discussion above implies that $\exp(2i\pi \, a/b)$ is a continuous eigenvalue. Also, observe that $(b,p_n)$ is a non decreasing sequence of integers bounded by $b$, so $b/(b,p_n)$ is eventually constant, say equal to $\mathbf{b}$. Since we are considering proper representations, the fact that $\mathbf{b}>1$ implies that $(q_n;n\geq 1)$ tends to infinity with $n$. Otherwise, the system will be linearly recurrent, and thus all eigenvalues will be continuous, which implies that $b/(b,p_n)=1$ for some $n>1$.
\medskip
Now we state our main result,
\medskip
\begin{theo}
\label{theo:principal}
Let $(X,T)$ be a minimal Cantor system with a Toeplitz type proper and clean Bratteli-Vershik representation of rank $d$ and characteristic sequence $(q_n;n\geq 1)$. Let $\mu$ be an ergodic probability measure.
Then, $\lambda=\exp(2i\pi a/b)$, with $a,b$ integers such that $(a,b)=1$, is a non continuous eigenvalue of $(X,T)$ for $\mu$ if and only if
\begin{enumerate}
\item
$b/(b,p_n)=\mathbf{b}$ for all $n$ large enough and some $1<\mathbf{b}\leq d$, and
\item\label{mainsecond}
for all $t_2\in I_{\mu}$
$$\sum_{t_{1}\in V_{m}}\frac{\abs{\sum_{s\in\sufijocbb{m}{n}{t_{1}}{t_2}}\lambda^{-p_ms}}}{q_{m,n}}\xrightarrow[m,n\to\infty]{}1,$$
uniformly in $m,n \in {\mathbb N}$ with $m<n$.
\end{enumerate}
\end{theo}
As was mentioned in the introduction, even if this condition looks ``heavy'' to check, in fact it is easy to verify and construct examples fulfilling it. This will be illustrated in Section \ref{examples}. The main tool is provided by the following corollary that follows from the construction in the proof of Theorem \ref{theo:principal}.
\smallskip
\begin{coro}
\label{coro:cocycle}
Let $(X,T)$ be a minimal Cantor system with a Toeplitz type proper and clean Bratteli-Vershik representation of rank $d$
and characteristic sequence $(q_n;n\geq 1)$. Let $\mu$ be an ergodic probability measure.
Let $(q_n;n\geq 1)$ be its characteristic sequence.
Then, $\lambda=\exp(2i\pi a/b)$, with $a,b$ integers such that $(a,b)=1$, is a non continuous eigenvalue of $(X,T)$ for $\mu$ if and only if up to a telescoping of the diagram we have
\begin{enumerate}
\item
$p_{n} = p \mod b$ for some $p \in \{0,\ldots,b-1\}$ and for all $n\geq 2$,
\item
$b/(b,p_n)=\mathbf{b}$ for all $n$ large enough and some $1<\mathbf{b}\leq d$,
\item
there exists a map
$k(\cdot,\cdot): \{1,\ldots,d \}\times \{1,\ldots,d \} \to \{0,\ldots,\mathbf{b}-1\}$
such that
\begin{align*}
p \cdot k (t_{1},t_{3}) &= p \cdot k (t_{1},t_{2}) + p \cdot k (t_{2},t_{3}) \mod b, \\
p \cdot k (t_{1},t_{1}) &= 0 \mod b, \ \ p \cdot k (t_{1},t_{2}) = -p \cdot k (t_{2},t_{1}) \mod b,
\end{align*}
for all $t_{1},t_{2}, t_{3}\in I_{\mu}$,
\item for $\mu$--almost every point $x\in X$ the equality $\bar s_{n}(x)=k(\tau_{n}(x),\tau_{n+1}(x)) \mod \mathbf{b}$ holds
for all large enough $n\in {\mathbb N}$.
\end{enumerate}
\end{coro}
In what follows we provide a number of reformulations and corollaries of the main theorem.
Some proofs are left to the reader since they can be easily deduced from a direct computation or Lemmas \ref{lemm:suma_matriz_coef_equivalentes} and \ref{lemm:indice_nivelmenor_sobre} provided below, others will be proved near the end of Section \ref{mainproofs} after proving the main theorem.
We start by a natural reformulation of Theorem \ref{theo:principal}. It says that we can replace $V_{m}$ by $I_{\mu}$ in the sum of statement (2) of the theorem. In other words, we only need to consider the vertices of the diagram determining the measure $\mu$.
We will need the following observation: for $t_{1}\not\in I_{\mu}$ and $t_2\in I_{\mu}$ one has
\begin{equation}\label{eq:p_tiende_a_cero_fuera_de_I}
\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}\xrightarrow[m,n\to\infty]{}0
\end{equation}
uniformly in $m,n \in {\mathbb N}$ with $m<n$. Indeed, since the diagram is clean,
$\mu(\torref{n}=t_2) \geq \delta > 0$ and $\lim_{m\to \infty}\mu(\torref{m}=t_1)=0$. These facts, together with the
|
following inequalities
$$\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}} \cdot \delta \leq
\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}\mu(\torref{n}=t_2)=\mu(\torref{m}=t_1, \torref{n}=t_{2})\leq \mu(\torref{m}=t_{1}),
$$
allow to deduce \eqref{eq:p_tiende_a_cero_fuera_de_I}. Since the cardinality of $\sufijocbb{m}{n}{t_{1}}{t_2}$ is equal to
$\matrizp{m}{n}{t_{1}}{t_2}$, we also deduce that
$$\sum_{t_{1}\in V_{m}\setminus I_{\mu}}\frac{\abs{\sum_{s\in\sufijocbb{m}{n}{t_{1}}{t_2}}\lambda^{-p_ms}}}{q_{m,n}}
\leq \sum_{t_{1}\in V_{m}\setminus I_{\mu}}\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}.$$
Therefore, a direct application of \eqref{eq:p_tiende_a_cero_fuera_de_I} in the last inequality allows to reformulate Theorem \ref{theo:principal} as follows.
\begin{coro}[Variation on Theorem \ref{theo:principal}]\label{theo:principal_var1}
The complex number $\lambda=\exp(2i\pi a/b)$, with $a,b$ integers such that $(a,b)=1$, is a non continuous eigenvalue of $(X,T)$ for $\mu$ if and only if
\begin{enumerate}
\item
$b/(b,p_n)=\mathbf{b}$ for all $n$ large enough and some $1<\mathbf{b}\leq d$, and
\item
for all $t_2\in I_{\mu}$
$$\sum_{t_{1}\in I_{\mu}}\frac{\abs{\sum_{s\in\sufijocbb{m}{n}{t_{1}}{t_2}}\lambda^{-p_ms}}}{q_{m,n}}\xrightarrow[m,n\to\infty]{}1,$$
uniformly in $m,n \in {\mathbb N}$ with $m<n$.
\end{enumerate}
\end{coro}
\bigskip
The following corollary is a reformulation of the main condition of Theorem \ref{theo:principal} and the corresponding one in Corollary \ref{theo:principal_var1}. It follows almost directly by combining Lemmas \ref{lemm:suma_matriz_coef_equivalentes} and \ref{lemm:indice_nivelmenor_sobre} in the next section, so its proof is left to the reader.
\begin{coro}\label{cor:condwithpartition}
The main condition in Theorem \ref{theo:principal} (resp. Corollary \ref{theo:principal_var1}) is equivalent to:
for all $t_2\in I_{\mu}$ and $m\geq 1$ there exists a sequence of partitions $({\mathcal H}_{m,n,t_2}; m<n)$ of
$V_{m}$ (resp. of $I_{\mu}$) with $\#{\mathcal H}_{m,n,t_2}=\mathbf{b}$ such that
$$\sum_{t_{1}\in A}\frac{\abs{\sum_{s\in\sufijocbb{m}{n}{t_{1}}{t_2}}\lambda^{-p_ms}}}{q_{m,n}}\xrightarrow[m,n\to\infty]{}\frac{1}{\mathbf{b}},$$
uniformly in $m,n \in {\mathbb N}$ with $m<n$ for any $A\in{\mathcal H}_{m,n,t_2}$.
\end{coro}
\medskip
This formulation pinpoints to the possible local orders that accept a Bratteli-Vershik representation to have non continuous eigenvalues. Part (3) of Lemma \ref{lemm:suma_matriz_coef_equivalentes} states that the main condition of Theorem \ref{theo:principal} (or its equivalent formulations) implies that the local order of most of the arrows from a vertex in an atom $A \in {\mathcal H}_{m,n,t_2}$ to $t_{2} \in I_{\mu}$ at level $n$ must be congruent modulo $\mathbf{b}$.
This condition is one of the main tools to explore non continuous rational eigenvalues of Toeplitz systems.
\medskip
Another interesting fact is that we can relate non continuous eigenvalues with the number of ergodic invariant measures of a Toeplitz system. Let $(X,T)$ be a minimal Cantor system and $\mu$ an ergodic measure as in Theorem \ref{theo:principal}.
Define,
$$
\mathbf{B}_{\mu}=\{\lim_{m\to \infty}b/(b,p_m) ; b \in {\mathbb N}, \ \exp(2i\pi/b) \textrm{ is a non continuous eigenvalue for } \mu \}
$$
and endow it with the divisibility (partial) order. Recall that $\lim_{m\to \infty}b/(b,p_m)$ is equal to $\mathbf{b}=b/(b,p_n)$ for a large $n \in {\mathbb N}$.
Denote by $\mathcal{M}_{erg} (X,T)$ the set of ergodic measures of $(X,T)$ and consider the set $\mathcal{M}$ defined by:
$$
\mathcal{M}=\left\{\mu \in \mathcal{M}_{erg}(X,T) \ ; \ \mathbf{B}_{\mu}\not = \emptyset \right \}.
$$
\begin{coro}
\label{cor:alternative} The following properties hold:
\begin{enumerate}
\item For any $\mu \in \mathcal{M}$ and $\mathbf{b} \in \mathbf{B}_{\mu}$, $\mathbf{b} \leq \# I_{\mu}$.
\item For any $\mu\in \mathcal{M}$, $\mathbf{B}_{\mu}$ has a unique divisibility-maximal element $\mathbf{b}_{\mu}$.
\item
$\sum_{\mu \in \mathcal{M}} \mathbf{b}_{\mu}\leq d$.
\item
$\# \mathcal{M} \leq \# \mathcal{M}_{erg} (X,T)\leq d-\sum_{\mu \in \mathcal{M}} (\mathbf{b}_{\mu}-1)$.
\end{enumerate}
\end{coro}
\medskip
The proof of this corollary will be given at the end of Section \ref{mainproofs}.
\medskip
Fix an ergodic measure $\mu$. To understand better the last corollary let us suppose the $p_{n}$'s are powers of the same prime number. In this case, for all integers $b$ such that $\lambda=\exp(2i\pi/b)$ is a non continuous eigenvalue for $\mu$ one has $(b,p_{n})=1$ and parts (1) and (2) of last corollary tell us that there is a unique
$b=\mathbf{b}_{\mu} \leq \#I_{\mu} \leq d$ which is maximal in $\mathbf{B}_{\mu}$. All other non continuous eigenvalues for $\mu$
are powers of $\lambda$. If $\mathbf{B}_{\mu}$ is empty, no non continuous eigenvalues exist for $\mu$. Notice that property (1) implies that we need at least $\mathbf{b}_{\mu}$ vertices to have the non continuous eigenvalue $\lambda$. Since $I_{\mu} \cap I_{\nu}=\emptyset$ for different ergodic measures $\mathbf{b}_{\nu} \leq d-\#I_{\mu} \leq d -\mathbf{b}_{\mu}$. We will see in some examples of Section \ref{examples} that these inequalities can be strict.
\medskip
In the particular case when $\mathbf{b}_{\mu}=d$ for some ergodic measure $\mu$ we get the following corollary.
\begin{coro}
\label{coro:ue}
Consider $\lambda=\exp{(2 i \pi a/b)}$, with $a,b$ integers such that $(a,b)=1$ and $b/(b,p_n)=d$ for all $n$ large enough. Then $\lambda$ is a non continuous eigenvalue of $(X,T)$ for the invariant measure $\mu$ if and only if
for all $t_1, t_{2} \in \{1,\ldots,d\}$
\begin{equation}\label{eq:coro7}
\frac{\abs{\sum_{s\in\sufijocbb{m}{n}{t_1}{t_2}}\lambda^{-p_ms}}}{q_{m,n}}\xrightarrow[m,n\to\infty]{}\frac{1}{d}
\end{equation}
uniformly in $m,n \in {\mathbb N}$ with $m<n$.
If $\lambda$ is an eigenvalue, then:
\begin{enumerate}
\item\label{uergodic}
the system $(X,T)$ is uniquely ergodic and $\mu$ is the unique invariant measure,
\item
\label{theo:principal1}
for all $t\in \{1,\ldots, d\}$, $\displaystyle\lim_{n\to\infty}\mu(\tau_n=t)=1/d$.
\end{enumerate}
\end{coro}
Condition \eqref{eq:coro7} and statement \eqref{uergodic} follow almost directly from Corollaries \ref{cor:condwithpartition} and \ref{cor:alternative}. Nevertheless, we provide a complete proof of the corollary at the end of Section \ref{mainproofs}.
\medskip
An analogous result to Corollary \ref{coro:ue} can be obtained
when the system is uniquely ergodic and $b/(b,p_n)=\#I_{\mu}$ for all $n$ large enough. The statement is obtained by replacing $d$ by $\#I_{\mu}$ and the set $\{1,\ldots,d\}$ by $I_{\mu}$ in the last corollary.
\section{Main technical lemmas}
In this section we will provide the main ingredients we need to prove Theorem \ref{theo:principal} and its corollaries.
\subsection{A geometric lemma}
The next lemma can be stated in a much more general situation and its proof follows from general facts of convex analysis. Nevertheless, since we consider a particular case, we provide a simple self-contained proof.
\begin{lemm}\label{lemm:geometrico}
Let $N$ be a positive integer. Then there exists a constant $C$ such that for any convex combination
$w=\sum_{j=0}^{N-1} \alpha_{j}\xi^{j}$ of the $N$-th roots of unity $1,\xi,\ldots,\xi^{N-1}$ verifying $1-\varepsilon<\abs{w}\leq 1$ for some $\varepsilon >0$ one has
$$1-C\varepsilon< \alpha_{i}\leq 1$$ for some $0\leq i\leq N-1$.
\end{lemm}
\begin{proof}
A proof is given only in the case when $\abs{w}\neq 1$. Write $w$ in the following way $$w=\alpha_{i}\xi^{i}+\beta\zeta,$$ where
$\alpha_{i}\geq 1/N$ and $\alpha_{i}+\beta=1$ (note that $\zeta$ belongs to the convex hull of the $N$-th roots of unity different from
$\xi^{i}$).
The function $F(z)=\alpha_{i}\xi^{i}+\beta z$ has maximal absolute value at $z\in \{\xi^{i-1}, \xi^{i+1}\}$ when restricted to the convex
hull of the $N$-th roots of unity different from $\xi^{i}$. Hence
\begin{eqnarray*}
1-\varepsilon &<& \abs{w}\ \; \; (=\abs{F(\zeta)}) \\
&\leq& \abs{F(\xi^{i+1})} \\
&= & \abs{1+\beta(\xi-1)} \\
&= & \sqrt{1-2\beta(1-\beta)(1-\cos{2\pi/N})}\\
&\leq & {1-\beta(1-\beta)(1-\cos{2\pi/N})}\\
&\leq& 1-\beta\left(\frac{1-\cos(2\pi/N)}{N}\right)
\end{eqnarray*}
and $$\alpha_{\ell}>1-\left(\frac{N}{1-\cos(2\pi/N)}\right)\varepsilon.$$
\end{proof}
\subsection{Special telescoping of a Bratteli-Vershik system}
At some point of the proof of Theorem \ref{theo:principal} we will need to telescope an ordered Bratteli diagram in the following particular way.
\begin{lemm}
\label{lemma:contraction-partition}
Let $B=\left(V,E,\preceq \right)$ be an ordered Bratteli diagram such that
$\#V_n = d$ for all $n\geq 1$ and identify $V_{n}$ with $\{1,\ldots,d \}$.
For all $1\leq m<n $ and $t\in \{1,\ldots,d \}$ consider $(\mathcal{G}_{m,n,t},\leq_{m,n,t})$, where
$\mathcal{G}_{m,n,t}$ is a partition of $V_m$ and $\leq_{m,n,t} $ is a total ordering on the atoms of
$\mathcal{G}_{m,n,t}$.
Then, there exists a strictly increasing sequence $(n_k)_{k\geq 0}$ in ${\mathbb N}$
such that for all $k_0\geq 0$, $k> k_0$ and $t\in \{1,\ldots,d \}$ we have
$$(\mathcal{G}_{n_{k_0},n_k,t} ,\leq_{n_{k_0},n_k,t}) = (\mathcal{G}_{n_{k_0},n_{k+1},t} , \leq_{n_{k_0},n_{k+1},t} ).$$
\end{lemm}
\begin{proof}
It suffices to remark that there are finitely many such structures on $\{ 1, \ldots, d \}$
(partitions endowed with total orderings). Then, one proceeds by induction using the pigeon hole principle.
Let us give some details. Take $n_0 = 1$.
By the pigeon hole principle, there exists a strictly increasing sequence $(n^{(0)}_k)_{k\geq 0}$,
with $n^{(0)}_0>n_0$, such that for all $k\geq 0$ and $t\in \{ 1, \ldots , d\}$ we have
$$(\mathcal{G}_{n_0,n^{(0)}_k ,t} , \leq_{n_0,n^{(0)}_k ,t}) = (\mathcal{G}_{{n_0},n^{(0)}_{k+1},t} , \leq_{{n_0},n^{(0)}_{k+1},t} ).$$
Now, let $n_1 = n^{(0)}_0$.
Using the same argument, there exists a strictly increasing subsequence $(n^{(1)}_k)_{k\geq 0}$ of $(n^{(0)}_k)_{k\geq 0}$, with $n^{(1)}_0>n^{(0)}_0$, such that for all $k\geq 0$ and $t\in \{ 1, \ldots , d\}$ we have $(\mathcal{G}_{n_1,n^{(1)}_k , t} , \leq_{n_1,n^{(1)}_k , t} ) = (\mathcal{G}_{{n_1},n^{(1)}_{k+1},t} , \leq_{{n_1},n^{(1)}_{k+1},t} )$.
Observe that we also have $(\mathcal{G}_{n_0,n^{(1)}_k , t} , \leq_{n_0,n^{(1)}_k , t} ) =
(\mathcal{G}_{{n_0},n^{(1)}_{k+1},t} , \leq_{{n_0},n^{(1)}_{k+1},t} ) $ for all $k \geq 0$ and $t\in \{ 1, \ldots , d\}$ by construction.
Proceeding in this way we obtain the desired sequence $(n_k)_{k\geq 0}$.
\end{proof}
\subsection{Uniform lower bound for consecutive towers in $I_{\mu}$}
\begin{lemm}
\label{lemma:low-independence}
Let $(X,T)$ be a minimal Cantor system with a Toeplitz type proper and clean Bratteli-Vershik representation of rank $d$ and $\mu$ be an ergodic probability measure. Let $(q_n;n\geq 1)$ be its characteristic sequence. For all $m\geq 1$,
there exists $n_0 > m$ such that for all $n\geq n_0$ and $t_1,t_2\in I_{\mu}$
$$
\frac{\matrizp{m}{n}{t_{1}}{t_{2}}}{q_{m,n}} \geq \frac{\delta}{3},
$$
where $\delta >0$ is such that $\mu(\tau_{n}=t)\geq \delta$ for any $t \in I_{\mu}$ and $n \in {\mathbb N}$ (coming from the cleanliness property of the diagram).
\end{lemm}
\begin{proof}
Fix $m\geq 1$ and $0< \epsilon < \delta^2/3$.
From Egorov's theorem and the ergodic theorem, there exists a measurable subset
$A_\epsilon$ with $\mu (A_\epsilon )\geq 1-\epsilon$ and a positive integer $M_0$ such that
for all $x\in A_\epsilon$ and $M\geq M_0$ we have
\begin{equation}
\label{Egorov}
\left |\frac{1}{M}\sum_{k=0}^{M-1} 1_{\{ \tau_m = t_1\}} (T^k x) - \mu ( \tau_m = t_1) \right | <\epsilon .
\end{equation}
Let $n > m$ be such that $p_n\geq M_0$ (recall that $p_{n}$ is the number of paths from $v_{0}$ to any vertex of $V_{n}$). There exists $x\in A_\epsilon \cap T^{-p_n-j+1} B_n(t_2)$ for some $0\leq j\leq \lfloor\frac{\epsilon p_n}{\delta}\rfloor < p_{n}$.
Indeed,
$$\mu\left(\bigcup_{j=0}^{\lfloor\frac{\epsilon p_n}{\delta} \rfloor} T^{-(p_{n}+j-1)}B_n({t_{2}})\right )=
\left (\left \lfloor\frac{\epsilon p_n}{\delta} \right \rfloor+1 \right ) \mu(B_n({t_2}) ) > \frac{\epsilon}{\delta} \mu(\tau_{n}=t_{2}) \geq \epsilon,$$
since $\mu(\tau_{n}=t_{2})= p_{n}\mu(B_n({t_2}))$ and $t_{2} \in I_{\mu}$. Hence,
$\bigcup_{j=0}^{\lfloor\frac{\epsilon p_n}{\delta} \rfloor} T^{-(p_{n}+j-1)}B_n({t_{2}})$ must intersect $A_{\epsilon}$.
Notice that the iterates $T^{j}x,\ldots, T^{j+p_{n}-1}x$ cross completely tower $t_{2} \in V_{n}$, from the lowest to the highest level. So those iterates enter to tower $t_{1} \in V_{m}$ exactly ${\matrizp{m}{n}{t_{1}}{t_{2}}} p_m$ times.
Then, since $t_{1} \in I_{\mu}$, $p_{n}+j \geq M_{0}$ and $x \in A_{\epsilon}$, we can use \eqref{Egorov} to get
\begin{align*}
\delta - \epsilon & \leq \mu ( \tau_m = t_1 ) - \epsilon \leq
\frac{1}{p_n+j}\sum_{k=0}^{p_n+j-1} 1_{\{ \tau_m = t_1\}} (T^k x) \\
& \leq \frac{j}{p_n+j} + \frac{1}{p_n+j}\sum_{k=0}^{p_n-1} 1_{\{ \tau_m = t_1\}} (T^k (T^{j}x))\\
& \leq \frac{\epsilon}{\delta} + \frac{{\matrizp{m}{n}{t_{1}}{t_{2}}} p_m}{p_n+j}
\leq
\frac{\epsilon}{\delta} + \frac{{\matrizp{m}{n}{t_{1}}{t_{2}}} }{q_{m,n}} \leq
\frac{\delta}{3}+ \frac{{\matrizp{m}{n}{t_{1}}{t_{2}}} }{q_{m,n}},
\end{align*}
which ends the proof.
\end{proof}
\subsection{Equivalent conditions for Theorem \ref{theo:principal}}
We follow the same notations as in Theorem \ref{theo:principal}: $\lambda=\exp(2i\pi a/b)$, with $(a,b)=1$,
and $\mathbf{b}$ is the limit in $n$ of $b/(b,p_n)$, which is attained from some large $n \in {\mathbb N}$.
In the sequel, equality modulo $\mathbf{b}$ and $b$ will be written $\equiv_\mathbf{b}$ and $\equiv_b$ respectively.
To make the text lighter, we need to introduce some extra notations.
For $t_1, t_{2}\in \{1,\ldots,d\}$, $k \in \{0,\ldots,\mathbf{b}-1\}$ and integers $1\leq m <n$, set
\medskip
\begin{eqnarray}
\label{eq:def_suma}\suma{m}{n}{t_1}{t_2} &=& \sum\nolimits_{s\in\sufijocbb{m}{n}{t_1}{t_2}}\lambda^{-p_ms} \textrm{\ \; \; \; }\\
\label{eq:def_coef}\sumacoef{m}{n}{t_1}{t_2}{k} &=& \#\set{s\in\sufijocbb{m}{n}{t_1}{t_2}; \ s\equiv_{\mathbf{b}} k }
\end{eqnarray}
\medskip
Notice that for $s,s'\in\sufijocbb{m}{n}{t_1}{t_2}$, $\lambda^{-p_m s}=\lambda^{-p_m s'}$ if and only if $s\equiv_{\mathbf{b}} s'$. Then,
\begin{eqnarray}
\label{eq:suma_en_sumacoef}\suma{m}{n}{t_1}{t_2} &=& \sum_{k=0}^{\mathbf{b}-1}\lambda^{-p_m k}\sumacoef{m}{n}{t_1}{t_2}{k}, \\
\label{eq:matriz_en_sumacoef}\matrizp{m}{n}{t_1}{t_2} &=& \sum_{k=0}^{\mathbf{b}-1}\sumacoef{m}{n}{t_1}{t_2}{k}, \\
\label{eq:desigualdad_suma_matriz}\abs{\suma{m}{n}{t_1}{t_2}}&\leq & \matrizp{m}{n}{t_1}{t_2}, \\
\label{eq:descomposicion_qmn}q_{m,n} &=& \sum_{k=0}^{\mathbf{b}-1}\sum_{t_{1}\in V_m}\sumacoef{m}{n}{t_{1}}{t_{2}}{k}, \\
\label{eq:sumacoef_torres_todas}\sum_{t_{1}\in V_m}\sumacoef{m}{n}{t_{1}}{t_2}{k} &=& \left\lfloor\frac{q_{m,n}}{\mathbf{b}}\right\rfloor \textrm{\ \; or\ \; } \left\lfloor\frac{q_{m,n}}{\mathbf{b}}\right\rfloor + 1.
\end{eqnarray}
\medskip
\begin{lemm}\label{lemm:suma_matriz_coef_equivalentes}
For any $t_2 \in \{1, \ldots, d\}$ the following conditions are equivalent:
(1) $\displaystyle \sum_{t_{1}\in V_m}\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q_{m,n}}\xrightarrow[m,n\to\infty]{} 1 \textrm{\; \; \; uniformly in $m,n \in {\mathbb N}$ with $m<n$\;}$ (this is condition \eqref{mainsecond} of Theorem \ref{theo:principal} stated for any $t_{2}$).
\smallskip
(2) For all $t_{1}\in \{ 1,\ldots, d\}$,
\smallskip
$$\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q_{m,n}}-\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}\xrightarrow[m,n\to\infty]{} 0 \textrm{\; \; \; \;}$$
uniformly in $m,n \in {\mathbb N}$ with $m<n$.
\smallskip
(3) For all integers $1\leq m<n$ and $t_{1} \in \{1,\ldots,d\}$, there exists $\indice{k}{m}{n}{t_{1}}{t_2}$ in $\{0,\ldots,\mathbf{b}-1\}$ such that
\smallskip
$$\frac{\sumacoef{m}{n}{t_{1}}{t_2}{\indice{k}{m}{n}{t_{1}}{t_2}}}{q_{m,n}}-\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}\xrightarrow[m,n\to\infty]{} 0
\textrm{\; \; \; \;}$$
uniformly in $m,n \in {\mathbb N}$ with $m<n$.
\end{lemm}
\begin{proof}
(1)$\Longrightarrow$(2). We proceed by contradiction. Suppose there exists $\overline{t_{1}}\in \{1,\ldots,d\}$ such that
for infinitely many positive integers $m,n$ with $m<n$
\begin{equation}\label{badthing}
\frac{\matrizp{m}{n}{\overline{t_{1}}}{t_2}}{q_{m,n}}-\frac{\abs{\suma{m}{n}{{\overline t_{1}}}{t_2}}}{q_{m,n}}\ge 2\varepsilon >0,
\end{equation}
where $\varepsilon$ is a positive real.
From (1) we have that for any large enough positive integers $m,n$ with $m<n$
\begin{equation}
\label{eq:auxiliar}
1-\varepsilon<\sum_{t_{1}\in V_m}\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q_{m,n}}<1+\varepsilon.
\end{equation}
Consider a pair of large integers $m,n$ with $m<n$ verifying \eqref{badthing}.
Then, from \eqref{eq:desigualdad_suma_matriz}, \eqref{badthing} and \eqref{eq:auxiliar} we get
$$
1 = \sum_{t_{1}\in V_m}\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}
\geq 2\varepsilon + \sum_{t_{1}\in V_m}\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q_{m,n}}
\geq 1+\varepsilon,
$$
which is impossible. Condition (2) follows.
\smallskip
(2)$\Longrightarrow$(3).
Take $\varepsilon>0$. By hypothesis and \eqref{eq:desigualdad_suma_matriz}, there exists a positive integer $N$ such that for all $n>m>N$ and $t_{1}\in \{1,\ldots,d\}$
$$0\leq\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}-\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q_{m,n}}<\varepsilon.$$
Alternatively, the last inequality can be written as
$$1-\frac{\varepsilon q_{m,n}}{\matrizp{m}{n}{t_{1}}{t_2}}<\left|\sum_{k=0}^{\mathbf{b}-1}\frac{\sumacoef{m}{n}{t_{1}}{t_2}{k}}{\matrizp{m}{n}{t_{1}}{t_2}}\lambda^{-kp_m}\right|\leq 1.$$
Notice that $\set{1,\lambda^{-p_m},\ldots,\lambda^{-(\mathbf{b}-1)p_m}}$ is the complete set of $\mathbf{b}$-th roots of unity if $m$ is large enough, and we
have a convex combination of them. Applying Lemma \ref{lemm:geometrico} we deduce that there exists $\indice{k}{m}{n}{t_{1}}{t_2} \in \{0,\ldots,\mathbf{b}-1\}$ such that
$$1-\frac{C\varepsilon q_{m,n}}{\matrizp{m}{n}{t_{1}}{t_2}}<\frac{\sumacoef{m}{n}{t_{1}}{t_2}{\indice{k}{m}{n}{t_{1}}{t_2}}}{\matrizp{m}{n}{t_{1}}{t_2}}\leq 1,$$ or equivalently,
$$0\leq\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}-\frac{\sumacoef{m}{n}{t_{1}}{t_2}{\indice{k}{m}{n}{t_{1}}{t_2}}}{q_{m,n}}<C\varepsilon.$$
The constructed sequence depends on $\varepsilon$. Taking a sequence $(\varepsilon_{\ell};\ell\in {\mathbb N})$ tending to zero and using a diagonal process one obtains the desired sequence
$$(\indice{k}{m}{n}{t_{1}}{t_2}; m,n\in {\mathbb N}, m<n).$$
\smallskip
(3)$\Longrightarrow$(1). Fix $\varepsilon>0$. There exists a positive integer $N$ large enough such that for any
$n>m>N$ and $t_{1}\in \{1,\ldots,d\}$,
\begin{equation}\label{hypothesisthree}
0\leq\frac{\matrizp{m}{n}{t_{1}}{t_2}}{q_{m,n}}-\frac{\sumacoef{m}{n}{t_{1}}{t_2}{\indice{k}{m}{n}{t_{1}}{t_2}}}{q_{m,n}}=\sum_{\substack{k=0 \\ k\neq \indice{k}{m}{n}{t_{1}}{t_2}}}^{\mathbf{b}-1}\frac{\sumacoef{m}{n}{t_{1}}{t_2}{k}}{q_{m,n}}<\varepsilon.
\end{equation}
So, using relations \eqref{eq:suma_en_sumacoef} and \eqref{hypothesisthree} we deduce that
\begin{align*}
\abs{\suma{m}{n}{t_{1}}{t_2}}&=
\abs{
\sumacoef{m}{n}{t_1}{t_2}{\indice{k}{m}{n}{t_1}{t_2}} \lambda^{-p_{m}\indice{k}{m}{n}{t_1}{t_2}}+
\sum_{\substack{k=0 \\ k\neq {\indice{k}{m}{n}{t_1}{t_2}}}}^{\mathbf{b}-1} \sumacoef{m}{n}{t_{1}}{t_2}{k} \lambda^{-p_{m}k}
}\\
&\geq \sumacoef{m}{n}{t_1}{t_2}{\indice{k}{m}{n}{t_1}{t_2}} - \sum_{\substack{k=0 \\ k\neq {\indice{k}{m}{n}{t_1}{t_2}}}}^{\mathbf{b}-1} \sumacoef{m}{n}{t_{1}}{t_2}{k}\\
&\geq \sumacoef{m}{n}{t_1}{t_2}{\indice{k}{m}{n}{t_1}{t_2}} - \epsilon q_{m,n}.
\end{align*}
From this inequality, \eqref{eq:desigualdad_suma_matriz} and \eqref{hypothesisthree} we get
$$ \frac{\sumacoef{m}{n}{t_{1}}{t_2}{\indice{k}{m}{n}{t_{1}}{t_2}}}{q_{m,n}}-\varepsilon \leq
\frac{\abs{\suma{m}{n}{t_{1}}{t_2}}}{q
|
the definition). By \Cref{rem:product agrees with one in the analytic setting}, when $X=\widehat{U}_K$ is the Raynaud generic fiber of a smooth affine $\mathcal O_K$-scheme $U$, this complex agrees with the "de Rham cohomology over $B_{\mathrm{dR}}$" $R\Gamma_{\mathrm{dR}}(\widehat{U}_K/B_{\mathrm{dR}})$ that we have defined in \Cref{constr:Hodge and de Rham cohomology over BdR}.
\end{rem}
We can now pass to the construction of the $B_{\mathrm{dR}}$-comparison map.
\begin{construction}\label{constr:map theta}
Let $X$, $i\colon D\hookrightarrow X$ be as in the beginning of this section and let $U\coloneqq X \setminus D$. Let $R\Gamma_{\mathrm{\acute et}}(U_C,\mathbb Q_p)\coloneqq R\Gamma_{\mathrm{\acute et}}(U_C,\mathbb Z_p)[\frac{1}{p}]$ and $R\Gamma((X_C)_{D,{\mathrm{k\acute et}}},\mathbb Q_p)\coloneqq R\Gamma((X_C)_{D,{\mathrm{k\acute et}}},\mathbb Z_p)[\frac{1}{p}]$. By \Cref{eq:kummer-etale vs etale} we have an equivalence
$$
R\Gamma_{\mathrm{\acute et}}(U_C,\mathbb Q_p){\xymatrix{\ar[r]^\sim &}} R\Gamma((X_C)_{D,{\mathrm{k\acute et}}},\mathbb Q_p).
$$
Having this (with a further identification $R\Gamma((X_C)_{D,{\mathrm{k\acute et}}},\mathbb Z_p)\simeq R\Gamma((X_C)_{D,{\mathrm{prok\acute et}}},\mathbb Z_p)$), the map $\mathbb Z_p \rightarrow \mathbb B_{\mathrm{dR}}$ of sheaves on $X_{D,{\mathrm{prok\acute et}}}$ induces a natural map
$$
R\Gamma_{\mathrm{\acute et}}(U_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{dR}} \xymatrix{\ar[r]&}R\Gamma(X_{D,{\mathrm{prok\acute et}}},\mathbb B_{\mathrm{dR}}).
$$
Composing further with the equivalence $R\Gamma(X_{D,{\mathrm{prok\acute
|
et}}},\mathbb B_{\mathrm{dR}}) {\xymatrix{\ar[r]^\sim &}} R\Gamma(X_{\mathrm{an}}, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\widehat\otimes B_{\mathrm{dR}})$ coming from \Cref{eq:pushfwrd as log de Rham} we get a natural map
$$
\Theta_{X,D}\colon R\Gamma_{\mathrm{\acute et}}(U_C,\mathbb Q_p)\otimes_{\mathbb Q_p}B_{\mathrm{dR}} \xymatrix{\ar[r]&} R\Gamma(X_{\mathrm{an}}, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\widehat\otimes B_{\mathrm{dR}})
$$
which we will call the \textit{$B_{\mathrm{dR}}$-comparison map}.
\end{construction}
\begin{ex}\label{ex: log-BdR vs usual dR}
By \cite[Theorem 3.2.3(3)]{DiaoKaiWenLiuZhu_LogarithmicRH} if $X$ is proper the map $\Theta_{X,D}$ is an equivalence for any normal crossings divisor $D$. Moreover, by \cite[Lemma 3.6.2]{DiaoKaiWenLiuZhu_LogarithmicRH} one can make an identification
$$
R\Gamma_{\log {\mathrm{dR}}}(X/K)\otimes_{K}B_{\mathrm{dR}} {\xymatrix{\ar[r]^\sim &}} R\Gamma(X, \Omega^\bullet_{X,D,\log {\mathrm{dR}}}\widehat\otimes B_{\mathrm{dR}}).
$$
In the case of the analytification $X^{\mathrm{an}}, D^{\mathrm{an}}$ of a smooth proper $K$-scheme $X$ with a normal crossings divisor $D$, one can make further natural identifications
$$
R\Gamma_{\log {\mathrm{dR}}}(X^{\mathrm{an}}/K)\xymatrix{\ar[r]^{\mathrm{GAGA}}_\sim&} R\Gamma_{\log {\mathrm{dR}}}(X/K) {\xymatrix{\ar[r]^\sim &}} R\Gamma_{\mathrm{dR}}(U/K),
$$
where the first equivalence is induced by GAGA and the second one by the restiction of the algebraic log de Rham complex (as a complex of sheaves) to $U\coloneqq X\setminus D$.
\end{ex}
|
\section{Introduction}
Schur--Weyl duality is a classical result in representation theory due to Issai Schur~\cite{Schur27} and publicized by Hermann Weyl~\cite{Weyl39} that relates the irreducible representations of the symmetric group $\Sigma_n$ with that of the general linear group $\operatorname{GL}_m(\mathbb{C})$.
If we take the natural representation $\mathbf{V} := \mathbb{C}^m$ of $\operatorname{GL}_m(\mathbb{C})$, then there is a natural $\Sigma_n$-action on $\mathbf{V}^{\otimes n}$ that permutes the factors.
Since $\operatorname{GL}_m(\mathbb{C})$ acts diagonally on $\mathbf{V}^{\otimes n}$, this commutes with the $\Sigma_n$-action.
Schur--Weyl duality is the statement that the image of the representation afforded by the $\Sigma_n$-action is everything that commutes with the $\operatorname{GL}_m(\mathbb{C})$-action and vice versa.
Consequently, for $m \geq n$, we have that
\[
\mathbf{V}^{\otimes n} \cong \bigoplus_{\lambda \vdash n} S^{\lambda} \boxtimes V(\lambda)
\]
as $(\Sigma_n \times \operatorname{GL}_m(\mathbb{C}))$-representations, where $S^{\lambda}$ (resp.~$V(\lambda)$) is the irreducible $\Sigma_n$ (resp.~$\operatorname{GL}_m(\mathbb{C})$) representation indexed by the partition $\lambda$.
In particular, the decomposition is multiplicity free.
For more information, we refer the reader to~\cite{EGHLSVY11,FH91,Howe95}.
(Additionally, the actual endomorphism algebra $S(m,n) = \End_{\Sigma_n} \mathbf{V}^{\otimes n}$ is the well-studied Schur algebra; see, \textit{e.g.},~\cite{Green07}.)
If we have a subgroup $H \subseteq \operatorname{GL}_m(\mathbb{C})$, then $\mathbf{V}$ is an $H$-representation by restriction.
Subsequently, the endomorphism algebra $\End_H \mathbf{V}^{\otimes n}$ could potentially be larger (and usually is).
As a first case, we take $H = \Sigma_m$.
To describe the endomorphism algebra, we will use the partition algebra $\mathcal{P}_n(\beta)$ introduced independently by Jones~\cite{Jones94} and Martin~\cite{Martin94} and is connected to the study of the Potts model from statistical mechanics (see also, \textit{e.g.},~\cite{Martin91,Martin96,Martin00}).
The partition algebra has been well-studied from different perspectives and applications; see, \textit{e.g.},~\cite{BDVO15,HR05,HL05,HJ20,GL96,MR98,Xi99}.
In particular, it is the prototypical diagram algebra whose diagrams are a graphical representations of set partitions of $\{1,\dotsc,n\} \sqcup \{1',\dotsc,n'\}$.
This leads to a combinatorial description of the centralizer algebras for groups $\Sigma_m \subseteq H \subseteq \operatorname{GL}_m(\mathbb{C})$.
One well-studied group is $H = \operatorname{O}_n(\mathbb{C})$ the orthogonal group.
The subalgebra of the partition algebra is named the Brauer algebra after Brauer for his initial work on it and showing the corresponding Schur--Weyl duality~\cite{Brauer37}.
This also (essentially) covers the case of the symplectic group $H = Sp_{2n}(\mathbb{C})$ (see, \textit{e.g.},~\cite[Thm.~B6.3]{BR99}).
Another case is $H = G(r,p,m)$ by Tanabe~\cite{Tanabe97}, where $G(r,p,m)$ is the infinite family of complex reflection groups in the Shepard--Todd classification~\cite{ST54}.
In particular $G(2,1,m)$ is the Weyl group of $\operatorname{O}_{2m+1}(\mathbb{C})$ consisting of signed permutations, with the centralizer algebra studied by Orellana~\cite{Orellana05,Orellana07}.
Another case that has been studied is the centralizer algebra of $G(r,1,m)$ for $r \geq n$~\cite{AMM21,Kosuda00,Kosuda06,OSSZ21}.
On the other hand, we can construct various subalgebras of $\mathcal{P}_n(\beta)$ by imposing combinatorial restrictions on the diagrams and study the corresponding algebras.
One of the most famous subalgebras is the Temperley--Lieb algebra first introduced by Temperley and Lieb in~\cite{TL71} in the context of the Potts model, which was rediscovered by Jones in his work on subfactors (\textit{e.g.},~\cite{Jones83,Jones85,Jones87}) and linked to knot theory.
The Temperley--Lieb algebra is a well-studied algebra with an extensive literature (see, \textit{e.g.},~\cite{BdGN01,BM05,BW89,GdlHJ89,GW93,Kauffman87,Martin91,PRdGN02,PRZ06,RSA14}) and generalizations (such as~\cite{BSA18,DG22,MS94,ILZ18,MW00,LRH20}).
Some facts include an analog of Schur--Weyl duality with the quantum group $U_q(\mathfrak{gl}_2)$ and that it is a quotient of the Hecke algebra of $\Sigma_m$~\cite{Jones87}.
Furthermore, it is (essentially) equivalent to the planar partition algebra (see, \textit{e.g.},~\cite{Jones94,HR05} and Conjecture~\ref{conj:TL_planar_zero}).
Other subalgebras that have been considered, such as the half partition algebra~\cite{HR05}, quasi-partition algebra~\cite{DO14}, rook algebra~\cite{Munn57,Solomon02}, and planar rook algebra~\cite{FHH09}.
For most of these cases, an analog of Schur--Weyl duality has been constructed.
Let us digress slightly by looking at the diagram algebras where blocks have size at most $2$.
For these cases, a new phenomenon can appear, where we can have a second independent parameter $\gamma$ that counts the number of simply-connected interior components when composing diagrams and $\beta$ counts the number of loops.
Specifically, these algebras are the rook Brauer algebra~\cite{DEG17,DG22II,HdM14,MM14}, Motzkin algebra~\cite{BH14,DEG17}, and partial Temperley--Lieb algebra~\cite{DG22}.
When $\gamma = \beta$, these are naturally a subalgebra of $\mathcal{P}_n(\beta)$, and they are isomorphic for all $\gamma \neq 0$ by rescaling the diagrams (see, \textit{e.g.},~\cite[Lemma~7.3]{DG22II}).
Hence, generically we can consider them to be diagrammatic subalgebras of $\mathcal{P}_n(\beta)$ up to rescaling.
Nothing seems to be known for the degenerate cases of $\beta = 0$ (resp.~$\beta = 0$) and $\gamma \neq 0$ (resp.~$\gamma \neq 0$).
In this note, we will survey a number of results on the representation theory of the partition algebra $\mathcal{P}_n(\beta)$ and some of its diagram subalgebras.
Our main tool will be using is the theory of cellular algebras introduced by Graham and Lehrer~\cite{GL96}.
We will then use the decomposition of the partition algebra of Xi~\cite{Xi99}, which is particularly amiable to combinatorics and diagrammatic subalgebra.
Thus, the representation theory of the algebras in large part reduces the study to that of the symmetric group.
In particular, we will be following the approach of half diagrams, given for the specific case of the Temperley--Lieb algebra under the name of (quotients of) link modules (see, \textit{e.g.},~\cite{GL96,Martin90,Martin91,RSA14}.
We remark that many of the algebras presented here (see Table~\ref{table:algebras} for many examples) are known to be cellular~\cite{BH14,DG22,GL96,Xi99}.
For nearly all of the other cases considered, they are possibly known to be by experts to be cellular even if they have not been explicitly written down.
More specifically, the centralizer algebra of $G(r,1,m)$ for $r \geq 2$ and $m \geq n$, the (planar) rook algebra, and half partition algebra can be shown to be cellular as a consequence of the Proposition~\ref{prop:subcellular} (which is immediate from the definition of a cellular algebra) and the partition algebra being cellular.
Yet not every cases follows quite so easily.
In particular, to show that the (planar) quasi-partition algebra is cellular (Theorem~\ref{thm:quasi_cellular} and Theorem~\ref{thm:quasi_partition_cellular}) requires more technical analysis, which the author believes to be new.
Additionally, we remark that the techniques used in other papers (\textit{e.g.}~\cite{DO14,HJ20,MM14,OSSZ21}) are no less important than the approach taken here; on the contrary, they often provide more refined descriptions and can be better adapted to addressing other questions such as characters.
Yet, the diagrammatic approach with cellular algebras taken here allows us to describe the cell modules (and, in principle, the simple modules) and the algebra action naturally in terms of tableau more uniformly in terms of inducing representations of Young subgroups of $\Sigma_n$.
On the other hand, the approach taken here allows us to study these diagram algebras over arbitrary fields $\mathbf{k}$ (not necessarily $\mathbb{C}$) and discuss the semisimplicity of a number of these algebras over arbitrary fields and describe their irreducible modules.
However, we do not perform this analysis here as each such algebra not previously treated deserves its own detailed paper.
It is the hope of the author that this paper helps facilitate easier translations between the combinatorial and algebraic information.
\begin{table}
\begin{center}
\begin{tabular}{ccccl}
\toprule
name & notation & planar & propagating & block sizes \\
\midrule
partition & $\mathcal{P}_n(\beta)$ & no & no & any size \\
half partition & $\mathcal{P}_{n-1/2}(\beta)$ & no & $n \sim n'$ & any size \\
quasi-partition & $\mathcal{Q}\mathcal{P}_n(\beta)$ & no & no & $> 1$ \\
$G(r,d,m)$-centralizer & $\mathcal{G}_n^{(r,d,m)}(\beta)$ & no & no & based on $r$ \\
uniform block & $\mathcal{U}_k$ & no & yes & top equals bottom \\
rook Brauer & $\mathcal{R}\mathcal{B}_n(\beta)$ & no & no & $\leq 2$ \\
rook & $\mathcal{R}_n(\beta)$ & no & if size $2$ & $\leq 2$ \\
Brauer & $\mathcal{B}_n(\beta)$ & no & no & $= 2$ \\
symmetric group & $\mathbf{k} [\Sigma_n]$ & no & yes & $= 2$ \\
planar partition & $\mathcal{P}\mcP_n(\beta)$ & yes & no & any size \\
Temperley--Lieb & $\mathcal{T}\mathcal{L}_n(\beta)$ & yes & no & $= 2$ \\
Motzkin & $\mathcal{M}_n(\beta)$ & yes & no & $\leq 2$ \\
partial TL & $\mathcal{P}\mathcal{T}\mathcal{L}_n(\beta)$ & yes & no & $\leq 2$ and balanced \\
planar quasi-partition & $\mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta)$ & yes & no & $> 1$ \\
planar rook & $\mathcal{P}\mathcal{R}_n(\beta)$ & yes & if size $2$ & $\leq 2$ \\
planar $r$-color & $\mathcal{P}\mathcal{C}_{r,n}(\beta)$ & yes & no & based on $r$ \\
\bottomrule
\end{tabular}
\end{center}
\caption{A list of some subalgebras of the partition algebra and a summary of the conditions imposed on the indexing diagrams.}
\label{table:algebras}
\end{table}
To illustrate this, we consider the uniform block (permutation) algebra $\mathcal{U}_n$ studied in~\cite{OSSZ21}.
This is spanned by the set diagrams that represent permutations of blocks of equal size.
We show that a number of the constructions given in~\cite{OSSZ21} can be described as coming from the cellular structure of the partition algebra $\mathcal{P}_n(\beta)$ restricted to $\mathcal{U}_n$.
As a consequence, we are able to show that $\mathcal{U}_n$ is a cellular algebra and describe its simple representations (as far as we understand the representations of the symmetric group) over an arbitrary field (in~\cite{OSSZ21}, they only considered it as a $\mathbb{C}$-algebra).
For another example, let us consider the description using multiset-valued tableaux, which are tableau filled with multisets with a given total ordering under the usual standard condition,\footnote{
The multiset-valued tableaux here are different than the those in the K-theory of the Grassmannian considered in~\cite{HS20,LP07,PP16}, which have different semistandard conditions, weights, and generating functions.}
for the $\mathcal{P}_n(\beta)$ irreducible representations.
This has appeared (sometimes implicitly) under a few different names in many different papers~\cite{BH19,BHH17,HJ20,MR98,OSSZ21,OZ21}.
We can see this as a rephrasing of the decomposition of~\cite{Xi99}, which breaks the diagrams into an upper part, lower part, and a middle permutation part using the propagating blocks, using a simple generalization of the blocks-with-defects approach (see Section~\ref{sec:partition_algebra}).
In both of these cases, we are just inducing a Specht module of a Young subgroup $\Sigma_k$ and the exactly $k$ defects in the half diagrams corresponds to the multiset that we use to fill the tableau.
The restriction of this decomposition was given for the Temperley--Lieb algebra and Brauer algebra in~\cite{GL96}, which appeared as early as the work of Brown~\cite{Brown56}.
For more detailed treatments of (most of) the general approach in this paper, see~\cite{BH14} for the Motzkin algebra and~\cite{DG22} for the partial Temperley--Lieb algebra.
Let us discuss the new results in this paper.
As previously mentioned, we show a number of diagram algebras are cellular and describe their cell modules, many of which are likely already known to experts but not written down, but we believe the (planar) quasi-partition algebra is new (Theorem~\ref{thm:quasi_cellular} and Theorem~\ref{thm:quasi_partition_cellular}).
A combinatorially interesting fact is that the cell modules of the planar quasi-partition algebra are given by the triangle Riordan numbers.
We provide a number of new formulas for the dimensions of the cell modules of the (planar) $G(r,1,m)$-centralizer algebra, including an appearance of the Fuss--Catalan numbers (Proposition~\ref{prop:planar_color_zero_defect_dim}); see Section~\ref{sec:planar_even} and Section~\ref{sec:planar_color}.
Another novel result in this paper (Theorem~\ref{thm:wreath_product}) is a general construction to construct new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra.
We call this the wreath product of cellular algebras.
This is the common generalization of the papers~\cite{RX04,ZC06} and also yields another proof of the cellularity of $\mathbf{k} [G(r,1,m)]$ (assuming the $r = 1$ case for the symmetric group).
We expect this can be extended to having the ``base'' be an algebra with Hecke type relations (\textit{e.g.}, BMW algebras), giving another proof of the cellularity of the Ariki--Koike algebra~\cite{GL96}.
In the process of trying to describe the various algebras and their irreducible modules, we came across a number of questions that we have included for the interested reader to pursue.
These include corner cases where the behavior seems to differ at an algebraic level, but perhaps not within their representation theory (Problem~\ref{prob:two_param_rook_brauer_semisimple}, Conjecture~\ref{conj:TL_planar_zero}, Problem~\ref{prob:two_param_motzkin}).
Other examples include combinatorial questions that might lead to interesting relations (\textit{e.g.}, Problem~\ref{prob:irrep_dim_CE}).
There could also be new diagram-type algebras constructed by mixing different constraints and understood using the techniques in this paper.
For example, a blob algebra~\cite{MS94} (see also Section~\ref{sec:blob_algebra}) version of the Motzkin algebra, or putting other cellular algebras on the leftmost strands (which could be seen as a subalgebra of the wreath product).
Next, we mention a number of known Schur--Weyl duality statements in Table~\ref{table:schur_weyl}, although this is likely not exhaustive of those involving subalgeras of the partition algebra.
Let us mention some additional cases not previously discussed.
The first is a Schur--Weyl duality for the Burau representations, both reduced and unreduced, of the braid group $\mathbf{B}_m$ given in~\cite{DG21}.
In this case, the centralizer algebra is given by the rook algebra $\mathcal{R}_n([m]_q)$, where $[m]_q = \frac{q^m-1}{q-1}$ are the $q$-analogs of $m$.
Next is the so-called tangle algebra $\mathcal{T}_n(1)$ studied in~\cite{BE18}, which we conjecture is equivalent to the planar quasi-partition algebra (Conjecture~\ref{conj:tangle_pqp}).
The centralizer of the twin group $TW_m$ on the natural (type $A$) Hecke algebra representation, which can be considered as the Burau representation, was shown to be the rook Brauer algebra (see Section~\ref{sec:rook_brauer}) in~\cite{DG22II}.
The general Lie superalgebra $\mathfrak{gl}(1|1)$ has a Schur--Weyl duality with the planar rook algebra~\cite{BM13}.
The last is the partition algebra Schur--Weyl duality has been extended to the non-semisimple setting over general commutative rings in~\cite{BDM22} with the kernel of the symmetric group action given a cellular basis in~\cite{BDM22II}.
Similar and some more general results using different techniques were shown in~\cite{Donkin22}.
\begin{table}
\begin{center}
\begin{tabular}{ccc}
\toprule
tensor $T$-action & module $V$ & module $G$-action\\
\midrule
$\mathbb{C} [\Sigma_n]$ & natural $\mathbf{V} \cong \mathbb{C}^m$ & $\operatorname{GL}_m(\mathbb{C})$\\
$\mathcal{G}_n^{(r,p,m)}(m)$ & natural $\mathbf{V} \cong \mathbb{C}^m$ & $\mathbb{C} [G(r,p,m)]$ \\
$\mathcal{P}_{n-1/2}(m)$ & natural $\mathbb{C} [\Sigma_m]$-module & $\mathbb{C} [\Sigma_{m-1}]$ \\
$\mathcal{Q}\mathcal{P}_n(m)$ & Sphect $S^{(n-1.1)}$ & $\mathbb{C} [\Sigma_m]$ \\
$\mathcal{R}_n(m)$ & $V(1) \oplus V(0)$ & $\operatorname{GL}_m(\mathbb{C})$ \\
$\mathcal{R}_n([m]_q)$ & unreduced Burau & $\mathbb{C} [\mathbf{B}_m]$ \\
$\mathcal{R}\mathcal{B}_n(m+1)$ & $V(1) \oplus V(0)$ & $\operatorname{O}_m(\mathbb{C})$ \\
$\mathcal{R}\mathcal{B}_n(m, \beta')$ & Barau $\mathbb{C}^m$ & $\mathbb{C} [TW_m]$ \\
$\mathcal{T}\mathcal{L}_n\bigl(\pm(q+q^{-1})\bigr)$ & natural $V(1) \cong \mathbb{C}^2$ & $U_q(\mathfrak{sl}_2)$ \\
$\mathcal{M}_n\bigl(1\pm(q+q^{-1})\bigr)$ & adjoint $V(1) \oplus V(0)$ & $U_q(\mathfrak{sl}_2)$ \\
$\mathcal{P}\mathcal{R}_n(\beta)$ & natural $\mathbf{V}^{\otimes n}$ & $U\bigl(\mathfrak{gl}(1|1)\bigr)$ \\
$\mathcal{P}\mathcal{T}\mathcal{L}_n\bigl(1\pm(q+q^{-1})\bigr)$ & $V(1) \oplus V(0)$ & $U_q(\mathfrak{gl}_2)$ \\
$\mathcal{T}_n(1)$ & $V(2)$ & $U(\mathfrak{sl}_2)$ \\
\bottomrule
\end{tabular}
\end{center}
\caption{Some known Schur--Weyl duality statements on the module $V^{\otimes n}$, where $V$ is a (left) $G$-module.}
\label{table:schur_weyl}
\end{table}
We conclude by mentioning some additional references for the interested reader.
There is the survey on combinatorial representation theory by Barcelo and Ram~\cite{BR99} that discusses questions on diagram algebras (among others) from a different perspective.
A different perspective on diagram algebras was considered by Cox, Martin, Parker, and Xi in~\cite{CMPX06}, where they studied the diagram algebras as a tower of algebras through an abstract framework.
Furthermore, they introduce a generalization of the Temperley--Lieb algebra by assigning arrows to each line, which also generalizes the blob algebra.
By considering Schur--Weyl duality with $\operatorname{GL}_m(\mathbb{C})$ on $\mathbf{V}^{\otimes n} \otimes (\mathbf{V}^*)^{\otimes k}$, we obtain the walled Brauer algebra, which is a subalgebra of the Brauer algebra originating in the work of Tureav~\cite{Turaev89} and Koike~\cite{Koike89} with its own rich literature; see \textit{e.g.},~\cite{BCHLLS94,BJSH21,CDVDM08,Halverson96,JK20} and references therein.
In particular, Brundan and Stroppel in~\cite{BS12} make a link with their previous work on studying Khovanov's arc algebra and generalizations~\cite{BS10,BS11,BS11III,BS12IV}.
In turn, this is connected with another diagrammatic algebra known as the Khovanov--Lauda--Rouquier (KLR) algebra or quiver Hecke algebra~\cite{KL09,Rouquier08} that has a vast literature too large to even begin to list here, but we simply mention the generalized Schur--Weyl duality functors of Kang, Kashiwara, and Kim~\cite{KKK15,KKK18} (see~\cite{KKOP20,KKOP21} for some recent results related to these functors).
A broader framework of sandwiched cellular algebras was introduced by Tubbenhauer and coauthors~\cite{TM21,Tubbenhauer22,TV21} as a way to generalize Kazhdan--Lusztig cells to general algebras.
\subsection*{Acknowledgements}
The author thanks Andrew Mathas for suggesting to show the uniform block algebra is cellular by using the partition algebra and~\cite{Xi99}, numerous conversions over the years about cellular algebras, and comments on an earlier draft of this paper.
The author thanks Stephen Doty for catching a number of mistakes in an earlier drafts of this paper and providing numerous suggestions, historical comments, and references and deeply appreciates his proofreading.
The author thanks Georgia Benkart, Rosa Orellana, Tom Halverson, Arun Ram, J{\o}rgen Rasmussen, David Ridout, Franco Saliola, Anne Schilling, and Mike Zabrocki for useful conversations and suggestions.
The author was partially supported by Grant-in-Aid for JSPS Fellows 21F51028.
This work was partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).
\section{Background}
\label{sec:background}
Consider a positive integer $m \in \mathbb{Z}_{>0}$.
Let $[m] := \{1 < 2 < \cdots < n\}$ and $[m'] := \{1' < 2' < \cdots < n'\}$.
Let $\mathbf{k}$ denote a field of characteristic $p$ (possibly $p = 0$).
All of the $\mathbf{k}$-algebras considered in this paper will be associative and unital.
Unless otherwise specified, tensor products will be over~$\mathbf{k}$.
Let $\mathbb{Z}_r := \mathbb{Z} / r \mathbb{Z}$, which we will often consider as a cyclic abelian group.
Let $\Sigma_m$ denote the symmetric group on $[m]$.
For $r, d \in \mathbb{Z}_{>0}$ such that $d \mid r$, let $G(r,d,m)$ denote the complex reflection group given by $r$ colored permutations (where multiplication is constructed as the wreath product $\mathbb{Z}_r \wr \Sigma_m$) such that the sum of the colors is equivalent to $0 \pmod{d}$.
This has a natural representation $\mathbf{V}$ on $\mathbb{C}^m$ corresponding to products of permutation matrices and diagonal matrices $D(\zeta_r^{i_1}, \dotsc, \zeta_r^{i_m})$ such that $i_1 + \cdots + i_m \equiv 0 \pmod{d}$, where $\zeta_r$ is a primitive $r$-th root of unity.
In particular, the matrices all are generalized permutation matrices, with each row and column having exactly one nonzero entry of the form $\zeta_r^k$.
Note that $G(1,1,m) \cong \Sigma_m$ and $G(r,1,m) \cong \mathbb{Z}_r \wr \Sigma_m$.
A (integer) \defn{partition} $\mu$ of $n$ is a weakly decreasing sequence of positive numbers $(\mu_1 \geq \mu_2 \geq \cdots \geq \mu_{\ell} > 0)$ whose sum $\mu_1 + \cdots + \mu_{\ell} = n$.
We write this as $\mu \vdash n$, and let $\ell(\mu) = \ell$ denote its length.
For nonnegative integers $m,n \in \mathbb{Z}_{\geq 0}$ and partitions $\mu \in m$ and $\nu \in n$, we say $\mu \leq \nu$ in graded dominance order if $m \leq n$ or if $m = n$ then $\sum_{i=1}^k \mu_i \leq \sum_{i=1}^k \nu_i$ for all $k$ (\textit{i.e.}, usual dominance order), where we extend $\mu$ and $\nu$ with an infinite number of trailing $0$'s.
A \defn{standard tableau} of $\mu$ by a totally ordered alphabet $A$ is a filling of the Young diagram of $\mu$ such that each letter appears exactly once and rows and columns are (strictly) increasing.
We draw our partitions and tableaux using English convention.
Let $f_{\lambda}$ denote the number of standard tableaux for a partition $\lambda$.
It is a classical fact that
\begin{equation}
\label{eq:fla=n2}
\sum_{\lambda \vdash k} f_{\lambda}^2 = k!
\end{equation}
with many proofs; for example, this is a consequence of the Robinson--Schensted--Knuth (RSK) bijection (see, \textit{e.g.},~\cite[Ch.~7]{ECII}).
\medskip
\noindent
\textbf{Caution:} It can be the case that $\lambda$ is \emph{not} a partition in the sequel.
\subsection{Partition algebras}
\label{sec:partition_algebra}
Fix some $\beta \in \mathbf{k}$.
Let $\mathcal{P}_n(\beta)$ denote the \defn{partition algebra}, whose basis is indexed by set partitions of $[n] \sqcup [n]'$, which we represent as diagrams from $[n]$ on the top to $[n]'$ on the bottom and identify the parts of the set partition with connected components.
We often identify the basis elements of $\mathcal{P}_n(\beta)$ with their defining set partition.
Following composition conventions, multiplication $\rho \cdot \sigma$ of basis elements (diagrams) $\rho$ and $\sigma$ as the set partition formed by stacking the diagram of $\sigma$ on top of $\rho$ and removing the $D$ interior components times $\beta^D$.
We say a block (or part) $\rho_i$ of a basis element $\rho$ is \defn{propagating} if $\rho_i \cap [n] \neq \emptyset$ and $\rho_i \cap [n]' \neq \emptyset$, \textit{i.e.}, it connects to both sides of the diagram.
\begin{ex}
\label{ex:diagram_multiplication}
For the diagrams
\begin{align*}
\rho & = \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,blue] (G1) -- (G-2) .. controls +(0, .9) and +(0, .9) .. (G-5) .. controls +(0, .5) and +(0, .5) .. (G-6);
\draw[-,color=darkred] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.75) and +(0, -.75) .. (G5);
\draw[-] (G-1) .. controls +(0, .75) and +(0, .75) .. (G-3) .. controls +(0, .5) and +(0, .5) .. (G-4);
\draw[-,color=OCUsapphire] (G6) -- (G-8);
\end{tikzpicture}
\; = \{{\color{blue}\{1,2',5',6'\}}, {\color{darkred}\{2,3,5\}}, \{4\}, {\color{OCUsapphire}\{6,8'\}}, \{7\}, \{8\}, \{1',3',4'\}, \{7'\}\},
\\[5pt]
\tau & = \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-2) .. controls +(-.2, 0.5) and +(0, -1.5) .. (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4);
\draw[-,blue] (G-1) -- (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=OCUenji] (G-3) .. controls +(0, .75) and +(0, .75) .. (G-5) -- (G8);
\draw[-,color=UQpurple] (G-4) .. controls +(0, .75) and +(0, .75) .. (G-7);
\end{tikzpicture}
\; = \{{\color{darkred}\{1,2,4,2'\}}, {\color{blue}\{3,5,6,1'\}}, \{7\}, {\color{OCUenji}\{8,3',5'\}}, {\color{UQpurple}\{4',7'\}}, \{6'\}, \{8'\}\},
\end{align*}
we have the multiplication
\begin{align*}
\rho \cdot \tau & = \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-3.5ex)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
\node[vertex] (Gp-\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,blue] (G-1) -- (Gp-2) .. controls +(0, .9) and +(0, .9) .. (Gp-5) .. controls +(0, .5) and +(0, .5) .. (Gp-6);
\draw[-,color=darkred] (G-2) .. controls +(0, -.5) and +(0, -.5) .. (G-3) .. controls +(0, -.75) and +(0, -.75) .. (G-5);
\draw[-] (Gp-1) .. controls +(0, .75) and +(0, .75) .. (Gp-3) .. controls +(0, .5) and +(0, .5) .. (Gp-4);
\draw[-,color=OCUsapphire] (G-6) -- (Gp-8);
\draw[-,color=darkred] (G-2) .. controls +(-.2, 0.5) and +(0, -1.5) .. (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4);
\draw[-,blue] (G-1) -- (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=OCUenji] (G-3) .. controls +(0, .75) and +(0, .75) .. (G-5) -- (G8);
\draw[-,color=UQpurple] (G-4) .. controls +(0, .75) and +(0, .75) .. (G-7);
\end{tikzpicture}
\, \Longleftrightarrow \,
\beta^2 \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.8) and +(0, -.8) .. (G4) .. controls +(0, -.9) and +(0, -.9) .. (G8);
\draw[-,blue] (G-6) .. controls +(0, .5) and +(0, .5) .. (G-5) .. controls +(0, .75) and +(0, .75) .. (G-2) .. controls +(0, 1.5) and +(0, -1.5) .. (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-] (G-1) .. controls +(0, .75) and +(0, .75) .. (G-3) .. controls +(0, .5) and +(0, .5) .. (G-4);
\end{tikzpicture}
\\ & = \beta^2 \, \{{\color{darkred}\{1,2,4,8\}}, {\color{blue}\{3,5,6,2',5',6'\}}, \{7\}, \{1',3',4'\}, \{7'\}, \{8'\}\}.
\end{align*}
We note that $\rho$ (resp.~$\tau$ and $\rho \cdot \tau$) has $2$ (resp.~$3$ and $1$) propagating blocks.
Likewise, we compute
\begin{align*}
\tau \cdot \rho & = \beta \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,blue] (G1) -- (G-1);
\draw[-,color=darkred] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.75) and +(0, -.75) .. (G5);
\draw[-,color=OCUenji] (G-3) .. controls +(0, .75) and +(0, .75) .. (G-5) -- (G6);
\draw[-,color=UQpurple] (G-4) .. controls +(0, .75) and +(0, .75) .. (G-7);
\end{tikzpicture}
\\ & = \beta \, \{{\color{blue}\{1,1'\}}, {\color{darkred}\{2,3,5\}}, \{4\}, \{6, 3', 5'\}, \{7\}, \{8\}, \{2'\}, \{4',7'\}, \{6'\}, \{8'\}\},
\end{align*}
which also has only $1$ propagating block.
\end{ex}
\begin{remark}
Our multiplication convention might be the reverse of some authors, \textit{e.g.},~\cite{HJ20}.
\end{remark}
By~\cite[Lemma~4.2]{Xi99}, we have a decomposition
\begin{equation}
\label{eq:partition_decomposition}
\mathcal{P}_n \cong \bigoplus_{k=0}^n V'_k \otimes \mathbf{k}[\Sigma_k] \otimes V_k
\end{equation}
as $\mathbf{k}$-modules, where $V_k$ (resp.~$V'_k$) is the free $\mathbf{k}$-module spanned by set partitions $\rho$ of $[n]$ (resp.~$[n]'$) with at least $k$ parts and chosen subset $S \subseteq \rho$ such that $\abs{S} = k$ called the \defn{defects}.
This means we can decompose our natural basis elements of $\mathcal{P}_n$ into two set partitions and a permutation encoding the crossings of the connecting blocks using the mapping from $\min(\rho_i \cap [n]) \mapsto \min(\rho_i \cap [n]')$.
Pictorially, this is dividing our diagram for $\rho$ into three parts, the lower set partition on $[n]$, the middle part with the crossings, and the upper set partition on $[n]'$.
Note that in the decomposition, the defects in the lower and upper parts correspond to the propagating blocks.
We call the basis elements of $V'_k$ (and $V_k$) \defn{half diagrams}.
\begin{ex}
\label{ex:decomposition}
If we consider the diagram $\tau$ from Example~\ref{ex:diagram_multiplication} (drawn below using a different realization where the propagation is done from smallest element to smallest element), then this decomposition is given by
\begin{align*}
\sigma & \longleftrightarrow \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]);
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -2) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-2) .. controls +(0, 1.5) and +(0, -1.5) .. (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.75) and +(0, -.75) .. (G4);
\draw[-,blue] (G-1) .. controls +(0, 1.5) and +(0, -1.5) .. (G3) .. controls +(0, -.75) and +(0, -.75) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=OCUenji] (G-5) .. controls +(0, .75) and +(0, .75) .. (G-3) .. controls +(0, 1.5) and +(0, -1.5) .. (G8);
\draw[-,color=UQpurple] (G-4) .. controls +(0, .75) and +(0, .75) .. (G-7);
\draw[dashed,black!50] (0,1) -- (9,1);
\draw[dashed,black!50] (0,-1) -- (9,-1);
\end{tikzpicture}
\hspace{30pt}
\begin{gathered}
(\{\{1,2,4\}, \{3,5,6\}, \{8\}\}, \{\{7\}\}) = v_{\tau}
\\
\begin{bmatrix} 1 & 3 & 8 \\ 2' & 1' & 3' \end{bmatrix} = \sigma_{\tau}
\\
(\{\{1'\},\{2'\},\{3',5'\}\},\{\{4',7'\},\{6'\},\{8'\}\}) = v'_{\tau}
\end{gathered}
\\ & \longleftrightarrow v_{\tau} \otimes v'_{\tau} \otimes \sigma_{\tau} \in V'_3 \otimes \mathbf{k} [\Sigma_3] \otimes V_3,
\end{align*}
where the set partition is the union of the pair and the first part is the defects and the permutation is written in two-line notation.
\end{ex}
The following is a classical fact due to Jones.
\begin{thm}[{\cite{Jones94}}]
\label{thm:partition_double_centralizer}
There exists a surjection $\mathcal{P}_n(m) \to \End_{\Sigma_m} \mathbf{V}^{\otimes n}$.
Furthermore, this map is a bijection if and only if $m \geq 2n$.
\end{thm}
An explicit bijection between the Bratteli diagram approach for the general theory of $\End_{\Sigma_m} \mathbf{V}^{\otimes n}$ and the partition algebra when $m \geq 2n$ was constructed in~\cite{COSSZ20}.
\begin{remark}
The partition algebra has a nice set of generators (and relations) known that involve fairly simple diagrams.
While these are useful to prove a number of facts (such as isomorphisms), we generally will not use them.
Consequently, we do not describe such presentations, but they can be found in the references.
\end{remark}
\subsection{Cellular algebras}
We give the necessary definitions following~\cite{GL96}.
\begin{dfn}[{Cellular algebra~\cite{GL96}}]
\label{defn:cellular_algebra}
Let $\mathcal{A}$ be a (unital associative) $\mathbf{k}$-algebra with an anti-involution~$\iota$.
Let $\Lambda$ be a finite poset.
Let $M = (M(\lambda) \mid \lambda \in \Lambda)$, where $M(\lambda)$ is a finite set.
Let
\[
C = \{C^{\lambda}_{ST} \mid \lambda \in \Lambda; S,T \in M(\lambda)\}
\]
be a $\mathbf{k}$-basis for $\mathcal{A}$.
We say $\mathcal{A}$ is a \defn{cellular algebra} with cell datum $(\Lambda, \iota, M, C)$ if
\begin{enumerate}
\item $\iota(C^{\lambda}_{ST}) = C^{\lambda}_{TS}$ and
\item \label{cell_basis_triangular} for every $\lambda \in \Lambda$, $S,T \in M(\lambda)$, and $a \in \mathcal{A}$, we can write
\[
a C^{\lambda}_{ST} = \sum_{U \in M(\lambda)} r_a(U,S) C^{\lambda}_{UT} + \mathcal{A}^{<\lambda},
\]
where $r_a(U,S) \in \mathbf{k}$ do not depend on $T$ and $\mathcal{A}^{<\lambda} := \Span_{\mathbf{k}}\{C^{\mu}_{UV} \mid \mu < \lambda; U, V \in M(\mu)\}$ is the module of lower order terms.
\end{enumerate}
The basis $C$ is called a \defn{cell basis} of $\mathcal{A}$.
\end{dfn}
From the definition, we have the following result, which is likely well-known to experts but the author could not find in the literature, about certain subalgebras of cellular algebras.
\begin{prop}
\label{prop:subcellular}
Let $\mathcal{A}$ be a cellular algebra with cell datum $(\Lambda, \iota, M, C)$.
Let $\overline{\mathcal{A}} \subseteq \mathcal{A}$ be a subalgebra with a basis $\overline{C} \subseteq C$ invariant under $\iota$.
Then $\overline{\mathcal{A}}$ is a cellular algebra with cell datum $(\overline{\Lambda}, \iota|_{\overline{\mathcal{A}}}, \overline{M}, \overline{C})$ with $\overline{\Lambda}$ and $\overline{M}$ being the indices that appear in $\overline{C}$.
\end{prop}
From the triangularity property~(\ref{cell_basis_triangular}) in the definition, we have the following way to construct new cellular bases that, similar to Proposition~\ref{prop:subcellular}, is likely well-known to experts.
\begin{prop}
\label{prop:cellular_triangle_basis}
Let $\mathcal{A}$ be a cellular algebra with cell datum $(\Lambda, \iota, M, C)$.
Then any basis of the form
\[
\widetilde{C} = \{ \widetilde{C}_{ST}^{\lambda} \in C_{ST}^{\lambda} + \mathcal{A}^{< \lambda} \mid \lambda \in \Lambda; S,T \in M(\lambda) \}
\]
is a cell basis of $\mathcal{A}$ and defines new cell datum $(\Lambda, \widetilde{\iota}, M, \widetilde{C})$ with $\widetilde{\iota}(\widetilde{C}_{ST}^{\lambda}) = \widetilde{C}_{TS}^{\lambda}$.
\end{prop}
\begin{proof}
We only need to show Definition~\ref{defn:cellular_algebra}(\ref{cell_basis_triangular}) holds.
We have
\[
a \widetilde{C}_{ST}^{\lambda} = a C_{ST}^{\lambda} + a \mathcal{A}^{<\lambda} = \sum_{U \in M(\lambda)} r_a(U,S) C_{UT} + \mathcal{A}^{<\lambda} = \sum_{U \in M(\lambda)} r_a(U,S) \widetilde{C}_{UT} + \mathcal{A}^{<\lambda},
\]
since $\mathcal{A}^{<\lambda}$ is a left ideal by Definition~\ref{defn:cellular_algebra} (see also~\cite{KX96}) and the change of basis $C \to C'$ is unitrangular.
\end{proof}
For a cellular algebra $\mathcal{A}$ with cell datum $(\Lambda, \iota, M, C)$, the \defn{cell module} (or \defn{standard module}) indexed by $\lambda \in \Lambda$ is the free $\mathbf{k}$-module
\[
W(\lambda) := \Span_{\mathbf{k}} \{ C_S \mid S \in M(\lambda) \}
\]
with the natural action $a C_S = \sum_U r_a(U,S) C_U$.
Roughly speaking, the action is given by fixing a $\lambda$ and forgetting one of the indices of the basis.
Hence, $\dim W(\lambda) = \abs{M(\lambda)}$.
An immediate consequence of these definitions is
\begin{equation}
\label{eq:dim_formula}
\dim \mathcal{A} = \sum_{\lambda \in \Lambda} \bigl( \dim W(\lambda) \bigr)^2.
\end{equation}
We define a bilinear form $\Phi_{\lambda} \colon W(\lambda) \times W(\lambda) \to \mathbf{k}$ by
\[
C^{\lambda}_{ST} C^{\lambda}_{UV} \equiv \Phi_{\lambda}(C_T, C_U) C^{\lambda}_{SV} \pmod{\mathcal{A}^{<\lambda}}
\]
and extended bilinearly.
The bilinear form $\Phi_{\lambda}$ is symmetric and $\mathcal{A}$-invariant in the sense $\Phi_{\lambda}(a v, w) = \Phi_{\lambda}(v, \iota(a) w)$ for all $a \in \mathcal{A}$ and $v,w \in W(\lambda)$.
This allows us to construct the simple modules.
\begin{thm}[{\cite{GL96}}]
The simple modules of a cellular algebra with cell datum $(\Lambda, \iota, M, C)$ are parameterized by $\{ \lambda \in \Lambda \mid \Phi_{\lambda} \neq 0 \}$.
Moreover, the (absolutely) simple module
\[
L(\lambda) \cong W(\lambda) / R(\lambda),
\]
where $R(\lambda) := \{w \in W(\lambda) \mid \Phi_{\lambda}(v,w) = 0 \text{ for all } v \in W(\lambda)\}$ is the radical of $\Phi_{\lambda}$.
\end{thm}
\begin{remark}
\label{rem:algebraic_closure}
A particular feature of this result is that for a cellular algebra, it is equivalent to work over the algebraic closure of $\mathbf{k}$, which is also necessarily cellular.
In other words, the semisimplicity and the set of simple modules is the same for a cellular algebra over $\mathbf{k}$ as its algebraic closure, which we can get by extension of scalars.
Note that this is not mean we can consider any subfield of $\mathbf{k}$ and the algebra remains cellular.
In particular, the group algebra $\mathbf{k} [\mathbb{Z}_n]$ when $\zeta_n \in \mathbf{k}$ (\textit{i.e.}, $\mathbf{k}$ has all $n$-th roots of unity) has $n$ irreducible representations (all of dimension $1$), but otherwise we can find fewer (nonsplit) irreducible representations of higher dimension.
For example, consider $n = 4$ and compare when $\mathbf{k} = \mathbb{C}$ and when $\mathbf{k} = \mathbb{R}$ (which has a simple $2$ dimensional module with the generator of $\mathbb{Z}_4$ acting as rotation by $\pi/2$).
Consequently, $\mathbb{R} [\mathbb{Z}_4]$ is not cellular but $\mathbb{C} [\mathbb{Z}_4]$ is (see Section~\ref{sec:wreath} below).
\end{remark}
\begin{remark}
\label{rem:assoc_graded}
From the definition of a cellular algebra, there is a natural $\Lambda$-filtration of two-sided ideals on any cellular algebra $\mathcal{A}$; that is $\mathcal{A} = \bigcup_{\lambda \in \Lambda} \mathcal{A}^{\leq \lambda}$ such that $\mathcal{A}^{\leq \lambda} \subseteq \mathcal{A}^{\leq \mu}$ whenever $\lambda \leq \mu$ in the poset $\Lambda$ and $\mathcal{A} \cdot \mathcal{A}^{\leq \lambda} \subseteq \mathcal{A}^{\leq \lambda}$ for any fixed $\lambda \in \Lambda$.
The cell modules are the ``square root by $\iota$'' of the direct summands of the associated graded module of the (left) regular representation with respect to this filtration; more precisely, $\mathcal{A} = \bigoplus_{\lambda \in \Lambda} \mathcal{A}^{\leq \lambda} / \mathcal{A}^{< \lambda}$ and $\mathcal{A}^{\leq \lambda} / \mathcal{A}^{< \lambda} \cong W(\lambda) \otimes \iota\bigl( W(\lambda) \bigr)$.
This is essentially the basis-free definition of cellular algebras given in~\cite{KX96} as the cell modules are left ideals of $\mathcal{A}^{\leq \lambda}$.
The construction of the link modules for the Temperley--Lieb algebra (see, \textit{e.g.},~\cite{GL96,RSA14}) is similar by looking at a maximal $\Lambda$-indexed chain of left ideals.
\end{remark}
We note that $\mathbf{k} [\Sigma_k]$ has an explicit construction of a cellular basis with $M(\lambda)$ given as the set of standard Young tableaux of shape $\lambda \vdash k$; see~\cite{GL96}.
By using a cellular basis for $\mathbf{k} [\Sigma_k]$ and the decomposition~\eqref{eq:partition_decomposition}, we have the following.
\begin{thm}[{\cite[Thm.~4.1]{Xi99}}]
\label{thm:partition_cellular}
The partition algebra $\mathcal{P}_n(\beta)$ is a cellular algebra with cell datum $(\Lambda, \iota, M, C)$ given by
\begin{itemize}
\item $\Lambda = \{\lambda \vdash k \mid k \in \{0,1,2,\dotsc,n\}\}$ under graded dominance order;
\item $\iota$ reflecting the diagram vertically;
\item $M(\lambda)$ is all pairs $(\rho, T)$, where $\rho$ is a set partition of $[n]$ such that $\abs{\rho} \geq \abs{\lambda}$ and $T$ is a standard tableau of shape $\lambda$ in the alphabet $A \subseteq \rho$ such that $\abs{A} = \abs{\lambda}$ under some total order on the subsets of $[n]$;
\item there exists a cellular basis $C$.
\end{itemize}
\end{thm}
\subsection{Cellular partition theory}
\label{sec:crt}
Recall that the characteristic of $\mathbf{k}$ is $p$.
An important consequence~\cite[Cor.~4.11]{Xi99} is we can parameterize the simple modules of $\mathcal{P}_n(\beta)$ as those $\lambda \in \Lambda$ that are a $p$-partition of size $\delta_{\beta0} \leq k \leq n$.
Next, we reinterpret the description of the cell modules from~\cite[Cor.~4.10]{Xi99} using interpretation given in~\cite{HL05} with the half diagram description.
Indeed, the basis for $W(\lambda)$ is given by $M(\lambda)$, where we use the total order given on subsets of $[n]$ by comparing the smallest value in each part.\footnote{Some authors use the largest value, such as in~\cite{BH19,HJ20,OSSZ21}. This is inconsequential and can be considered as using a different cellular basis.}
Note that the elements in the standard tableau are the defects, and the cell module comes with a natural $\mathcal{P}_n(\beta)$-action.
We note that the $\mathcal{P}_n(\beta)$-action cannot increase the number of defects, and since we want the number of defects to remain fixed, any element of $\mathcal{P}_n(\beta)$ that decreases the number of defects acts by $0$.
This translates to an action of the natural basis on the set partition and tableau pair, where we can apply the Garnir straightening relations.
Hence, two alternative descriptions~\cite{BDVK15,Xi99} of the cell module for $\lambda \vdash k$ are
\begin{equation}
\label{eq:specht_decomposition}
W(\lambda) \cong V'_k \otimes S^{\lambda} \otimes v_k \cong V'_{n,k} \otimes_{\mathbf{k} [\Sigma_k]} S^{\lambda},
\end{equation}
where $S^{\lambda}$ is the Specht module of $\Sigma_k$, $v_k$ is a fixed vector in $V_k$, and $V'_{n,k}$ is the $\mathbf{k}$-span of the diagrams with $k$ propagating blocks and $i$, for all $1 \leq i \leq n-k$, a singleton block.
From the half diagram description, we can write some formulas for the dimension of $V_k'$, which then yields $\dim W(\lambda) = f_{\lambda} \dim V'_k$.
Let $\stirling{a}{b}$ denote the Stirling number of the second kind counting the number of set partitions of $[a]$ into $b$ parts.
Define $B_a = \sum_{b=0}^a \stirling{a}{b}$ as the $a$-th Bell number that counts the number of set partitions of $[a]$,
We claim
\[
\dim V'_k = \sum_{j=k}^n \binom{n}{j} \stirling{j}{k} B_{n-j} = \sum_{j=k}^n \binom{j}{k} \stirling{n}{j}.
\]
The first formula comes from choosing a subset of size $j$ for all of the defect blocks and then taking a set partition on the remaining elements, which was proven bijectively in~\cite[Thm.~2.4]{CDDSY07} (see also~\cite{CD02,DS95,Roby91,Roby95}).
The second formula is from choosing $k$ parts of a set partition of $n$ (with exactly $j$ parts) to be the defects.
When $\mathbf{k}$ is a field of characteristic $0$ and $\beta \notin \{0,1,2,\dotsc,2n-1\}$, the cell modules are the set of simple modules from the double-centralizer theory with $\Sigma_m$ (see, \textit{e.g.},~\cite{HR05,MS94II}).
Furthermore, there is an alternative presentation for these modules using vacillating tableaux or, equivalently, chains in the Bratelli diagram.
We can give a necessary pictorial condition for the bilinear form $\Phi_{\lambda}(C_T, C_U) = 0$ as the diagram formed by reflecting $U$ vertically and connecting it with $T$ does not induce a bijection between the defects of $T$ and $U$.
Indeed, this will cause the number of propagating blocks (which equals the number of defects on the top and bottom) in the result to decrease and corresponds to a multiplication by $0$; see also Remark~\ref{rem:assoc_graded}.
This bijection then induces a permutation in $\Sigma_k$, which
Furthermore, it is equal to $\beta^D$, where $D$ is the number of interior components not containing a defect, times the bilinear form induced from the module $S^{\lambda}$ of $\mathbf{k} [\Sigma_k]$ evaluated at the corresponding induced permutation.
\begin{ex}
Consider the elements $v_{\tau}$ and $v'_{\tau}$ from Example~\ref{ex:decomposition}, which identify with elements $C_U$ and $C_T$, respectively, in a cell module $W(\lambda)$, where $\abs{\lambda} = 3$.
Then the corresponding pairing $\Phi_{\lambda}(C_T, C_U) = \Phi_{\lambda}(C_U, C_T) = 0$ as we do not have a bijection between the defects:
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline=10pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) + (0,-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4);
\draw[-,blue] (G3) + (0,-1) -- (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=blue] (G5) .. controls +(0, .75) and +(0, .75) .. (G3) -- ++(0,1);
\draw[-,color=darkred] (G4) .. controls +(0, .75) and +(0, .75) .. (G7);
\draw[-,color=darkred] (G1) -- ++(0,1);
\draw[-,color=darkred] (G2) -- ++(0,1);
\draw[-,color=UQpurple] (G8) -- ++(0,-1);
\end{tikzpicture}
\]
where we have reflected $T$ instead of $U$.
However, if we instead took
\[
U' = (\{\{2'\},\{3',5'\},\{8'\}\},\{\{1'\},\{4',7'\},\{6'\}\}),
\]
then for $\Phi_{\lambda}(C_T, C_{U'})$ we do obtain a bijection:
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline=10pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) + (0,-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4);
\draw[-,blue] (G3) + (0,-1) -- (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=blue] (G5) .. controls +(0, .75) and +(0, .75) .. (G3) -- ++(0,1);
\draw[-,color=darkred] (G4) .. controls +(0, .75) and +(0, .75) .. (G7);
\draw[-,color=UQpurple] (G8) -- ++(0,1);
\draw[-,color=darkred] (G2) -- ++(0,1);
\draw[-,color=UQpurple] (G8) -- ++(0,-1);
\end{tikzpicture}
\]
which induces the identity permutation and is scaled by $\beta^0$.
Finally, if we take
\[
U'' = (\{\{3',5',6'\},\{4'\},\{8'\}\},\{\{1'\},\{2'\},\{7'\}),
\]
then for $\Phi_{\lambda}(C_T, C_{U'})$ obtain
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline=10pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,8} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) + (0,-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4);
\draw[-,blue] (G3) + (0,-1) -- (G3) .. controls +(0, -.6) and +(0, -.6) .. (G5) .. controls +(0, -.5) and +(0, -.5) .. (G6);
\draw[-,color=blue] (G6) .. controls +(0, .5) and +(0, .5) .. (G5) .. controls +(0, .75) and +(0, .75) .. (G3) -- ++(0,1);
\draw[-,color=darkred] (G4) + (0,1) -- (G4);
\draw[-,color=UQpurple] (G8) -- ++(0,1);
\draw[-,color=UQpurple] (G8) -- ++(0,-1);
\end{tikzpicture}
\]
which induces the simple transposition $(1 \; 2)$ and scaled by $\beta^1$.
\end{ex}
\section{Cellular subalgebras}
For the remainder of this paper $\iota$ will be the automorphism of the partition algebra that reflects the diagram vertically (\textit{i.e.}, sends $i \leftrightarrow i'$ within the set partition) restricted to the subalgebra we are considering.
\subsection{Half integer partition algebra}
We begin with the \defn{half integer partition algebra} $\mathcal{P}_{n-1/2}(\beta)$ studied in~\cite{HR05}, which is defined as the subalgebra spanned by all diagrams such that $n$ and $n'$ are in the same part.
As such, the analysis of the representation theory is similar to that of the usual partition algebra with a few small differences.
We have $\dim \mathcal{P}_{n-1/2}(\beta) = B_{2n-1}$ since we chose the set partition for $[n-1] \sqcup [n]'$ and the part $n$ belongs to is the same as $n`$.
This is analogous to $\dim \mathcal{P}_n(\beta) = B_{2n}$.
By Proposition~\ref{prop:subcellular}, we have the following.
\begin{prop}
The half integer partition algebra $\mathcal{P}_{n-1/2}(\beta)$ is cellular.
\end{prop}
Since $n$ and $n'$ need to be in the same block, the half diagrams on $[n]'$ are now required to have $n'$ as an element of a defect, and so $\Lambda$
Thus, we see that $\Lambda$ is the set of all partitions $\lambda$ such that $1 \leq \abs{\lambda} \leq n$ and
\[
\dim W(\lambda) = (k \widetilde{V}'_k + \widetilde{V}'_{k-1}) f_{\lambda},
\]
where $k:= \abs{\lambda}$ and $\widetilde{V}'_k$ is the half diagram module for $\mathcal{P}_{n-1}(\beta-1)$ (from~\eqref{eq:specht_decomposition}), by choosing the defect block to add $n'$ to or if $n'$ is a defect on its own.
\subsection{Quasi-partition algebra}
The quasi-partition algebra $\mathcal{Q}\mathcal{P}_n(\beta)$ was defined in~\cite{DO14} as a centralizer algebra.
However, we instead define it by as a subalgebra of $\mathcal{P}_n(\beta - 1)$ using~\cite[Lemma~2.3]{DO14} with the basis given by all diagrams without any isolated vertices (\textit{cf.}~\cite[Sec.~2.4]{DO14}).
(Due to the differences in multiplication conventions, we need to flip all diagrams by $\iota$.)
However, the basis is not given by these diagrams alone, but instead as a sum over certain subsets.
In order for this algebra to be well-defined, we require $\beta \neq 0$.
For brevity, we do not include the explicit description of the basis here as we only need the properties given by~\cite[Lemma~2.2]{DO14}.
\begin{thm}
\label{thm:quasi_cellular}
The quasi-partition algebra is a cellular algebra.
\end{thm}
\begin{proof}
By~\cite[Lemma~2.2]{DO14}, the basis of $\mathcal{Q}\mathcal{P}_n(\beta)$ is triangular with respect to refinement of set partitions.
Furthermore, when we refine any block on the top, we must decrease the number of propagating blocks.
Thus, by the argument of Proposition~\ref{prop:cellular_triangle_basis}, we can ignore these.
Therefore, we see that the top half diagram (the element of $V_k$ in the decomposition~\eqref{eq:partition_decomposition}) does not change for the terms we need to consider.
We use the map $\widetilde{\iota}$ from Proposition~\ref{prop:cellular_triangle_basis} as the anti-involution, which simply reflects the diagram indexing the basis (like for the usual partition algebra).
Hence, this satisfies the conditions for an interated inflation~\cite{GP18,Xi99}.
\end{proof}
\begin{ex}
The map $\widetilde{\iota}$ is not simply the restriction of $\iota$ for $\mathcal{P}_n(\beta-1)$ since
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2) .. controls +(0, .5) and +(0, .5) .. (G-1) -- (G1);
\end{tikzpicture}
\quad \longleftrightarrow \quad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2) .. controls +(0, .5) and +(0, .5) .. (G-1) -- (G1);
\end{tikzpicture}
- \beta^{-1}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2);
\end{tikzpicture}
- \beta^{-1}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2);
\end{tikzpicture}
+ \beta^{-2}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2);
\end{tikzpicture}
+ \beta^{-2}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\end{tikzpicture}\, \in \mathcal{P}_n(\beta-1).
\]
Indeed, this element is invariant under $\widetilde{\iota}$ but not $\iota$.
\end{ex}
Note that $\Lambda$ is given by all partitions of size at most $n$ just like for $\mathcal{P}_n(\beta)$, recovering~\cite[Cor.~4.3]{DO14}.
We recover~\cite[Thm.~4.6]{DO14} from a straightforward counting of the half diagrams of the cell modules.
\begin{cor}[{\cite[Thm.~4.6]{DO14}}]
Let $\lambda \in \Lambda$ and $k = \abs{\lambda}$.
Then we have
\[
\dim W(\lambda) = f_{\lambda} \sum_{s=0}^k \binom{n}{s} \sum_{j=k-s}^{\lfloor \frac{n-s}{2} \rfloor} \binom{j}{k-s} \stirling{n-s}{j}_{\geq2},
\]
where $\stirling{a}{b}_{\geq2}$ is the number of set partitions of $a$ into $b$ parts with each part having size at least $2$.
\end{cor}
\begin{proof}
From Theorem~\ref{thm:quasi_cellular}, it is sufficient to count the half diagrams by selecting the defect blocks as the $f_{\lambda}$ comes from the decomposition~\eqref{eq:specht_decomposition}.
We chose exactly $s$ singleton defects, then we chose the remaining $k - s$ defects from the parts of the set partition, which necessarily has no singletons.
\end{proof}
The dimension of the quasi-partition algebra was given in~\cite[Cor.~2.9]{DO14} as
\[
\dim \mathcal{Q}\mathcal{P}_n(\beta) = \sum_{j=1}^{2n} (-1)^{j-1} B_{2n-j} + 1;
\]
see also~\cite[A000296]{OEIS}.
We can also perform the same analysis like the half partition algebra to get the odd sized base sets, but it would require showing that the restricted basis is closed under multiplication.
If that is true, the Schur--Weyl duality for the half partition algebra suggests a Schur--Weyl duality for the half quasi-partition algebra.
\subsection{Complex reflection group centralizer}
We begin this subsection with the subalgebra $\mathcal{G}^{(r,d,m)}_n(\beta) \subseteq \mathcal{P}_n(\beta)$ Tanabe studied in~\cite{Tanabe97} and its relationship with $\End_{G(r,d,m)} \mathbf{V}^{\otimes n}$, the centralizer algebra of $G(r,d,m)$ under the natural diagonal action.
Following~\cite[Lemma~2.1]{Tanabe97}, we \emph{define} $\mathcal{G}^{(r,d,m)}_n$ as the subalgebra of $\mathcal{P}_n(\beta)$ with basis given by the set partitions $\rho = \{\rho_1, \rho_2, \dotsc, \rho_{\ell}\}$ such that $\ell \leq m$ and satisfy either of the following conditions:
Denote $N(\rho_i) := \abs{\rho_i \cap [n]}$ and $N'(\rho_i) := \abs{\rho_i \cap [n]'}$
\begin{enumerate}
\item \label{cond:blocks} $N(\rho_i) \equiv N'(\rho_i) \pmod{r}$ for all $i$; or
\item \label{cond:d} $\ell = m$ and
\begin{enumerate}
\item \label{cond:diff} $N(\rho_i) \equiv N'(\rho_i) \pmod{r/d}$ for all $i$, and
\item \label{cond:constant_level} there exists $s \in \{1, 2, \dotsc, r-1\}$ such that $N(\rho_i) - N'(\rho_i) \equiv s \pmod{r}$ for all $i$.
\end{enumerate}
\end{enumerate}
Note that Condition~(\ref{cond:constant_level}) is stronger than $N(\rho_i) - N'(\rho_i) \not\equiv 0 \pmod{r}$ for all $i$.
We remark that Condition~(\ref{cond:d}) can never be satisfied if $d = 1$ or if $m > 2n$.
Similarly, Condition~(\ref{cond:diff}) becomes vacuous if $d = r$.
In~\cite[Lemma~2.1]{Tanabe97}, it is claimed that $\mathcal{G}^{(r,d,m)}_n(m) \cong \End_{G(r,d,m)} \mathbf{V}^{\otimes n}$, but his claim of linear independence relies on the faithfulness of the $\mathcal{P}_n(m)$-representation on $\End_{\Sigma_m} \mathbf{V}^{\otimes n}$ from Theorem~\ref{thm:partition_double_centralizer}.
Thus his proof that this does form a basis of the centralizer of the $G(r,d,m)$-action only holds when $m \geq 2n$.
In fact, for $m < n$, $\mathcal{G}_n^{(r,1,m)}$ does not necessarily have a unit as the next example shows, and hence it cannot be isomorphic to the endomorphism algebra (which is unital).
\begin{ex}
Consider $\mathcal{G}_3^{(2,1,2)}(\beta)$ for $\beta \neq 0$, and it can be verified that $\dim \mathcal{G}_3^{(2,1,2)}(\beta) = 25$ and $\mathcal{G}_3^{(2,1,2)}(\beta)$ is closed under the usual diagram multiplication.
All of its diagrams have either one or two propagating blocks, and we denote the set of diagrams with $i$ propagating blocks by $P_i$.
We have $\abs{P_1} = 16$ and $\abs{P_2} = 9$, and one such diagram in $P_1$ and $P_2$ are
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-2) .. controls +(0,1) and +(0,-1) .. (G3) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1);
\draw[-] (G-3) .. controls +(0., .65) and +(0, .65) .. (G-1);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1) .. controls +(0,-1) and +(0,1) ..
(G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\draw[-] (G3) .. controls +(0,-1) and +(0,1) .. (G-1);
\end{tikzpicture}\,.
\]
Note that $\mathcal{G}_3^{(2,1,2)}(\beta)$ does not contain $1 \in \mathcal{P}_n(\beta)$ as this diagram has $3$ parts.
The product of diagrams $P_1 \cdot P_2 \subseteq P_1$ and $P_2 \cdot P_1 \subseteq P_1$ by a parity argument.
Hence, $\mathcal{G}_3^{(2,1,2)}(\beta)$ is not a unital algebra.
\end{ex}
However, his main result describing the generators yields the following.
\begin{thm}[{\cite{Tanabe97}}]
\label{thm:complex_centralizer}
Suppose $\mathcal{G}_n^{(r,d,m)}$ is an algebra over $\mathbf{k}$.
There exists a surjection
\[
\phi \colon \mathcal{G}_n^{(r,d,m)}(m) \to \End_{G(r,d,m)} \mathbf{V}^{\otimes n}.
\]
Furthermore, the surjection $\phi$ is an isomorphism if $m \geq 2n$.
\end{thm}
\begin{conj}
For $r > 1$ and $d = 1$, the map $\psi$ is an isomorphism if and only if $m \geq n$.
\end{conj}
Let us further discuss the assumption that $\mathcal{G}_n^{(r,d,m)}(\beta)$ is an algebra.
Indeed, the number of blocks could potentially grow larger than $m$.
In~\cite{Tanabe97}, it was said diagrams with strictly more than $m$ parts are $0$ in $\mathcal{G}_n^{(r,d,m)}(\beta)$, which means such diagrams should \emph{span} an ideal $\mathcal{I}$ of the partition algebra such that $\mathcal{G}_n^{(r,d,m)}(\beta) \cong \mathcal{P}_n(\beta) / \mathcal{I}$.
However, this is not the case, as the next example shows.
This is a byproduct of the fact that the diagrams should be in a quotient of the algebra $\mathcal{G}_n^{(r,d,2n)}(\beta)$ for small $m$.
\begin{ex}
\label{ex:G2222_diagrams}
Consider $\mathcal{G}_2^{(2,2,2)}(\beta)$.
We have the diagrams coming from Condition~(\ref{cond:d}):
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) -- (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2) -- (G2);
\end{tikzpicture}\,,
\]
and using these elements, we can form the elements
\begin{equation}
\label{eq:max2_3parts}
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-2) -- (G2);
\end{tikzpicture}\,.
\end{equation}
Hence, the ``basis'' for $\mathcal{G}_2^{(2,2,2)}(\beta)$ with $\beta \neq 0$ (such as $\beta = m = n = 2$) is not closed under multiplication unless we consider the product to be $0$.
As above, we must have the diagrams with $3$ or $4$ parts spanning an ideal of $\mathcal{P}_2(\beta)$.
However, we can take one of the products
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2) -- (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1);
\end{tikzpicture}
\; \cdot \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-2) -- (G2);
\end{tikzpicture}
\; = \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2) -- (G2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-2) -- (G2);
\end{tikzpicture}
\; \cdot \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2) -- (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1);
\end{tikzpicture}
\; = \;
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2);
\end{tikzpicture}\,,
\]
which is not in the supposed ideal (either left or right).
In fact, we note that the elements in~\eqref{eq:max2_3parts} plus $s_1$ and $\{\{1,2,1',2'\}\}$ (both are defined as ``basis'' elements of $\mathcal{G}_2^{(2,2,2)}(\beta)$) are the generators of the full partition algebra $\mathcal{P}_2(\beta)$.
Hence, under the usual multiplication, we have $\mathcal{G}_2^{(2,2,2)}(\beta) = \mathcal{P}_2(\beta)$.
\end{ex}
Another way to recover the important portion of Tanabe's results in~\cite{Tanabe97} would be to say there exists a surjection $\mathcal{G}_n^{(r,d,2n)}(m) \to \End_{G(r,d,m)} \mathbf{V}^{\otimes n}$.
Note that $\mathcal{G}_n^{(r,d,m)}(\beta) \cong \mathcal{G}_n^{(r,d,2n)}(\beta)$ for all $m \geq 2n$.
Furthermore, it is likely that the set of diagrams for $m < 2n$ is the indexing set for a basis, but the multiplication is more complicated than the usual concatenation of diagrams.
Consequently, we will be considering either when $r > n$ or when $m \geq 2n$.
Recall that in these cases, diagrams satisfying Condition~(\ref{cond:d}) can never be satisfied.
Furthermore, $\mathcal{G}_n^{(r,d,m)}(\beta)$ is always an algebra.
Now we present our first main result.
\begin{thm}
\label{thm:ccc}
Suppose that either $r > n$ or $m \geq 2n$.
The $G(r,d,m)$-partition algebra $\mathcal{G}^{(r,d,m)}_n(\beta)$ is a cellular algebra.
\end{thm}
\begin{proof}
The same proof as in~\cite[Thm.~4.1]{Xi99} (see also~\cite{GP18}) holds here since the additional conditions makes the allowed set of permutations isomorphic to a direct product of symmetric groups.
Note that the set of such diagrams is invariant under $\iota$.
This gives us the indexing set $\Lambda$ as a $k$-tuple of partitions, where $k$ is the number of factors in the direct product for those diagrams satisfying Condition~(\ref{cond:blocks}).
We take the ordering on $\Lambda$ as a product poset of the graded dominance order.
For any $\lambda \in \Lambda$, the set $M(\lambda)$ is the corresponding set of pairs consisting of a tuple of semistandard Young tableaux of shape $\lambda$ and a set partition.
\end{proof}
Recall that $G(1,1,m) \cong \Sigma_m$ and the centralizer is the full partition algebra, which was shown to be cellular for $m \geq 2n$ by Xi~\cite{Xi99} (with a small correction by~\cite{GP18}).
This likely extends to all cases such that $\mathcal{G}_n^{(r,d,m)}(\beta)$ is a $\mathbf{k}$-algebra with a potentially more complicated description of $\Lambda$.
Next, we will explore some particular cases in more detail and describe the cell datum and cell modules.
\subsubsection{Uniform block algebra}
\label{sec:uniform_block}
We consider the case of $G(r,d,m)$ when $d = 1$ and $r > n$.
Let $\mathcal{U}_n$ denote the \defn{uniform block algebra}, which is a subalgebra of the partition algebra such that the number of top elements equals the number of bottom elements.
We omitted $m$ from our notation as we are primarily interested in the case $m \geq n$, where the description is independent of $m$, and the case $m < n$ is not fundamentally different.
The construction of all simple modules of $\mathcal{U}_n$ over $\mathbb{C}$ was recently computed by Orellana, Saliola, Schilling, and Zabrocki~\cite{OSSZ21} when $m \geq n$.
Their representation theoretic results essentially becomes a corollary of Theorem~\ref{thm:ccc} using the framework of Section~\ref{sec:crt}.
Let us examine their results in detail.
We begin with the (sub)algebra of idempotents $\mathcal{I}\mathcal{U}_n$ whose structure was given by~\cite[Lemma~2.3]{OSSZ21}.
The following proposition is a straightforward consequence and forms the foundation of the constructions of~\cite{OSSZ21}.
\begin{prop}
\label{prop:uniform_idempotent_cellular}
The algebra $\mathcal{I}\mathcal{U}_k$ is a cellular algebra with cell datum $(\Lambda, \iota, M, C)$ given by
\begin{itemize}
\item $\Lambda$ is all set partitions of $[n]$ with at most $m$ parts under the refinement order with $\{[n]\}$ as the bottom element;
\item $\iota$ becomes the identity;
\item $M(\lambda) = \{\lambda\}$ (so we can ignore it);
\item $C$ is the natural diagram basis.
\end{itemize}
Moreover, the cell modules are all (nonzero) simple modules and one dimensional.
\end{prop}
The main observation is that we cannot have an arbitrary permutation $\sigma \in \Sigma_k$ for $k \leq \min(n,m)$, but instead have to preserve the block sizes.
Hence, we remark that~\cite[Prop.~2.5]{OSSZ21} is effectively the decomposition of~\cite[Lemma~4.2]{Xi99}.
Furthermore, we obtain a Young subgroup corresponding to the block sizes, which was the maximal group for the corresponding idempotent in~\cite{OSSZ21}.
This means we could use Proposition~\ref{prop:subcellular} to give an alternative proof via a product of symmetric group algebra cellular bases.
Within each Young subgroup, we have a permutation of multisets of the same size $k$ (contrast this with the partition algebra) representing what can happen with the blocks of size $k$.
This permutation can be represented by a pair of semistandard tableaux whose entries are (disjoint) sets of size $k$, where we use the total ordering given by comparing the smallest element in each set.
Consequently, we see that we can interpret $\Lambda$ as the set of partition tuples $\lambda = (\lambda^{(1)}, \dotsc, \lambda^{(n)})$ such that $\sum_{k=1}^n k \abs{\lambda^{(k)}} = n$.
Furthermore, $M(\lambda)$ becomes the set of semistandard tableau tuples, where the $k$-th element has shape $\lambda^{(k)}$ and is filled with subsets of $[n]$, and the union of all of the entries is $[n]$.
Compare this with Theorem~\ref{thm:partition_cellular}, where we are requiring for each pair $(\rho, T)$ that $\abs{rho} = \abs{\lambda}$ and are further separating into separate blocks based on the size of the parts.
From the above description, we see that the cell modules are precisely the modules given in~\cite{OSSZ21}.
Additionally, the bilinear form $\Phi_{\lambda}$ is given by the corresponding bilinear form of a product of symmetric groups, which is the product of the bilinear forms of the appropriate symmetric group.
Consequently, all of the cell modules are simple when $\mathbf{k}$ has characteristic $0$, which recovers the description in~\cite{OSSZ21}.
Furthermore, we obtain a set of simple modules corresponding over a field of characteristic $p > 0$ when each partition in the tuple $\lambda$ is a $p$-regular partition (\textit{cf.}~\cite[Cor.~4.11]{Xi99}).
\subsubsection{Parity matching algebra}
Recall that the group of signed permutations of rank $m$, which is the Weyl group of $\operatorname{O}_{2m+1}(\mathbb{C})$ or the Coxeter group of type $B_m$, and corresponds to $G(2,1,m)$.
The centralizer algebra was studied in more detail in~\cite{Orellana05,Orellana07}, where it was related to the colored partition algebra introduced by Bloss~\cite{Bloss03}.
From~\cite[Lemma~2.1]{Tanabe97}, the diagrams for this algebra are given by the set partitions of $[n]$ and $[n]'$ such that
\begin{enumerate}[(I)]
\item the blocks of even (resp.\ odd) size connect to blocks of even (resp.\ odd) size;
\item all odd blocks must be connected; and
\item there are at most $m$ such connected components in the resulting diagram. \label{cond:block_sizes}
\end{enumerate}
In particular, the number of odd sized blocks must be the same for $[n]$ and $[n]'$.
If we have $n \leq m$, the last condition~(\ref{cond:block_sizes}) is always satisfied and $\mathcal{G}_n^{(2,1,m)}(\beta)$ is a (unital) $\mathbf{k}$-algebra.
Hence, assuming $m \geq n$, we call the subalgebra $\mathcal{P}_n(\beta)$ spanned by these diagrams the \defn{parity matching algebra} and denote it by $\mathcal{P}\mathcal{M}_n(\beta)$ since it becomes independent of $m$ like the uniform block algebra.
Furthermore, $\mathcal{P}\mathcal{M}_n(\beta)$ is cellular by Theorem~\ref{thm:ccc}.
An alternative description of the basis of $\mathcal{P}\mathcal{M}_n(\beta)$ consists of all diagrams $\rho = \{\rho_1, \dotsc, \rho_{\ell}\}$ that are even set partitions, that is, we have $\abs{\rho_i} \equiv 0 \pmod{2}$ for all $i$.
Hence, another name for this algebra could be the ``even partition algebra.''
We can see this is indeed a $\mathbf{k}$-subalgebra (\textit{i.e.}, closed under multiplication) by a straightforward parity argument.
As a consequence, we obtain the recursion formula (see, \textit{e.g.},~\cite[Sec.~5]{Orellana07}) for its dimension
\[
\dim \mathcal{P}\mathcal{M}_n(\beta) = \sum_{i=1}^n \binom{2k-1}{2i-1} \dim \mathcal{P}\mathcal{M}_{n-i}(\beta)
\]
by removing the block containing $n$ from each diagram.
The sequence of dimensions is~\cite[A005046]{OEIS}, which contains further combinatorial interpretations and the closed formula
\[
\dim \mathcal{P}\mathcal{M}_n(\beta) = \sum_{k=1}^{2n} \sum_{i=0}^{k-1} (-1)^i \frac{(i-k)^{2n} }{2^{k-1} k!} \binom{2k}{i}.
\]
Another similarity to the uniform block algebra is the even and odd blocks cannot interact.
Hence we can use the Young subgroup of $\Sigma_{k_1} \times \Sigma_{k_2} \subseteq \Sigma_k$, where there are $k_1$ (resp.~$k_2$) odd (resp.\ even) with $k = k_1 + k_2$, to deconstruct our diagrams.
The cell modules of $\mathcal{P}\mathcal{M}_n(\beta)$ are parameterized by pairs of partitions $(\mu, \nu)$ such that
\begin{subequations}
\label{eq:PM_conditions}
\begin{align}
\abs{\mu} + 2\abs{\nu} & \leq n, \label{eq:PM_min_sizes}
\\ n - \abs{\mu} & \equiv 0 \pmod{2}. \label{eq:PM_even_disconnect}
\end{align}
\end{subequations}
Recall this is the set $\Lambda$.
The inequality~\eqref{eq:PM_min_sizes} comes from partitioning the blocks into the even and odd propagating blocks, which must have size at least $2$ and $1$ respectively.
The condition~\eqref{eq:PM_even_disconnect} is that the non-defect blocks must have even size.
The indexing set $M(\lambda)$ for the basis of the irreducible representation $W(\lambda)$ for $\lambda = (\mu, \nu) \in \Lambda$ is a pair of standard tableaux of shapes $(\mu, \nu)$ such that $\mu$ (resp.~$\nu$) are filled with subsets of $[n]$ of odd (resp.~even) size and a set partition of the remaining letters with all blocks having even size such that the disjoint union of all of the entries is $[n]$.
For the element $v_k$ in the construction~\eqref{eq:specht_decomposition}, we can have very propagating block (which is fixed by the choice of $\lambda$) have size $1$ or $2$ in some fixed order.
By the condition~\eqref{eq:PM_min_sizes}, such an element exists.
To describe the dimensions of the cell modules, we need some combinatorial data.
Let $O_{n,k}$ (resp.\ $E_{n,k}$) denote the number of set partitions $\rho = \{\rho_1, \dotsc, \rho_k\}$ (so exactly $k$ parts) of $[n]$ such that $\abs{\rho_i} \equiv 1 \pmod{2}$ (resp.\ $\abs{\rho_i} \equiv 0 \pmod{2}$) for all $i$.
These are the set of odd (resp.\ even) set partitions with exactly $k$ parts, and $O_{n,k}$ is the sequence~\cite[A136630]{OEIS} (resp.~\cite[A156289]{OEIS} for $E_{2n,k}$).
We note the formulas
\begin{align*}
O_{n,k} & = O_{n-2,k-2} + k^2 O_{n-2,k},
&
E_{2n,k} & = (2k-1) E_{2n-2,k-1} + k^2 E_{2n-2,k},
\\
O_{n,k} & = \frac{1}{2^k k!} \sum_{j=0}^k (-1)^{k-j} \binom{k}{j} (2j - k)^n,
&
E_{2n,k} & = \frac{2}{2^k k!} \sum_{j=1}^k (-1)^{k-j} \binom{2k}{k-j} j^n.
\end{align*}
Note that $E_{2n+1,k} = 0$ and $O_{n,k} = 0$ whenever $n + k \equiv 1 \pmod{2}$ by parity arguments.
The $O_{n,k}$ recurrence relation can be proven combinatorially from two cases:
We have a singleton $\{n\}$, which necessarily means there is another singleton, and removing both of these singletons yields $O_{n-2,k-2}$ (it does not affect the result which other singleton we remove by renaming).
Otherwise we need to remove another element from the part containing $n$.
To reconstruct such a part, for some fixed $\nu$ contributing to $O_{n-2,k}$, we choose a part $\nu_i$ in some $\nu \in O_{n-2,k}$ to add $n$ to and choose another part to correspond to $n-1 \leftrightarrow \min \nu_j - 1/2$ by renaming and reordering.
A combinatorial proof for the $E_{n,k}$ recurrence is similar.
While an explicit combinatorial proof does not seem to be in the literature, these proofs sketched above are likely known to experts.
For $\lambda = (\mu, \nu) \in \Lambda$, from the combinatorial description of $M(\lambda)$, we see that
\begin{subequations}
\label{eq:dim_cell_PM}
\begin{align}
\dim W(\lambda) & = f_{\mu} f_{\nu} \sum_{i=k_1}^n \binom{n}{i} O_{i,k_1} \sum_{j=k_2}^{\lfloor (n - i) / 2 \rfloor} \binom{j}{k_2} E_{n-i,j}
\\ & = f_{\mu} f_{\nu} \sum_{i=k_2}^{\lfloor (n-k_1)/2 \rfloor} \binom{n}{2i} \sum_{j=k_2}^i \binom{j}{k_2} E_{2i,j} \cdot O_{n-2i,k_1},
\end{align}
\end{subequations}
where $k_1 = \abs{\mu}$ and $k_2 = \abs{\nu}$.
Indeed, for the first equality, the first sum comes from choosing the elements for the odd sized blocks (all of which must be defects), the second sum comes from choosing exactly $k_2$ of the even sized set partitions to be defects.
The second equality is similar but first choosing the elements for the even sized blocks.
Furthermore, we have
\begin{align*}
\dim \mathcal{P}\mathcal{M}_n(\beta) & = \sum_{\lambda \in \Lambda} \bigl( \dim W(\lambda) \bigr)^2
\\ & = \sum_{k_1=0}^n \sum_{k_2=0}^{\rfloor (n-k_1) / 2 \lfloor} k_1! k_2! \left( \sum_{i=k_1}^n \binom{n}{i} O_{i,k_1} \sum_{j=k_2}^{\lfloor (n - i) / 2 \rfloor} \binom{j}{k_2} E_{n-i,j} \right)^2
\\ & = \sum_{k_1=0}^n \sum_{k_2=0}^{\rfloor (n-k_1) / 2 \lfloor} k_1! k_2! \left( \sum_{i=k_2}^{\lfloor (n-k_1)/2 \rfloor} \binom{n}{2i} \sum_{j=k_2}^i \binom{j}{k_2} E_{2i,j} \cdot O_{n-2i,k_1} \right)^2
\end{align*}
from Equation~\eqref{eq:dim_cell_PM}, where we have used Equation~\eqref{eq:fla=n2} to obtain the final two formulas.
By the double centralizer theorem (see, \textit{e.g.},~\cite{EGHLSVY11}), the irreducible representations of $\mathcal{P}\mathcal{M}_n(m)$ are in bijection with those irreducible $G(2,1,m)$-representations appearing in the decomposition of $\mathbf{V}^{\otimes n}$.
All irreducible representations of $G(2,1,m)$ which are indexed by pairs of partitions $(\mu, \widetilde{\nu})$ such that $\abs{\mu} + \abs{\widetilde{\nu}} = m$ (this was attributed to Specht~\cite{Specht32} in~\cite{Ram97}).
We can see by directly counting that we do not obtain all irreducible representations of $G(2,1,n)$ when $m \geq n$.
\begin{ex}
The basis for $\mathcal{P}\mathcal{M}_2(\beta)$ is given by four diagrams
\ytableausetup{boxsize=1.1em}
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) -- (G-1);
\draw[-,color=blue] (G2) -- (G-2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) -- (G-2);
\draw[-,color=blue] (G2) -- (G-1);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1) --
(G-1) .. controls +(0., .5) and +(0, .5) .. (G-2);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2);
\draw[-,color=blue] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2);
\end{tikzpicture}\,,
\]
which indeed forms a $\mathbf{k}$-algebra.
The set $\Lambda$ is given by the following pairs of partitions
\[
\left( \ydiagram{2}, \emptyset \right),
\qquad
\left( \ydiagram{1,1}, \emptyset \right),
\qquad
\left( \emptyset, \ydiagram{1} \right),
\qquad
(\emptyset, \emptyset),
\]
which are each one-dimensional modules with the corresponding semistandard tableaux
\[
\left( \ytableaushort{12}, \emptyset \right),
\qquad
\left( \ytableaushort{1,2}, \emptyset \right),
\qquad
\left( \emptyset, \ytableaushort{{12}} \right),
\qquad
(\emptyset, \emptyset),
\]
However, there are 5 irreducible representations of $G(2,1,2)$ corresponding to the pair of partitions
\[
\left( \ydiagram{2}, \emptyset \right),
\qquad
\left( \ydiagram{1,1}, \emptyset \right),
\qquad
\left( \emptyset, \ydiagram{1,1}\right),
\qquad
\left( \emptyset, \ydiagram{2} \right),
\qquad
\left( \ydiagram{1}, \ydiagram{1} \right),
\]
where the last one corresponds to a $2$ dimensional representation.
\end{ex}
\subsubsection{Colored permutation symmetrizer}
The remaining case corresponding to colored permutations $G(r,1,m)$ with $2 < r \leq n$ is essentially the same as $r = 2$.
The index set will be a subset of all $r$-tuples of partitions that satisfy the general form of the conditions~\eqref{eq:PM_conditions} for $\lambda =(\lambda^{(1)}, \dotsc, \lambda^{(r)})$:
There exists a $k_0 \in \mathbb{Z}_{\geq 0}$ such that
\begin{align*}
r k_0 + \sum_{j=1}^r j \abs{\lambda^{(j)}} & = 0,
\end{align*}
with $\lambda$ again coming from encoding the propagating blocks and $k_0$ approximately counting the non-propagating blocks (which must necessarily be a multiple of $r$; hence $k_0$ is a sum of these factors).
Furthermore, such a diagram exists independent of $m$ as we can always combine the blocks of size $r$ with one of those for $\lambda^{(j)}$ for any $j$ and since all of these are connected with the defect blocks, they do not contribute any additional connected components.
The set $M(\lambda)$ is the analogous tuple of semistandard tableaux of shape $\lambda$ with $\lambda^{(j)}$ being filled with multisets of size $j$ and a set partition with all the blocks having size equivalent to $0$ modulo $r$.
As before, all entries must be disjoint.
\subsection{Brauer algebra}
The \defn{Brauer algebra} $\mathcal{B}_n(\beta)$ is the subalgebra of the partition algebra $\mathcal{P}_n(\beta)$ spanned by all diagrams with parts that have exactly size $2$.
It was introduced by Brauer~\cite{Brauer37} coming from Schur--Weyl duality using the orthognal group, and it is known to be a cellular algebra~\cite{GL96} (see also~\cite{Xi99}) of dimension
\[
\dim \mathcal{B}_n(\beta) = \frac{(2n)!}{2^n n!} = (2n - 1)!!.
\]
Furthermore, from our description, we can easily see that the cell modules (which are Brauer's Specht modules) have dimension
\[
\dim W(\lambda) = \binom{n}{\abs{\lambda}} (n - \abs{\lambda} - 1)!! \dim V_{\Sigma_{\abs{\lambda}}}(\lambda),
\]
where $\lambda$ is a partition such that $\abs{\lambda} \leq n$ and $\abs{\lambda} \equiv n \pmod{2}$.
Indeed, we select $\abs{\lambda}$ propagating (necessarily) singletons from $[n]$, then we are left choosing a Brauer diagram on the renaming $n - \abs{\lambda}$ such nodes; the usual Specht module $V_{\Sigma_{\abs{\lambda}}}(\lambda)$ comes from Equation~\eqref{eq:specht_decomposition}.
When $\mathbf{k}$ has characteristic $0$, Wenzl showed~\cite{Wenzl1988} it is semisimple whenever $\beta \in \mathbf{k} \setminus \{0, \pm 1, \dotsc, \pm n\}$.
The representations of $\mathcal{B}_n(\beta)$ when it is not semisimple were studied by Doran, Hanlon, and Wales~\cite{DWH99} and Cox, De Visscher, and Martin~\cite{CDVM09,CDVM09II}.
The classification of when $\mathcal{B}_n(\beta)$ is semisimple for the general characteristic case was shown by Rui and Si~\cite{Rui05,RS06}.
\subsection{Rook Brauer algebra}
\label{sec:rook_brauer}
The \defn{rook Brauer algebra} $\mathcal{R}\mathcal{B}_n(\beta)$ (also known as the partial Brauer algebra) is defined as the subalgebra of $\mathcal{P}_n(\beta)$ spanned by all diagrams such that the parts have size at most $2$.
This has been studied in~\cite{HdM14,MM14} with two different descriptions of their irreducible representations.
The monoid algebra version (so $\beta = 1$) was also studied in~\cite{DEG17}.
Furthermore, we have the following result.
\begin{thm}[{\cite{MM14}}]
\label{thm:rook_Brauer}
The rook Brauer algebra is a cellular algebra. Moreover, it is Morita equivalent to
\[
\mathcal{R}\mathcal{B}_n(\beta) \simeq \mathcal{B}_n(\beta - 1) \oplus \mathcal{B}_{n-1}(\beta - 1)
\]
for $\beta - 1, \beta \neq 0$.
\end{thm}
Theorem~\ref{thm:rook_Brauer} gives us one method to describe all of the irreducible representations.
However, we could also undertake the same half diagram analysis like for the Brauer algebra to determine the dimensions of its cell modules:
\[
\dim W(\lambda) = f_{\lambda} \binom{n}{k} \sum_{m=0}^{\lfloor (n-k)/2 \rfloor} \binom{n-k}{2m} (2m-1)!!,
\]
where $k = \abs{\lambda}$, by choosing the positions of the singletons.
We can allow $\beta = 0$ in Theorem~\ref{thm:rook_Brauer} if we slightly change the multiplication to only count the number of loops removed in the product, not paths contractible to a point.
In fact, this leads to a more general definition of the rook Brauer algebra (and its subalgebras) using two parameters $\mathcal{R}\mathcal{B}_n(\beta,\gamma)$, where $\beta$ counts the number of loops removed and $\gamma$ counts the number of contractible paths.
Thus $\mathcal{R}\mathcal{B}_n(\beta) = \mathcal{R}\mathcal{B}_n(\beta, \beta)$, and furthermore, we have isomorphic algebras $\mathcal{R}\mathcal{B}_n(\beta, \beta) \cong \mathcal{R}\mathcal{B}_n(\beta, \gamma)$ whenever $\beta, \gamma \neq 0$~\cite{HdM14,MM14}.
\begin{ex}
\label{ex:nonassoc_product}
There is a misprint in the definition of the diagram multiplication in~\cite{HdM14}, as it is not sufficient to only consider singletons for the second parameter $\gamma$.
Indeed, if we take this as the definition, then we do not have an associative algebra:
\[
\gamma^2 \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (Gp\i) at (\i, 3) [shape=circle, draw] {};
\node[vertex] (Gm\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,color=darkred] (Gp1) .. controls +(0, -.5) and +(0, -.5) .. (Gp2);
\draw[-,color=UQpurple] (Gm1) .. controls +(0, .5) and +(0, .5) .. (Gm2);
\end{tikzpicture}
\quad = \quad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (Gp\i) at (\i, 3) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
\node[vertex] (Gm\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,color=darkred] (Gp1) .. controls +(0, -.5) and +(0, -.5) .. (Gp2);
\draw[-,color=UQpurple] (Gm1) .. controls +(0, .5) and +(0, .5) .. (Gm2);
\end{tikzpicture}
\quad = \quad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (Gp\i) at (\i, 3) [shape=circle, draw] {};
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
\node[vertex] (Gm\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,color=darkred] (Gp1) .. controls +(0, -.5) and +(0, -.5) .. (Gp2);
\draw[-,color=blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2) -- (G-1);
\draw[-,color=UQpurple] (Gm1) .. controls +(0, .5) and +(0, .5) .. (Gm2);
\end{tikzpicture}
\quad = \quad \gamma \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (Gp\i) at (\i, 3) [shape=circle, draw] {};
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (Gm\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,color=darkred] (Gp1) .. controls +(0, -.5) and +(0, -.5) .. (Gp2);
\draw[-,color=blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2);
\draw[-,color=UQpurple] (Gm1) .. controls +(0, .5) and +(0, .5) .. (Gm2);
\end{tikzpicture}
\quad = \quad \beta \gamma \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (Gp\i) at (\i, 3) [shape=circle, draw] {};
\node[vertex] (Gm\i) at (\i, -3) [shape=circle, draw] {};
}
\draw[-,color=darkred] (Gp1) .. controls +(0, -.5) and +(0, -.5) .. (Gp2);
\draw[-,color=UQpurple] (Gm1) .. controls +(0, .5) and +(0, .5) .. (Gm2);
\end{tikzpicture}
\]
\end{ex}
Given the two parameters, we have the following statement and problem (\textit{cf.}~\cite{MM14,DG22II}).
\begin{prop}
\label{prop:two_param_rook_brauer_iso}
For all $\gamma \neq 0$, we have $\mathcal{R}\mathcal{B}_n(0, \gamma) \cong \mathcal{R}\mathcal{B}_n(0, 1)$.
\end{prop}
\begin{proof}
Same as the case $\beta \neq 0$.
\end{proof}
\begin{problem}
\label{prob:two_param_rook_brauer_semisimple}
Determine when $\mathcal{R}\mathcal{B}_n(0, \gamma)$ and $\mathcal{R}\mathcal{B}_n(\beta, 0)$ are semisimple.
\end{problem}
We remark that the second parameter can never occur in the Brauer algebra as we can only have loops in the product.
\subsection{Rook algebra}
The \defn{rook algebra} $\mathcal{R}_n(\beta) \subseteq \mathcal{P}_n(\beta)$ is the set of diagrams $\rho$ such that every block has size at most $2$ and $N(\rho), N'(\rho) \leq 1$.
In other words, every block of size $2$ must be propagating.
An alternative description is in terms of partial permutations of $[n]$, which are injective maps $\pi \colon D \to [n]'$ for some $D \subseteq [n]$.
Furthermore, by rescaling by $\beta^{-I}$, where $I$ is the number of isolated in one half of the diagram (note that we necessarily have the number of isolated vertices on each side being equal), we can see $\mathcal{R}_n(\beta) \cong \mathcal{R}_n(1)$ for all $\beta \neq 0$.
This was first introduced and the irreducible representations studied by Munn~\cite{Munn57}\footnote{Munn in~\cite{Munn57} considered it as a monoid algebra, and so $\beta = 1$.} and later refined by Solomon~\cite{Solomon02}.
Like for the (rook) Brauer algebra, we have $\Lambda$ as for the partition algebra, but $M(\lambda)$ consists only of the standard Young tableau with entries consisting of a single entry.
This is just like the classical case when we restrict to a specific subset of $[n]$ of size $\abs{\lambda}$, which correspond to the propagating blocks.
As a consequence, we obtain that
\begin{equation}
\label{eq:rook_dim}
\dim \mathcal{R}_n(\beta) = \sum_{k=0}^n \sum_{\mu \vdash k} \binom{n}{\abs{\mu}}^2 f_{\mu}^2 = \sum_{k=0}^n \binom{n}{k}^2 k!,
\end{equation}
where $f_{\mu}$ is equal to the number of standard Young tableau on $[k]$ of shape $\mu \vdash k$.
We have factored out the choice of subset in the binomial coefficient and used Equation~\eqref{eq:fla=n2}.
The dimension formula in Equation~\eqref{eq:rook_dim} is also known from the original definition of the rook monoid through a straightforward combinatorial argument; see, \textit{e.g.},~\cite[Eq.~(1.2)]{Solomon02}.
We remark that we cannot have any loops to remove in the product of the rook algebra.
As such, we have $\mathcal{R}_n(\gamma) \subseteq \mathcal{R}\mathcal{B}_n(\beta,\gamma)$ and $\beta$ is not involved.
In~\cite{DG21}, it was shown that the centralizer algebra of the $n$-fold tensor power of the (unreduced) Burau representation $\mathbf{F}$ of the braid group $\mathbf{B}_m$ with $m$ strands is isomorphic to $\mathcal{R}_n([m]_q)$ for $q$ not a root of unity and $m > 2n$.
It is generated by $\sigma_i$ being the simple crossing of the $i$-th strand over the $(i+1)$-th strand, and these satisfy the braid relations $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}$.
However, the braid group has a number of interesting subgroups:
\begin{itemize}
\item The affine braid group $\widetilde{\mathbf{B}}_{m-1}$ of $m- 1 $ braids a cylinder by considering the center to be an additional (fixed) strand.
\item If $m = 2M - 1$ is odd, the type $B_M$ Artin group is $\langle \sigma_1 \sigma_m, \sigma_2 \sigma_{m-1}, \dotsc, \sigma_{M-1} \sigma_{M+1}, \sigma_M \rangle$ as it can be easily checked that $\tau_{M-1} \sigma_M \tau_{M-1} \sigma_M = \sigma_M \tau_{M-1} \sigma_M \tau_{N-1}$ for $\tau_{M-1} := (\sigma_{M-1} \sigma_{M+1})$.
\item The pure braid group $\mathbf{PB}_m$, which can be defined by the short exact sequence $0 \to \mathbf{PB}_m \to \mathbf{B}_m \to \Sigma_m \to 0$ using the natural projection.
\item The subgroup $\mathbf{R}_m = \langle \sigma_i^2 \rangle$, which is a right-angled Artin group (also known as a partially commutative group) of a line\footnote{Some conventions in the literature use the complement graph.} by~\cite{Collins94,CP01,Humphries94}.
\end{itemize}
There are other subgroups of these subgroups.
For example, the affine braid group $\widetilde{\mathbf{B}}_{m-1}$ has the pure affine braid group $\widetilde{\mathbf{PB}}_{m-1}$ with the short exact sequence $0 \to \widetilde{\mathbf{PB}}_{m-1} \to \widetilde{\mathbf{B}}_{m-1} \to \widetilde{\Sigma}_{m-1} \to 0$, where $\widetilde{\Sigma}_{m=1}$ is the affine symmetric group; the right-angled Artin group of a circle generated by the square of generators of $\widetilde{\mathbf{B}}_{m-1}$; and the affine type $\widetilde{C}_M$ braid group (constructed similarly to the type $B_M$ inside $\mathbf{B}_{2M-1}$).
Right-angled Artin groups are known to have many interesting subgroups~\cite{BB97}, and $\mathbf{R}_m$ naturally projects onto the twin group studied in~\cite{DG22II}.
(Some surveys on general (right-angled) Artin groups are~\cite{Charney07,McCammond17}.)
\begin{problem}
Determine the centralizer algebra for the subgroups of the braid group mentioned above acting on $\mathbf{F}^{\otimes n}$.
Furthermore, classify the subgroups where the centralizer algebra is isomorphic to a subalgebra of $\mathcal{P}_n([m]_q)$.
\end{problem}
It is possible that the techniques used in~\cite{DG22II} can be applied to any projection from an Artin group to its corresponding Coxeter group, where we consider the Barau representation as Hecke algebra representation.
\section{Planar algebras}
A diagram is \defn{planar} if it can be drawn without crossings.
A \defn{planar algebra} is a subalgebra of the partition algebra if it has a basis given by planar diagrams.
We will sometimes use the term ``noncrossing'' instead of ``planar'' in this manuscript.
For any planar algebra, we necessarily have for each diagram the corresponding permutation $\sigma = 1$, but this does not completely classify planar diagrams in general.
Furthermore, since the diagrams are noncrossing, the bilinear form of~\cite[Lemma~4.3]{Xi99} maps to the $\mathbf{k}$-span of the trivial permutation.
Thus, each diagram is an element of the cellular basis for $\mathcal{P}_n(\beta)$ given in Theorem~\ref{thm:partition_cellular}.
Proposition~\ref{prop:subcellular} yields the following.
\begin{thm}
\label{thm:planar_cellular}
Any planar algebra on $[n] \sqcup [n]'$ fixed under $\iota$ is cellular with a cell basis given by the natural diagram basis and $\Lambda$ being a graded poset, where the grading is the number of defects.
Moreover, there is a unique top element in $\Lambda$ of rank $n$ corresponding to $1 \in \mathcal{P}$, and if there exists an element of rank $0$ in $\Lambda$, it is unique.
\end{thm}
Note that the element of rank $n$ corresponds to half of the identity element.
For the element of rank $0$, fix some set partition
$\rho_0$ (resp. $\rho_0'$) of $[n]$ (resp.~$[n]'$) with $0$ defects.
Then we can compose any $0$-defect set partition $\nu$ of $[n]'$ with $\{\rho_0, \rho_0'\}$, and the result is $\rho_0'$.
Consequently, we will always take the trivial $\mathbf{k} [\Sigma_k]$ module to build the cell modules.
Thus, in each case where we restrict to the planar subalgebra, we can define the basis of the cell modules by the corresponding noncrossing set partitions (with some fixed number of designated propagating blocks).
Furthermore, for generic $\beta$, the pictorial description of bilinear form being $0$ is also a sufficient condition as in the resulting diagram $\Delta$, the result is simply $\beta^D$, where $D$ is the number of interior components (\textit{i.e.}, not containing a matching pair of defects).
\begin{ex}
If we have the pairing of $\Phi_{\lambda}(C_U, C_T)$ given pictorally as
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline=10pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,21} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) + (0,-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.6) and +(0, -.6) .. (G4) .. controls +(0, 1.1) and +(0, 1.1) .. (G9);
\draw[-,color=darkred] (G2) -- ++(0,1);
\draw[-,color=UMNgold] (G5) + (0,-1) -- (G5) .. controls +(0, -.6) and +(0, -.6) .. (G7) .. controls +(0, -1.3) and +(0, -1.3) .. (G12);
\draw[-,color=UMNgold] (G12) .. controls +(0, .6) and +(0, .6) .. (G10) -- ++(0,1);
\draw[-,color=blue] (G6) .. controls +(0, .6) and +(0, .6) .. (G8) .. controls +(0, -.75) and +(0, -.75) .. (G11);
\draw[-,color=UQpurple] (G13) + (0,1) -- (G13) .. controls +(0, 1.3) and +(0, 1.3) .. (G21) .. controls +(0, -.9) and +(0, -.9) .. (G18) .. controls +(0, .5) and +(0, .5) .. (G17) .. controls +(0, -1.1) and +(0, -1.1) .. (G13);
\draw[-,color=UQpurple] (G18) -- ++(0,-1);
\draw[-,color=OCUenji] (G15) .. controls +(0, .5) and +(0, .5) .. (G14) .. controls +(0, -.5) and +(0, -.5) .. (G15) .. controls +(0, .9) and +(0, .9) .. (G19);
\end{tikzpicture}
\]
then we have $\Phi_{\lambda}(C_U, C_T) = \beta^5$.
\end{ex}
We note that the description of the cell modules is essentially the same as the construction given in~\cite[Thm.~3.21]{HJ20} for the planar algebras considered there.
In turn, this yields the tableaux description by following the construction in Section~\ref{sec:crt}.
\subsection{Temperley--Lieb and planar partition algebras}
We begin with the Temperley--Lieb algebra $\mathcal{T}\mathcal{L}_n(\beta)$~\cite{TL71}, which was surveyed in~\cite{RSA14}.
This is spanned by the set of diagrams such that all parts in the set partition have size $2$, which is also called a perfect matching and it counted by the even Catalan numbers
\begin{equation}
\label{eq:TL_dim}
\dim \mathcal{T}\mathcal{L}_n(\beta) = C_N = \frac{1}{N+1}\binom{2N}{N},
\end{equation}
where $N = 2n$.
Thus, we see the cell modules built from the partition algebra are a reformulation of the link modules~\cite{GL96} (see also Remark~\ref{rem:assoc_graded}) with the simple modules indexed by $\Lambda = \{n - 2k \mid k \in \langle n \rangle \}$ under the natural order, where $\langle n \rangle := \{0,1,\dotsc,\lfloor n/2 \rfloor\}$.\footnote{In~\cite{GL96,RSA14}, instead of the $\Lambda$ given here, the authors used the set $\langle n \rangle$ to index the cell modules, which correspond to the number of defects instead of the number of non-defect entries as described here.}
Precise conditions on $\beta$ when $\mathcal{T}\mathcal{L}_n(\beta)$ is semisimple can be found in, \textit{e.g.},~\cite[Thm~4.7]{RSA14} and~\cite[Thm.~A.1]{DG22} (which interprets results in~\cite{GdlHJ89}).
For any half diagram for $M(\lambda)$, the number of defects is equal to $\lambda$.
Furthermore, we see that the dimensions of the cell modules are the triangle Catalan numbers
\[
\dim W(\lambda) = \abs{M(\lambda)} = C_{N,k} = \binom{N+k}{k} - \binom{N+k}{k-1} = \frac{N+k-1}{N+1} \binom{N+k}{N},
\]
where $\lambda = 2n - k$ and $N = n - k$.
In particular, when $\lambda = 0,1$, this is precisely the $\lfloor n/2 \rfloor$-th Catalan number.
We also have that $\dim W(\lambda)$ counts the number of Dyck paths of length $n$ that end at height $\lambda$.
Furthermore, Equation~\eqref{eq:dim_formula} yields the identity
\begin{equation}
\label{eq:catalan_square}
C_{2n} = \sum_{k=0}^{\lfloor n/2 \rfloor} C_{n-k,k}^2.
\end{equation}
Next, we consider the planar partition algebra $\mathcal{P}\mcP_n(\beta)$.
For $\beta \neq 0$, this reduces to the case of the Temperley--Lieb algebra by the following well-known result (see, \textit{e.g.},~\cite{Jones94,HR05}).
\begin{thm}
\label{thm:TL_planer_iso}
For $\beta \neq 0 $, we have
\[
\mathcal{P}\mcP_n(\beta^2) \cong \mathcal{T}\mathcal{L}_{2n}(\beta).
\]
\end{thm}
Roughly speaking, the isomorphism is to consider the outline of a thickened version of the planar partition algebra.
Making this precise, for $\beta \neq 0$, the isomorphism is given by
\begin{align*}
p_i =
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,3,5,6,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
\draw[-] (G\i) -- (G-\i);
}
\node[vertex] (G4) at (4, 1) [shape=circle, draw] {};
\node[vertex] (G-4) at (4, -1) [shape=circle, draw] {};
\draw (2,1) node {$\cdots$};
\draw (2,-1) node {$\cdots$};
\draw (7,1) node {$\cdots$};
\draw (7,-1) node {$\cdots$};
\end{tikzpicture}
& \mapsto
\beta \cdot
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, fill=red, inner sep=1pt]
\foreach \i in {1,3,5,6,8} {
\node[vertex] (G\i) at (\i-.2, 1) [shape=circle, draw] {};
\node[vertex] (Gp\i) at (\i+.2, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i-.2, -1) [shape=circle, draw] {};
\node[vertex] (Gp-\i) at (\i+.2, -1) [shape=circle, draw] {};
\draw[-] (G\i) -- (G-\i);
\draw[-] (Gp\i) -- (Gp-\i);
}
\node[vertex] (G4) at (4-.2, 1) [shape=circle, draw] {};
\node[vertex] (Gp4) at (4+.2, 1) [shape=circle, draw] {};
\node[vertex] (G-4) at (4-.2, -1) [shape=circle, draw] {};
\node[vertex] (Gp-4) at (4+.2, -1) [shape=circle, draw] {};
\draw[-] (G4) .. controls +(0.1, -.25) and +(-0.1, -.25) .. (Gp4);
\draw[-] (G-4) .. controls +(0.1, .25) and +(-0.1, .25) .. (Gp-4);
\draw (2,1) node {$\cdots$};
\draw (2,-1) node {$\cdots$};
\draw (7,1) node {$\cdots$};
\draw (7,-1) node {$\cdots$};
\end{tikzpicture}
= \beta e_{2i-1},
\\
b_i =
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,3,4,5,6,8} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\foreach \i in {1,3,6,8} {
\draw[-] (G\i) -- (G-\i);
}
\draw (2,1) node {$\cdots$};
\draw (2,-1) node {$\cdots$};
\draw (7,1) node {$\cdots$};
\draw (7,-1) node {$\cdots$};
\draw[-] (G4) .. controls +(0, -.5) and +(0, -.5) .. (G5) -- (G-5) .. controls +(0, .5) and +(0, .5) .. (G-4) -- (G4);
\end{tikzpicture}
& \mapsto
\beta^{-1} \cdot
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, fill=red, inner sep=1pt]
\foreach \i in {1,3,4,5,6,8} {
\node[vertex] (G\i) at (\i-.2, 1) [shape=circle, draw] {};
\node[vertex] (Gp\i) at (\i+.2, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i-.2, -1) [shape=circle, draw] {};
\node[vertex] (Gp-\i) at (\i+.2, -1) [shape=circle, draw] {};
}
\foreach \i in {1,3,6,8} {
\draw[-] (G\i) -- (G-\i);
\draw[-] (Gp\i) -- (Gp-\i);
}
\draw[-] (Gp4) .. controls +(0.1, -.25) and +(-0.1, -.25) .. (G5);
\draw[-] (Gp-4) .. controls +(0.1, .25) and +(-0.1, .25) .. (G-5);
\draw[-] (G4) -- (G-4);
\draw[-] (Gp5) -- (Gp-5);
\draw (2,1) node {$\cdots$};
\draw (2,-1) node {$\cdots$};
\draw (7,1) node {$\cdots$};
\draw (7,-1) node {$\cdots$};
\end{tikzpicture}
= \beta^{-1} e_{2i}.
\end{align*}
It is a simple direct check to see this is a morphism as it satisfies the Temperley-Lieb algebra relations.
Since the image is the generating set of $\mathcal{T}\mathcal{L}_{2n}(\beta)$ and the dimensions of the two algebras are equal, this is an isomorphism.
(Alternatively, the map is clearly invertible that maps to the generating set of $\mathcal{P}\mcP_n(\beta)$.)
For $\beta = 0$ (or if we wanted to work over more general rings, when $\beta$ is not a unit), the map above does not make sense.
However, based upon computations done using \textsc{SageMath}~\cite{sage} with $\mathbf{k} = \mathbb{Q}$ (this should have no dependence on the field since all structure coefficients are either $1$ or $0$) and vague claims in the literature, there should still be an isomorphism at $\beta = 0$.
The author does not know of any proof and could not construct an explicit isomorphism for $n = 2$ over $\mathbb{C}$.
\begin{conj}
\label{conj:TL_planar_zero}
Theorem~\ref{thm:TL_planer_iso} holds at $\beta = 0$.
\end{conj}
Therefore, by Theorem~\ref{thm:TL_planer_iso} and assuming Conjecture~\ref{conj:TL_planar_zero}, the semisimplicity of $\mathcal{P}\mcP_n(\beta)$ is determined by the Temperley--Lieb algebra $\mathcal{T}\mathcal{L}_{2n}(\sqrt{\beta})$ description from~\cite{RSA14} as we can always extend $\mathbf{k}$ to include $\sqrt{\beta}$ by Remark~\ref{rem:algebraic_closure}.
It is possible these algebras are not isomorphic at $\beta = 0$ but satisfy the weaker statement of being Morita equivalent, which would be sufficient for the purposes of this article since we are only interested in properties of the representations.
A proof of the Morita equivalence should be a consequence of the description of the bilinear forms $\Phi_{\lambda}$ from~\cite[Sec.~4]{RSA14} with suitable modifications for the natural basis of $\mathcal{P}\mcP_n(\beta)$.
It is known that the $\beta = 0$ case is often one of the more interesting cases (see, \textit{e.g.},~\cite{RSA14}).
For example, in~\cite{ILZ19}, the Temperley--Lieb algebra $\mathcal{T}\mathcal{L}_n\bigl(\pm(q+q^{-1})\bigr)$ at $q$ being a root of unity was related to the fusion category generated by tilting modules of $U_q(\mathfrak{sl}_2)$; see also~\cite{AST18}.
\subsection{Planar uniform block algebra}
The planar subalgebra $\mathcal{P}\mathcal{U}_n$ of $\mathcal{I}\mathcal{U}_n \subseteq \mathcal{U}_n$ is simply the subalgebra of planar idempotents of $\mathcal{U}_n$ (see Section~\ref{sec:uniform_block}).
Therefore, the basis is indexed by compositions of $n$.
For such a composition $\mu = (\mu_1, \dotsc, \mu_{\ell})$, the basis element is indexed by the idempontents of the form
\[
\bigl\{ \{1, \dotsc, \mu_1\}, \{\Psi_1+1, \dotsc, \Psi_1+\mu_2\}, \dotsc, \{\Psi_{\ell-1}+1, \dotsc, n\} \bigr\},
\]
where $\Psi_k = \mu_1 + \cdots + \mu_k$, to the same primed set partition.
Consequently, we see that
\begin{equation}
\label{eq:planar_uniform_dim}
\dim \mathcal{P}\mathcal{U}_n = 2^{n-1},
\end{equation}
and by Proposition~\ref{prop:uniform_idempotent_cellular}, all of the cell modules are one dimensional irreducible representations.
\subsection{Planar rook algebra}
The planar rook algebra $\mathcal{P}\mathcal{R}_n(\beta)$ was studied in~\cite{FHH09} and consists of planar diagrams with either singletons or propagating blocks.
As a consequence, the planar condition is now equivalent to having a corresponding permutation $\sigma = 1$.
\begin{cor}
The planar rook algebra $\mathcal{P}\mathcal{R}_n(\beta)$ is a cellular algebra with cell data given by $\Lambda = \{0,1,\dotsc,n\}$ and $M(\lambda)$ being the subsets of $[n]$ of size $\lambda$.
The Gram matrix for the bilinear form $\Phi_{\lambda}$ with respect to the natural diagram basis is a diagonal matrix with entries being $\beta^{\lambda}$.
It is always semisimple unless $\beta = 0$, in which case there is a unique simple one dimensional module $M(0)$.
\end{cor}
\begin{proof}
The first statement is Theorem~\ref{thm:planar_cellular} with the subsets of $[n]$ corresponding to the singleton blocks.
The remaining claims are straightforward from the combinatorial description.
\end{proof}
For $\beta = 0$, we can describe the action on $M(0)$ explicitly by $a v = c_a v$ for all $a \in \mathcal{P}\mathcal{R}_n(\beta)$ and $v \in M(0)$, where $c_a$ is the coefficient of $1$ in $a$.
We can also easily see from Equation~\eqref{eq:dim_formula} that
\[
\dim \mathcal{P}\mathcal{R}_n(\beta) = \binom{2n}{n} = \sum_{\lambda=0}^n \binom{n}{\lambda}^2.
\]
\subsection{Motzkin algebra}
The Motzkin algebra $\mathcal{M}(\beta)$ introduced in~\cite{BH14} can be described as the Temperley--Lieb algebra but with singletons allowed.
The dimension of $\mathcal{M}(\beta)$ is the $2n$-th Motzkin number $M_{2n}$.
For the cell datum, we have $\Lambda = [n]$ with $M(\lambda)$ being the set of noncrossing matchings with allowing singletons with exactly $\lambda$ defects, a set of non-nested singletons.
The dimensions of the cell modules are counted by the Motzkin triangle numbers, which are known to be computed by~\cite[Eq.~(3.22)]{BH14} and~\cite{Lando03} (see also~\cite[A026300]{OEIS}):
\begin{align*}
M_{n,k} & = \sum_{i=0}^{\lfloor (n-k)/2 \rfloor} \binom{n}{k-2i} \stirling{k+2i}{i}
\\ & = \sum_{i=0}^{\lfloor (n-k)/2 \rfloor} \binom{n}{2i+k} \left[ \binom{2i+k}{i} - \binom{2i+k}{i-1} \right]
\\ & = M_{n-1,k-1} + M_{n-1,k} + M_{n-1,k+1},
\end{align*}
with $M_{n,-1} = M_{n,n+1} = 0$ and $M_{0,0} = 1$.
The bijection from a half diagram in $M(\lambda)$ to a Motzkin path of length $n$ that ends at height $\lambda$ is by reading the diagram from left-to-right and treating each pair $\{a,b\}$ as an arc from $a$ to $b$, where each outgoing (resp.\ incoming) edge is $+1$ (resp.~$-1$), non-defect singletons are $0$, and defect singletons are $+1$.
This is the cell datum shown in Benkart and Halverson~\cite[Thm.~4.16]{BH14}.
We also have the analog of Equation~\eqref{eq:catalan_square} by Equation~\eqref{eq:dim_formula}:
\begin{equation}
\label{eq:motzkin_square}
M_{2n} = \sum_{\lambda=0}^n M_{n,\lambda}^2.
\end{equation}
\begin{thm}[{\cite[Thm.~5.14]{BH14}}]
The Motzkin algebra $\mathcal{M}_n(\beta)$ is semisimple if and only if $\beta - 1$ is not the root of the rescaled Chebyshev polynomials of the second kind\footnote{The rescaled Chebyshev polynomials have coefficients in $\mathbb{Z}$, so they can still be evaluated when $x$ is an element of a field of characteristic $2$.}
defined by $u_k(x) = U_k(x/2)$ for all $1 \leq k < n$.
\end{thm}
The first definition in~\cite{BH14} has the Motzkin algebra with two parameters $\mathcal{M}(\beta, \gamma)$ as it is a subalgebra of the rook Brauer algebra.
There is a similar misprint in~\cite{BH14} as in~\cite{HdM14} as described in Example~\ref{ex:nonassoc_product}.
We have the analogs of Proposition~\ref{prop:two_param_rook_brauer_iso} and Problem~\ref{prob:two_param_rook_brauer_semisimple}.
\begin{prop}
For all $\gamma \neq 0$, we have $\mathcal{R}\mathcal{B}_n(0, \gamma) \cong \mathcal{R}\mathcal{B}_n(0, 1)$.
\end{prop}
\begin{problem}
\label{prob:two_param_motzkin}
Determine when the two-parameter Motzkin algebra $\mathcal{M}(\beta, 0)$ is semisimple.
Are the points where it is not semisimple described as roots to some specialization of a higher level generalization of Chebyshev polynomials in the Askey scheme, more specifically with a specialization of Jacobi polynomials?
\end{problem}
We remark on another curious appearance of Chevyshev polynomials of the second kind with the branching rule of the Brauer algebra $\mathcal{B}_n(\beta)$ to $\mathcal{T}\mathcal{L}_n(\beta)$ that was given in~\cite{BM05}.
\begin{problem}
Determine if there is a relationship between the branching rule from the Motzkin algebra to the Brauer algebra and the semisimplicity of the Motzkin algebra.
\end{problem}
One way to define a half integer Motzkin algebra would be to mandate that $1$ is always a singleton.
Indeed, this is a subalgebra of dimension $M_{2n-1}$, but it is not immediately clear it is cellular since the basis is not invariant under $\iota$.
\subsection{Partial Temperley--Lieb algebra}
The partial Temperley--Lieb algebra $\mathcal{P}\mathcal{T}\mathcal{L}_n(\beta)$ was recently introduced in~\cite{DG22} as a subalgebra of the Motzkin algebra $\mathcal{M}_n(\beta)$.
Specifically, its basis is indexed by \defn{balanced} diagrams: For a diagram $\rho$, the number of pairs of $\rho \cap [n]$ equals those of $\rho \cap [n]'$.
Each balanced diagram corresponds to a basis element given as an alternating sum of diagrams by removing edges (but leaving the vertices).
The proof of cellularity by Doty and Giaquinto is using the following stronger result.
\begin{thm}[{\cite[Thm.~4.2]{DG22}}]
\label{thm:PTL_Morita}
We have the Morita equivalence
\[
\mathcal{P}\mathcal{T}\mathcal{L}_n(\beta) \simeq \bigoplus_{k=1}^n \mathcal{T}\mathcal{L}_k(\beta - 1).
\]
\end{thm}
As a consequence of Theorem~\ref{thm:PTL_Morita}, we have a complete classification of the semisimplicity of the partial Temperley--Lieb algebra from the Temperley--Lieb one.
Furthermore, their description of the modules of $\mathcal{P}\mathcal{T}\mathcal{L}_n(\beta)$ is essentially the same as given here by using half diagrams with the cellular algebra structure.
We remark that Proposition~\ref{prop:subcellular} and Proposition~\ref{prop:cellular_triangle_basis} give an alternative simple proof that $\mathcal{P}\mathcal{L}_n(\beta)$ is a cellular algebra (using that $\mathcal{M}_n(\beta)$ is a cellular algebra).
\subsection{Planar even algebra}
\label{sec:planar_even}
If we take the planar version of the parity matching algebra, we obtain a cellular algebra with interesting combinatorial properties.
We call the planar subalgebra of $\mathcal{P}\mathcal{M}_n(\beta)$ the \defn{planar even algebra} and denote it by $\mathcal{P}\mathcal{E}_n(\beta)$.
The dimension of this subalgebra is known~\cite[A001764]{OEIS}:
\begin{equation}
\label{eq:even_planar_set_partitions}
\dim \mathcal{P}\mathcal{E}_n(\beta) = \frac{1}{2n+1}\binom{3n}{n},
\end{equation}
as well as many other combinatorial interpretations.
It would be interesting to see what the product structure in $\mathcal{P}\mathcal{E}_n(\beta)$ is on other such interpretations, and if these have other natural algebraic structures, what is there intrepretation in terms of $\mathcal{P}\mathcal{E}_n(\beta)$.
Next, we explicitly describe a parameterization of the cell modules for $\beta \neq 0$.
\begin{thm}
\label{thm:planar_even_cell_modules}
The set $\Lambda$ for $\mathcal{P}\mathcal{E}_n(\beta)$ is given by all words $\lambda = \lambda_1 \dotsm \lambda_{\ell}$ in the alphabet $\{1, 2\}$ such that the sum $S = \sum_{i=1}^{\ell} \lambda_i \leq n$ and $n \equiv S \pmod{2}$.
Furthermore, the bilinear form $\Phi_{\lambda} \neq 0$ for all $\lambda \in \Lambda$.
\end{thm}
\begin{proof}
We first assume $\beta \neq 0$.
The basis elements of all cell modules are given by all planar set partitions of $[n]'$, where the choice of defect blocks must not be nested.
Our first claim that for any such basis diagram $\nu \in W$, we can multiply it multiplied by a diagram $\rho \in \mathcal{P}\mathcal{E}_n(\beta)$ such that we get some sequence $\lambda$ of defect blocks of sizes $1$ and $2$ in some order followed by a sequence of caps $\{i', (i+1)'\}$.
We denote such an element by $\rho_{\lambda}$.
If there are no defects (necessarily, we must have $n$ being even), we can simply multiply by the cap and cup element:
\begin{equation}
\label{eq:cap_cup_element}
\begin{gathered}
\{\{1,2\},\dotsc,\{2n-1,2n\}, \{1',2'\},\dotsc,\{(2n-1)',(2n)'\} = b_1 b_3 \dotsm b_{2n-1}
\\ =
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,3,4,5,7} {
\node[vertex] (G\i) at (2*\i, 1) [shape=circle, draw] {};
\node[vertex] (Gp\i) at (2*\i+1, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (2*\i, -1) [shape=circle, draw] {};
\node[vertex] (Gp-\i) at (2*\i+1, -1) [shape=circle, draw] {};
\draw[-] (G\i) .. controls +(0, -.5) and +(0, -.5) .. (Gp\i);
\draw[-] (G-\i) .. controls +(0, .5) and +(0, .5) .. (Gp-\i);
}
\draw (4.5,1) node {$\cdots$};
\draw (4.5,-1) node {$\cdots$};
\draw (12.5,1) node {$\cdots$};
\draw (12.5,-1) node {$\cdots$};
\end{tikzpicture}
\end{gathered}
\end{equation}
We can do a similar multiplication but reordering all elements in the defects of $\nu$ so that they become $[m']$ for some $m$.
In particular, if the defects are on the elements $i_1 < \cdots i_m$, then we can multiply by the diagram in $\mathcal{P}\mathcal{E}_n(\beta)$ consisting of $\{i_1,1'\}, \dotsc, \{i_m,m'\}$ and all remaining elements are cups or caps.
Thus, we can assume every element in $[n]'$ belongs to a defect of $\nu$.
It is sufficient to prove it for when $\nu = \{[n]'\}$ with a single defect block.
If $n$ is odd, the we use the same diagram in Equation~\eqref{eq:cap_cup_element} except we make the last block a propagating block $\{n, n'\}$.
If $n$ is odd, we make the rightmost block a propagating block instead of a cap-cup pair.
Thus, we have shown the first claim.
Next, we claim that an even propagating block cannot cross an odd propagating block.
This follows from a straightforward parity argument.
Hence, the elements $\rho_{\lambda}$ uniquely determine the cell modules up to isomorphism, which is clearly in bijection with the claimed $\Lambda$.
This also naturally extends to a description of the cellular basis by connecting a pair of such cell module basis elements (essentially undoing the decomposition~\eqref{eq:partition_decomposition}).
For $\beta = 0$, we have some additional terms equal to $0$, which does not change the proof above that the resulting basis is cellular.
To see that $\Phi_{\lambda} \neq 0$ for some fixed $\lambda \in \Lambda$, consider the element $\widetilde{\rho}_{\lambda}$, which connects all of the caps in $\rho_{\lambda}$ to the last defect block.
The pairing $\Phi_{\lambda}(\widetilde{\rho}_{\lambda}, \widetilde{\rho}_{\lambda}) = 1$.
\end{proof}
Define $E_P(m) := \dim \mathcal{P}\mathcal{E}_{m/2}$ given in Equation~\eqref{eq:even_planar_set_partitions} if $m \in 2\mathbb{Z}$ and $0$ otherwise.
Thus, $E_P(m)$ counts the number of planar even set partitions of $[m]$.
Note $E_P(0) = 1$ by Equation~\eqref{eq:even_planar_set_partitions}.
\begin{prop}
\label{prop:even_planar_cell_dim}
Fix $\lambda \in \Lambda$ such that $\lambda$ is a permutation of $1^{k_1} 2^{k_2}$.
Then we have
\begin{equation}
\label{eq:planar_even_cell_dim}
\dim W(\lambda) = \sum_{\{i_1 < \cdots < i_{\ell-1}\} \in \binom{[n-1]}{\ell-1}} \prod_{j=1}^{k_1} (2 - \delta_{i_j,i_{j-1}+1}) E_P(i_j - i_{j-1} - 1) \prod_{j=k_1+1}^{\ell} E_P(i_j - i_{j-1}),
\end{equation}
where $i_0 = 0$.
\end{prop}
\begin{proof}
Since we cannot cross propagating blocks, we divide $[n]'$ up into $\ell = k_1 + k_2$ sets
\[
\{1', \dotsc, i_1'\} \sqcup \{(i_1+1)', \dotsc, i_2'\} \sqcup \cdots \sqcup \{(i_{\ell-1}+1)', \dotsc, n'\},
\]
where $i_1 < \cdots < i_{\ell-1}$, such that the smallest entry in each block part of the propagating block.
Without loss of generality, we can look at the first set $\{1', \dotsc, i_1'\}$.
If $\lambda_1 = 2$, then if we consider all even set partitions on $[i_1']$, we simply consider the lexicographic smallest part as the propagating block.
If $\lambda_1 = 1$, then we consider all even set partitions on $\{2', \dotsc, i_1'\}$, but we have two cases.
The first is $\{1'\}$ is an isolated propagating block, and the second is we join $1'$ to the lexicographic smallest part.
Note that the second case is impossible when $i_{j-1} + 1 = i_j$.
The claim follows as we can chose these even set partitions independently in each set.
\end{proof}
Consequently, we see that the dimension only depends on the number of $1$'s and $2$'s in the word $\lambda$, and there are $\binom{k_1 + k_2}{k_1}$ such sequences.
This gives the following identity of binomial coefficients
\[
E_P(n) = \sum_{k_1 + k_2 \leq n} \binom{k_1+k_2}{k_1} \bigl( \dim W(1^{k_1}2^{k_2}) \bigr)^2
\]
after substituting in Equation~\eqref{eq:planar_even_cell_dim}.
We give some examples of the dimensions in Table~\ref{table:even_dim}.
We note that $W(\emptyset) = E_P(n)$ since it consists of all noncrossing even set partitions on $[n]'$.
\begin{table}
\begin{center}
\begin{tabular}[t]{cccc}
\toprule
$n=1$ & $n=2$ & $n=3$ & $n=4$
\\ \midrule
\begin{tabular}[t]{r|cc}
& 0 & 1
\\ \hline 0 & 0 & 0
\\ 1 & 0 & 1
\end{tabular}
&
\begin{tabular}[t]{r|ccc}
& 0 & 1 & 2
\\ \hline 0 & 1 & 0 & 0
\\ 1 & 1 & 0 & 0
\\ 2 & 0 & 0 & 1
\end{tabular}
&
\begin{tabular}[t]{r|cccc}
& 0 & 1 & 2 & 3
\\ \hline 0 & 0 & 0 & 0 & 0
\\ 1 & 0 & 3 & 0 & 0
\\ 2 & 0 & 1 & 0 & 0
\\ 3 & 0 & 0 & 0 & 1
\end{tabular}
&
\begin{tabular}[t]{r|ccccc}
& 0 & 1 & 2 & 3 & 4
\\ \hline 0 & 3 & 0 & 0 & 0 & 0
\\ 1 & 4 & 0 & 0 & 0 & 0
\\ 2 & 1 & 0 & 5 & 0 & 0
\\ 3 & 0 & 0 & 1 & 0 & 0
\\ 4 & 0 & 0 & 0 & 0 & 1
\end{tabular}
\\ \bottomrule
\end{tabular}
\end{center}
\caption{The dimension of $W(1^{k_1}2^{k_2})$ in $\mathcal{P}\mathcal{E}_n(\beta)$ for $n \leq 4$. The value $k_1$ (resp.\ $k_1 + k_2$ or the number of propagating blocks) is given by the column (resp.\ row) in each table.}
\label{table:even_dim}
\end{table}
\begin{problem}
\label{prob:irrep_dim_CE}
Find a recurrence relation, generating function, and more compact (closed) formula for $\dim(1^{k_1} 2^{k_2})$.
\end{problem}
\begin{ex}
The basis diagrams for the cell modules of $\mathcal{P}\mathcal{E}_4(\beta)$ are
\begin{gather*}
W(\emptyset):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2) .. controls +(0, .5) and +(0, .5) .. (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2); \draw[-,blue] (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G1) .. controls +(0, 1) and +(0, 1) .. (G4); \draw[-,blue] (G3) .. controls +(0, .5) and +(0, .5) .. (G2);
\end{tikzpicture}\,,
\\
W(1):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, .5) and +(0, .5) .. (G2) .. controls +(0, .5) and +(0, .5) .. (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, .5) and +(0, .5) .. (G2); \draw[-,blue] (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2); \draw[-,darkred] (G3) + (0,1) -- (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, 1) and +(0, 1) .. (G4); \draw[-,blue] (G3) .. controls +(0, .5) and +(0, .5) .. (G2);
\end{tikzpicture}\,,
\\
W(22):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, .5) and +(0, .5) .. (G2);
\draw[-,darkred] (G3) + (0,1) -- (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\end{tikzpicture}\,,
\quad
W(11):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, .5) and +(0, .5) .. (G2) .. controls +(0, .5) and +(0, .5) .. (G3);
\draw[-,darkred] (G4) + (0,1) -- (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G2) + (0,1) -- (G2) .. controls +(0, .5) and +(0, .5) .. (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\draw[-,darkred] (G1) + (0,1) -- (G1);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G3) .. controls +(0, .5) and +(0, .5) .. (G4); \draw[-,darkred] (G1) + (0,1) -- (G1);
\draw[-,darkred] (G2) + (0,1) -- (G2);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G2) .. controls +(0, .5) and +(0, .5) .. (G3); \draw[-,darkred] (G1) + (0,1) -- (G1);
\draw[-,darkred] (G4) + (0,1) -- (G4);
\end{tikzpicture}\,,
\quad
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,blue] (G1) .. controls +(0, .5) and +(0, .5) .. (G2); \draw[-,darkred] (G3) + (0,1) -- (G3);
\draw[-,darkred] (G4) + (0,1) -- (G4);
\end{tikzpicture}\,,
\\
W(211):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G1) + (0,1) -- (G1) .. controls +(0, .5) and +(0, .5) .. (G2);
\draw[-,darkred] (G3) + (0,1) -- (G3);
\draw[-,darkred] (G4) + (0,1) -- (G4);
\end{tikzpicture}\,,
\quad
W(121):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G2) + (0,1) -- (G2) .. controls +(0, .5) and +(0, .5) .. (G3);
\draw[-,darkred] (G1) + (0,1) -- (G1);
\draw[-,darkred] (G4) + (0,1) -- (G4);
\end{tikzpicture}\,,
\quad
W(112):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
}
\draw[-,darkred] (G3) + (0,1) -- (G3) .. controls +(0, .5) and +(0, .5) .. (G4);
\draw[-,darkred] (G1) + (0,1) -- (G1);
\draw[-,darkred] (G2) + (0,1) -- (G2);
\end{tikzpicture}\,,
\\
W(1^4):
\begin{tikzpicture}[scale = 0.5,thick, baseline=30pt]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,...,4} {
\node[vertex] (G\i) at (\i, 2) [shape=circle, draw] {};
\draw[-,darkred] (G\i) + (0,1) -- (G\i);
}
\end{tikzpicture}\,,
\end{gather*}
We can see that $\dim W(211) = \dim W(121) = \dim W(112)$ and $\dim W(\emptyset) = \dim \mathcal{P}\mathcal{E}_2(\beta)$.
\end{ex}
\subsection{Planar \texorpdfstring{$r$}{r}-color algebra}
\label{sec:planar_color}
Like $\mathcal{P}\mathcal{E}_n(\beta)$ as a planar version of $\mathcal{P}\mathcal{M}_n(\beta)$, we can consider a planar version of $\mathcal{G}_n^{(r,1,m)}(\beta)$ for any $m \geq n$.
This is again independent of $m$ whenever $m \geq n$.
Let $\mathcal{P}\mathcal{C}_{r,n}(\beta)$ denote the corresponding algebra, which we call the \defn{planar $r$-color algebra}.
The set $\Lambda$ is similar to the case $r = 2$: We take all compositions $\lambda$ with parts in $[r]$ (\textit{i.e.}, $\lambda_i \in [r]$) such that $\abs{\lambda} \leq n$ and $\abs{\lambda} \equiv n \pmod{r}$.
Note that compositions are equivalent to words.
Recall that $\mathcal{T}\mathcal{L}_{2n}(\beta) \cong \mathcal{P}\mcP_n(\beta^2)$ has dimension equal to the Catalan numbers given by Equation~\eqref{eq:TL_dim}, which is the planar version of $\mathcal{G}_n^{(1,1,m)}(\beta^2)$.
Additionally recall Equation~\eqref{eq:even_planar_set_partitions}, which is for the planar version of $\mathcal{G}_n^{(1,1,m)}(\beta^2)$.
Thus, a natural guess is $\dim \mathcal{P}\mathcal{C}_{r,n}(\beta)$ is a \defn{Fuss--Catalan number} (of type $A_n$; see, \textit{e.g.},~\cite{STW15} and references therein or~\cite[A137211]{OEIS}):
\[
C_n^{(r)} := \frac{1}{rn + 1} \binom{(r+1)n}{n},
\]
Indeed, Edelman showed in~\cite{Edelman80} that the Fuss--Catalan numbers count the number of noncrossing set partitions of $[nr]$ with block sizes that are divisible by $r$, with the enumeration dating back to the work of Fuss~\cite{Fuss91}.
Unfortunately, this is not the case, as Table~\ref{table:dim_planar_color} indicates when compared with
\[
\bigl( C_n^{(3)} \bigr)_{n=0}^{\infty} = (1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, \ldots).
\]
However, we do have that the dimension of the unique cell module with zero defects is equal to the Fuss--Catalan numbers from the description given by Edelman~\cite{Edelman80}.
Indeed, the following is just a rephrasing of this result.
\begin{prop}[{\cite{Edelman80}}]
\label{prop:planar_color_zero_defect_dim}
There is a diagram of $\mathcal{P}\mathcal{C}_{r,n}(\beta)$ with zero propagating blocks if and only if $r \mid n$.
Moreover, if $r \mid n$, then let $\lambda = \emptyset$ be the unique index for the cell module corresponding to zero defects in $\mathcal{P}\mathcal{C}_{r,n}(\beta)$.
Then
\[
\dim W(\emptyset) = C_{n/r}^{(r)}.
\]
\end{prop}
We can compute $\dim W(\lambda)$ by using the analog of Proposition~\ref{prop:even_planar_cell_dim}, where $E_P(j)$ is replaced by the corresponding Fuss--Catalan number.
Thus, we obtain a closed, if somewhat complicated, formula for $\dim \mathcal{P}\mathcal{C}_{r,n}$ by Equation~\eqref{eq:dim_formula}.
Moreover, given Equation~\eqref{eq:catalan_square} and Equation~\eqref{eq:motzkin_square} (as well as Theorem~\ref{thm:quasi_partition_cellular} below for the Riordan numbers), we have the following.
\begin{problem}
Determine if the dimensions of the cell modules $W(\lambda)$ can be used to define Fuss--Catalan triangle, or an $r+1$ dimensional simplex, numbers.
\end{problem}
For the remainder of this section, we will focus on the case $n \geq r > n/2$ and can give some explicit compact formulas for the dimensions of the planar $r$-color algebra and its cell modules $W(\lambda)$ (as well as $\abs{\Lambda}$).
From the definitions and Equation~\eqref{eq:planar_uniform_dim}, we have $\dim \mathcal{P}\mathcal{C}_{r,n}(\beta) = \dim \mathcal{P}\mathcal{U}_n = 2^{n-1}$ for all $r > n$.
It is easy to see that $\dim \mathcal{P}\mathcal{C}_{n,n}(\beta) = 2^{n-1} + 1$.
Indeed, we have all of the basis elements of $\mathcal{P}\mathcal{C}_{n+1,n}(\beta) = \mathcal{P}\mathcal{U}_n$ (where any such block must be propagating) plus the diagram $\{[n], [n]'\}$ (the unique element in $\mathcal{P}\mathcal{C}_{n,n}(\beta)$ with no propagating blocks).
Similarly, we can show $\dim \mathcal{P}\mathcal{C}_{n-1,n}(\beta) = 2^{n-1} + 8$ by noting that there are additional diagrams of the form
\[
\begin{array}{c@{\qquad\qquad}c@{\qquad\qquad}c@{\qquad\qquad}c}
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) -- (G-1);
\draw[-,color=blue] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.5) and +(0, -.5) .. (G4);
\draw[-,color=blue] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G4) -- (G-1);
\draw[-,color=blue] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3);
\draw[-,color=blue] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) -- (G-4);
\draw[-,color=blue] (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.5) and +(0, -.5) .. (G4);
\draw[-,color=blue] (G-1) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G4) -- (G-4);
\draw[-,color=blue] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3);
\draw[-,color=blue] (G-1) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\end{tikzpicture}\,,
\\[20pt]
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-4) -- (G4) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.5) and +(0, -.5) .. (G1);
\draw[-,color=blue] (G-1) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) .. controls +(0, -.5) and +(0, -.5) .. (G3) .. controls +(0, -.5) and +(0, -.5) .. (G4);
\draw[-,color=blue] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G4) -- (G-4) .. controls +(0, .5) and +(0, .5) .. (G-3) .. controls +(0, .5) and +(0, .5) .. (G-2) .. controls +(0, .5) and +(0, .5) .. (G-1);
\draw[-,color=blue] (G1) .. controls +(0., -.5) and +(0, -.5) .. (G2) .. controls +(0., -.5) and +(0, -.5) .. (G3);
\end{tikzpicture}\,,
&
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G1) -- (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2) .. controls +(0, .5) and +(0, .5) .. (G-3) .. controls +(0, .5) and +(0, .5) .. (G-4);
\draw[-,color=blue] (G2) .. controls +(0., -.5) and +(0, -.5) .. (G3) .. controls +(0., -.5) and +(0, -.5) .. (G4);
\end{tikzpicture}\,,
\end{array}
\]
generalized to arbitrary $n$.
Extending this argument, we obtain the following lemma.
\begin{lemma}
\label{lemma:pc_dim}
There exists a sequence $(a_j)_{j=0}^{\infty}$ such that for $n \geq r > n / 2$, we have
\[
\dim \mathcal{P}\mathcal{C}_{r,n}(\beta) = 2^{n-1} + a_{n-r}.
\]
\end{lemma}
\begin{proof}
Since $n \geq r > n/2$, there can be at most one additional block of size $r$ on either side that is not propagating or adjoined to a propagating block.
Considering half diagrams, if every block is a defect, it is a half diagram of $\mathcal{P}\mathcal{U}_n$, which is our base case since $\mathcal{P}\mathcal{C}_{r,n} = \mathcal{P}\mathcal{U}_n$ for $r > n$.
Thus, we can assume that every half diagram has exactly one non-propagating block which occupies consecutive positions $\{j', \dotsc, (j+r)'\}$ for $j \in [n-r+1]$.
On the remaining $n-r$ elements, we must have defect blocks.
Therefore, the number of additional diagrams not in $\mathcal{P}\mathcal{U}_n \subseteq \mathcal{P}\mathcal{C}_{r,n}$ depends only on $n-r$.
\end{proof}
Expanding slightly on the proof of Lemma~\ref{lemma:pc_dim}, we can obtain explicit dimension formulas for the cell modules.
\begin{prop}
\label{prop:dim_large_r_PC_cell}
Let $r > n/2$.
Then the dimensions of the cell modules of $\mathcal{P}\mathcal{C}_{r,n}$ is given by
\[
\dim W(\lambda) = \begin{cases}
1 & \text{if } \abs{\lambda} = n, \\
n - r + 1 + \ell(\lambda) & \text{if } \abs{\lambda} = n - r.
\end{cases}
\]
\end{prop}
\begin{proof}
If $\abs{\lambda} = n$, then this is the same as $\mathcal{P}\mathcal{U}_n$; thus we assume $\abs{\lambda} = n - r$.
As mentioned above, there are $n - r + 1$ places to place the consecutive positions of the nondefect block of size $r$.
Then there are also $\ell(\lambda)$ to attach the additional (consecutive) $r$ elements.
\end{proof}
Furthermore, we can obtain a simple formula for $\dim \mathcal{P}\mathcal{C}_{r,n}(\beta)$ when $r > n/2$.
We separate the cases when $r = n$ and $r > n$ as the formula we give breaks down and these cases have already been given above.
\begin{cor}
Let $n > r > n / 2$.
Then, we have
\[
\dim \mathcal{P}\mathcal{C}_{r,n} =
2^{n-1} + (9(n-r)^2 + 17(n-r) + 6) 2^{n-r-3}.
\]
\end{cor}
\begin{proof}
Note that number of compositions of $N$ of length $\ell$ are $\binom{N-1}{\ell-1}$.
Every composition of $n - r$ corresponds to a composition of $r$ containing a part of size strictly greater than $r$ and choosing a part of the composition.
So we remove these from our count of $2^{n-1}$.
Then from Proposition~\ref{prop:dim_large_r_PC_cell} and Equation~\eqref{eq:dim_formula}, we have
\begin{equation}
\label{eq:first_dim_PC}
\dim \mathcal{P}\mathcal{C}_{r,n} = 2^{n-1} - \sum_{\ell=1}^{n-r} \binom{n-r-1}{\ell-1} \ell + \sum_{\ell=1}^{n-r} \binom{n-r-1}{\ell-1} (n - r + 1 + \ell)^2.
\end{equation}
Then by well-known binomial coefficient sums (from the binomial theorem), we have
\begin{align*}
\sum_{\ell=0}^N \binom{N}{\ell} (M + \ell) = (N + 2M) 2^{N-1},
\quad
\sum_{\ell=0}^N \binom{N}{\ell} (M + \ell)^2
& = (4M^2 + 4MN + N + N^2) 2^{N-2},
\end{align*}
which applied to Equation~\eqref{eq:first_dim_PC} yields the claim after some simple manipulations.
\end{proof}
\begin{table}
\[
\begin{array}{c|ccccccc}
& 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\hline
1& 2 & 14 & 132 & 1430 & 16796 & 208012 & 2674440 \\
2& 1 & 3 & 12 & 55 & 273 & 1428 & 7752\\
3& \cdot & 2 & 5 & 16 & 54 & 186 & 689 \\
4& \cdot & \cdot & 4 & 9 & 24 & 70 & 202 \\
5& \cdot & \cdot & \cdot & 8 & 17 & 40 & 102 \\
6& \cdot & \cdot & \cdot & \cdot & 16 & 33 & 72 \\
7& \cdot & \cdot & \cdot & \cdot & \cdot & 32 & 65 \\
8& \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & 64 \\
\end{array}
\]
\caption{The $(r,n)$-th entry is equal to $\dim \mathcal{P}\mathcal{C}_{r,n}(\beta)$. Note that every entry marked with a $\cdot$ is equal to the entry above it, which equals $\dim \mathcal{P}\mathcal{U}_n = 2^{n-1}$ (and also holds for the subdiagonal entries; see Equation~\eqref{eq:planar_uniform_dim}).}
\label{table:dim_planar_color}
\end{table}
\begin{ex}
Consider $\mathcal{P}\mathcal{C}_{3,5}$.
We have
\[
\dim W(\lambda) = 1
\quad (\abs{\lambda} = 5),
\qquad
\dim W(2) = 4,
\qquad
\dim W(11) = 5,
\]
with the half diagrams spanning $W(2)$ and $W(11)$ being
\begin{align*}
W(2) : & \qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-1) -- ++ (0,1);
\draw[-,color=blue] (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4) .. controls +(0., .5) and +(0, .5) .. (G-5);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-5) .. controls +(0., 1.0) and +(0, 1.0) .. (G-1) -- ++ (0,1);
\draw[-,color=blue] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-5) .. controls +(0., .5) and +(0, .5) .. (G-4) -- ++ (0,1);
\draw[-,color=blue] (G-1) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\end{tikzpicture}\,,
\qquad\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-5) .. controls +(0., .5) and +(0, .5) .. (G-4) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-1) -- ++ (0,1);
\end{tikzpicture}\,,
\\
W(11) : & \quad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-1) -- ++ (0,1);
\draw[-,color=darkred] (G-2) -- ++ (0,1);
\draw[-,color=blue] (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4) .. controls +(0., .5) and +(0, .5) .. (G-5);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-1) -- ++ (0,1);
\draw[-,color=darkred] (G-5) -- ++ (0,1);
\draw[-,color=blue] (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-4);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-4) -- ++ (0,1);
\draw[-,color=darkred] (G-5) -- ++ (0,1);
\draw[-,color=blue] (G-1) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-3);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-5) -- ++ (0,1);
\draw[-,color=darkred] (G-4) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-2) .. controls +(0., .5) and +(0, .5) .. (G-1) -- ++ (0,1);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,2ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2,3,4,5} {
\node[vertex] (G-\i) at (\i, 0) [shape=circle, draw] {};
}
\draw[-,color=darkred] (G-1) -- ++ (0,1);
\draw[-,color=darkred] (G-5) .. controls +(0., .5) and +(0, .5) .. (G-4) .. controls +(0., .5) and +(0, .5) .. (G-3) .. controls +(0., .5) and +(0, .5) .. (G-2) -- ++ (0,1);
\end{tikzpicture}\,.
\end{align*}
We see that there are $13 = 2^4 - 1 - 2$ compositions of $5$ with parts in $\{1,2,3\}$, and we verify that
\[
\dim \mathcal{P}\mathcal{C}_{3,5} = 54 = 13 + 1 \cdot 4^2 + 1 \cdot 5^2 = \sum_{\lambda \in \Lambda} \bigl( \dim W(\lambda) \bigr)^2.
\]
\end{ex}
Additional examples of the dimensions of $\mathcal{P}\mathcal{C}_{r,n}(\beta)$ is given in Table~\ref{table:dim_planar_color}, which was computed by directly counting the number of diagrams in the basis using \textsc{SageMath}.
Furthermore, we see that the sequence in Lemma~\ref{lemma:pc_dim}
|
is given by
\[
a_j = (9j^2 + 17j + 6) 2^{j-3}
\]
for $j > 0$ with $a_0 = 1$.
Some initial terms are
\[
(1, 8, 38, 138, 436, 1264, 3456, 9056, 22976, \ldots).
\]
\subsection{Planar quasi-partition algebra}
By~\cite[Lemma~2.2]{DO14}, the basis of the quasi-partition algebra $\mathcal{Q}\mathcal{P}_n(\beta)$ is given inside of $\mathcal{P}_n(\beta-1)$ by looking at certain refinements of the indexing diagram.
From this and~\cite[Cor.~2.7]{DO14}, we can restrict to the set of planar diagrams without singletons and this can be constructed as a subalgebra of $\mathcal{P}\mcP_n(\beta-1)$.
We call this the \defn{planar quasi-partition alagebra} and denote it by $\mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta)$.
By~\cite[Thm.~5.12]{BE18} (see also~\cite[A099251]{OEIS}), we have
\[
\dim \mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta) = R_N = \frac{1}{N+1} \sum_{k=1}^{\lfloor N/2 \rfloor} \binom{N+1}{k} \binom{N-k-1}{k-1} = \frac{N-1}{N+1} \left(2 R_{N-1} + 3 R_{N-2} \right),
\]
where $N = 2n$ and $R_N$ are the Riordan numbers~\cite{Riordan75} (see also~\cite[A005043]{OEIS}), with $R_0 = 1$ and $R_1 = 1$.
This can be described as the number of Motzkin paths from $(0,0)$ to $(n,0)$ that do not have any horizontal steps on the $y=0$ line.
Let us consider the algebra given in~\cite{BE18}, which we call the \defn{tangle algebra} and denote it by $\mathcal{T}_n(\gamma)$; here we add the parameter $\gamma$ that counts the number of interior components in the product.
As noted in~\cite{BE18}, the tangle algebra has the same dimension as $\mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta)$.
Furthermore, the tangle algebra for $\gamma = 1$ has a Schur--Weyl duality property~\cite[Thm.~5.7]{BE18} (see also~\cite[Rem.~5.13]{BE18}) similar to the cases considered here.
This leads to the following conjecture.
\begin{conj}
\label{conj:tangle_pqp}
Let $\mathbf{k}$ be a field such that $\abs{\mathbf{k}} > n + 1$.
Let $\gamma \neq 0$.
There exists a $\beta \in \mathbf{k}$ such that
\[
\mathcal{T}_n(\gamma) \cong \mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta).
\]
\end{conj}
By a brute-force computation, we can see that
\[
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1);
\draw[-] (G-2) -- (G2);
\end{tikzpicture}
\, \mapsto \,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt, fill=OCUenji]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1);
\draw[-] (G-2) -- (G2);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2);
\draw[-] (G-1) .. controls +(0, .5) and +(0, .5) .. (G-2);
\end{tikzpicture}
\, \mapsto \,
\frac{\gamma}{\beta - 1}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt, fill=OCUenji]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) -- ++(.5, -.5) -- (G2);
\draw[-] (G-1) -- ++(.5, .5) -- (G-2);
\end{tikzpicture}\,,
\qquad
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G-1) -- (G1) .. controls +(0, -.5) and +(0, -.5) .. (G2) -- (G-2) .. controls +(0, .5) and +(0, .5) .. (G-1);
\end{tikzpicture}
\, \mapsto \,
- \frac{2 \beta}{\beta - 2}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt, fill=OCUenji]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) -- ++(.5, -.5) -- (G2);
\draw[-] (G-1) -- ++(.5, .5) -- (G-2);
\draw[-] (1.5,-.5) -- (1.5,.5);
\end{tikzpicture}
+ \frac{2}{\beta - 2}\,
\begin{tikzpicture}[scale = 0.5,thick, baseline={(0,-1ex/2)}]
\tikzstyle{vertex} = [shape=circle, minimum size=2pt, inner sep=1pt, fill=OCUenji]
\foreach \i in {1,2} {
\node[vertex] (G\i) at (\i, 1) [shape=circle, draw] {};
\node[vertex] (G-\i) at (\i, -1) [shape=circle, draw] {};
}
\draw[-] (G1) -- ++(.5, -.5) -- (G2);
\draw[-] (G-1) -- ++(.5, .5) -- (G-2);
\end{tikzpicture}\,,
\]
defines an isomorphism $\mathcal{P}\mathcal{Q}\mathcal{P}_2(\beta) \to \mathcal{T}_2(\gamma)$.
It would be interesting to determine for which values $\gamma$ and $\beta$ we have an isomorphism.
We define the \defn{Riordan triangle numbers} recursively by
\begin{equation}
\label{eq:riordan_triangle_recursion}
R_{n,\lambda} = \begin{cases}
R_{n-1,\lambda+1} + R_{n-1,\lambda} + R_{n-,\lambda-1} & \text{if } \lambda > 0, \\
R_{n-1,1} & \text{if } \lambda = 0,
\end{cases}
\end{equation}
with $R_{n,n} = 1$ and $R_{1,0} = 0$~\cite{Bernhart97,MRSV97}.
\begin{thm}
\label{thm:quasi_partition_cellular}
The planar quasi-partition algebra $\mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta)$ is a cellular algebra of dimension $R_{2n}$ with $\Lambda = \{0, 1, \dotsc, n\}$.
Furthermore, for any $\lambda \in \Lambda$, we have
\[
\dim W(\lambda) = R_{n,\lambda}.
\]
Moreover, we have
\[
R_{2n} = \sum_{\lambda=0}^n R_{n,\lambda}^2.
\]
\end{thm}
\begin{proof}
From Theorem~\ref{thm:quasi_cellular} and Theorem~\ref{thm:planar_cellular}, the planar quasi-partition algebra is cellular.
We note that any singleton in a half diagram must be a propagating block.
We can also move all of the propagating blocks to the front and have size $1$.
This proves the indexing set $\Lambda = \{0, 1, \dotsc, n\}$.
The last claim is simply Equation~\eqref{eq:dim_formula}.
Therefore, we only need to prove the dimension of the cell modules.
We show that the half diagrams satisfy the same recurrence relation as the Riordan triangle numbers.
Let $\rho$ be a half diagram, and suppose $n \in \rho_1$.
If $\abs{\rho_1} > 1$, then construct a new half diagram by having $\rho_1 \setminus \{n\}$ be a defect block and keeping the other parts the same.
Thus the number of defects has increased by one if and only if $\rho_1$ is a defect (otherwise the number of defects does not change).
This is bijective as we simply take the rightmost defect block in the half diagram of $[n-1]$ to reconstruct $\rho$.
Lastly, if $\abs{\rho_1} = 1$, then it necessarily must be a defect.
We simply remove $\rho_1$ from the half diagram to form the new half diagram.
This decreases the number of defects and is clearly bijective.
\end{proof}
For some other interesting appearances of the Riordan triangle numbers, see~\cite{KLO17,OS19} (there is an unfortunate misprint in the definition of the Riordan triangle numbers in~\cite{OS19}).
One potential approach to proving Conjecture~\ref{conj:tangle_pqp} would be to show they are both satisfy the same Schur--Weyl duality.
To this, we believe there is a minor misprint in~\cite{BE18}, as the module should be $V(2)$ instead of the adjoint representation.
In particular, the dimension of the irreducible module $V(\lambda)$, which equals the cell module $W(\lambda)$ in this case by semisimplicity, is the multiplicity of $V(2\lambda)$ in the decomposition of $V(2)^{\otimes n}$ (as $U(\mathfrak{sl}_2)$-modules).
This can be seen by counting the multiplicities inductively on $n$, where the Pieri rule yields the Riordian triangle number recursion relation~\eqref{eq:riordan_triangle_recursion}.
Alternatively, if it was built using the adjoint representation, then~\cite{BH14} implies that the tangle algebra is isomorphic to the Motzkin algebra, but they (and their cell modules) have different dimensions.
\begin{conj}
The tangle algebra $\mathcal{T}_n(\gamma)$ and the quasi-partition algebra $\mathcal{P}\mathcal{Q}\mathcal{P}_n(\beta)$ satisfy Schur--Weyl duality with $U_q(\mathfrak{sl}_2)$ for the module $V(2)^{\otimes n}$ for some $\gamma$ and $\beta$ when $\mathbf{k}$ is a field of characteristic $0$.
\end{conj}
Given that the half (integer) quasi-partition algebra exists, we can also do the same for the planar version.
This would lead to an algebra $\mathcal{P}\mathcal{Q}\mathcal{P}_{n-1/2}(\beta)$ whose dimension is $R_{2n-1}$.
\section{Alternative perspectives and generalizations}
In this section, we discuss just a few generalizations of the partition algebra, although there are indubitably many more than we discuss here.
We also give an alternative perspective using tensor categories, which are often strongly linked to combinatorics (which can be seen in, \textit{e.g.},~\cite{BMT21}).
\subsection{Blob algebra}
\label{sec:blob_algebra}
The blob algebra $\mathcal{B}_n(\beta, \gamma, \delta)$, for parameters $\beta,\gamma,\delta \in \mathbf{k}$, defined by Martin and Saleur~\cite{MS94} can be considered as the type $B$ analog the Temperley--Lieb algebra, where we can put idempotent blobs on strands that can escape out the left boundary of the diagram.
Alternative, if we unfold the diagram (along the right side), then these are the strands that are not nested in the noncrossing perfect matching.
Multiplication is given as for the Temperley--Lieb algebra except loops with a blob contribute $\delta$ instead of $\beta$ and the blobs are idempotent (which is resolved before loops are removed): when we combine two blobs together, we have a blob remaining and multiply by a factor of $\gamma$.
The blob algebra has also been well-studied (see, \textit{e.g.},~\cite{ILZ18} and references therein) and is known to be cellular~\cite{GL03} (along with some generalizations, such as in~\cite{LRH20} using the version of Martin and Woodcock~\cite{MW00}).
The classical blob algebra also fits into our framework, but now
\[
\Lambda = \{n-2k, \overline{n-2k} \mid k \in \langle n \rangle\}
\]
under the ordering $\overline{1} < 1 < \overline{2} < 2 < \cdots$.
For $\lambda \in \Lambda$ with $\lambda = k$ or $\overline{k}$, the number of defects is equal to $k$.
Furthermore, the barred values indicate that the leftmost defect has a blob on it and unbarred entries have no blobs on the defects.
Otherwise, $M(\lambda)$ is the expected set of blobbed half diagrams.
We can see this is a cellular algebra since blobs are idempotent, and so we cannot remove a blob from a strand/defect once it has been added.
\subsection{Other Schur--Weyl duality algebras}
We briefly remark on some other variations of the partition algebra that have appeared coming from a Schur--Weyl duality.
The first is the rook partition algebra $\mathcal{R}\mathcal{P}_n(\beta)$ introduced by Grood~\cite{Grood06} that comes from Schur--Weyl duality involving the $\operatorname{GL}_m(\mathbb{C})$ module $V = V(1) \oplus V(0)$ restricted to the corresponding $\Sigma_m$ action.
This version involves the usual diagrams except we color singletons by two different colors.
Alternatively, we can color the nodes by two different colors, call them red and green, and if a node is colored red, then it must be a singleton.
Since the only difference is coloring singletons, the proof that it is a cellular algebra is the same as for the classical partition algebra.
\begin{prop}
The rook partition algebra is a cellular algebra with the same cell datum as $\mathcal{P}_n(\beta)$ except $M(\lambda)$ consists of all half diagrams with singletons colored one of two colors.
\end{prop}
\begin{cor}
Let $\lambda \in \Lambda$ be a partition of $k$.
Then we have
\[
\dim W(\lambda) = f_{\lambda} \sum_{i=0}^{n-k} \binom{n}{i} \sum_{j=k}^{n-i} \binom{j}{k} \stirling{n-i}{j}.
\]
\end{cor}
\begin{proof}
This comes from choosing $i$ nodes to first color red, and the rest is just the usual partition algebra formula.
\end{proof}
Grood also showed~\cite{Grood06} that $\dim \mathcal{R}\mathcal{P}_n(\beta) = B_{2n+1}$ through an indirect combinatorial argument, but we can give a more straightforward argument.
The red colored nodes are simply one special block, which how we mark it as special is say it contains an extra node $\{0\}$.
This perspective gives us an alternative formula for the cell module dimensions as
\[
\dim W(\lambda) = f_{\lambda} \sum_{j=k}^n \binom{n}{j} \stirling{j}{k} B_{n-j+1}.
\]
Ly in~\cite{Ly19} studied a Schur--Weyl type duality using the supercharacter theory of $U_m(\mathbb{F}_q)$ of upper triangular matrices from~\cite{Thiem10}, which is used to approximate its ``wild'' type representation theory.
However, it is not clear how to apply the techniques of this paper to that construction.
\begin{problem}
Determine if the centralizer algebra in~\cite{Ly19} is cellular.
\end{problem}
Lastly, there is the \defn{walled Brauer algebra} $\mathcal{B}_{n,k}(\beta)$ coming from Schur--Weyl duality with $\operatorname{GL}_m(\mathbb{C})$ on $\mathbf{V}^{\otimes n} \otimes (\mathbf{V}^*)^{\otimes k}$ that was initially studied in~\cite{Turaev89,Koike89,BCHLLS94}.
This is a subalgebra of $\mathcal{B}_{n+k}(\beta)$, where we do not have any propagating strands $\{a, \overline{b}\}$ or $\{b, \overline{a}\}$ for $1 \leq a < n + \frac{1}{2} < b \leq n+k$.
That is, there is a ``wall'' between positions $k$ and $k+1$ that does not allow propagating blocks to pass it (although caps and cups are fine).
The walled Brauer algebra was shown to be cellular in~\cite{CDVDM08}, where they also classified when it is semisimple.
This could also be seen from Proposition~\ref{prop:subcellular}, where we are restricting $\Sigma_{n+k}$ that dictates the decomposition~\eqref{eq:partition_decomposition} to the Young subgroup $\Sigma_n \times \Sigma_k$.
Hence, $\Lambda$ now given by a pair of partitions $(\lambda, \mu)$ of sizes at most $k$ and $n$, respectively (\textit{cf.}~\cite{Halverson96}).
Generalizations have also been considered, such as~\cite{RS15,Sartori14}.
\subsection{Diagram algebras as categories}
Another perspective on the diagram algebras is to view them as (graded) monoidal categories, where the algebra is isomorphic to the Grothendieck ring of the category.
This is known as categorification, and it can bring out new properties both for the category and Grothendieck ring.
One example the Temperley--Lieb category $\mathsf{TL}$, which was first introduced by Graham and Lehrer~\cite{GL98}, with objects $\mathbb{Z}_{>0}$ and morphisms $n \to m$ corresponding to Temperley--Lieb diagrams from $[n] \to [m']$ (with $m$ not necessarily equal to $n$).
The composition of diagrams is the composition of morphisms (as per our multiplication convention).
If we restrict to the subcategory corresponding to the object $n$, then the Grothendieck ring is isomorphic to $\mathcal{T}\mathcal{L}_n(\beta)$.
The category $\mathsf{TL}$ and its representations have been studied with expected applications to conformal field theory; see, \textit{e.g.},~\cite{BSA18II} and references therein.
In~\cite{KMY19}, an interpolation category was constructed between $\mathsf{TL}$ and the corresponding categorification of the Brauer algebra.
A different categorification of $\mathcal{T}\mathcal{L}_n(1)$ was given by Bernstein, Frenkel, and Khovanov~\cite{BFK99}, although it is just the Karubi envelope of the additive closure the previous construction.
Their construction was later extended to $\mathcal{T}\mathcal{L}_n(\beta)$ and other types by Stroppel~\cite{Stroppel05}.
Recently, a two parameter analog of $\mathsf{TL}$ was given in~\cite{KS21} as a method to categorify Chebyshev polynomials of the second kind (yet another appearance of these polynomials).
A related but different construction for the invariant spaces of the $U_q(\mathfrak{sl}_2)$-action on $\mathbf{V}^{\otimes n}$ and generalizations was given diagrammatically by Kuperberg webs and spiders~\cite{Kuperberg96}.
These have seen some attention, such as in~\cite{Tymoczko12,Scherer21} with many open questions remaining, such as a basis for $\mathfrak{sl}_m$ for $m \geq 4$.
\section{Wreath products}
\label{sec:wreath}
Recall that $G(r,1,m) \cong \mathbb{Z}_r \wr \Sigma_m$, whose group algebra has a cellular basis~\cite{GL96} roughly speaking by taking ``product'' of the cellular basis of $\mathbf{k} [\mathbb{Z}_r]$ and $\mathbf{k} [\Sigma_m]$.
The cellularity of $\mathbf{k} [\mathbb{Z}_r]$ when $\zeta_r \in \mathbf{k}$ (alternatively $\mathbf{k}$ splits $x^r - 1$) follows from a special case of the the Ariki--Koike algebras from~\cite{GL96}, which is done over $\mathbb{Z}[q,u_1,\dotsc,u_r]$ but the group algebra is under the specialization $q = 1$ and $u_k = \zeta_r^k$.
A more direct construction was done in, \textit{e.g.},~\cite[Sec.~4]{RX04} (see also~\cite[Ex.~4.14]{ZC06}).
There is also the cyclotomic blob (resp.\ Temperley--Lieb) algebra introduced in~\cite{ZC06} (resp.~\cite{RX04}), which can be constructed as the blob (resp.\ Temperley--Lieb) algebra with each strand carrying a copy of $\mathbb{Z}_r$, where it was shown to be cellular (again, assuming $\mathbf{k}$ splits $x^r - 1$).
Furthermore, the cellular basis of the cyclotomic blob/Termperley--Lieb algebra has the same ``product'' of cellular bases structure.
In this section, we generalize this construction as a method to produce new cellular algebras by taking a wreath product of an arbitrary cellular algebra with certain subalgebras of the partition algebra.
Let $\mathcal{A}$ be any finite dimensional $\mathbf{k}$-algebra with basis $B$.
Let $\mathcal{S}$ be any subalgebra of the partition algebra such that blocks have size at most $2$.
Define the wreath product $\mathcal{A} \wr \mathcal{S}$ as the $\mathbf{k}$-span of diagrams of $\mathcal{S}$ with an element of $B$ attached to each block.
The multiplication is the natural concatenation of diagrams: where we take the product in $\mathcal{S}$, then multiply the elements of $\mathcal{A}$ along any strand and expanding this as a linear combination of basis elements in the natural way before removing cycles.
This has a natural involution $\iota_{\wr}$ induced from the involution $\iota$ of $\mathcal{S}$.
We call this the \defn{cellular wreath product} of the base $\mathcal{S}$ by the algebra $\mathcal{A}$ and denote this by $\mathcal{A} \wr \mathcal{S}$.
\begin{thm}
\label{thm:wreath_product}
Let $\mathcal{A}$ be a cellular algebra with cell datum $(\widetilde{\Lambda}, \widetilde{\iota}, \widetilde{M}, \widetilde{C})$.
Let $\mathcal{S}$ be any subalgebra of the partition algebra such that blocks have size at most $2$ with cell datum $(\Lambda, \iota, M, C)$.
Then the wreath product $\mathcal{A} \wr \mathcal{S}$ is a cellular algebra.
If $\mathcal{S} = \mathcal{R}\mathcal{B}_n(\beta, \gamma)$ (resp.~$\mathcal{M}_n(\beta, \gamma)$), then the cell datum $(\Lambda_{\wr}, \iota_{\wr}, M_{\wr}, C_{\wr})$ is given by
\begin{itemize}
\item $\Lambda_{\wr} = \{(\lambda, L) \mid \lambda \in \Lambda, L \in \langle \widetilde{\Lambda}, D(\lambda) \rangle \}$, where $D(\lambda)$ is the number of defects corresponding to $\lambda$ and $\langle \widetilde{\Lambda}, \ell \rangle$ is the number of multisets (resp.\ sequences) of size $\ell$ with elements in $\widetilde{\Lambda}$, under the natural lexicographic order;
\item $\iota_{\wr}$ is the natural involution induced from $\iota$ on $\mathcal{S}$;
\item $M_{\wr}(\lambda, L) = M(\lambda) \times \widetilde{M}(L) \times (\bigsqcup \widetilde{M})^{N(\lambda)}$, where $N(\lambda)$ equals the number of non-defect blocks corresponding to~$\lambda$, $\widetilde{M}(L) := \prod_{i=1}^{D(\lambda)} \widetilde{M}(L_i)$, and $\bigsqcup \widetilde{M} := \bigsqcup_{\lambda \in \widetilde{\Lambda}} \widetilde{M}(\lambda)$;
\item $C_{\wr}$ is formed by writing the cellular basis element $C_{ST}^{\lambda}$ in terms of the natural diagram basis and attaching the corresponding element in $C_{\widetilde{S}_i\widetilde{T}_i}^{\widetilde{\lambda}_i}$ to the $i$-th strand for all such $i$ determined by (inverse) RSK.
\end{itemize}
Otherwise it is given by the appropriate restriction.
\end{thm}
\begin{proof}
Consider some $\lambda_{\wr} \in \Lambda_{\wr}$ and $S_{\wr}, T_{\wr} \in M_{\wr}(\lambda, L)$.
We have $\iota_{\wr}(C^{\lambda}_{S_{\wr}T_{\wr}}) = C^{\lambda}_{T_{\wr}S_{\wr}}$ by construction.
For every $\lambda \in \Lambda$, $S,T \in M(\lambda)$, and $a \in \mathcal{A}$, we have
\[
a C^{\lambda_{\wr}}_{S_{\wr}T_{\wr}} = \sum_{U,V \in M(\lambda, L)} r_a(U,S_{\wr},V) C^{\lambda_{\wr}}_{UV} + \mathcal{A}^{<\lambda_{\wr}},
\]
since we attach elements of the cellular basis of $\mathcal{A}$ to each strand and that multiplication in $\mathcal{S}$ cannot increase the number of propagating blocks.
It remains to show that we must have $V = T_{\wr}$ and $r_a(U,S_{\wr},V)$ does not depend on $T$.
Indeed, this follows from the fact that we can think of the (partial) permutation as a (partial) mapping from $\bigsqcup \widetilde{M} \to \bigsqcup \widetilde{M}$, the multiplication properties of cellular basis elements of $\mathcal{A}$ to each strand, and properties of the Kazhdan--Lusztig basis (see, \textit{e.g.},~\cite[Ex.~1.2]{GL96}).
The restriction is Proposition~\ref{prop:subcellular}.
\end{proof}
Our proof is essentially a mild generalization of the proof of~\cite[Thm.~5.5]{GL96}.
In fact, this suggests that there should be a good set of commutative elements that would play the role of the Jucys--Murphy elements in a Hecke algebra analog (\textit{cf.}~\cite[Sec.~2.2]{Mathas04}).
\begin{problem}
Show this can be extended to the Hecke algebra action on the diagrams of $\mathcal{S}$ (instead of $\mathbf{k} [\Sigma_n]$).
Furthermore, construct a set of elements $L_2, \dotsc, L_N$ such that:
\begin{itemize}
\item they generate an abelian subalgebra,
\item there is a basis given by monomials in $L_i$ times a Hecke algera basis element $T_w$, and
\item the center contains all symmetric polynomials in $L_2, \dotsc, L_N$.
\end{itemize}
\end{problem}
For the product to be well-defined, it is necessary that the blocks have the same size.
We have used the particular case that each block has size $2$, but this could be extended for the block partition algebra by associating to each block of size $k$ a bundle of $k$ (ordered) strands.
In this case, we get a natural Hecke algebra analog by taking the corresponding product of (type $A$) Hecke algebras, analogous to how we have used a product of symmetric group algebras (equivalently the group algebra of a product of symmetric groups).
Let us remark on two other constructions based on wreath products.
The first is the coloring of the partition algebras by a finite group $G$ in the work by Bloss~\cite{Bloss03} to understand Schur--Weyl duality with the wreath product $G \wr \Sigma_m$.
When we restrict to subalgebras of the rook Brauer algebra, we obtain our construction.
The second is the ramified partition algebra of Martin and Elgamal~\cite{ME04}, which was further studied in~\cite{Martin11} using the symmetric group and the partition algebra.
However, those constructions are distinct from $\mathbf{k} [\Sigma_n] \wr \mathcal{P}_n(\beta)$ since this has the same dimension as $\mathbf{k} [\Sigma_n] \otimes \mathcal{P}_n(\beta)$ but they are not isomorphic for $n > 1$ since there are more irreducible representations of $\mathbf{k} [\Sigma_n] \wr \mathcal{P}_n(\beta)$ (\textit{e.g.}~$12$ for $n = 2$) than $\mathbf{k} [\Sigma_n] \otimes \mathcal{P}_n(\beta)$ (resp. $6$).
Moreover, the algebra $\mathcal{P}_n^{\ltimes}(\beta)$ in~\cite{Martin11} clearly has smaller dimension than $\mathbf{k} [\Sigma_n] \otimes \mathcal{P}_n(\beta)$.
It is clear that the wreath product construction extends to graded cellular algebras defined in~\cite{HM10}.
It is expected that the wreath product construction will extend to generalizations of cellular algebras, such as affine cellular algebras~\cite{KX12}, the recently defined skew graded cellular algebras~\cite{HMR21}.
It should also work by replacing the underlying diagram algebra; for example, the base algebra could use the blob algebra $\mathcal{B}_n(\beta, \gamma, \delta)$, generalizations of the Temperley--Lieb algebra (see, \textit{e.g.},~\cite{BCF22} and references therein), the higher genus diagram algebras~\cite{TV21}, the BMW algebra~\cite{BW89,Murakami87} or the associated tangle algebra~\cite{FG95}, or quiver Hecke algebras (also known as Khovanov--Lauda--Rouquier (KLR) algebras~\cite{KL09,Rouquier08})~\cite{HM10}.
One important algebra could be the cellular wreath product of the symmetric group with itself.
Indeed, the cell modules for this are given by composing a $\Sigma_n$-representation with a $\Sigma_k$-representation.
This would be a restriction of the corresponding $\operatorname{GL}_n$-representation with a $\operatorname{GL}_k$-representation that defines the plethysm $s_{\lambda}[s_{\mu}]$.
This leads to the following problem.
\begin{problem}
\label{prob:plethysm}
Determine the relationship between the representation theory of $\mathbf{k} [\Sigma_n] \wr \mathbf{k} [\Sigma_k]$ and the plethysm coefficients $a_{\lambda\mu}^{\nu}$ given by $s_{\lambda}[s_{\mu}] = \sum_{\nu} a_{\lambda\mu}^{\nu} s_{\nu}$.
\end{problem}
We refer the reader to~\cite{COSSZ22} for some recent information on plethysm coefficients.
\bibliographystyle{alpha}
|
\section{Introduction}
Let $G$ be a finite abelian group written with additive notation, let $m$ be a positive integer with $m \leq |G|$, and let $h$ be a nonnegative integer. In \cite{BajMat:2014a}, we introduced the function
$$\rho_{\pm}(G, m, h) = \min \{ |h_{\pm}A| \; : \; A \subseteq G, |A|=m\},$$ where
$$h_{\pm} A=\{ \Sigma_{i=1}^m \lambda_i a_i \; : \; (\lambda_1,\dots,\lambda_m) \in \mathbb{Z}^m, \; \Sigma_{i=1}^m |\lambda_i|=h\}$$ is the
$h$-fold {\em signed sumset} of an $m$-subset $A=\{a_1, \dots, a_m\}$ of $G$
(as usual, $|S|$ denotes the size of the finite set $S$). The function $\rho_{\pm}(G, m, h)$ is the analogue of the well-known
$$\rho(G, m, h) = \min \{ |hA| \; : \; A \subseteq G, |A|=m\},$$ where
$$hA=\{ \Sigma_{i=1}^m \lambda_i a_i \; : \; (\lambda_1,\dots,\lambda_m) \in \mathbb{N}_0^m, \; \Sigma_{i=1}^m \lambda_i=h\}$$ is the usual $h$-fold {\em sumset} of $A$.
Signed sumsets have already been studied in the past: For example, in \cite{BajRuz:2003a}, the first author and Ruzsa investigated the {\em independence number} of a subset $A$ of $G$, defined as the maximum value of $t \in \mathbb{N}$ for which $$0 \not \in \cup_{h=1}^t h_{\pm}A$$ (see also \cite{Baj:2000a} and \cite{Baj:2004a}); and in \cite{KloLev:2003a}, Klopsch and Lev discussed the {\em diameter} of $G$ with respect to $A$, defined as the minimum value of $s \in \mathbb{N}$ for which $$\cup_{h=0}^s h_{\pm}A=G$$ (see also \cite{KloLev:2009a}). The independence number of $A$ in $G$ quantifies the ``degree'' to which $A$ is linearly independent in $G$, while the diameter of $G$ with respect to $A$ measures how ``effectively'' $A$ generates $G$ (if at all). While research on minimum sumset size goes back to the work of Cauchy and is now known for all $G$, $m$, and $h$, to the best of our knowledge, \cite{BajMat:2014a} is the first systematic study of the minimum size of signed sumsets. In this paper we continue our work and consider $\rho_{\pm}(G, m, h)$ for elementary abelian groups $G$.
Let us review what we need to know about $\rho(G, m, h)$. It has been over two hundred years since Cauchy \cite{Cau:1813a} found the minimum possible size of
$$A+B=\{a+b \; : \; a \in A,\; b \in B \}$$ among subsets $A$ and $B$ of the cyclic group $\mathbb{Z}_p$ of given sizes. (Here and elsewhere in the paper $p$ denotes a positive prime.) Over a hundred years later, Davenport \cite{Dav:1935a} (cf.~\cite{Dav:1947a}) rediscovered Cauchy's result, which is now known as the Cauchy--Davenport Theorem:
\begin{thm}[Cauchy--Davenport Theorem] \label{Cauchy--Davenport}
If $A$ and $B$ are nonempty subsets of the group $\mathbb{Z}_p$ of prime order $p$, then
$$|A+B| \geq \min \{p, |A|+|B|-1\}.$$
\end{thm}
It can easily be seen that the bound is tight for all values of $|A|$ and $|B|$, and thus
$$ \rho (\mathbb{Z}_p, m, 2)=\min\{p,2m-1\}.$$
Relatively recently, $ \rho (G, m, h)$ was finally evaluated for all parameters by Plagne \cite{Pla:2006a} (see also \cite{Pla:2003a}, \cite{EliKer:2007a}, and \cite{EliKerPla:2003a}) in 2003. To state the result, we introduce the function
$$u(n,m,h)=\min \{f_d (m,h) \; : \; d \in D(n)\},$$ where $n$, $m$, and $h$ are positive integers, $D(n)$ is the set of positive divisors of $n$, and
$$f_d(m,h)=\left(h\left \lceil m/h \right \rceil-h +1 \right) \cdot d.$$
(Here $u(n,m,h)$ is a relative of the Hopf--Stiefel function used also in topology and bilinear algebra; see, for example, \cite{EliKer:2005a}, \cite{Kar:2006a}, \cite{Pla:2003a}, and \cite{Sha:1984a}.)
\begin{thm} [Plagne; cf.~\cite{Pla:2006a}] \label{value of u}
Let $n$, $m$, and $h$ be positive integers with $m \leq n$. For any abelian group $G$ of order $n$ we have
$$\rho (G, m, h)=u(n,m,h).$$
\end{thm}
Let us turn now to $\rho_{\pm} (G, m, h)$. It is easy to see that $\rho_{\pm} (G,1,h)$ and $\rho_{\pm} (G,m,0)$ both equal $1$ and that $\rho_{\pm} (G,m,1)$ equals $m$ for all $G$, $m$, and $h$. (To see the last equality, it suffices to verify that one can always find a {\em symmetric} subset of size $m$ in $G$, that is, an $m$-subset $A$ of $G$ for which $A=-A$.) Therefore, from now on, we assume that $m \geq 2$ and $h \geq 2$.
Perhaps surprisingly, we find that, while the $h$-fold signed sumset of a given set is generally much larger than its sumset, $\rho_{\pm} (G, m, h)$ often agrees with $\rho (G, m, h)$; in particular, this is always the case when $G$ is cyclic:
\begin{thm} [Cf.~\cite{BajMat:2014a}] \label{cyclic} For all positive integers $n$, $m$, and $h$, we have
$$\rho_{\pm} (\mathbb{Z}_n, m, h)= \rho (\mathbb{Z}_n, m, h).$$
\end{thm}
The situation seems considerably more complicated for noncyclic groups: in contrast to $\rho (G, m,h)$, the value of $\rho_{\pm} (G, m,h)$ depends on the structure of $G$ rather than just the order $n$ of $G$.
Observe that by Theorem \ref{value of u}, we have the lower bound
$$\rho_{\pm} (G, m,h) \geq u(n,m,h)=\min \{f
|
_d (m,h) \; : \; d \in D(n)\}.$$ In \cite{BajMat:2014a}, we proved that with a certain subset $D(G,m)$ of $D(n)$, we have
$$\rho_{\pm} (G, m,h) \leq u_{\pm} (G,m,h)=\min \{f_d (m,h) \; : \; d \in D(G,m)\};$$ here $D(G,m)$ is defined in terms of the {\em type} $(n_1,\dots,n_r)$ of $G$, that is, via integers $n_1,\dots,n_r$ such that $n_1 \geq 2$, $n_i$ divides $n_{i+1}$ for each $i \in \{1,\dots, r-1\}$, and for which $G$ is isomorphic to the invariant product
$$\mathbb{Z}_{n_1} \times \cdots \times \mathbb{Z}_{n_r}.$$
Namely, we proved the following result:
\begin{thm} [Cf.~\cite{BajMat:2014a}] \label{u pm with f} The minimum size of the $h$-fold signed sumset of an $m$-subset of a group $G$ of type $(n_1,\dots,n_r)$ satisfies
$$\rho_{\pm} (G, m,h) \leq u_{\pm} (G,m,h),$$
where $$u_{\pm} (G,m,h)=\min \{f_d (m,h) \; : \; d \in D(G,m) \}$$ with $$D(G,m)=\{d \in D(n) \; : \; d= d_1 \cdots d_r, d_1 \in D(n_1), \dots, d_r \in D(n_r), dn_r \geq d_rm \}.$$
\end{thm}
Observe that, for cyclic groups of order $n$, $D(G,m)$ is simply $D(n)$.
Additionally, we believe that $u_{\pm} (G,m,h)$ actually yields the exact value of $\rho_{\pm} (G,m,h)$ in all cases except for one very special situation (which occurs only when $h=2$). In particular, we made the following conjecture:
\begin{conj} [Cf.~\cite{BajMat:2014a}] \label{conj for rho pm}
Suppose that $G$ is an abelian group of order $n$ and type $(n_1, \dots, n_r)$.
If $h \geq 3$, then $$\rho_{\pm} \left(G, m, h \right) =u_{\pm}(G,m,h).$$
If each odd divisor of $n$ is less than $2m$, then $$\rho_{\pm} \left(G, m, 2 \right) =u_{\pm}(G,m,2).$$
If there are odd divisors of $n$ greater than $2m$, let $d_m$ be the smallest one. We then have
$$\rho_{\pm} \left(G, m, 2 \right) = \min\{u_{\pm}(G,m,2), d_m-1\}.$$
\end{conj}
We will need to use the following ``inverse type'' result from \cite{BajMat:2014a} regarding subsets that achieve $\rho_{\pm} \left(G, m, h \right)$. Given a group $G$ and a positive integer $m \leq |G|$, we define a certain collection ${\cal A}(G,m)$ of $m$-subsets of $G$.
We let
\begin{itemize}
\item $\mathrm{Sym}(G,m)$ be the collection of {\em symmetric} $m$-subsets of $G$, that is, $m$-subsets $A$ of $G$ for which $A=-A$;
\item $\mathrm{Nsym}(G,m)$ be the collection of {\em near-symmetric} $m$-subsets of $G$, that is, $m$-subsets $A$ of $G$ that are not symmetric, but for which $A\setminus \{a\}$ is symmetric for some $a \in A$;
\item $\mathrm{Asym}(G,m)$ be the collection of {\em asymmetric} $m$-subsets of $G$, that is, $m$-subsets $A$ of $G$ for which $A \cap (-A)=\emptyset$.
\end{itemize} We then let
$${\cal A}(G,m)=\mathrm{Sym}(G,m) \cup \mathrm{Nsym}(G,m)\cup \mathrm{Asym}(G,m).$$ In other words, ${\cal A}(G,m)$ consists of those $m$-subsets of $G$ that have exactly $m$, $m-1$, or $0$ elements whose inverse is also in the set.
\begin{thm} [Cf.~\cite{BajMat:2014a}] \label{symmetry thm}
For every $G$, $m$, and $h$, we have
$$\rho_{\pm} (G,m,h)= \min \{|h_{\pm} A| \; : \; A \in {\cal A}(G,m)\}.$$
\end{thm}
We should add that each of the three types of sets are essential as can be seen by examples (cf.~\cite{BajMat:2014a}).
Our goal in this paper is to investigate $\rho_{\pm} (G,m,h)$ for elementary abelian groups $G$. In particular, we wish to classify all cases for which
$$\rho_{\pm} (\mathbb{Z}_p^r, m, h) = \rho (\mathbb{Z}_p^r, m, h),$$
where $p$ denotes a positive prime and $r$ is a positive integer. By Theorem \ref{cyclic}, we assume that $r \geq 2$, and, since obviously
$$\rho_{\pm} (\mathbb{Z}_2^r, m, h) = \rho (\mathbb{Z}_2^r, m, h)$$ for all $m$, $h$, and $r$, we will also assume that $p \geq 3$.
Let us first exhibit a sufficient condition for $\rho_{\pm} (\mathbb{Z}_p^r, m, h)$ to equal $\rho(\mathbb{Z}_p^r, m, h)$. When $p \leq h$, our result is easy to state; we will prove the following:
\begin{thm} \label{p leq h}
If $p \leq h$, then for all values of $1 \leq m \leq p^r$ we have $$\rho_{\pm} (\mathbb{Z}_p^r, m, h) = \rho (\mathbb{Z}_p^r, m, h).$$
\end{thm}
The case $h \leq p-1$ is more complicated and delicate. In order to state our results, we will need to introduce some notations. Suppose that $m \geq 2$ is a given positive integer. First, we let $k$ be the maximal integer for which
$$p^k +\delta \leq hm-h+1,$$ where
$\delta=0$ if $p-1$ is divisible by $h$, and $\delta=1$ if it is not. Second, we let $c$ be the maximal integer for which
$$(hc+1) \cdot p^k + \delta \leq hm-h+1.$$ Note that $k$ and $c$ are nonnegative integers and $c \leq p-1,$ since for $c \geq p$ we would have
$$(hc+1) \cdot p^k \geq p^{k+1} +\delta > hm-h+1.$$
|
appa+1\right)^2}\right)$ and any choice of $\boldsymbol{\phi}$ and ${\bf{f}}$ does not impact the distribution of $\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}^{small}{\bf{f}}\sim \mathcal{CN}\left({ 0},\frac{2\kappa MP}{\left(\kappa+1\right)^2}\right)$.
\end{prop}
\vspace{0ex}
\begin{proof}
See Appendix D.
\end{proof}
Similarly, we denote $\gamma_B \triangleq \frac{\kappa^2 M^2 N_tP}{\left(\kappa+1\right)^2}+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}^{small}{\bf{f}}|^2-\frac{2\kappa M \sqrt{N_tP}}{\kappa+1}|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}^{small}{\bf{f}}|$,
denote the cumulative distribution function of $\gamma_B$ as $F_B(x)$, and $\alpha_B \triangleq F^{-1}_B(p_{out})$. It is not hard to see that
the constraint $(\ref{outB})\leq p_{out}$ can be equivalently transformed as $\alpha_B \geq \frac{\left(2^{R_B}-1\right)\sigma^2}{L_{AI}L_{IB}}$, which could be calculated numerically in an off-line manner once $p_{out}$ is given. Note that if this constraint holds, the original constraint ${\rm{Pr}}\left\{C_B(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi})\leq R_B \right\} \leq p_{out}$ in (\ref{out}) must hold.
After transforming (\ref{out}) as $\alpha_E\leq \frac{\left(2^{R_E}-1\right)\sigma^2}{L_{AI}L_{IE}}$ and $\alpha_B\geq \frac{\left(2^{R_B}-1\right)\sigma^2}{L_{AI}L_{IB}}$, we have following proposition.
\begin{prop}
(\ref{AO_method1}) can be equivalently transformed as
\begin{align}
\label{max_location}
\max \limits_{\boldsymbol{\omega}_I} \ \frac{\sigma^2+\alpha_B L_{AI}L_{IB}}{\sigma^2+\alpha_E L_{AI}L_{IE}}, \ s.t.\ \boldsymbol{\omega}_I\in {\Omega}_I,
\end{align}
where $L_{AI}=\frac{L_0 }{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{A}\|^{\rho_{AI}}}$, $L_{IB}=\frac{L_0}{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{B}\|^{\rho}}$ and $ L_{IE}=\frac{L_0}{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{E}\|^{\rho}}$ are functions of $\boldsymbol{\omega}_{I}$.
\end{prop}
\begin{proof}
See Appendix E.
\end{proof}
Problem (\ref{max_location}) is non-convex due to the objective function. To solve it, we can globally search over $\boldsymbol{\omega}_I$ to obtain the optimal solution. However the computation complexity for this method is high, so we propose a successive convex approximation (SCA) method with a low complexity to solve (\ref{max_location}) and take this global search method as a benchmark.
First we find a lower bound $\frac{\alpha_BL_{AI}L_{IB}}{\sigma^2+\alpha_EL_{AI}L_{IE}}$ of the objective function by ignoring the noise in numerator and this lower bound is approximately equal to the original objective function because the path loss much larger than noise by appropriately optimizing the location of IRS. Then, (\ref{max_location}) is transformed as
\begin{align}
\label{min_location}
\min \limits_{\boldsymbol{\omega}_I} \ \frac{\sigma^2+\alpha_EL_{AI}L_{IE}}{\alpha_BL_{AI}L_{IB}}, \ {\rm{s.t.}}\ \boldsymbol{\omega}_I \in \Omega_{I}.
\end{align}
To further solve the problem, we introduce auxiliary variables ${\bf a}=[a_{AI},a_{IB},a_{IE},a_{AB},a_{BE}]^T$,
where $ a_{AI}=\frac{1}{L_{AI}}$, $ a_{IE}=\frac{1}{L_{IE}}=$, $ a_{IB}=\frac{1}{L_{IB}}$, $a_{AB}= a_{AI}a_{IE}$, $a_{BE}= \frac{a_{IB}}{a_{IE}}$, and transform
(\ref{min_location}) as
\begin{subequations}\label{auxiliary}
\begin{align}
\min \limits_{\boldsymbol{\omega}_I,{\bf a}} &\ \frac{\sigma^2}{\alpha_B}a_{AB}+\frac{\alpha_E}{\alpha_B}a_{BE}, \label{auxiliarya}\\
{\rm{s.t.}} &\
a_{AI}\geq \frac{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{A}\|^{\rho_{AI}}}{L_0}, \
a_{IB}\geq \frac{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{B}\|^{\rho}}{L_0},\label{auxiliaryb}\\
& a_{IE} \leq \frac{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{E}\|^{\rho}}{L_0}, \ a_{AB}\geq a_{AI}a_{IB},\ a_{BE}\geq \frac{a_{IB}}{a_{IE}}. \label{auxiliaryc}
\end{align}
Note that in problem (\ref{auxiliary}), the inequalities in constraints (\ref{auxiliaryb}) and (\ref{auxiliaryc}) will be active at the optimum. This could be proved by the contradiction. Assume that at the optimum one of the corresponding constraints in (\ref{auxiliary}) is a strict inequality. Then, we can always decrease $a_{AI}$ to satisfy the constraint with equality, which decreases the objective value. Therefore, for the optimal solution of (\ref{auxiliary}), $a_{AI}$ must be satisfied with equality. Similarly, all the other auxiliary variables must be active at the optimum. Now, the objective function and constraint (\ref{auxiliaryb}) are convex but constraints in (\ref{auxiliaryc}) are still non-convex. In the following, we propose an SCA method \cite{T-16} to solve (\ref{auxiliary}) iteratively by exploiting the first-order Taylor expansion of all the constraints.
First, by given a feasible point $\boldsymbol{\omega}_{I}^{(l)}$, the upper bound for $a_{IE} \leq \frac{\|\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{E}\|^{\rho}}{L_0}$ in (\ref{auxiliaryc}) is given by
\begin{align}
\notag
L_0a_{IE}&\leq \| \boldsymbol{\omega}_{I}^{(l)}-\boldsymbol{\omega}_{E}\|^{\rho}+\rho \left( \| \boldsymbol{\omega}_{I}^{(l)}-\boldsymbol{\omega}_{E}\|^2\right)^{\frac{\rho}{2}-1}\\
\label{taylor}
&\times\left( \boldsymbol{\omega}_{I}^{(l)}-\boldsymbol{\omega}_{E}\right)^T\left(\boldsymbol{\omega}_{I}-\boldsymbol{\omega}_{I}^{(l)} \right),
\end{align}
Next, we equivalently transform the constraints $ a_{AB}\geq a_{AI}a_{IB}$ and $\ a_{BE}\geq \frac{a_{IB}}{a_{IE}}$ as
\begin{align}
\notag
&\ a_{AB}\geq a_{AI}a_{IB}=\frac{1}{2}\left[\left(a_{AI}+a_{IB} \right)^2-a_{AI}^2-a_{IB}^2\right],\\
\notag
&\ a_{BE}\geq \frac{a_{IB}}{a_{IE}}\Leftrightarrow\\
& a_{IB}\leq a_{IE}a_{BE}=\frac{1}{2}\left[\left(a_{IE}+a_{BE} \right)^2-a_{IE}^2-a_{BE}^2\right].
\end{align}
Then, the convex upper and lower bounds at given points $a_{AI}^{(l)},a_{IE}^{(l)},a_{IB}^{(l)}$ and $a_{BE}^{(l)}$ are given by (\ref{taylor1}) and (\ref{taylor2}) at the top of next page.
\end{subequations}
\begin{figure*}
\begin{align}
\label{taylor1}
a_{AI}a_{IB}\leq b_1 &\triangleq\frac{1}{2}\left[\left(a_{AI}+a_{IB} \right)^2-\left({a_{AI}^{(l)}}^2+2a_{AI}^{(l)}\left(a_{AI}-a_{AI}^{(l)} \right)\right)-\left({a_{IB}^{(l)}}^2+2a_{IB}^{(l)}\left(a_{IB}-a_{IB}^{(l)} \right)\right)\right],\\
\label{taylor2}
a_{IE}a_{BE}\geq b_2&\triangleq \frac{1}{2}\left(a_{IE}^{(l)}+a_{BE}^{(l)} \right)^2+\left(a_{IE}^{(l)}+a_{BE}^{(l)} \right)\left(a_{IB}-a_{IB}^{(l)} \right)
+\left(a_{IE}^{(l)}+a_{BE}^{(l)} \right)\left(a_{BE}-a_{BE}^{(l)} \right)-\frac{1}{2}(a_{IE}^2+a_{BE}^2).
\end{align}
{\noindent} \rule[-6pt]{18cm}{0.05em}\\
\end{figure*}
Finally, the problem $(\ref{auxiliary})$ is transformed as
\begin{align}
\notag
\min \limits_{\boldsymbol{\omega}_I,{\bf a}} &\ \frac{\sigma^2}{\alpha_B}a_{AB}+\frac{\alpha_E}{\alpha_B}a_{BE}, \\
\label{auxiliary_final}
{\rm{s.t.}} &\ (\ref{auxiliaryb}),\ (\ref{taylor}), \ a_{AB}\geq b_1, \ a_{IB}\leq b_2,
\end{align}
where all the constraints are convex, so this problem can be conveniently solved. The overall algorithm to maximize $R$ by optimizing $\boldsymbol{\omega}_I$ is summarized in Algorithm 1.
\begin{algorithm}[h]
\caption{SCA method To Solve Problem (\ref{auxiliary_final})}
\begin{algorithmic}
\State 1. set the convergence precision $\epsilon$ and $\boldsymbol{\omega}_I^{(0)}$, $l=0$;
\Repeat
\State 2. solve problem \ref{auxiliary_final}) to obtain $\boldsymbol{\omega}_I^{(l)}$;
\State 3. $l=l+1$;
\Until{$\|\boldsymbol{\omega}_I^{(l+1)}-\boldsymbol{\omega}_I^{(l)}\|^2\leq \epsilon$;}
\State 4. output $\boldsymbol{\omega}_I$.
\end{algorithmic}
\end{algorithm}
\subsection{ Optimize Beamformer and Phase Shift}\label{BeamPhase}
After obtain the optimized location of IRS, the next step is to optimize the beamformer ${\bf{f}}$ and phase shifts $\boldsymbol{\phi}$. Note that once a deployed location is fixed, the instantaneous CSI of Alice-IRS and IRS-Bob are available by channel training, and LoS components of IRS-Eve is also available, but the NLoS components of IRS-Eve link are unknown random variables. The CSI model of Eve now is expressed as (\ref{EveError}). The sub-problem to optimize $\left({\bf{f}},\boldsymbol{\phi}\right)$ is expressed as
\begin{subequations}\label{AO}
\begin{align}
\max \limits_{{\bf{f}},\boldsymbol{\phi}} &\ R, \label{AOa} \\
{\rm{s.t.}} \ &\ {\rm{Pr}}\left\{R_s({\bf{f}},\boldsymbol{\phi})\geq R\right\} \geq 1-p_{out}, \label{AOb}\\
&\ \|{\bf{f}}\|^2 \leq P, |\boldsymbol{\phi}_{i}|=1,\ i=1,2,\cdots,M,\label{AOc}
\end{align}
\end{subequations}
This problem is non-convex due to the outage probability constraint (\ref{AOb}) so we first handle this constraint. To make it tractable, we rewrite the inequality $R_s({\bf{f}},\boldsymbol{\phi})\geq R$ as
\begin{align}
\notag
&\log\left(1+\frac{|\boldsymbol{\phi}^H{\bf{G}}_{AB}{\bf{f}}|^2}{{\sigma}^2}\right) - \log\left(1+\frac{|\boldsymbol{\phi}^H
{\bf{G}}_{AE}{\bf{f}}|^2}{{\sigma}^2}\right)\geq R,\\
\label{Trans}
&\Leftrightarrow {\rm Tr}\left({{\bf{G}}}_{AE}{\bf{F}}{{\bf{G}}}_{AE}^H{\bf{Q}}\right)\leq 2^{-R}\left({\sigma}^2+
{\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)\right)-{\sigma}^2,
\end{align}
where ${\bf{F}}\triangleq {\bf{f}}{\bf{f}}^H$ and ${\bf{Q}}\triangleq \boldsymbol{\phi}\boldsymbol{\phi}^H$. Here, ${\bf G}_{AB}$ is known, ${\bf G}_{AE}=\overline{\bf G}_{AE} + \widetilde{\bf G}_{AE}$ is modeled in (\ref{EveError}) with determinate and random components $\overline{\bf G}_{AE}$ and $\widetilde{\bf G}_{AE}$, where each element of $\widetilde{{\bf{G}}}_{AE} $ follows $\mathcal{CN}\left({0},\frac{\kappa L_{AI}L_{IE}} {(\kappa +1)^2}\right)$ based on proposition 1. By substituting ${\bf G}_{AE}$ into (\ref{Trans}), we could further obtain
\begin{align}
\notag
&{\rm Tr}\left(\left(\overline{\bf{G}}_{AE}+\widetilde{\bf G}_{AE}\right){\bf{F}}\left(\overline{\bf{G}}_{AE}^H
+\widetilde{\bf G}_{AE}^H\right){\bf{Q}}\right)\\
\notag
&\leq 2^{-R}\left({\sigma}^2+
{\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)\right)-{\sigma}^2,\\
\label{trans}
\Leftrightarrow
&\underbrace{{\rm Tr}\left(\widetilde{\bf G}_{AE}{\bf{F}}\widetilde{\bf G}_{AE}^H{\bf{Q}}\right)}_{f_1}+
2{\rm{Re}}\underbrace{\left\{{\rm Tr}\left(\overline{{{\bf{G}}}}_{AE}{\bf{F}}\widetilde{\bf G}_{AE}^H{\bf{Q}}\right)\right\}}_{f_2}\leq c_1,
\end{align}
where $c_1 \triangleq2^{-R}\left({\sigma}^2+
{\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)\right)-{\sigma}^2-{\rm Tr}\left( \overline{\bf{G}}_{AE}{\bf{F}}\overline{\bf{G}}_{AE}^H{\bf{Q}}\right)$ does not include the random variable $\widetilde{{\bf{G}}}_{AE}$, but both $f_1$ and $f_2$ include it.
To handle $f_1$ and $f_2$, we first denote ${\bf{g}}_{AE}\triangleq{\rm vec}\left( \widetilde{{\bf{G}}}_{AE}\right) \in \mathbb{C}^{MN_t\times 1} \sim \mathcal{CN}\left({\bf{0}},\delta_{AE}^2{\bf{I}}_{MN_t}\right)$ as ${\bf{g}}_{AE}= \delta_{AE}
{\bf{u}}$, where $\delta_{AE}=\sqrt{\frac{\kappa L_{AI}L_{IE}} {(\kappa +1)^2}}$, and ${\bf{u}} \in \mathbb{C}^{MN_t\times 1}\sim \mathcal{CN}\left({\bf0},{{\bf{I}}}_{MN_t}\right)$. Then $f_1$ in $(\ref{trans})$ can be reformulated as
\begin{align}
\label{f1}
f_1
\overset{(c)}{=} {\bf{g}}_{AE}^H\left( {\bf{F}}^T\otimes {\bf{Q}}\right){\bf{g}}_{AE}
=\delta_{AE}^2 {\bf{u}}^H\left( {\bf{F}}^T\otimes {\bf{Q}}\right){\bf{u}}
\triangleq {\bf{u}}^H{\bf{A}}_{AE}{\bf{u}},
\end{align}
where $(c)$ is obtained by invoking the identity ${\rm Tr}\left({\bf{A}}^H\bf{BCD}\right)={\rm vec}^H\left({\bf{A}}\right)\left( \bf{D}^T\otimes {\bf{B}}\right)\rm vec(\bf{C})$ and ${\bf{A}}_{AE}=\delta_{AE}^2 \left( {\bf{F}}^T\otimes {\bf{Q}}\right)$. Similarly, the expression $f_2$ in $(\ref{trans})$ can be reformulated as
\begin{align}
\label{f2}
f_2 =& \delta_{AE}{\bf{u}}^H\left( {\bf{F}}^T\otimes {\bf{Q}}\right){\rm vec}\left(\overline{\bf{G}}_{AE}\right)
\triangleq {\bf{u}}^H{\bf{a}}_{AE},
\end{align}
where ${\bf{a}}_{AE}=\delta_{AE}\left( {\bf{F}}^T\otimes {\bf{Q}}\right){\rm vec}\left(\overline{\bf{G}}_{AE}\right)$. By substituting $(\ref{f1})$ and$(\ref{f2})$ into $(\ref{trans})$, we have
\begin{align}
(\ref{trans}) \Leftrightarrow {\bf{u}}^H{\bf{A}}_{AE}{\bf{u}}+2{\rm{Re}}\left\{{\bf{u}}^H{\bf{a}}_{AE}\right\}-c_1\leq 0,
\end{align}
and the outage constraint $(\ref{AOb})$ can be reformulated as
\begin{align}
\label{pout_final}
{\rm{Pr}}\left\{{\bf{u}}^H{\bf{A}}_{AE}{\bf{u}}+2{\rm{Re}}\left\{{\bf{u}}^H{\bf{a}}_{AE}\right\}-c_1\geq 0\right\} \leq p_{out}.
\end{align}
With the probability constraint in form of $(\ref{pout_final})$, we could exploit Bernstein-Type Inequality-I (BTI-I) in Lemma 1 to handle it which has quadratic forms of Gaussian variables matrix.
\begin{lemma}
(Bernstein-Type Inequality-I \cite{Ma-14}): Let ${\bf{G}} = {\bf{x}}^H{\bf{Cx}}+2{\rm Re}\left\{{\bf{x}}^H{\bf{c}}\right\}$, where ${\bf{C}}\in \mathbb{C}^{N\times N}$
is a complex Hermitian matrix, ${\bf{c}} \in \mathbb{C}^{N\times 1}$, and ${\bf{x}}\sim \mathcal{CN}(\bf{0},{{\bf{I}}})$. Then for any $\varrho \geq 0$, we have
\begin{align}
\notag
{\rm{Pr}} \left\{{\bf G}\geq \rm Tr({\bf C}) +\sqrt{2\varrho}\sqrt{||\rm vec({\bf C})||^2+2||{\bf c}||^2}+\varrho \lambda^+({\bf C})\right\} \leq e^{-\varrho},
\end{align}
where $\lambda^+({\bf C})=max\left\{ \lambda_{max}({\bf C}),0\right\}$, and $\lambda_{max}({\bf C})$ represents the maximum eigenvalue of ${\bf C}$.
\end{lemma}
With BTI-I, $(\ref{pout_final})$ is transformed to a deterministic form as
\begin{align}
\notag
&{\rm Tr}({\bf{A}}_{AE}) +\sqrt{2\varrho}\sqrt{\|{\rm vec}({\bf{A}}_{AE})\|^2++2\|{\bf{a}}_{AE}\|^2}\\
\label{BTI}
&+\varrho \lambda^+({\bf{A}}_{AE})-c_1\leq 0,
\end{align}
where $\varrho = -{\rm ln}(p_{out})$. If $(\ref{BTI})$ is true, $(\ref{pout_final})$ must hold true. Consequently, $(\ref{AO})$ is transformed as
\begin{subequations}\label{ NextPerfectLocation}
\begin{align}
\max \limits_{{\bf{F}} ,{\bf{Q}}} &\ R, \label{ NextPerfectLocationa} \\
\notag
\ {\rm{s.t.}} & \ (\ref{BTI}),\ {\rm Tr}({\bf{F}})\leq P,\ {\rm rank}({\bf{F}})=1,{\bf F} \succeq {\bf 0}, \\
\ &\ {\rm Diag}\left({\bf{Q}}\right)= {\bf{1}}_M,\ {\rm rank}({\bf{Q}})=1,\ {\bf{Q}} \succeq {\bf 0} \label{ NextPerfectLocationc},
\end{align}
\end{subequations}
where ${\rm Diag}\left({\bf{Q}}\right)= {\bf{1}}_M$ ensures the phase shifts $\boldsymbol{\phi}$ with unit modulus.
Although ${\bf{F}}$ and ${\bf{Q}}$ are coupled in (\ref{BTI}), fortunately, this constraint is convex for ${\bf{F}}$ given ${\bf{Q}}$, and is convex for ${\bf{Q}}$ given ${\bf{F}}$ as well, respectively. All the other constraints by dropping rank-one constraint in $(\ref{ NextPerfectLocation})$ are also convex. Hence, an AO algorithm is developed to optimize ${\bf{F}}$ and ${\bf{Q}}$ iteratively.
\subsubsection{Optimize Beamformer}\label{Beam}
Firstly, we optimize ${\bf{F}}$ by fixed ${\bf{Q}}$, which is expressed as
\begin{align}
\label{beam}
\max \limits_{{\bf{F}}\succeq {\bf 0}} \ R, \ {\rm{s.t.}} \ (\ref{BTI}), \ {\rm Tr}({\bf{F}})\leq P, \ {\rm rank}({\bf{F}})=1.
\end{align}
In fact, to maximize $R$ in (\ref{beam}) is equivalently to first solve a power minimization (PM) problem and then take a bisection search over $R$ to obtain the optimal $R^*$ because the optimal value of PM problem is monotonically increasing with respect to $R$ \cite{Ma-14}. Thus, solving PM problem with different $R$ and using a bisection search over $R$, $R^{*}$ can be obtained.
For given a target SR $R>0$, PM problem is expressed as
\begin{align}
\label{beam2} \min \limits_{{\bf{F}}} \ {\rm Tr}({\bf F}), \ {\rm{s.t.}} \ (\ref{BTI}), \ {\rm rank}({\bf{F}})=1,\ {\bf{F}}\succeq {\bf 0}.
\end{align}
(\ref{beam2}) is equivalent to the following relaxed problem by dropping the constraint ${\rm{rank}}({\bf{F}})=1$.
\begin{subequations}\label{beam4}
\begin{align}
\min \limits_{{\bf{F}}\succeq {\bf 0},\zeta,\upsilon} &\ {\rm Tr}\left({\bf{F}}\right) \label{BTIa}\\
\notag
{\rm{s.t.}} &\ {\rm Tr}\left({\bf{A}}_{AE}\right) +\sqrt{2\varrho}\zeta+\varrho \upsilon+{\sigma}^2+ {\rm Tr}\left( \overline{\bf{G}}_{AE}{\bf{F}}\overline{\bf{G}}_{AE}^H{\bf{Q}}\right)\\
\notag
&-2^{-R}\left({\sigma}^2+
{\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)\right)\leq 0,\\
&\ \begin{Vmatrix}
{\rm vec}\left({\bf{A}}_{AE}\right)\\
2{\bf{a}}_{AE}
\end{Vmatrix}\leq \zeta, \ \upsilon{{\bf{I}}}-{\bf{A}}_{AE}\succeq {\bf{0}},\ \upsilon\geq 0, \label{BTId}
\end{align}
\end{subequations}
where $\upsilon$ and $\zeta$ are the slack variables. Since ${\bf{A}}_{AE}$ and ${\bf{a}}_{AE}$ are liner for ${\bf F}$, so
${\rm Tr}\left({\bf{A}}_{AE}\right)$, ${\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)$ and ${\rm Tr}\left( \overline{\bf{G}}_{AE}{\bf{F}}\overline{\bf{G}}_{AE}^H{\bf{Q}}\right)$ are liner constraints, $\begin{Vmatrix}
{\rm vec}\left({\bf{A}}_{AE}\right)\\
2{\bf{a}}_{AE}
\end{Vmatrix}$ is a second cone (SOC) constraint and $\upsilon{{\bf{I}}}-{\bf{A}}_{AE}\succeq {\bf{0}}$, ${\bf{F}}\succeq {\bf{0}}$ are linear matrix inequality (LMI) constraints. Therefore, it is a convex problem which can be solved \cite{Boyd-04}. Considering the rank-1 constraint, in the following proposition we prove that we can always obtain a rank-one optimal ${\bf{F}}$ if the problem is feasible.
Thus, the optimal ${\bf{f}}$ can be obtained by eigen-decomposition of ${\bf{F}}$
\begin{prop}
A rank-one solution in $(\ref{beam4})$ can always be obtained if $(\ref{beam4})$ is feasible.
\end{prop}
\begin{proof}
Please refer to Appendix F.
\end{proof}
The overall algorithm to maximize $R$ by optimizing ${\bf F}$ is summarized in Algorithm 1.
\begin{algorithm}[h]
\caption{Bisection Method To Solve Problem $(\ref{beam})$}
\begin{algorithmic}
\State 1. set $\epsilon=10^{-3}$, the upper $\overline{R}_u$ and lower bound $\overline{R}_l$;
\State 2. let $\overline{R}_{mid}=\frac{\overline{R}_u+\overline{R}_l}{2}$ and solve (\ref{beam4}) with $R=\overline{R}_{mid}$;
\State 3. if (\ref{beam4}) is feasible, check the power constraint ${\rm Tr}({\bf F})\leq P$, if power constraint satisfies, set $\overline{R}_l = \overline{R}_{mid}$ and go to step 5; otherwise, set $\overline{R}_u = \overline{R}_{mid}$ and go to step 2;
\State 4. if (\ref{beam4}) is infeasible, let $\overline{R}_u = \overline{R}_{mid}$ and go to step 2;
\State 5. if $\overline{R}_u-\overline{R}_l\leq \epsilon$, stop and $R^*=\overline{R}_l$; else go to step 2.
\end{algorithmic}
\end{algorithm}
\subsubsection{Optimize Phase Shift}\label{Phase}
The sub-problem of optimizing ${\bf Q}$ under fixed ${\bf{F}}$ is expressed as
\begin{align}
\label{shift}
\max \limits_{ {\bf{Q}}\succeq {\bf 0}} \ R,\
{\rm{s.t.}} \ (\ref{BTI}), \ {\rm Diag}\left({\bf{Q}}\right)= {\bf{1}}_M,\ {\rm rank}({\bf{Q}})=1.
\end{align}
The optimal value of (\ref{shift}) can be obtained by first solving a feasibility check problem and then using bisection search.
For given a target $R>0$, the feasibility check problem is
\begin{align}
\label{shift2}
\mathop{\rm{Find}}\limits_{{\bf Q}\succeq {\bf 0}}\ {\bf Q}, \ {\rm{s.t.}}\ (\ref{BTI}), \ {\rm Diag}\left({\bf{Q}}\right)= {\bf{1}}_M,\ {\rm rank}({\bf{Q}})=1.
\end{align}
Obviously, the feasible area is monotonically decreasing with respect to $R$. Thus, an iterative bisection search over $R$ can obtain $R^{*}$. To solve (\ref{shift2}), we first reformulate it as (\ref{shift3}) by dropping the rank one constraint.
\begin{subequations}\label{shift3}
\begin{align}
\mathop{\rm{Find}}\limits_{{\bf Q}\succeq {\bf 0}} &\ {\bf{Q}} \label{finda}\\
\notag
{\rm{s.t.}} \ &\ {\rm Tr}\left({\bf{A}}_{AE}\right) +\sqrt{2\varrho}\alpha+\varrho \beta+{\sigma}^2+ {\rm Tr}\left( \overline{\bf{G}}_{AE}{\bf{F}}\overline{\bf{G}}_{AE}^H{\bf{Q}}\right)\\
\notag
&-2^{-R}\left({\sigma}^2+
{\rm Tr}\left({{\bf{G}}}_{AB}{\bf{F}}{{\bf{G}}}_{AB}^H{\bf{Q}}\right)\right)\leq 0, \ {\rm Diag}\left({\bf{Q}}\right)= {\bf{1}}_M,\\
& \begin{Vmatrix}
{\rm vec}\left({\bf{A}}_{AE}\right)\\
2{\bf{a}}_{AE}
\end{Vmatrix}\leq \alpha, \ \beta{{\bf{I}}}-{\bf{A}}_{AE}\succeq {\bf{0}}, \ \beta\geq 0, \label{findd}
\end{align}
\end{subequations}
If dropping the rank-one constraint of ${\rm{rank}}({\bf{Q}})=1$, problem $(\ref{shift3})$ is convex with the same analysis of $(\ref{beam4})$. Therefore, we use SDR method to find a solution and then use sequential rank-one constraint relaxation algorithm to recover the rank-one constraint. In such a way, $\boldsymbol{\phi}$ is a KKT stationary solution
of (\ref{shift}) \cite{Cao-17}.
The algorithm is summarized in Algorithm 2.
\begin{algorithm}[h]
\caption{Bisection Method To Solve Problem $(\ref{shift})$}
\begin{algorithmic}
\State 1. set $\epsilon=10^{-3}$, the upper $\overline{R}_u$ and lower bound $\overline{R}_l$ of $R$;
\Repeat
\State 2. let $\overline{R}_{mid}=\frac{\overline{R}_u+\overline{R}_l}{2}$ and solve $(\ref{shift3})$ with $R=\overline{R}_{mid}$;
\State 3. if infeasible, let $\overline{R}_u = \overline{R}_{mid}$; otherwise, let $\overline{R}_l = \overline{R}_{mid}$;
\Until{$\overline{R}_u-\overline{R}_l\leq \epsilon$};
\State 4. output the optimal the $R^*$ and recover rank-one constraint to obtain the local optimal $\boldsymbol{\phi}^*$;
\end{algorithmic}
\end{algorithm}
\subsection{Algorithm Analysis}
The overall two-stage algorithm to solve the joint optimization problem (\ref{ PerfectLocation}) of IRS location, beamformer and reflection coefficient is summarized in Algorithm 3.
\begin{algorithm}[h]
\caption{Two-Stage Method To Solve Problem (\ref{ PerfectLocation})}
\begin{algorithmic}
\State 1. set the convergence precision $\epsilon=10^{-3}$, initial $\boldsymbol{\phi}^{(0)}$, $n=0$;
\State 2. solve problem (\ref{auxiliary_final}) to obtain the sub-optimal $\boldsymbol{\omega}_I$;
\Repeat
\State 3. solve $(\ref{beam})$ with Algorithm 1 to obtain ${\bf f}^{(n+1)}$ ;
\State 4. solve $(\ref{shift})$ with Algorithm 2 to obtain ${\boldsymbol{\phi}}^{(n+1)}$ and $R^{(n+1)}$;
\State 5. $n=n+1$;
\Until{$R^{(n+1)}-R^{(n)}\leq \epsilon$;}
\end{algorithmic}
\end{algorithm}
Based on the algorithm description, we provide a brief analysis on the convergence and complexity. For optimizing $\boldsymbol{\omega}_I$ in $(\ref{auxiliary})$, this problem only involve vector multiplication so the computational
complexity is $\mathcal{O}\left\{ n\right\}$. For given IRS location, by iteratively solving $(\ref{beam})$ with optimal and solving $(\ref{shift})$ with local optimal, the target SR $R$ can be monotonically increased with guaranteed convergence. Note that for given $\boldsymbol{\omega}_I$, $R\left({{\bf{f}}^{(n)}},\boldsymbol{\phi}^{(n)}\right)$ is a feasible solution for $(\ref{ NextPerfectLocation})$ in the $n$-th iteration. For the next iteration, we solve $(\ref{beam})$ optimally by fixed $\boldsymbol{\phi}^{(n)}$
so that $R\left({{\bf{f}}^{(n)}},\boldsymbol{\phi}^{(n)}\right)\leq R\left({{\bf{f}}^{(n+1)}},\boldsymbol{\phi}^{(n)}\right)$. Then, a local optimal solution
$\boldsymbol{\phi}^{(n+1)}$ can be obtained by solve $(\ref{shift})$ under fixed ${{\bf{f}}^{(n+1)}}$, so that
$R\left({{\bf{f}}^{(n+1)}},\boldsymbol{\phi}^{(n)}\right)\leq R\left({{\bf{f}}^{(n+1)}},\boldsymbol{\phi}^{(n+1)}\right)$. Therefore, the two iterative steps
follow $R\left({{\bf{f}}^{(n)}},\boldsymbol{\phi}^{(n)}\right)\leq R\left({{\bf{f}}^{(n+1)}},\boldsymbol{\phi}^{(n+1)}\right)$ thus guarantee the convergence.
Since both the resulting convex problem $(\ref{beam4})$ and $(\ref{shift3})$ involve two LMI, one SOC and there linear constraints that can be solved
by a standard interior point method, the general expression for computational complexity has been given in \cite{Zhou-20}. The computational
complexities of proposed method for (\ref{beam4}) and (\ref{shift3}) in per iteration are listed in Table I.
\begin{table}[h]
\centering
\caption{computational complexities}
\label{tab:Margin_settings}
\begin{tabular}{|l|c|}\hline
\multirow{2}{*}{(\ref{beam4})}&$\mathcal{O}\left\{\left(N_tM+N_t+2\right)^{1 / 2} N_t^2\left[N_t^{4}+N_t^2\left(N_t^2M^2+N_t^2\right)\right. \right.$\\
&$\left. \left. +N_t^3M^3+N_t^3+N_t^2\left(N_t^2M^2+N_tM\right)^2\right]\right\}$ \\ \hline
\multirow{2}{*}{(\ref{shift3})}&$\mathcal{O}\left\{\left(N_tM+M+2\right)^{1 / 2} M^2\left[M^4+M^2\left(N_t^2M^2+M^2\right)\right. \right.$ \\
&$\left. \left. +N_t^3M^3+M^3+M^2 \left(N_t^2M^2+N_tM\right)^2\right]\right\}$ \\ \hline
\end{tabular}
\end{table}
\section{ Problem Solution With Location Region Of Eve}
In this section, we aim to maximize SR without exact location of Eve but only a suspicious area where an Eve may exist. In this case, both large-scale and small-scale fadings of IRS-Eve link are unknown, which prevents the optimization of IRS location. To guarantee the security, we consider the worst case that the information leakage to Eve is maximum, which means we will maximize the minimum secrecy rate. The problem is expressed as
\begin{subequations}\label{worst_Eve}
\begin{align}
\max \limits_{{\bf{f}},\boldsymbol{\phi},\boldsymbol{\omega}_I} \min \limits_{\boldsymbol{\omega}_E}&\ R \label{worst_Evea}\\
{\rm{s.t.}} &\ {\rm{Pr}}\left\{R_s(\boldsymbol{\omega}_I,\boldsymbol{\omega}_E,{\bf{f}},\boldsymbol{\phi})\leq R\right\} \leq p_{out},\label{worst_Eveb}\\
\notag
&\ \|{\bf{f}}\|^2 \leq P, |\boldsymbol{\phi}_{i}|=1,\ i=1,\cdots,M,\\
& \boldsymbol{\omega}_I\in \Omega_I, \ \boldsymbol{\omega}_E\in \Omega_E.\label{worst_Evec}
\end{align}
\end{subequations}
Since $\boldsymbol{\omega}_I$ is relevant to $\boldsymbol{\omega}_E$, if $\boldsymbol{\omega}_E$ is taken as an optimization variable, $\boldsymbol{\omega}_I$ is also an optimization variable, so the instantaneous CSI NLoS components ${\bf{H}}_{AI}^{NLoS}, {\bf{h}}_{IB}^{NLoS}, {\bf{h}}_{IE}^{NLoS}$ are unavailable for Alice, which should be taken as random variables. In this case the CSI model is expressed as (3). However, once $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$ are fixed and
for optimizing ${\bf f}$ and $\boldsymbol{\phi}$, only ${\bf{h}}_{IE}^{NLoS}$ are unavailable and the CSI model now is expressed as (4). Therefore, the problem is similar to the previous problem (\ref{ PerfectLocation}) and the AO method could not be applied directly to solve this problem.
Hence, we still utilize the two-stage method to solve problem (\ref{worst_Eve}), where it jointly optimizes $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$ to obtain $\boldsymbol{\omega}_I$ and worst $\boldsymbol{\omega}_E$ in the first stage rather than optimize them in two independently subproblems, and then to optimize ${\bf f}$ and $\boldsymbol{\phi}$ to maximize $R$ by fixed IRS and Eve at these locations. Therefore, the objective function could be rewritten as:
\begin{align}
\max \limits_{{\bf{f}},\boldsymbol{\phi}} \left( \max_{\boldsymbol{\omega}_I} \min \limits_{\boldsymbol{\omega}_E} R \right),
\end{align}
where in the first stage the sub-optimization problem in the bracket is solved, where both the worst location of Eve and IRS deployment location are obtained, and in the second stage the out bracket subproblem is solved, where the optimal beamformer and IRS phase shifts are obtained.
With the same analysis in Sec. \ref{Two-Stage}, in the first stage, we still transform the outage constraints (\ref{worst_Eveb}) with its upper bound based on proposition 3 and 4 to make it only related to $\left(\boldsymbol{\omega}_I,\boldsymbol{\omega}_E\right)$ but not to ${\bf{f}}$ and $\boldsymbol{\phi}$. We denote $\gamma_E \triangleq\frac{\kappa^2 M^2 N_tP}{\left(\kappa+1\right)^2}+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}^{small}{\bf{f}}|^2+\frac{2\kappa M \sqrt{N_tP}}{\kappa+1}|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}^{small}{\bf{f}}|$ and $\gamma_B\triangleq \frac{\kappa^2 M^2 N_tP}{\left(\kappa+1\right)^2}+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}^{small}{\bf{f}}|^2-\frac{2\kappa M \sqrt{N_tP}}{\kappa+1}|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}^{small}{\bf{f}}|$ in proposition 3 and 4, and denote cumulative distribution functions of $\gamma_E$ and $\gamma_B$ as as $F_E(x)$ and $F_B(x)$, which are determinate. Then, by denoting $\alpha_E \triangleq F^{-1}_E(1-p_{out})$ and $\alpha_B \triangleq F^{-1}_B(p_{out})$, where $F^{-1}$ is the inverse cumulative distribution function, (\ref{worst_Eveb}) is transformed as $\alpha_E\leq \frac{\left(2^{R_E}-1\right)\sigma^2}{L_{AI}L_{IE}}$ and $\alpha_B\geq \frac{\left(2^{R_B}-1\right)\sigma^2}{L_{AI}L_{IB}}$, so $\left(\boldsymbol{\omega}_I,\boldsymbol{\omega}_E\right)$ only impact $L_{AI}, L_{IB}, L_{IE}$. Then, based on proposition 5, the problem for optimizing $\left(\boldsymbol{\omega}_I,\boldsymbol{\omega}_E\right)$ can be transformed as
\begin{align}
\label{NL_worst}
\max \limits_{\boldsymbol{\omega}_I} \min \limits_{\boldsymbol{\omega}_E} \ \frac{\sigma^2+\alpha_BL_{AI}L_{IB}}{\sigma^2+\alpha_EL_{AI}L_{IE}}, \ {\rm{s.t.}} \ \boldsymbol{\omega}_I\in {\Omega}_I,\ \boldsymbol{\omega}_E\in \Omega_E.
\end{align}
where $L_{AI}, L_{IB}$ are functions of $\boldsymbol{\omega}_I$, and $L_{IE}$ is a function of both $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$.
Since $L_{AI}, L_{IB}$ are only related to $\boldsymbol{\omega}_I$, but $L_{IE}$ in numerator is related to both $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$, $\boldsymbol{\omega}_I$ is interacted with $\boldsymbol{\omega}_E$, so this Max-Min problem is hard to be solved directly. Since this problem is a maximum problem for $\boldsymbol{\omega}_I$, but a minimum problem for $\boldsymbol{\omega}_E$, the monotonicity cannot be guaranteed by exploited the AO algorithm. To solve it, we transform the original Max-Min problem into several parallel Min-problems by given different $\boldsymbol{\omega}_I$ and then global search for the maximum value over $\boldsymbol{\omega}_I$.
The Min-problem for optimized $\boldsymbol{\omega}_E$ by given any $\boldsymbol{\omega}_I$ is expressed as
\begin{align}
\label{Eve_location}
\min \limits_{\boldsymbol{\omega}_E} \ \frac{\sigma^2+\alpha_BL_{AI}L_{IB}}{\sigma^2+\alpha_EL_{AI}L_{IE}}, \ {\rm{s.t.}} \ \boldsymbol{\
|
omega}_E\in \Omega_E.
\end{align}
where $\boldsymbol{\omega}_E$ only impacts $L_{IE}$ but does not impact $L_{AI}, L_{IB}$. Hence, we only optimize $\boldsymbol{\omega}_E$ to maximize $L_{IE}$ and this problem is convex which can be solved directly.
After obtaining $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$, in the second stage, we optimize $\left({\bf{f}},\boldsymbol{\phi}\right)$ to maximize SR subject to all the constraints with the same method as in Sec. \ref{BeamPhase}.
\section{Simulation Results}
To validate the performance of proposed two stage algorithms, extensive simulation results have been carried out in this section. The system parameters are listed in Table II. Note that all the simulation results illustrated except Fig.2 are averaged over 100 randomly generated channels.
\begin{table}[h]
\centering
\caption{simulation parameter}
\label{tab:Margin_settings}
\begin{tabular}{|l|c|}\hline
Carrier center frequency & 2.4GHz\\ \hline
Path loss exponents & $\rho_{AI}=2.2$, $\rho_{IE}=\rho_{IB}=3$\\ \hline
Noise power at Bob and Eve & $\sigma_b^2 =\sigma_e^2= -95$ dBm\\ \hline
Outage probability & $p_{out}=0.05$ \\ \hline
Number of transmit antenna & $N_t=4$ \\ \hline
\multirow{2}{*}{ Location of Bob and Eve} & $\boldsymbol{\omega}_B=(100,15)$, \\
& $\boldsymbol{\omega}_E=(95,13)$ \\ \hline
\multirow{2}{*}{ Location area of IRS} & $x_I\in \Delta_{x_I}=\left[0, 105\right]$, \\
& $ y_I \in \Delta_{y_I}=\left[20,30\right]$ \\ \hline
\end{tabular}
\end{table}
We demonstrate the advantage of the proposed algorithm by comparing its performance with the following three benchmark schemes: 1) Random Location: $\boldsymbol{\omega}_I$ is randomly selected and ${\bf{f}}$ and $\boldsymbol{\phi}$ are optimized by solving $(\ref{beam})$ and $(\ref{shift})$; 2) Global Search Method: $\boldsymbol{\omega}_I$ is optimized by global search solving problem (\ref{max_location}); 3) MRT Method: $\boldsymbol{\omega}_I$ is optimized by proposed method, but ${\bf{f}}$ and $\boldsymbol{\phi}$ are only designed as ${\bf{f}}= \sqrt{\frac{1}{N_t}}{\boldsymbol{\alpha}}_A(\varphi_{AI})$ and $\boldsymbol{\phi}={\rm diag}\left( \boldsymbol{\alpha}_I(\varphi_{IB})\right)\boldsymbol{\alpha}_{I}\left(\theta_{AI}\right)$; 4) Gaussian Random method: We use global search method to optimize the location of IRS, then use the SDR+Guassian Random method to solve the two subproblems for optimizing ${\bf f}$ and $\boldsymbol{\phi}$.
\begin{figure}
\centering
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=3.2in]{fig2.eps}
\caption{Convergence of the proposed two stage scheme with different values of $M$. We set $P=30$dBm, $\kappa=2$.}
\end{minipage}
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=3.2in]{fig3.eps}
\caption{Secrecy rate versus number of IRS with different values of $\kappa$. We set $P=30$dBm.}
\end{minipage}
\end{figure}
\begin{figure*}
\centering
\subfigure[ With the location of Eve $\boldsymbol{\omega}_{E}=(95,13)$. ]{\includegraphics[width=3.2in]{fig4a.eps}}
\subfigure[ With the suspicious area of Eve $x_E \in \left(50, 98\right), y_E\in\left(5,13\right)$.]{\includegraphics[width=3.2in]{fig4b.eps}}
\caption{Secrecy rate versus the transmit power with different values of $M$. We set $\kappa=2$.}
\end{figure*}
\begin{figure*}
\centering
\subfigure[ $\boldsymbol{\omega}_I$ versus $\boldsymbol{\omega}_E$ via fixed $\boldsymbol{\omega}_B=(100,15)$ m.]{\includegraphics[width=3.2in]{fig5a.eps}}
\subfigure[ $\boldsymbol{\omega}_I$ versus $\boldsymbol{\omega}_B$ via fixed $\boldsymbol{\omega}_{E}=(95,13)$ m.]{\includegraphics[width=3.2in]{fig5b.eps}}
\caption{The location of IRS versus location of Eve and Bob with two schemes. We set $P=30$dBm, $\kappa=2$.}
\end{figure*}
\begin{figure*}
\centering
\subfigure[ The location of IRS and Eve with suspicious areas of Eve.]{\includegraphics[width=3.2in]{fig6a.eps}}
\subfigure[ The secrecy rate with different suspicious areas of Eve.]{\includegraphics[width=3.2in]{fig6b.eps}}
\caption{The location of IRS and Eve and secrecy rate with different suspicious areas of Eve via fixed $\boldsymbol{\omega}_B=(100,15)$ m.}
\end{figure*}
The convergence of the proposed two stage method with different numbers of IRS is investigated in Fig.2. It is seen that the proposed algorithm converges for different element of IRS. With increasing $M$, the dimensions of optimization variables $\boldsymbol{\phi}$ increase, resulting in the computation time increasing.
Fig.3 shows the SR versus number of IRS with different values of $\kappa$. With increased IRS elements, more additional reflecting power can be applied to
transmit signal, thus increasing SR. In addition, we find that SR decreases with decreased $\kappa$. This is because the randomness of NLoS component of Eve link increases with decreased $\kappa$, thus increasing information leakage to Eve.
In Fig.4, we show the maximum SR versus the transmit power with different number of IRS and with different schemes in two cases. As observed from Fig.4(a) and Fig.4(b), we find that the maximum SR increases monotonically with the transmit power and elements of IRS. This is because larger SR is required more transmit power and more additional reflecting power can be applied to transmit signal with increased IRS elements. Compared to the random location scheme, our scheme can significantly enhance SR. This is due to the fact that if the location of IRS is randomly chosen, the path loss of both Eve and Bob may be large and LoS components of Eve and Bob may be more similar, so it is hard to guarantee security. Therefore, an optimized IRS location not only reduce the path loss, but enhance the superiority of legitimate channels. In addition, from Fig.4(a), we find that the proposed scheme with low complexity and the global search method with high complexity have the same performance.
Fig.5 depicts the location of IRS $\boldsymbol{\omega}_I$ versus the location of Eve $\boldsymbol{\omega}_E$ and Bob $\boldsymbol{\omega}_B$. In Fig.5(a) and Fig.5(b), we show the $\boldsymbol{\omega}_I$ optimized by our proposed method and use global search method as benchmarks under four different $\boldsymbol{\omega}_E$ by fixed $\boldsymbol{\omega}_B=(100,15)$ and under six different $\boldsymbol{\omega}_B$ by fixed $\boldsymbol{\omega}_E=(95,13)$. From these two figures, we find that the location of IRS optimized by proposed method is very close to that optimized by global search method. Moreover, the figures show that IRS should be deployed as close to Bob as possible wherever Eve is located. This is because that when IRS is deployed closely to Bob, the large-scale path loss of Alice-IRS-Bob link is small and the quality of legitimate channel is enhanced significantly.
Fig.6 shows the locations of IRS and Eve and secrecy rate versus suspicious areas of Eve. Fig.6(a) shows the $\boldsymbol{\omega}_I$ and $\boldsymbol{\omega}_E$ optimized by SCA and global search method under four different suspicious areas of Eve by fixed $\boldsymbol{\omega}_B=(100,15)$ m. We find that the worst $\boldsymbol{\omega}_E$ is the location closest to IRS in each the suspicious area of Eve. This is due to the fact that when Eve is closest to IRS, the path loss of IRS-Eve is minimum and information leakage to Eve is maximum. In Fig.6(b), by fixed $M=5$, we find that when suspicious area of Eve is closer to Bob, the SR is smaller. This is because that when the suspicious area of Eve is closer to Bob, the worst $\boldsymbol{\omega}_E$ is closer to Bob and IRS, thus increasing the information leakage to Eve and decreasing the SR.
\section{Conclusion}
In this paper, we investigated the robust secrecy transmission in the IRS-aided multiple antennas wireless communications. For the first time the location optimization was considered in this work and we aim to maximize the secrecy rate by optimizing the location of IRS, transmit beamformer at Alice and phase shifts at IRS under two
different cases with the location of Eve or not. We show the joint optimization problem could be solved thought a two-stage optimization framework. In the first stage, IRS location could be optimized via exploiting successive convex approximation method. In the second stage, an AO algorithm is proposed to optimize beamformer and phase shifts iteratively.
Similar idea has also been developed to solve the case where only a suspicious area of Eve is known. Simulation results have verified the effectively of the proposed algorithm and shown the importance of IRS location optimization for enhancing secrecy performance.
\section{Appendix}
\subsection{ Proof of Proposition 1}
First, for ${\rm diag}\left(\overline{ \bf{h}}_{IJ} \right)\widetilde{\bf{H}}_{AI}$ in (\ref{AllError}), each element of $\widetilde{\bf{H}}_{AI}$ follows $\mathcal{CN}\left({ 0},\frac{L_{AI}}{\kappa+1}\right)$ and each element of $\overline{\bf{h}}_{IJ}$ is determinate whose modulus is $\frac{\kappa L_{IJ}}{\kappa+1}$, so each element of ${\rm diag}\left(\overline{ \bf{h}}_{IJ} \right)\widetilde{\bf{H}}_{AI}$ follows $ \mathcal{CN}\left({ 0},\frac{\kappa L_{IJ}L_{AI}}{\left(\kappa+1\right)^2}\right)$. Then, similarly to ${\rm diag}\left(\widetilde{\bf{h}}_{IJ}\right)\overline{\bf{H}}_{AI}$, $\widetilde{\bf{h}}_{IJ}\sim \mathcal{CN}\left({\bf 0},\frac{L_{IJ}}{\kappa+1}{\bf I}_M\right)$ and each element of $\overline{\bf{H}}_{AI}$ is determinate with constant modulus $\frac{\kappa L_{AI}}{\kappa+1}$, so each element of ${\rm diag}\left(\widetilde{\bf{h}}_{IJ}\right)\overline{\bf{H}}_{AI}$ approximately follows $ \mathcal{CN}\left({0},\frac{\kappa L_{IJ}L_{AI}}{\left(\kappa+1\right)^2}{}\right)$. Hence, each element of $\widetilde{\bf{G}}_{AJ}$ follows $\mathcal{CN}\left({0},\frac{2\kappa L_{AI}L_{IJ}}{(\kappa+1)^2}\right)$.
\subsection{ Proof of Proposition 2}
First, according to (\ref{rate}), we re-write $\boldsymbol{\phi}^H \widetilde{\bf{G}}_{AJ} {\bf{f}}=\widetilde{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\overline{\bf{H}}_{AI}{\bf{f}} + \overline{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\widetilde{\bf{H}}_{AI}{\bf{f}}$. Based on proposition 1, we have $\widetilde{\bf{h}}_{IJ}\sim \mathcal{CN}\left({\bf 0},\frac{L_{IJ}}{\kappa+1}{\bf I}_{M}\right)$ and the each element of $\widetilde{\bf{H}}_{AI}$ follows $\mathcal{CN}\left({ 0},\frac{L_{AI}}{\kappa+1}\right)$. For any unitary matrices $\boldsymbol{\Phi}$, we have $
\widetilde{\bf h}_{IJ}^H\boldsymbol{\Phi} \sim \mathcal{CN}\left({\bf 0},\frac{L_{IJ}}{\kappa+1}{{\bf I}_{M}}\right)$ and each element of $\ \boldsymbol{\Phi}\widetilde{\bf H}_{AI}$ also follows $ \mathcal{CN}\left({0},\frac{L_{AI}}{\kappa+1}{}\right),$
so that $\widetilde{\bf h}_{IJ}^H\boldsymbol{\Phi}$ and $\boldsymbol{\Phi}\widetilde{\bf H}_{AI}$ have the same distributions with $\widetilde{\bf h}_{IJ}^H$ and $\widetilde{\bf H}_{AI}$. Then, we have $\widetilde{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\overline{\bf{H}}_{AI} \sim \overline{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\widetilde{\bf{H}}_{AI} \sim \mathcal{CN}\left({\bf 0},\frac{\kappa L_{IJ}L_{AI}M}{\left(\kappa+1\right)^2}{\bf I}_{N_t}\right)$. For brief, we denote ${\bf x}=\widetilde{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\overline{\bf{H}}_{AI}$. For any ${\bf f}$,
$\widetilde{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\overline{\bf{H}}_{AI}{\bf{f}}=\sum \limits_{i=1}^{N_t} x_if_i$, where $x_i\sim \mathcal{CN}\left({ 0},\frac{\kappa L_{IJ}L_{AI}M}{\left(\kappa+1\right)^2}\right)$, so $\sum \limits_{i=1}^{N_t} x_if_i \sim \mathcal{CN}\left({0},\sum\limits_{i=1}^{N_t}\frac{\kappa L_{IJ}L_{AI}M|f_i|^2}{\left(\kappa+1\right)^2}\right)$. Since $\| {\bf f}\|^2=P$, we have $\widetilde{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\overline{\bf{H}}_{AI}{\bf{f}}\sim \mathcal{CN}\left({ 0},\frac{\kappa L_{IJ}L_{AI}MP}{\left(\kappa+1\right)^2}\right)$. Similarly, $\overline{\bf{h}}_{IJ}^H \boldsymbol{\Phi}\widetilde{\bf{H}}_{AI}{\bf{f}}$ follows $\mathcal{CN}\left({ 0},\frac{\kappa L_{IJ}L_{AI}MP}{\left(\kappa+1\right)^2}\right)$. Hence, for any $\boldsymbol{\phi}$ and ${\bf f}$, $\boldsymbol{\phi}^H \widetilde{\bf{G}}_{AJ}{\bf{f}}\sim \mathcal{CN}\left({ 0},\frac{2\kappa L_{IJ}L_{AI}MP}{\left(\kappa+1\right)^2}\right)$.
\subsection{ Proof of Proposition 3}
For the left constraint of (\ref{out}), according to the triangle inequality, we find
\begin{align}
\notag
&\big |\boldsymbol{\phi}^H \left(\overline{\bf{G}}_{AE}+\widetilde{\bf{G}}_{AE} \right){\bf{f}}\big |^2 \\
\notag
&\leq |\boldsymbol{\phi}^H\overline{\bf{G}}_{AE} {\bf{f}}|^2+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}{\bf{f}}|^2+2|\boldsymbol{\phi}^H\overline{\bf{G}}_{AE} {\bf{f}}||\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}{\bf{f}}|\\
\notag
&\overset{(d)}\leq L_{AI}L_{IE}\left(\frac{\kappa M\sqrt{N_tP}}{\kappa+1}+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}^{small}{\bf{f}}|\right)^2,
\end{align}
where $\widetilde{\bf{G}}_{AE}^{small}\triangleq\frac{\sqrt{\kappa}}{\kappa+1}\left({\rm diag}\left({\bf{h}}_{IE}^{NLoS}\right){\bf{H}}_{AI}^{LoS} + {\rm diag}\left({\bf{h}}_{IE}^{LoS}\right){\bf{H}}_{AI}^{NLoS}\right)\sim \mathcal{CN}\left({ 0},\frac{2\kappa M}{\left(\kappa+1\right)^2}\right)$ based on proposition 1 and any choice of $\boldsymbol{\phi}$ and ${\bf{f}}$ does not impact the distribution of $\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}^{small}{\bf{f}}\sim \mathcal{CN}\left({ 0},\frac{2\kappa MP}{\left(\kappa+1\right)^2}\right)$ based on proposition 2. $(d)$ is due to $|\boldsymbol{\phi}^H\overline{\bf{G}}_{AE} {\bf{f}}|^2\leq \|\boldsymbol{\phi}^H\|^2\|\overline{\bf{G}}_{AE}\|^2_F \|{\bf{f}}\|^2= \frac{\kappa^2 L_{AI}L_{IE} M^2 N_tP}{\left(\kappa+1\right)^2}$, where $\|\boldsymbol{\phi}^H\|^2=M$, $\|{\bf{f}}\|^2=P$ and each element of $\overline{ \bf{G}}_{AE}$ has constant modulus $\frac{\kappa \sqrt{L_{AI}L_{IE}}}{\kappa+1}$ analyzed in (\ref{AllError}) so $\|\overline{\bf{G}}_{AE}\|^2_F=\frac{\kappa^2 L_{AI}L_{IE} M N_t}{\left(\kappa+1\right)^2}$. When ${\bf{f}}$ and $\boldsymbol{\phi}$ are adopted to the LoS components in Alice-IRS-Eve channel link $\overline{\bf{G}}_{AE}$, the equation $|\boldsymbol{\phi}^H\overline{\bf{G}}_{AE} {\bf{f}}|^2= \frac{\kappa^2 L_{AI}L_{IE} M^2 N_tP}{\left(\kappa+1\right)^2}$ in $(b)$ holds. It implies that the information leakage for Eve is maximum.
Then, we have the upper bound as
\begin{align}
\notag
&{\rm{Pr}}\left\{C_E(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi})\geq R_E \right\}\\
\notag
&\leq {\rm{Pr}}\left\{ \left(\frac{\kappa M \sqrt{N_tP}}{\kappa+1}+|\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AE}^{small}{\bf{f}}|\right)^2\geq \frac{\left(2^{R_E}-1\right)\sigma^2}{L_{AI}L_{IE}} \right\}.
\end{align}
This constraint is irrelevant to $\boldsymbol{\phi}$ and ${\bf{f}}$.
\subsection{ Proof of Proposition 4}
For the right constraint of (\ref{out}), with the triangle inequality, we obtain
\begin{align}
\notag
&\big |{\boldsymbol{\phi}}^H \left(\overline{\bf{G}}_{AB}+\widetilde{\bf{G}}_{AB} \right){\bf{f}}\big|^2 \\
\notag
&\geq |{\boldsymbol{\phi}}^H\overline{\bf{G}}_{AB}
{\bf{f}}|^2+|{\boldsymbol{\phi}}^H\widetilde{\bf{G}}_{AB}{\bf{f}}|^2-2|{\boldsymbol{\phi}}^H\overline{\bf{G}}_{AB} {\bf{f}}||{\boldsymbol{\phi}}^H\widetilde{\bf{G}}_{AB}{\bf{f}}|,
\end{align}
Then, we transform the constraint ${\rm{Pr}}\left\{C_B(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi}) \leq R_B\right\}$ with its upper bound as
\begin{align}
\notag
&{\rm{Pr}}\left\{C_B(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi})\leq R_B \right\}\\
\notag
&\leq {\rm{Pr}}\left\{\log\left(1+\frac{\left(|{\boldsymbol{\phi}}^H\overline{\bf{G}}_{AB} {\bf{f}}|-|{\boldsymbol{\phi}}^H\widetilde{\bf{G}}_{AB}{\bf{f}}|\right)^2 }{{\sigma}^2}\right)
\leq R_B \right\}.
\end{align}
Although the choice of ${\bf{f}}$ and $\boldsymbol{\phi}$ will not change the distribution of $\boldsymbol{\phi}^H \widetilde{\bf{G}}_{AB}{\bf{f}}$ based on proposition 2, but ${\bf{f}}$ and $\boldsymbol{\phi}$ impact the value of $\boldsymbol{\phi}^H \overline{\bf{G}}_{AB}{\bf{f}}$, so ${\bf{f}}$ and $\boldsymbol{\phi}$ impact this constraint together.
The main challenge is still how to adjust ${\bf{f}}$ and $\boldsymbol{\phi}$. Fortunately, the existing work \cite{Yan-16} implies that for a Rician fading, if ${\bf{f}}$ and $\boldsymbol{\phi}$ will not impact the communication quality of Eve, they should be adopted to the LoS component in Alice-IRS-Bob link for achieving the lowest outage probability. This is because that the outage probability is decreasing with Rician factor. When ${\bf{f}}$ and $\boldsymbol{\phi}$ are adopted to the LoS component in Alice-IRS-Bob link, the power of the deterministic component is maximum, so the Rician factor is maximum.
Therefore, if we can prove that ${\boldsymbol{\phi}}^H \left(\overline{\bf{G}}_{AB}+\widetilde{\bf{G}}_{AB} \right){\bf{f}}$ follows Rician fading and any ${\bf{f}}$ and $\boldsymbol{\phi}$ will not impact SNR of Eve, ${\bf{f}}$ and $\boldsymbol{\phi}$ should adopt to the LoS component $\overline{\bf{G}}_{AB}$ to obtain the lowest outage probability.
In fact, ${\boldsymbol{\phi}}^H \left(\overline{\bf{G}}_{AB}+\widetilde{\bf{G}}_{AB} \right){\bf{f}}$ is a Rician channel based on the analysis of (\ref{rate}) with determinate component $\boldsymbol{\phi}^H\overline{\bf{G}}_{AB}{\bf{f}}$ and random component $\boldsymbol{\phi}^H\widetilde{\bf{G}}_{AB}{\bf{f}}$. And based on proposition 4, the upper bound of ${\rm{Pr}}\left\{C_E(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi}) \geq R_E\right\}$ is irrelevant to $\boldsymbol{\phi}$ and ${\bf{f}}$.
Hence, by adjusting ${\bf{f}}^*= \sqrt{\frac{P}{N_t}}{\boldsymbol{\alpha}}_A(\varphi_{AI})$ and $\boldsymbol{\phi}^*={\rm diag}\left(
\boldsymbol{\alpha}_I(\varphi_{IB})\right)\boldsymbol{\alpha}_{I}\left(\theta_{AI}\right)$, the lowest outage probability is achieved. By setting the optimal ${\bf{f}}^*$ and ${\boldsymbol{\phi}}^*$, we obtain
\begin{align}
\notag
&{\rm{Pr}}\left\{\log\left(1+\frac{\left(|{\boldsymbol{\phi}}^H\overline{\bf{G}}_{AB} {\bf{f}}|-|{\boldsymbol{\phi}}^H\widetilde{\bf{G}}_{AB}{\bf{f}}|\right)^2 }{{\sigma}^2}\right)
\leq R_B \right\}\\
\label{PB}
&={\rm{Pr}}\left\{ \left( \frac{\kappa M\sqrt{N_tP}}{\kappa+1}-|\boldsymbol{\phi}^{H*}\widetilde{\bf{G}}_{AB}^{small}{\bf{f}^*}| \right)^2\leq \frac{\left(2^{R_B}-1\right)\sigma^2}{L_{AI}L_{IB}} \right\}.
\end{align}
where $\widetilde{\bf{G}}_{AB}^{small}\triangleq\frac{\sqrt{\kappa}}{\kappa+1}\left({\rm diag}\left({\bf{h}}_{IB}^{NLoS}\right){\bf{H}}_{AI}^{LoS} + {\rm diag}\left({\bf{h}}_{IB}^{LoS}\right){\bf{H}}_{AI}^{NLoS}\right) \sim \mathcal{CN}\left({ 0},\frac{2\kappa M}{\left(\kappa+1\right)^2}\right)$ based on proposition 1. Since the choice of ${\bf{f}}$ and $\boldsymbol{\phi}$ will not change distribution of $\boldsymbol{\phi}^H \widetilde{\bf{G}}_{AB}^{small}{\bf{f}}$, so ${\boldsymbol{\phi}^H}^* \widetilde{\bf{G}}_{AB}^{small}{\bf{f}}^*$ still follows $\mathcal{CN}\left({ 0},\frac{2\kappa MP}{\left(\kappa+1\right)^2}\right)$ based on proposition 2.
After transforming ${\rm{Pr}}\left\{C_B(\boldsymbol{\omega}_I,{\bf{f}},\boldsymbol{\phi}) \leq R_B\right\}$ with (\ref{PB}), this constraint is irrelevant to ${\bf{f}}$ and ${\boldsymbol{\phi}}$.
\subsection{Proof of Proposition 5}
First, we transform the two constraints $\alpha_E\leq \frac{\left(2^{R_E}-1\right)\sigma^2}{L_{AI}L_{IE}}$ and $\alpha_B\geq \frac{\left(2^{R_B}-1\right)\sigma^2}{L_{AI}L_{IB}}$ as $R_E\geq \log\left( 1+\frac{\alpha_EL_{AI}L_{IE}}{\sigma^2}\right)$
and $R_B\leq \log\left( 1+\frac{\alpha_BL_{AI}L_{IB}}{\sigma^2}\right)$. Then, (\ref{AO_method1}) is transformed as
\begin{align}
\notag
&\max \limits_{\boldsymbol{\omega}_I \in \Omega_{I},R_B,R_E} \ R_B-R_E, \\
\notag
&\ s.t. \ R_E\geq \log\left( 1+\frac{\alpha_EL_{AI}L_{IE}}{\sigma^2}\right),\ R_B\leq \log\left( 1+\frac{\alpha_BL_{AI}L_{IB}}{\sigma^2}\right).
\end{align}
To maximize $R_B-R_E$ is equivalent to maximize $\log\left( 1+\frac{\alpha_BL_{AI}L_{IB}}{\sigma^2}\right)-\log\left( 1+\frac{\alpha_EL_{AI}L_{IE}}{\sigma^2}\right)$. Hence, the above problem is equivalently transformed as
\begin{align}
&\max \limits_{\boldsymbol{\omega}_I\in {\Omega}_I} \ \frac{\sigma^2+\alpha_BL_{AI}L_{IB}}{\sigma^2+\alpha_EL_{AI}L_{IE}}.
\end{align}
\subsection{ Proof of Proposition 6}
For briefly, we denote ${\bf{h}}_b\triangleq\left({\bf{h}}_{IB}^H\boldsymbol{\Phi}{{\bf{H}}}_{AI}\right)^H\in \mathbb{C}^{M\times 1}$ and ${\bf{h}}_e \triangleq \left({\bf{h}}_{IE}^H\boldsymbol{\Phi}
{{\bf{H}}}_{AI}\right)^H\in \mathbb{C}^{M\times 1}$. Then, we assume that ${\bf{F}}^*$ is the optimal solution of $(\ref{beam2})$ and construct a new solution $\widetilde{\bf{F}}^*\triangleq{\bf{F}}^{*\frac{1}{2}}{\bf{P}}{\bf{F}}^{*\frac{1}{2}}$, where ${\bf{P}}\triangleq\frac{{\bf{F}}^{*\frac{1}{2}}
{\bf{h}}_b{\bf{h}}_b^H{\bf{F}}^{*\frac{1}{2}}}{\left\|{\bf{F}}^{*\frac{1}{2}} {\bf{h}}_b\right\|^2}$ is the projection matrix.
Obviously, $\widetilde{\bf{F}}^*$ is a rank-one matrix. From the value of function ${\bf{F}}^*-\widetilde{\bf{F}}^*={\bf{F}}^{*\frac{1}{2}}\left(
{\bf{I}}-{\bf{P}}\right){\bf{F}}^{*\frac{1}{2}}\succeq {\bf{0}}$, we find ${\rm Tr}\left( \widetilde{\bf{F}}^* \right)\leqslant {\rm Tr}\left({\bf{F}}^* \right)$,
which means that the objective value of $(\ref{beam2})$ obtained by $\widetilde{\bf{F}}^*$ is not worse than that obtained by ${\bf{F}}^*$. Finally, we check whether $\widetilde{\bf{F}}^*$ is satisfying constraint (\ref{BTI}). Since it is computationally
intractable to check whether $\widetilde{\bf{F}}^*$ satisfies the constraint (\ref{BTI}) directly, we instead consider the equivalent constraint (\ref{AOb}), which can be equivalently reformulated as
\begin{align}
{\rm{Pr}}\left\{ \underbrace{\log\left(1+\frac{{\bf{h}}_b^H{\bf{F}}{\bf{h}}_b}{{\sigma}^2}\right)}_{t_1}- \underbrace{\log\left(1+\frac{{\bf{h}}_e^H{\bf{F}}{\bf{h}}_e}{{\sigma}^2}\right)}_{t_2} \geq R \right\} \geq 1- p_{out}.
\end{align}
By substituting $\widetilde{\bf{F}}^*$ into $t_1$, we have
${\bf{h}}_b^H\widetilde{\bf{F}}^*{\bf{h}}_b={\bf{h}}_b^H {\bf{F}}^{*\frac{1}{2}}{\bf{P}}{\bf{F}}^{*\frac{1}{2}} {\bf{h}}_b=\frac{{\bf{h}}_b^H {\bf{F}}^{*\frac{1}{2}}{\bf{F}}^{*\frac{1}{2}}{\bf{h}}_b{\bf{h}}_b^H{\bf{F}}^{*\frac{1}{2}}{\bf{F}}^{*\frac{1}{2}} {\bf{h}}_b}{ {\bf{h}}_b^H {\bf{F}}^{*\frac{1}{2}} {\bf{F}}^{*\frac{1}{2}} {\bf{h}}_b}={\bf{h}}_b^H {\bf{F}}^* {\bf{h}}_b$. Hence, the value of $t_1$ is the same by replacing ${\bf{F}}^*$ with $\widetilde{\bf{F}}^*$. Moreover, we have
${\bf{h}}_e^H{\bf{F}}^*{\bf{h}}_e-{\bf{h}}_e^H\widetilde{\bf{F}}^*{\bf{h}}_e= {\bf{h}}_e^H\left({\bf{F}}^*-{\bf{F}}^{*\frac{1}{2}}{\bf{P}}{\bf{F}}^{*\frac{1}{2}}\right){\bf{h}}_e = {\bf{h}}_e^H\left( {\bf{F}}^{*\frac{1}{2}}\left({\bf{I}}-{\bf{P}}\right){\bf{F}}^{*\frac{1}{2}}\right){\bf{h}}_e\geqslant 0$. Thus, the value of $t_2$ is not increase by substituting $\widetilde{\bf{F}}^*$ into it. Therefore, $\widetilde{\bf{F}}^*$ still satisfies $(\ref{AOb})$ and then satisfies (\ref{BTI}). Hence,
we must obtain a rank-one solution of $(\ref{beam2})$ if the problem is feasible.
|
\section{Introduction}
\label{sec:introduction}
Particle swarm optimization (PSO) is a metaheuristic optimization approach that imitates the dynamics of biological systems such as the swarm of birds. Kennedy and
Eberhart \cite{kennedy1995particle} published the first seminal paper on PSO. As a result, the variety of research articles ascribed to the development of particle swarm optimization and swarm intelligence grown massively. The PSO algorithm is simple to implement and has a low memory requirements. Consequently, it has been used in a multitude of complex applications that involve optimization, control, machine learning, and so on.
PSO is widely used in the design, sizing, control and maximum power point tracking (MPPT) of renewable energy systems; such as photovoltaic (PV) systems \cite{harrag2019pso,eltamaly2020performance,shaqarin2021modified}, wind turbines \cite{aguilar2020multi,kamarzarrin2020intelligent,sushmitha2021novel} and Hybrid PV-wind systems \cite{ghorbani2018optimizing,saad2021implementation,el2022sizing}. The authors in \cite{yifei2018research,wu2021motion,zhang2022pso} implemented PSO in the control of robotics for various applications. PSO is also popular in unmanned vehicles in performing path planning \cite{guo2020global,tavoosi2020optimized} and path tracking \cite{al2015path,amer2018path,jiang2022model}.
PSO has been implemented widley in the sizing, design,
modeling, and control of various typres of refrigeration system, such as reducing power consumption of multiple chiller systems \cite{beghi2012pso}, optimizing cascade refrigeration cycles \cite{ghorbani2014optimization}, standing wave thermoacoustic refrigerators \cite{rahman2019single} and vapor compression refrigeration cycles \cite{kong2021global}.
Nevertheless, while PSO is efficient in optimizing multidimensional complexities \cite{huang2019memetic}, its drawbacks involve premature convergence and stagnation to local optima \cite{jiao2008elite,chen2010simplified,liu2016topology,pahnehkolaei2022analytical,ramirez2022pso,he2022semi}, as well as a slow convergence rate \cite{jiao2008elite,chen2010simplified,he2022semi,liu2018particle,ramirez2022pso}. According to Jiao \emph{et al.} and Chen \cite{jiao2008elite,chen2010simplified}, the drawbacks of PSO originates from its own structure. The social model readily falls into local optima, practically, each particle in PSO follows rather dynamically the path of the global best position. This global best could be influenced by a local minimum, which leads all particles to a position with poor fitness. Whereas, the cognitive model has the drawback of slow convergence especially in the final phases of the evolving iterations when tackling various complexities.
This motivates the particles' classifications in order to treat each category with different updating strategy to achieve better convergence rates and to enhance global explorations capabilities \cite{angeline1998using,parrott2006locating,chen2010simplified,zheng2018fault}.
The authors in \cite{angeline1998using,jiao2008elite,yang2015improved,alshammari2020elitist,yang2021quantum} implemented the particle elitism approach to speed up the convergence of PSO. Elitism is a process in which particles with poor fitness values are replaced with particles with the best fitness value (elite particles) after a certain number of iterations, resulting in the production of a new swarm with a better particle's average fitness. The elitism process in nature requires a preliminary step which is the particle classification, in order to distinguish poor and elite particles. This process will indeed speed up the convergence but on the other hand, it will lessen the diversity of the particles and may increase the probability of falling into a local optimum.
Beneficial to boost particle swarm diversity and mitigate the risk of falling into local optimum. References \cite{higashi2003particle,jiao2008elite,zhang2019differential,wang2021exergoeconomic,lu2022comprehensive} proposed the concept of mutation in PSO algorithm to improve the convergence accuracy and speed. In the literature there are mainly two mutation's types implemented in the PSO algorithm. The first one is the global best mutation as proposed by \cite{jiao2008elite,lu2022comprehensive}, in which the particles' position are mutated indirectly in response to the change of global best. The second type is the direct position mutation as reported by \cite{wang2021exergoeconomic}, which is better emulating the mutation process inspired by genetic algorithms (GA). The mutation process will enhance the diversity of the swarm which results in better exploration capabilities but it may slow down the convergence as reported by \cite{wang2021exergoeconomic}. This can be explained by the fact that mutation will indeed be useful to poor or moderate particles but it may also deteriorate the fitness of good particle.
In this work, the basic PSO will be highlighted along with several PSO variants that involve the three key elements affecting the convergence speed and global exploration capabilities; particle's classification, elitism and mutation. The proposed PSO method, particle swarm optimization with targeted position-mutated elitism (PSO-TPME), will benefits from these key features in order to enhance both convergence speed and the global exploration capabilities, by introducing an alternative classification technique, elitism and targeted position mutation in the PSO algorithm. The proposed algorithm will be tested on multidimensional benchmark functions with various complexities.
\section{Materials and Methods}
\subsection{Particel swarm optimization (PSO)}
PSO's method generates a swarm of particles at random positions to represent possible solutions to an optimization task throughout the parameter space. The particle locations are hence iterated with the target of achieving a global optimum of a fitness value. The algorithm then examines each particle's position and stores the best solution for each particle, which is called personal best ($P_b$). Moreover, during every iteration ($It$), PSO records the best solution throughout the swarm, which is called global best ($G_b$). For every subsequent iteration, the particles' position ($x$) and velocity ($v$) are determined as a function of the swarm's best position (social component), the particle's best personal position (the cognitive component), and its prior velocity (memory component). In general, PSO has several versions, linearly decreasing weight particle swarm optimization is one of the basic versions of PSO, known as LDW-PSO, that reads;
\begin{equation}
\label{eq::pso_v}
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})+c_{2}r_{2}({G_b}_{j}-x_{ij}^{k})
\end{equation}
\begin{equation}
\label{eq::pso_x}
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1}
\end{equation}
\begin{equation}
\label{eq::pso_ldw}
w=w_{max}-It\times\frac{(w_{max}-w_{min} ) }{It_{max}}
\end{equation}
The subscript $_i$ is ranging from 1 to $N$ number of particles and the subscript $_j$ is ranging from 1 to $n$ dimensions. The factor $w$ is known as the inertia weight, $w_{max}$ and $w_{min}$ are the maximum and the minimum inertia weight, respectively, and the product $wv$ represents the particle's momentum. The acceleration factors are $c_1$ and $c_2$, while $r_1$ and $r_2$ are the output of random generators ranging from 0 to 1.
\subsection{PSO with classification}
Chen \cite{chen2010simplified} proposed PSO-M, a study on particle classification that improves the convergence rate and convergence accuracy of PSO. The classification is based on observing the particles' fitness value at each iteration, identifying the particles' fitness mean ($aver$) and also the best and worst fitness values, ($f_{max}$) and ($f_{min}$), respectively. Then, two averages are calculated, $aver1$ and $aver2$, between $f_{max}$ and $aver$ and between $f_{min}$ and $aver$, respectively. Hence, the fitness space ($f_{min}$ -- $f_{max}$) is divided into three categories. Considering maximization problem, the classification will be as follows; particles with fitness values in the range ($aver1$ -- $f_{max}$) are updated using the cognitive PSO component, particles with fitness values in the range ($f_{min}$ -- $aver2$) are updated using the social PSO component, and the remaining particles are updated using the basic PSO model. One drawback of this classification is that all particles are classified starting from the initialization and can even last after the convergence, that is, no classification termination criteria, as seen in Fig. \ref{fig::pso-m}.
\begin{figure}[H]
\includegraphics[width=10.5 cm]{fig/shaqa1.pdf}
\caption{Particles' evolution and classification with evolving iterations using PSO-M.\label{fig::pso-m}}
\end{figure}
\unskip
\subsection{PSO with mutation}
Influenced by genetic algorithms (GA), Jiao \emph{et al.} \cite{jiao2008elite} proposed the concept of elite PSO with mutation (EPSOM) to improve the convergence accuracy and speed. The global best was mutated to boost particle swarm diversity and mitigate the risk of falling into local optimum. The mutation is performed as follows;
\begin{equation}
\label{eq::pso_EPSOM}
{G_b}^{\prime}=G_b(1+0.5\eta)
\end{equation}
where $\eta$ is a randomly generated number ranging from 0 to 1.
Another recent study by L{\"u} \emph{et al.} \cite{lu2022comprehensive} that relies on the global best mutation in a similar manner as in \eqref{eq::pso_EPSOM}, proposed an adaptive weighted and mutation particle swarm optimization (AWMPSO) to enhance global search capabilities of the PSO. In which the mutation probability of the global best is now adaptive, where the mutation probability of the global best depend on the population fitness's variance. Apart from the significant improvement of this approach on the enhanced exploration ability and the convergence speed and accuracy, this approach still has limitations due to the fact that the mutation is implemented on the global best as seen in \eqref{eq::pso_EPSOM}. Hence, the particles' positions are mutated indirectly through the global best mutation. According to \eqref{eq::pso_EPSOM}, the change in the global best is limited and can vary gradually with the evolving iterations. Furthermore, the mutation is not targeted, since the mutation of the global best affect all particles. In this case, the mutation will indeed be useful to poor or moderate particles but it may also deteriorate the fitness of good particle.
Wang \emph{et al.} \cite{wang2021exergoeconomic} proposed a new method that relies on the position mutation (M-PSO).
The particles' position mutation process is activated conditionally as an intermediate step just before the evaluation of the personal best and the global of the PSO algorithm. The condition requires that a randomly generated number to be less than an adaptively generated threshold value ($TH$). The threshold value was defined as:
\begin{equation}
\label{eq::pso_threshold}
TH=\left(1-\frac{i-1}{It_{max}-1}\right)^{\frac{1}{mu}}
\end{equation}
where $mu$ is the mutation factor. Wang \emph{et al.} \cite{wang2021exergoeconomic} reported that increasing mutation factor can enhance the convergence accuracy but it reduces the convergence speed. As seen in \eqref{eq::pso_threshold}, the threshold value is updated by the evolving iterations, not the particles' fitness, which can be mainly considered as a termination criterion. The threshold value is crucial parameter in M-PSO approach, not only because it is dictating the activation of the mutation process, but because it also decides the upper and the lower bounds of the mutated position. In M-PSO approach, there is no particle classifications, hence, the mutation is performed on all the particles. That is, M-PSO does not use targeted mutation which can contribute negatively on the movement of the good particles and consequently affect the convergence speed as reported by Wang \emph{et al.} \cite{wang2021exergoeconomic}.
\subsection{Proposed PSO (PSO-TPME)}
The initial concept of the proposed PSO, is automated-termination, adaptive particles' classification with elitism based on particles' fitness values ($f$). The classification is performed as follows; the mean ($m$) of the fitness values of all particles is calculated at each iteration. Then, a fixed percentage ($p$) around the mean is calculated to generate lower and upper bounds. Now the particles are divided into three categories: good, fair, and bad. Hence, considering a maximization problem, particles with fitness higher than the upper bound (good particles) can decrease their velocity to improve exploitation in the local domain via relying on their personal best (PSO cognitive component) instead of the global best (PSO social component). The particles with fitness that is located within the lower and upper bounds (fair particles), those have relatively average fitness can continue both exploration and exploitation while using the basic PSO algorithm. Particles with fitness that is less than the lower bound (bad particles) can initially increase their velocities to enhance exploration in the global search domain, via relaying on the global best instead of its personal best. If the bad particles after a number of iterations ($N_{e}$) are still classified as bad particles. Then the elitism process is activated to speed up the convergence rate. The elitism process is basically intended to cope with the so called "hopeless particles", those given the chance for exploring the search space and failed to level up to better category. Now the elitism simply locates the hopeless particles in the position of the particle with the maximum fitness ($f_{max}$).
This process speeds up the convergence rate significantly with proper choice of $N_{e}$ as shown in the Fig. \ref{fig::proposed-pso}.
\begin{figure}[H]
\includegraphics[width=10.5 cm]{fig/shaqa2.pdf}
\caption{Particles' evolution with evolving iterations using the proposed classification with elitism. \label{fig::proposed-pso}}
\end{figure}
Conducting PSO using the earlier approach will drastically reduce the diversity of the particle swarm during the preliminary stages of evolving iterations. As a result, the probability of entrapment in a region of the local optimum is high. Adding a mutation to the particle's position of the maximum fitness $(x_{j}(f_{max}))$ will boost the diversification of the elite particle and reduce the possibility of falling into a local optimum. Hence, the mutation is now targeting bad particles only, and the mutation is performed on the particle's position directly instead of indirect mutation of the position using the global best as proposed by Jiao \emph{et al.} \cite{jiao2008elite}. The details of the proposed PSO algorithm considering a maximization problem are in \eqref{eq::pso_proposed}, in which the elite particle's positions $(x_{j}(f_{max}))$ are mutated by $(2a\eta+(1-a))$, where $\eta$ is a randomly generated number ranging from 0 to 1 and $a$ is a presetting parameter that defines the mutation range. The basic idea is that the elite position $(x_{j}(f_{max}))$ is mutated in the range $(x_{j}(f_{max})(1\pm a))$, in which the mean is unity multiplied by $x_{j}(f_{max})$ which gives higher probability to $x_{j}(f_{max})$, that is, exploitation of the elite particle's position without neglecting the chances of exploring new particles' positions. The aforementioned range can be increased or decreased via varying $a$ for higher or lower dimension functions, respectively.
This classification is called automated-termination classification because, after some iterations, all the particles will fall into the middle category since, on one hand, the particles' fitness values will eventually be close and, on the other hand, the particles' mean fitness is gradually increasing which will expand the bounds of the middle category. Hence, the classification will stop automatically as depicted in Fig. \ref{fig::proposed-pso}. This classification is also considered adaptive classification because of particles' mean variation at each iteration. The particles' mean will change, therefore the upper and lower bound of the middle category will dynamically vary, as the calculation of these bounds depends only on a fixed percentage around the particles' mean. This type of particles' classification is motivated by its implantation simplicity, low memory requirements, and automated-termination criteria based on the particles convergence. The latter is very crucial since elitism with mutation takes part in the proposed PSO. Now, the automated termination criteria will stop the classification and the elitism with mutation as well, without manual or other termination criteria being involved, which otherwise may require extra processing.
\begin{equation}
\label{eq::pso_proposed}
\begin{cases}
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})& \text{if } f_{ij}^{k}> (1+p)m\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\noindent\rule{7.5cm}{0.4pt}\\
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})+ c_{2}r_{2}({G_b}_{j}-x_{ij}^{k})& \text{if }(1-p)m\leq f_{ij}^{k}\leq(1+p)m\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\noindent\rule{7.5cm}{0.4pt}\\
v_{ij}^{k+1}=wv_{ij}^{k}+c_{2}r_{2}({G_b}_{j}-x_{ij}^{k}) &\text{if } f_{ij}^{k}< (1-p)m \,\, \& \,\, It<N_e\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\hdashrule{7.6cm}{0.4pt}{4pt}\\
x_{ij}^{k+1}=x_{j}(f_{max})(2a\eta+(1-a)),& \text{if } f_{ij}^{k}< (1-p)m \,\, \& \,\, It\geq N_e
\end{cases}
\end{equation}
\subsection{Benchmark problems}
A group of recognized benchmark multi-dimensional functions, which are extensively adopted in the optimization field, were employed to assess the performance of EPSOM, LDW-PSO, PSO-M, M-PSO, and the suggested PSO algorithm in terms of convergence accuracy and convergence speed. The benchmark functions are used herein as minimization problems, each of which offers a distinct level of complexity to the tested algorithm to be assessed. Function complexity such as; unimodality or multimodality, symmetric or asymmetric and separable or inseparable in its variables. The benchmark multi-dimensional functions are:
Rosenbrock function is an unimodal function that is extensively used for local exploration, which was first used in optimization assessment of Genetic Algorithms (GA) by De Jong \cite{de1975analysis}. According to Shang and Qiu \cite{shang2006note}, the n-dimensional Rosenbrock function ($n>4$) can be a bimodal function, that makes it even a more complex minimization problem, this complexity also originates from the function's asymmetry and variables inseparability. The global minimum value of Rosenbrock function is zero, which is located at $(1,1,...)$. The multi-dimensional Rosenbrock function reads as:
\begin{equation}
\label{eq::f_Rosenbrock}
f_{1}(x)=\sum_{i=1}^{n}100(x_{i+1}^{2}-x_{i})^{2}+(1-x_{i})^{2}
\end{equation}
Rastrigin function is a multimodal function that is employed for the performance assessment of evolutionary algorithms \cite{varIslandNum07}. Although it is extremely multimodal, the positions of the minima are evenly dispersed, the function is symmetric and separable. The global minimum value of Rastrigin function is zero that is located at $(0,0,...)$. Rastrigin function is defined as:
\begin{equation}
\label{eq::f_Rastrigrin}
f_2(x)=\sum_{i=1}^{n}(x_{i}^{2}-10cos(2\pi x_{i}))+10
\end{equation}
Griewank function is also a multimodal and symmetric function that is widely used for global optimization. The global minimum value of Griewank function is zero that is located at $(0,0,...)$. Following Locatelli \cite{locatelli2003note}, the function contains a huge number of local minima, which increases exponentially with the number of dimensions. According to Jumonji \emph{et al.} \cite{varIslandNum07}, Griewank function is inseparable in its variables and is defined as follows:
\begin{equation}
\label{eq::f_Griewank}
f_3(x)=\sum_{i=1}^{n}\frac{x_{i}^{2}}{4000} - \prod_{i=1}^{n}cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1
\end{equation}
\section{Results}
Beneficial to evaluate the efficiency of the proposed PSO algorithm on large-scale problems, the benchmark functions, Griewank, Rastrigrin, and Rosenbrock, are set to have 30 dimensions.
To assess the performance of the proposed PSO algorithm in comparison to
EPSOM, LDW-PSO, PSO-M and M-PSO, the parameters of all the tested algorithms are identically selected for consistency.
The search and initialization space for all the functions are [-100 - 100]. The number of particles is set to 40, and a maximum iteration of 2000. Furthermore, $w$ is declining linearly from $w_{max} = 0.9$ to $w_{min} =0.1$, c1 is 1.4962, and c2 is 1.4962. Regarding the PSO-M, the mutation factor $mu$ is set to 0.05.
The proposed PSO method includes three more presetting parameters: one is the classification percentage around the fitness mean, denoted by p, and the other is the elitism with position mutation process initiation iteration, denoted by Ne, the last one is the mutation range, that is defined by $a$. In these simulations, these parameters are selected as $p=0.02$, $N_e=3$, and $a=0.5$. Consequently, the particles with fitness in the range of $\pm 2\%$ of mean particles' fitness value are classified in the middle category (``fair particles''), the elitism process starts after the third iteration, and the mutation range is $\pm50\%$ of $x_{j}(f_{max})$.
Twenty independent simulations were carried out on the three previously aforementioned functions for each minimization algorithm, which is beneficial to minimizing the statistical errors of the optimization performance of the previously mentioned algorithms. The outcomes of each algorithm were averaged across 20 simulations. The averaged fitness performance of PSO variants on 30-dimensional benchmark functions is depicted in Fig. \ref{fig::benchmark}. The optimization results show that the proposed PSO excels EPSOM, LDW-PSO, PSO-M and M-PSO in terms of convergence speed and global exploration capabilities, on all tested benchmark functions. Another important result from the figure, is that all the mutation based approaches have better global exploration capabilities, since their convergence accuracy is better than other PSO approaches, as seen from the optimization performance of Griewank, Rastringrin and Rosenbrock function.
\begin{figure}[ht ]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Griewank}
\label{fig::Griewank}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Rastrigrin }
\label{fig::Rastrigrin}
\end{subfigure}
\hfill
\newline
\newline
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Rosenbrock }
\label{fig::Rosenbrock}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3d.pdf}
\caption{Zoom on proposed PSO }
\label{fig::proposed_pso}
\end{subfigure}
\caption{Fitness performance of PSO variants on multi-dimensional benchmark functions. Figures (a-c) depict comparison of EPSOM, LDW-PSO, PSO-M and M-PSO with the proposed PSO on a 30-dimensional functions (Griewank, Rastringrin and Rosenbrock). Figure (d) shows a zoomed-in plot of the performance of the proposed PSO on the three functions}
\label{fig::benchmark}
\end{figure}
Figure \ref{fig::proposed_pso} shows a zoomed-in plot of the minimization performance of the proposed PSO on the 30-dimensional Griewank, Rastrigrin, and Rosenbrock function. The figure undeniably shows the gigantic improvement of the elitism with targeted position mutation on the proposed PSO after the iteration $N_e=3$. The bad particles are now exploiting the particle's position with maximum fitness with high probability of exploring new particles' positions as a consequence of position mutation.
It can be seen from Fig. \ref{fig::benchmark} that the proposed PSO attained the global minimum ($10^{-15}$) for the 30-dimensional Griewank function in 11 iterations, whereas the other techniques needed 1900, 501, 732 and 597 iterations to reach considerably greater local minima by orders of magnitude, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. The suggested PSO attained the global minimum ($10^{-12}$) for the 30-dimensional Rastrigrin function in 11 iterations, whereas the other techniques required 1713, 607, 716 and 919 iterations to reach significantly greater local minima by orders of magnitude, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. In terms of the optimization results of the Rosenbrock function utilizing the suggested PSO, the proposed technique required 6 iterations to attain a minimum that outperformed the other approaches, considering that the comparative approaches required 1850, 552, 835 and 980 iterations, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. For all tested function, the convergence rate for the proposed PSO is faster by orders of magnitude compared with all tested variants.
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/griewank_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Griewank function after 10 and 2000 iterations. \label{fig::Griewank-30-90}}
\end{figure}
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/rastrig_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Rastrig function after 10 and 2000 iterations.. \label{fig::Rastrig-30-90}}
\end{figure}
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/rosenbrock_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Rosenbrock function after 10 and 2000 iterations.. \label{fig::Rosenbrock-30-90}}
\end{figure}
Figures \ref{fig::Griewank-30-90}, \ref{fig::Rastrig-30-90}, and \ref{fig::Rosenbrock-30-90} show the optimization results for the PSO-TPME and the previously described PSO variants. The optimal fitness value in the figures is the average optimal one of the solutions in 20 trials. To evaluate the early and late exploration capabilities of the PSO variations, the optimum fitness is estimated after 10 and 2000 iterations, respectively. The optimization was carried out on the previously specified benchmark functions with dimensions of 30, 60, and 90 while keeping the same presetting settings for all PSO variations, including the number of particles, constant. The figures undeniably reveal that the PSO-TPME's early exploration performance outperforms the overall (early and late) exploration performance of the investigated PSO variants by orders of magnitude. This is also proven for all previously described benchmark functions for large dimensional problems of 30, 60, and 60 dimensions. This clearly shows that PSO-TPME has remarkably fast and accurate convergence characteristics, which are supported by benchmark functions with various levels of complexity and a large number of dimensions.
\section{Discussion}
Starting point are five popular variants of particle swarm optimization.
The proposed new PSO variant (PSO-TPME)
is shown to dramatically improve convergence speed and global exploration capabilities.
This variant comprises the three major factors affecting convergence speed and global exploration capabilities: particles' classification, elitism, and mutation, as well as the original PSO's cognitive and social models. This variation introduced an alternative classification approach, elitism, and targeted position mutation, all of which were integrated into the basic PSO algorithm. The introduced particle classification process is simple to apply, requires low memory, is adaptive, and provides automated termination criteria based on convergence. These qualities of the proposed classification technique permitted the implementation of targeted elitism and mutation, in terms of targeting just the poor particles, terminating the elitism and mutation process automatically in the event of convergence, and applying different updating models (social and/or cognitive) based on the particle's category.
A set of benchmark multi-dimensional functions widely used in the optimization problems were used to compare the performance of the proposed PSO-TPME to EPSOM, LDW-PSO, PSO-M and M-PSO, in terms of convergence accuracy and convergence speed. The 30, 60, and 90-dimensional Griewank, Rastrigrin, and Rosenbrock functions are employed. Each of them provides a different level of complexity, such as unimodality or multimodality, symmetry or asymmetry, and separability or inseparability. For each benchmark function, many minimization simulations were performed, repeated, and averaged to reduce statistical errors. The simulations revealed that PSO-TPME surpasses the aforementioned variants by orders of magnitude in terms of convergence rate and accuracy.
\vspace{6pt}
\authorcontributions{Conceptualization, T. Shaqarin and B. R. Noack; methodology, T. Shaqarin; software, T. Shaqarin; validation, T. Shaqarin and B. R. Noack; formal analysis, T. Shaqarin; investigation, T. Shaqarin.; resources, T. Shaqarin; data curation, T. Shaqarin; writing---original draft preparation, T. Shaqarin and B. R. Noack; writing---review and editing, T. Shaqarin and B. R. Noack; visualization, T. Shaqarin; supervision, T. Shaqarin and B. R. Noack; project administration, B. R. Noack; funding acquisition, B. R. Noack. All authors have read and agreed to the published version of the manuscript.}
\funding{This work is supported
by the National Science Foundation of China (NSFC) through grants 12172109 and 12172111
and by the Natural Science and Engineering grant 2022A1515011492
of Guangdong province, China.}
\dataavailability{Not applicable}
\conflictsofinterest{ The authors declare no conflict of interest.}
\begin{adjustwidth}{-\extralength}{0cm}
\reftitle{References
|
}
\section{Introduction}
\label{sec:introduction}
Particle swarm optimization (PSO) is a metaheuristic optimization approach that imitates the dynamics of biological systems such as the swarm of birds. Kennedy and
Eberhart \cite{kennedy1995particle} published the first seminal paper on PSO. As a result, the variety of research articles ascribed to the development of particle swarm optimization and swarm intelligence grown massively. The PSO algorithm is simple to implement and has a low memory requirements. Consequently, it has been used in a multitude of complex applications that involve optimization, control, machine learning, and so on.
PSO is widely used in the design, sizing, control and maximum power point tracking (MPPT) of renewable energy systems; such as photovoltaic (PV) systems \cite{harrag2019pso,eltamaly2020performance,shaqarin2021modified}, wind turbines \cite{aguilar2020multi,kamarzarrin2020intelligent,sushmitha2021novel} and Hybrid PV-wind systems \cite{ghorbani2018optimizing,saad2021implementation,el2022sizing}. The authors in \cite{yifei2018research,wu2021motion,zhang2022pso} implemented PSO in the control of robotics for various applications. PSO is also popular in unmanned vehicles in performing path planning \cite{guo2020global,tavoosi2020optimized} and path tracking \cite{al2015path,amer2018path,jiang2022model}.
PSO has been implemented widley in the sizing, design,
modeling, and control of various typres of refrigeration system, such as reducing power consumption of multiple chiller systems \cite{beghi2012pso}, optimizing cascade refrigeration cycles \cite{ghorbani2014optimization}, standing wave thermoacoustic refrigerators \cite{rahman2019single} and vapor compression refrigeration cycles \cite{kong2021global}.
Nevertheless, while PSO is efficient in optimizing multidimensional complexities \cite{huang2019memetic}, its drawbacks involve premature convergence and stagnation to local optima \cite{jiao2008elite,chen2010simplified,liu2016topology,pahnehkolaei2022analytical,ramirez2022pso,he2022semi}, as well as a slow convergence rate \cite{jiao2008elite,chen2010simplified,he2022semi,liu2018particle,ramirez2022pso}. According to Jiao \emph{et al.} and Chen \cite{jiao2008elite,chen2010simplified}, the drawbacks of PSO originates from its own structure. The social model readily falls into local optima, practically, each particle in PSO follows rather dynamically the path of the global best position. This global best could be influenced by a local minimum, which leads all particles to a position with poor fitness. Whereas, the cognitive model has the drawback of slow convergence especially in the final phases of the evolving iterations when tackling various complexities.
This motivates the particles' classifications in order to treat each category with different updating strategy to achieve better convergence rates and to enhance global explorations capabilities \cite{angeline1998using,parrott2006locating,chen2010simplified,zheng2018fault}.
The authors in \cite{angeline1998using,jiao2008elite,yang2015improved,alshammari2020elitist,yang2021quantum} implemented the particle elitism approach to speed up the convergence of PSO. Elitism is a process in which particles with poor fitness values are replaced with particles with the best fitness value (elite particles) after a certain number of iterations, resulting in the production of a new swarm with a better particle's average fitness. The elitism process in nature requires a preliminary step which is the particle classification, in order to distinguish poor and elite particles. This process will indeed speed up the convergence but on the other hand, it will lessen the diversity of the particles and may increase the probability of falling into a local optimum.
Beneficial to boost particle swarm diversity and mitigate the risk of falling into local optimum. References \cite{higashi2003particle,jiao2008elite,zhang2019differential,wang2021exergoeconomic,lu2022comprehensive} proposed the concept of mutation in PSO algorithm to improve the convergence accuracy and speed. In the literature there are mainly two mutation's types implemented in the PSO algorithm. The first one is the global best mutation as proposed by \cite{jiao2008elite,lu2022comprehensive}, in which the particles' position are mutated indirectly in response to the change of global best. The second type is the direct position mutation as reported by \cite{wang2021exergoeconomic}, which is better emulating the mutation process inspired by genetic algorithms (GA). The mutation process will enhance the diversity of the swarm which results in better exploration capabilities but it may slow down the convergence as reported by \cite{wang2021exergoeconomic}. This can be explained by the fact that mutation will indeed be useful to poor or moderate particles but it may also deteriorate the fitness of good particle.
In this work, the basic PSO will be highlighted along with several PSO variants that involve the three key elements affecting the convergence speed and global exploration capabilities; particle's classification, elitism and mutation. The proposed PSO method, particle swarm optimization with targeted position-mutated elitism (PSO-TPME), will benefits from these key features in order to enhance both convergence speed and the global exploration capabilities, by introducing an alternative classification technique, elitism and targeted position mutation in the PSO algorithm. The proposed algorithm will be tested on multidimensional benchmark functions with various complexities.
\section{Materials and Methods}
\subsection{Particel swarm optimization (PSO)}
PSO's method generates a swarm of particles at random positions to represent possible solutions to an optimization task throughout the parameter space. The particle locations are hence iterated with the target of achieving a global optimum of a fitness value. The algorithm then examines each particle's position and stores the best solution for each particle, which is called personal best ($P_b$). Moreover, during every iteration ($It$), PSO records the best solution throughout the swarm, which is called global best ($G_b$). For every subsequent iteration, the particles' position ($x$) and velocity ($v$) are determined as a function of the swarm's best position (social component), the particle's best personal position (the cognitive component), and its prior velocity (memory component). In general, PSO has several versions, linearly decreasing weight particle swarm optimization is one of the basic versions of PSO, known as LDW-PSO, that reads;
\begin{equation}
\label{eq::pso_v}
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})+c_{2}r_{2}({G_b}_{j}-x_{ij}^{k})
\end{equation}
\begin{equation}
\label{eq::pso_x}
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1}
\end{equation}
\begin{equation}
\label{eq::pso_ldw}
w=w_{max}-It\times\frac{(w_{max}-w_{min} ) }{It_{max}}
\end{equation}
The subscript $_i$ is ranging from 1 to $N$ number of particles and the subscript $_j$ is ranging from 1 to $n$ dimensions. The factor $w$ is known as the inertia weight, $w_{max}$ and $w_{min}$ are the maximum and the minimum inertia weight, respectively, and the product $wv$ represents the particle's momentum. The acceleration factors are $c_1$ and $c_2$, while $r_1$ and $r_2$ are the output of random generators ranging from 0 to 1.
\subsection{PSO with classification}
Chen \cite{chen2010simplified} proposed PSO-M, a study on particle classification that improves the convergence rate and convergence accuracy of PSO. The classification is based on observing the particles' fitness value at each iteration, identifying the particles' fitness mean ($aver$) and also the best and worst fitness values, ($f_{max}$) and ($f_{min}$), respectively. Then, two averages are calculated, $aver1$ and $aver2$, between $f_{max}$ and $aver$ and between $f_{min}$ and $aver$, respectively. Hence, the fitness space ($f_{min}$ -- $f_{max}$) is divided into three categories. Considering maximization problem, the classification will be as follows; particles with fitness values in the range ($aver1$ -- $f_{max}$) are updated using the cognitive PSO component, particles with fitness values in the range ($f_{min}$ -- $aver2$) are updated using the social PSO component, and the remaining particles are updated using the basic PSO model. One drawback of this classification is that all particles are classified starting from the initialization and can even last after the convergence, that is, no classification termination criteria, as seen in Fig. \ref{fig::pso-m}.
\begin{figure}[H]
\includegraphics[width=10.5 cm]{fig/shaqa1.pdf}
\caption{Particles' evolution and classification with evolving iterations using PSO-M.\label{fig::pso-m}}
\end{figure}
\unskip
\subsection{PSO with mutation}
Influenced by genetic algorithms (GA), Jiao \emph{et al.} \cite{jiao2008elite} proposed the concept of elite PSO with mutation (EPSOM) to improve the convergence accuracy and speed. The global best was mutated to boost particle swarm diversity and mitigate the risk of falling into local optimum. The mutation is performed as follows;
\begin{equation}
\label{eq::pso_EPSOM}
{G_b}^{\prime}=G_b(1+0.5\eta)
\end{equation}
where $\eta$ is a randomly generated number ranging from 0 to 1.
Another recent study by L{\"u} \emph{et al.} \cite{lu2022comprehensive} that relies on the global best mutation in a similar manner as in \eqref{eq::pso_EPSOM}, proposed an adaptive weighted and mutation particle swarm optimization (AWMPSO) to enhance global search capabilities of the PSO. In which the mutation probability of the global best is now adaptive, where the mutation probability of the global best depend on the population fitness's variance. Apart from the significant improvement of this approach on the enhanced exploration ability and the convergence speed and accuracy, this approach still has limitations due to the fact that the mutation is implemented on the global best as seen in \eqref{eq::pso_EPSOM}. Hence, the particles' positions are mutated indirectly through the global best mutation. According to \eqref{eq::pso_EPSOM}, the change in the global best is limited and can vary gradually with the evolving iterations. Furthermore, the mutation is not targeted, since the mutation of the global best affect all particles. In this case, the mutation will indeed be useful to poor or moderate particles but it may also deteriorate the fitness of good particle.
Wang \emph{et al.} \cite{wang2021exergoeconomic} proposed a new method that relies on the position mutation (M-PSO).
The particles' position mutation process is activated conditionally as an intermediate step just before the evaluation of the personal best and the global of the PSO algorithm. The condition requires that a randomly generated number to be less than an adaptively generated threshold value ($TH$). The threshold value was defined as:
\begin{equation}
\label{eq::pso_threshold}
TH=\left(1-\frac{i-1}{It_{max}-1}\right)^{\frac{1}{mu}}
\end{equation}
where $mu$ is the mutation factor. Wang \emph{et al.} \cite{wang2021exergoeconomic} reported that increasing mutation factor can enhance the convergence accuracy but it reduces the convergence speed. As seen in \eqref{eq::pso_threshold}, the threshold value is updated by the evolving iterations, not the particles' fitness, which can be mainly considered as a termination criterion. The threshold value is crucial parameter in M-PSO approach, not only because it is dictating the activation of the mutation process, but because it also decides the upper and the lower bounds of the mutated position. In M-PSO approach, there is no particle classifications, hence, the mutation is performed on all the particles. That is, M-PSO does not use targeted mutation which can contribute negatively on the movement of the good particles and consequently affect the convergence speed as reported by Wang \emph{et al.} \cite{wang2021exergoeconomic}.
\subsection{Proposed PSO (PSO-TPME)}
The initial concept of the proposed PSO, is automated-termination, adaptive particles' classification with elitism based on particles' fitness values ($f$). The classification is performed as follows; the mean ($m$) of the fitness values of all particles is calculated at each iteration. Then, a fixed percentage ($p$) around the mean is calculated to generate lower and upper bounds. Now the particles are divided into three categories: good, fair, and bad. Hence, considering a maximization problem, particles with fitness higher than the upper bound (good particles) can decrease their velocity to improve exploitation in the local domain via relying on their personal best (PSO cognitive component) instead of the global best (PSO social component). The particles with fitness that is located within the lower and upper bounds (fair particles), those have relatively average fitness can continue both exploration and exploitation while using the basic PSO algorithm. Particles with fitness that is less than the lower bound (bad particles) can initially increase their velocities to enhance exploration in the global search domain, via relaying on the global best instead of its personal best. If the bad particles after a number of iterations ($N_{e}$) are still classified as bad particles. Then the elitism process is activated to speed up the convergence rate. The elitism process is basically intended to cope with the so called "hopeless particles", those given the chance for exploring the search space and failed to level up to better category. Now the elitism simply locates the hopeless particles in the position of the particle with the maximum fitness ($f_{max}$).
This process speeds up the convergence rate significantly with proper choice of $N_{e}$ as shown in the Fig. \ref{fig::proposed-pso}.
\begin{figure}[H]
\includegraphics[width=10.5 cm]{fig/shaqa2.pdf}
\caption{Particles' evolution with evolving iterations using the proposed classification with elitism. \label{fig::proposed-pso}}
\end{figure}
Conducting PSO using the earlier approach will drastically reduce the diversity of the particle swarm during the preliminary stages of evolving iterations. As a result, the probability of entrapment in a region of the local optimum is high. Adding a mutation to the particle's position of the maximum fitness $(x_{j}(f_{max}))$ will boost the diversification of the elite particle and reduce the possibility of falling into a local optimum. Hence, the mutation is now targeting bad particles only, and the mutation is performed on the particle's position directly instead of indirect mutation of the position using the global best as proposed by Jiao \emph{et al.} \cite{jiao2008elite}. The details of the proposed PSO algorithm considering a maximization problem are in \eqref{eq::pso_proposed}, in which the elite particle's positions $(x_{j}(f_{max}))$ are mutated by $(2a\eta+(1-a))$, where $\eta$ is a randomly generated number ranging from 0 to 1 and $a$ is a presetting parameter that defines the mutation range. The basic idea is that the elite position $(x_{j}(f_{max}))$ is mutated in the range $(x_{j}(f_{max})(1\pm a))$, in which the mean is unity multiplied by $x_{j}(f_{max})$ which gives higher probability to $x_{j}(f_{max})$, that is, exploitation of the elite particle's position without neglecting the chances of exploring new particles' positions. The aforementioned range can be increased or decreased via varying $a$ for higher or lower dimension functions, respectively.
This classification is called automated-termination classification because, after some iterations, all the particles will fall into the middle category since, on one hand, the particles' fitness values will eventually be close and, on the other hand, the particles' mean fitness is gradually increasing which will expand the bounds of the middle category. Hence, the classification will stop automatically as depicted in Fig. \ref{fig::proposed-pso}. This classification is also considered adaptive classification because of particles' mean variation at each iteration. The particles' mean will change, therefore the upper and lower bound of the middle category will dynamically vary, as the calculation of these bounds depends only on a fixed percentage around the particles' mean. This type of particles' classification is motivated by its implantation simplicity, low memory requirements, and automated-termination criteria based on the particles convergence. The latter is very crucial since elitism with mutation takes part in the proposed PSO. Now, the automated termination criteria will stop the classification and the elitism with mutation as well, without manual or other termination criteria being involved, which otherwise may require extra processing.
\begin{equation}
\label{eq::pso_proposed}
\begin{cases}
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})& \text{if } f_{ij}^{k}> (1+p)m\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\noindent\rule{7.5cm}{0.4pt}\\
v_{ij}^{k+1}=wv_{ij}^{k}+c_{1}r_{1}({P_b}_{ij}-x_{ij}^{k})+ c_{2}r_{2}({G_b}_{j}-x_{ij}^{k})& \text{if }(1-p)m\leq f_{ij}^{k}\leq(1+p)m\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\noindent\rule{7.5cm}{0.4pt}\\
v_{ij}^{k+1}=wv_{ij}^{k}+c_{2}r_{2}({G_b}_{j}-x_{ij}^{k}) &\text{if } f_{ij}^{k}< (1-p)m \,\, \& \,\, It<N_e\\
x_{ij}^{k+1}=x_{ij}^{k}+v_{ij}^{k+1},& \\
\hdashrule{7.6cm}{0.4pt}{4pt}\\
x_{ij}^{k+1}=x_{j}(f_{max})(2a\eta+(1-a)),& \text{if } f_{ij}^{k}< (1-p)m \,\, \& \,\, It\geq N_e
\end{cases}
\end{equation}
\subsection{Benchmark problems}
A group of recognized benchmark multi-dimensional functions, which are extensively adopted in the optimization field, were employed to assess the performance of EPSOM, LDW-PSO, PSO-M, M-PSO, and the suggested PSO algorithm in terms of convergence accuracy and convergence speed. The benchmark functions are used herein as minimization problems, each of which offers a distinct level of complexity to the tested algorithm to be assessed. Function complexity such as; unimodality or multimodality, symmetric or asymmetric and separable or inseparable in its variables. The benchmark multi-dimensional functions are:
Rosenbrock function is an unimodal function that is extensively used for local exploration, which was first used in optimization assessment of Genetic Algorithms (GA) by De Jong \cite{de1975analysis}. According to Shang and Qiu \cite{shang2006note}, the n-dimensional Rosenbrock function ($n>4$) can be a bimodal function, that makes it even a more complex minimization problem, this complexity also originates from the function's asymmetry and variables inseparability. The global minimum value of Rosenbrock function is zero, which is located at $(1,1,...)$. The multi-dimensional Rosenbrock function reads as:
\begin{equation}
\label{eq::f_Rosenbrock}
f_{1}(x)=\sum_{i=1}^{n}100(x_{i+1}^{2}-x_{i})^{2}+(1-x_{i})^{2}
\end{equation}
Rastrigin function is a multimodal function that is employed for the performance assessment of evolutionary algorithms \cite{varIslandNum07}. Although it is extremely multimodal, the positions of the minima are evenly dispersed, the function is symmetric and separable. The global minimum value of Rastrigin function is zero that is located at $(0,0,...)$. Rastrigin function is defined as:
\begin{equation}
\label{eq::f_Rastrigrin}
f_2(x)=\sum_{i=1}^{n}(x_{i}^{2}-10cos(2\pi x_{i}))+10
\end{equation}
Griewank function is also a multimodal and symmetric function that is widely used for global optimization. The global minimum value of Griewank function is zero that is located at $(0,0,...)$. Following Locatelli \cite{locatelli2003note}, the function contains a huge number of local minima, which increases exponentially with the number of dimensions. According to Jumonji \emph{et al.} \cite{varIslandNum07}, Griewank function is inseparable in its variables and is defined as follows:
\begin{equation}
\label{eq::f_Griewank}
f_3(x)=\sum_{i=1}^{n}\frac{x_{i}^{2}}{4000} - \prod_{i=1}^{n}cos\left(\frac{x_{i}}{\sqrt{i}}\right)+1
\end{equation}
\section{Results}
Beneficial to evaluate the efficiency of the proposed PSO algorithm on large-scale problems, the benchmark functions, Griewank, Rastrigrin, and Rosenbrock, are set to have 30 dimensions.
To assess the performance of the proposed PSO algorithm in comparison to
EPSOM, LDW-PSO, PSO-M and M-PSO, the parameters of all the tested algorithms are identically selected for consistency.
The search and initialization space for all the functions are [-100 - 100]. The number of particles is set to 40, and a maximum iteration of 2000. Furthermore, $w$ is declining linearly from $w_{max} = 0.9$ to $w_{min} =0.1$, c1 is 1.4962, and c2 is 1.4962. Regarding the PSO-M, the mutation factor $mu$ is set to 0.05.
The proposed PSO method includes three more presetting parameters: one is the classification percentage around the fitness mean, denoted by p, and the other is the elitism with position mutation process initiation iteration, denoted by Ne, the last one is the mutation range, that is defined by $a$. In these simulations, these parameters are selected as $p=0.02$, $N_e=3$, and $a=0.5$. Consequently, the particles with fitness in the range of $\pm 2\%$ of mean particles' fitness value are classified in the middle category (``fair particles''), the elitism process starts after the third iteration, and the mutation range is $\pm50\%$ of $x_{j}(f_{max})$.
Twenty independent simulations were carried out on the three previously aforementioned functions for each minimization algorithm, which is beneficial to minimizing the statistical errors of the optimization performance of the previously mentioned algorithms. The outcomes of each algorithm were averaged across 20 simulations. The averaged fitness performance of PSO variants on 30-dimensional benchmark functions is depicted in Fig. \ref{fig::benchmark}. The optimization results show that the proposed PSO excels EPSOM, LDW-PSO, PSO-M and M-PSO in terms of convergence speed and global exploration capabilities, on all tested benchmark functions. Another important result from the figure, is that all the mutation based approaches have better global exploration capabilities, since their convergence accuracy is better than other PSO approaches, as seen from the optimization performance of Griewank, Rastringrin and Rosenbrock function.
\begin{figure}[ht ]
\centering
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Griewank}
\label{fig::Griewank}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Rastrigrin }
\label{fig::Rastrigrin}
\end{subfigure}
\hfill
\newline
\newline
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3a.pdf}
\caption{Rosenbrock }
\label{fig::Rosenbrock}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{fig/shaqa3d.pdf}
\caption{Zoom on proposed PSO }
\label{fig::proposed_pso}
\end{subfigure}
\caption{Fitness performance of PSO variants on multi-dimensional benchmark functions. Figures (a-c) depict comparison of EPSOM, LDW-PSO, PSO-M and M-PSO with the proposed PSO on a 30-dimensional functions (Griewank, Rastringrin and Rosenbrock). Figure (d) shows a zoomed-in plot of the performance of the proposed PSO on the three functions}
\label{fig::benchmark}
\end{figure}
Figure \ref{fig::proposed_pso} shows a zoomed-in plot of the minimization performance of the proposed PSO on the 30-dimensional Griewank, Rastrigrin, and Rosenbrock function. The figure undeniably shows the gigantic improvement of the elitism with targeted position mutation on the proposed PSO after the iteration $N_e=3$. The bad particles are now exploiting the particle's position with maximum fitness with high probability of exploring new particles' positions as a consequence of position mutation.
It can be seen from Fig. \ref{fig::benchmark} that the proposed PSO attained the global minimum ($10^{-15}$) for the 30-dimensional Griewank function in 11 iterations, whereas the other techniques needed 1900, 501, 732 and 597 iterations to reach considerably greater local minima by orders of magnitude, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. The suggested PSO attained the global minimum ($10^{-12}$) for the 30-dimensional Rastrigrin function in 11 iterations, whereas the other techniques required 1713, 607, 716 and 919 iterations to reach significantly greater local minima by orders of magnitude, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. In terms of the optimization results of the Rosenbrock function utilizing the suggested PSO, the proposed technique required 6 iterations to attain a minimum that outperformed the other approaches, considering that the comparative approaches required 1850, 552, 835 and 980 iterations, for EPSOM, LDW-PSO, PSO-M and M-PSO, respectively. For all tested function, the convergence rate for the proposed PSO is faster by orders of magnitude compared with all tested variants.
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/griewank_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Griewank function after 10 and 2000 iterations. \label{fig::Griewank-30-90}}
\end{figure}
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/rastrig_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Rastrig function after 10 and 2000 iterations.. \label{fig::Rastrig-30-90}}
\end{figure}
\begin{figure}[H]
\includegraphics[width=12 cm]{fig/rosenbrock_bar.pdf}
\caption{Fitness performance of PSO variants on 30, 60 and 90 dimensional Rosenbrock function after 10 and 2000 iterations.. \label{fig::Rosenbrock-30-90}}
\end{figure}
Figures \ref{fig::Griewank-30-90}, \ref{fig::Rastrig-30-90}, and \ref{fig::Rosenbrock-30-90} show the optimization results for the PSO-TPME and the previously described PSO variants. The optimal fitness value in the figures is the average optimal one of the solutions in 20 trials. To evaluate the early and late exploration capabilities of the PSO variations, the optimum fitness is estimated after 10 and 2000 iterations, respectively. The optimization was carried out on the previously specified benchmark functions with dimensions of 30, 60, and 90 while keeping the same presetting settings for all PSO variations, including the number of particles, constant. The figures undeniably reveal that the PSO-TPME's early exploration performance outperforms the overall (early and late) exploration performance of the investigated PSO variants by orders of magnitude. This is also proven for all previously described benchmark functions for large dimensional problems of 30, 60, and 60 dimensions. This clearly shows that PSO-TPME has remarkably fast and accurate convergence characteristics, which are supported by benchmark functions with various levels of complexity and a large number of dimensions.
\section{Discussion}
Starting point are five popular variants of particle swarm optimization.
The proposed new PSO variant (PSO-TPME)
is shown to dramatically improve convergence speed and global exploration capabilities.
This variant comprises the three major factors affecting convergence speed and global exploration capabilities: particles' classification, elitism, and mutation, as well as the original PSO's cognitive and social models. This variation introduced an alternative classification approach, elitism, and targeted position mutation, all of which were integrated into the basic PSO algorithm. The introduced particle classification process is simple to apply, requires low memory, is adaptive, and provides automated termination criteria based on convergence. These qualities of the proposed classification technique permitted the implementation of targeted elitism and mutation, in terms of targeting just the poor particles, terminating the elitism and mutation process automatically in the event of convergence, and applying different updating models (social and/or cognitive) based on the particle's category.
A set of benchmark multi-dimensional functions widely used in the optimization problems were used to compare the performance of the proposed PSO-TPME to EPSOM, LDW-PSO, PSO-M and M-PSO, in terms of convergence accuracy and convergence speed. The 30, 60, and 90-dimensional Griewank, Rastrigrin, and Rosenbrock functions are employed. Each of them provides a different level of complexity, such as unimodality or multimodality, symmetry or asymmetry, and separability or inseparability. For each benchmark function, many minimization simulations were performed, repeated, and averaged to reduce statistical errors. The simulations revealed that PSO-TPME surpasses the aforementioned variants by orders of magnitude in terms of convergence rate and accuracy.
\vspace{6pt}
\authorcontributions{Conceptualization, T. Shaqarin and B. R. Noack; methodology, T. Shaqarin; software, T. Shaqarin; validation, T. Shaqarin and B. R. Noack; formal analysis, T. Shaqarin; investigation, T. Shaqarin.; resources, T. Shaqarin; data curation, T. Shaqarin; writing---original draft preparation, T. Shaqarin and B. R. Noack; writing---review and editing, T. Shaqarin and B. R. Noack; visualization, T. Shaqarin; supervision, T. Shaqarin and B. R. Noack; project administration, B. R. Noack; funding acquisition, B. R. Noack. All authors have read and agreed to the published version of the manuscript.}
\funding{This work is supported
by the National Science Foundation of China (NSFC) through grants 12172109 and 12172111
and by the Natural Science and Engineering grant 2022A1515011492
of Guangdong province, China.}
\dataavailability{Not applicable}
\conflictsofinterest{ The authors declare no conflict of interest.}
\begin{adjustwidth}{-\extralength}{0cm}
\reftitle{References}
|
\section{Introduction} \label{sec-intro}
Few movements are as familiar and recognizable as human walking and
running. Almost any collection of dots, lines, or shapes attached to
an unseen walking figure is quickly identified and understood as
human. Studies in human perception have displayed walking motion
using only dots of light located at the joints and have found test
subjects quite adept at assessing the nature of the underlying
motion\cite{Johansson:1973:vpbmma}. In particular, subjects can
identify the gender of a walker and recognize specific individuals
from light-dot displays even when no other cues are
available\cite{Cutting:1977:rftw,Kozlowski:1977:rswdpd,Kozlowski:1978:rgwplmasst}.
In part because people are skilled at detecting subtleties in human
motion, the animation of human figures has long been regarded as an
important, but difficult, problem in computer animation. Recent
publications have presented a variety of techniques for creating
animations of human motion. Promising approaches include techniques
for manipulating keyframed or motion capture
data\cite{Bruderlin:1995:MSP,Rose:1996:EGMT,Unuma:1995:FPE,Witkin:1995:MW},
control systems for dynamic
simulations\cite{Hodgins:1995:AHA,Laszlo:1996:LCCAABW,Ngo:1993:SCR,Panne:1995:GOBL,Panne:1993:SAN},
and other procedural or hybrid
approaches\cite{Badler:1993:SHC,Bruderlin:1989:GDD,Cohen:1992:ISC,Girard:1985:CMC,Ko:1993:SLW,Laurent:1992:it,Perlin:1995:RTR}.
Each method has its own strengths and weaknesses, making the visual
comparison of results essential, especially for the evaluation of such
subjective qualities as naturalness and emotional expression.
The research community has not yet adopted a standard set of models and
there is currently enormous variety in the models and rendering styles
used to present results.
Our ability to make judgments about human motion from displays as
rudimentary as dot patterns raises an important question: Does the
geometric model used to render an animation affect a viewer's judgment
of the motion or can a viewer make accurate judgments independent of
the geometric model? There are three plausible but contradictory
answers to this question.
\vskip 0.0625in
{\bf Possibility 1. Simple representations may allow finer
distinctions when judging human motion.}
Simpler models may be easier to comprehend than more complex ones,
allowing the viewer's attention to focus more completely on the
details of the movement rather than on the details of the model. For
example, a stick figure is an obvious abstraction and rendering flaws
may be easily ignored. When more detailed models are used, subtle
flaws in rendering, body shape, posture, or expression may draw
attention away from the movements themselves. Complex models may also
obscure the motion. For example, the movement of a jacket sleeve might
hide subtle changes to the motion of the arm underneath.
\vskip 0.0625in
{\bf Possibility 2. Complex, accurate representations may allow finer
distinctions.}
People have far more experience judging the position and movement of
actual human shapes than they do judging abstract representations
such as stick figures. A viewer, therefore, may be able to make finer
distinctions when assessing the motion of more human-like
representations. Furthermore, complex representations provide more
features to identify and track. Each body segment in a polygonal
human model has a distinctive, familiar shape, making it
easier to gauge fine variations in both position and rotation.
\vskip 0.0625in
{\bf Possibility 3. Both simple and complex representations may allow
equally fine distinctions.}
The human visual system may use a displayed image only to maintain the
positions of a three-dimensional mental representation. Judgments
about the motion may be made from this mental representation rather
than directly from the viewed image. Displayed images must of course
supply enough cues to keep the mental representation accurate, but
additional detail and accuracy may be irrelevant. Just as joint
positions shown by light dots are sufficient to control the mental
representation, connecting the dots with a stick figure might not
improve the viewer's perception. Similarly, encasing a stick figure
within a detailed human body shape might likewise prove unnecessary.
\vskip .2in
Objective evidence is needed to determine which of these possibilities
is correct. We argue that definitive experiments to select between
possibilities~1 and~2 are impractical. The question of which style of
geometric model is more useful for judging motion is likely to be
highly complex and context dependent, affected by all of the variables
of both the motion and the rendering. If possibility~3 were correct
and model style were largely irrelevant, then we would be able to
perform critical comparisons of the motion synthesis techniques in the
literature by direct comparison of the substantially different
geometric models used in each publication. This paper provides
experimental evidence to disprove possibility~3 by showing that viewer
sensitivities to variations in motion are significantly different for
the stick figure model and the polygonal model shown in
Fig.~\ref{fig:polyAndStick}. In particular, for the types of motion
variation we tested, viewers were more sensitive to motion changes
displayed through the polygonal model than through the stick figure
model. This result suggests that stick figures may not always have
the required complexity to ensure that the subtleties of the motion
are apparent to the viewer.
\figureSingleW{
\centerline{\epsfysize=2.25in \epsfbox{./PsImages/examplePairs.eps}}
\caption{
Images of an animated human runner. The pair on the left compares
two running motions rendered using a polygonal model. On the
right, the same pair of motions are rendered with a stick figure
model. Modifications to the motion were controlled by a
normalized parameter,~$\lambda$, that varied between $\lambda=0$
and $\lambda=1$. These images are from the motion generated for
the additive noise test discussed in Section~\ref{subsec-noise}.
The difference in posture created by the additive noise can be
seen in the increased angle of the neck and waist in the right
image of each pair ($\lambda=1$).
} \label{fig:polyAndStick}
}
\section{Background} \label{sec-background}
Several researchers have used light-dot displays, also referred to as
biological motion stimuli, to study perception of human movements and
to investigate the possibility of dynamic mental
models\cite{Freyd:1987:dmr}. The light-dot displays show only dots or
patches of light that move with the main joints of walking figures
(Fig.~\ref{fig:dots}), but even these minimal cues have been shown
to be sufficient for viewers to make detailed assessments of the
nature of both the motion and the underlying figure.
\figureSingleN{
\centerline{\epsfxsize=3in \epsfbox{./PsImages/dotRunner.eps}}
\caption{
The dot pattern on the left shows the joint locations of a human
runner at a single point in time. On the right, these joint
locations are shown over the course of one step in the running
cycle. Although it is difficult to determine the nature of these
patterns from a still image, studies show that most people are
able to recognize the motion and even to make fine judgments when
shown moving sequences of similar images.
} \label{fig:dots}
}
The ability to perceive human gaits from light-dot displays has been
widely reported to be acute and robust. Early experiments by
Johansson reported that 10-12~light dots ``evoke a compelling
impression of human walking, running, dancing,
etc.''\cite{Johansson:1973:vpbmma}. Because such displays provide
motion cues independent of form or outline, other investigators have
used them to study human motion perception. Work by Cutting and
Kozlowski showed that viewers easily recognized friends by their
walking gaits on light-dot displays\cite{Cutting:1977:rftw}. They
also reported that the gender of unfamiliar walkers was readily
identifiable, even after the number of lights had been reduced to just
two located on the ankles\cite{Kozlowski:1977:rswdpd}. In a published
note, they later explained that the two light-dot decisions were
probably attributable to stride
length\cite{Kozlowski:1978:rgwplmasst}. Continuing this work,
Barclay, Cutting, and Kozlowski showed that gender recognition based
on walking gait required between 1.6~and 2.7~seconds of display, or
about two step cycles\cite{Barclay:1978:tasfigptigr}. Our experiments
used pairs of running stimuli 4~seconds in duration that displayed
about six strides. We noticed that test subjects often marked
their answer sheets near the midpoint of the second stimuli which is
consistent with Barclay's results.
Motion is apparently essential for identifying human figures on
light-dot displays. The Cutting studies reported that while moving
light-dot displays were recognized immediately, still light-dot
displays of a walking figure were not recognized as human. Poizner
and colleagues also noted that movement is required for accurately
reading American Sign Language gestures\cite{Poizner:1981:poaslidpld}.
This capacity to recognize moving figures was shown to be robust in
the presence of masking by additional light points. In a modified
experiment, subjects were shown light-dot displays of walkers facing
either left or right and asked to determine walking direction. Only
complex masks of extraneous light dots moving in patterns that were
similar to those of the walking figure were able to disrupt viewer
judgments\cite{Cutting:1988:mtmohg}.
Appropriate synthetic movements are easily accepted as human when
rendered as light-dot displays. Cutting and colleagues
found that apparent torso structure and rotation were strongly
correlated with judgments of walker gender\cite{Cutting:1978:abifgp}.
Cutting then constructed a simple mathematical model of light-dot
motion for human walkers and computed displays of synthetic walkers.
Viewers easily identified the synthetic displays as human walkers and
accurately determined the intended gender of the walkers. These
experiments clearly showed that variations in torso rotation are
important for gender judgments. Accordingly, we chose to measure
viewer sensitivity to torso rotations in one of our experiments.
Proffitt and colleagues found that occlusion of light dots by clothing
or human body segments plays an important role in gait judgment and
may also provide information about body
outlines\cite{Proffitt:1978:trooirmimpld}. Synthetic displays without
occlusion yielded poorer subject performance. These experimental
observations suggest that extremely simple models of human figures,
such as thin stick figures, may present similar difficulties for
the~viewer.
Surprisingly, the perception of rigid body segments between moving
light dots at joints does not generalize to movements of isolated
pairs of light dots. Ishiguchi showed test subjects one fixed light
dot and a second one that moved on an arc of $\pm15$ degrees as if it
were on the end of a pendulum with the first light dot as the pivot
joint\cite{Ishiguchi:1988:teoooiesfdm}. Viewers perceived the dots as
attached to a flexible bar held fixed at the first light dot rather
than as a rigid bar moving as a pendulum. Thus the perception of
rigid body segments in the largely pendulum-like movements of human
walking is exceptional; perhaps the ensemble of dots is important, or
perhaps the movements are so intimately familiar that the perception
of an assembly of flexible bars is overridden.
\section {Experimental Methods}
\figureSingleW{
\centerline{\epsfxsize=140.0mm \epsfbox{./PsImages/assemPoly.eps}}
\caption{
Examples from the motion sequences rendered with the polygonal model.
{\bf First Row:} Original motion sequence, $\lambda=0$, used in all tests.
{\bf Second Row:} Torso rotation motion sequence with $10\times$
magnification of the torso rotation, $\lambda=1$.
{\bf Third Row:} Dynamic arm motion sequence with maximum exaggeration,
$\lambda=1$.
{\bf Fourth Row:} Additive noise motion sequence with sinusoidal noise of
$\pm0.15$~radians, $\lambda=1$.
Images are spaced at intervals of $0.067$ seconds.
}\label{fig:motPoly}
}
\figureSingleW{
\centerline{\epsfxsize=140.0mm \epsfbox{./PsImages/assemStik.eps}}
\caption{
Examples from the motion sequences rendered with the stick figure
model.
{\bf First Row:} Original motion sequence, $\lambda=0$, used in all tests.
{\bf Second Row:} Torso rotation motion sequence with $10\times$
magnification of the torso rotation, $\lambda=1$.
{\bf Third Row:} Dynamic arm motion sequence with maximum exaggeration,
$\lambda=1$.
{\bf Fourth Row:} Additive noise motion sequence with sinusoidal noise of
$\pm0.15$~radians, $\lambda=1$.
Images are spaced at intervals of $0.067$ seconds.
} \label{fig:motStik}
}
While it is impossible to exhaustively test all of the variables that
may affect a perceived motion, we can use A/B comparison tests to form
a preliminary assessment of whether the geometric model affects a
viewer's perception of motion. We evaluated three different types of
motion variation in separate experiments described below: torso
rotation, dynamic arm motion, and additive noise. For each
experiment, the modifications to the motion were controlled by a
normalized parameter, $\lambda$, that varied between $\lambda=0$ and
$\lambda=1$. \mbox{Figs.~\ref{fig:motPoly}}
\mbox{and~\ref{fig:motStik}} show sequences of images excerpted
from the base motion sequence, $\lambda=0$, and the modified
sequences, $\lambda=1$, used in all three experiments. Joint angle
trajectories are shown in Fig.~\ref{fig:dplts} to illustrate the key
components of the base motion and the modified motions created by
setting $\lambda=1$.
In each of the three tests, subjects viewed pairs of animated
sequences rendered using the same geometric model and were asked
whether the motions in the two sequences were the same or different.
We then computed a sensitivity measure for each type of geometric
model. The difference between the sensitivity values is a measure of
whether a particular subject was better able to discriminate between
the motions when they were rendered with a polygonal model or with a
stick figure model.
\subsection {Experiment One: Torso Rotation}
This experiment measured whether a subject's ability to differentiate
between larger and smaller yaw rotations of a runner's torso was
affected by the geometric model used for rendering. The motion
sequences were generated by making kinematic modifications to data
obtained from a physically based dynamic simulation of a human
runner\cite{Hodgins:1995:AHA}. The torso's rotation about the
longitudinal axis, or yaw relative to the pelvis, was exaggerated
(Fig.~\ref{fig:showRot}--A). The neck was counter-rotated to
compensate for the torso rotation so that the facing direction of the
head remained unchanged.
\figureSingleN{
\centerline{
\epsfysize=2in \epsfbox{./PsImages/showRot.eps}
}
\caption{
Degrees of freedom for data plotted in Fig.~\ref{fig:dplts}.
{\bf A.}~Rotation of torso at waist about longitudinal axis relative to pelvis.
{\bf B.}~Rotation of arm at shoulder about transverse axis relative to torso.
{\bf C.}~Rotation of torso at waist about transverse axis relative to pelvis.
} \label{fig:showRot}
\vskip 0.05in
}
\figureSingleW{
\centerline{
\epsfxsize=2in \epsfbox{./PsImages/motionTorso.eps}
\epsfxsize=2in \epsfbox{./PsImages/motionElbows.eps}
\epsfxsize=2in \epsfbox{./PsImages/motionNoise.eps}
}
\caption{
Selected joint angle trajectories demonstrating motion differences
plotted for base motion, $\lambda = 0$, and for extremes of
modified motion, $\lambda = 1$.
{\bf A.}~Rotation of torso at waist about the longitudinal axis
(\textit{z}--axis) for torso rotation test.
{\bf B.}~Shoulder angle about the transverse axis
(\textit{y}--axis) for dynamic arm motion test.
{\bf C.}~One representative modification for the additive noise
test: the rotation of torso at waist about the transverse axis
(\textit{y}--axis).
} \label{fig:dplts}
}
The magnitude of the exaggeration in torso rotation was controlled by
a normalized parameter, $\lambda$. A value of $\lambda=0$ gave a
magnification factor of $1\times$ so that the modified motion was
identical to that of the original data. Larger values of $\lambda$
correspond linearly to higher magnification factors, with $\lambda =
1$ yielding a $10\times$ magnification of the torso rotation. The
motion of body segments below the waist was left unchanged
(Figs.~\ref{fig:motPoly},~\ref{fig:motStik},
and~\ref{fig:dplts}--A).
The test consisted of a series of 40~pairs of motion sequences divided
into two sets of 20~pairs each. One set was rendered with the stick
figure model and the other with the polygonal model
(Fig.~\ref{fig:polyAndStick}). All other parameters used to render
the animations, such as lighting, ground models, and camera motion,
were identical for the two sets. Within each set, half of the pairs
were randomly selected to show two different motion sequences
(different $\lambda$ values). Of those that were different, the pairs
with the largest disparity in $\lambda$ were placed toward the
beginning of each set so that the questions became progressively more
difficult. To minimize bias due to fatigue or learning effects, we
varied the order in which the two sets were presented. Asymmetric
learning effects would not necessarily be minimized by this ordering.
Twenty-six student volunteers who were not familiar with the animations
served as subjects. All had normal or
corrected-to-normal vision. Subjects were tested in groups of two or
three in a quiet room. They were instructed to remain silent and not to collaborate during the test.
The test stimulus was presented on a 20-inch
monitor approximately three feet from the subjects. All animations
were prerendered and shown at 30~frames per second in NTSC
resolution. These experimental conditions were selected because they
match the viewing conditions commonly encountered when watching
animated motion.
Subjects were told that they would be shown a series of 4-second
computer-generated animations of a human runner and that the
animations would be grouped in A/B~pairs with 5~seconds of delay
between the presentation of each pair. Subjects were asked to view
each pair and then indicate on a response sheet whether the two
motions were the same or different. They were also informed that the
variations would be confined to the motion of the runner's upper body
and that the questions would become progressively more difficult. A
monetary reward for the highest percentage of correct responses was
offered as an incentive to
|
all test subjects. Subjects were not told
the purpose of the experiment.
\subsection {Experiment Two: Dynamic Arm Motion}
This experiment measured whether a subject's ability to differentiate
between larger and smaller arm motions was affected by the geometric
model used for rendering. The motion sequences were generated by
modifying the desired fore-aft rotation about the transverse axis at
the shoulder joint in the dynamic simulation of the human runner
(Fig.~\ref{fig:showRot}--B). The control routines then computed
torques based on the desired value of the shoulder joint. These
torques were applied to the dynamic model. The resulting motion is
shown in Figs.~\ref{fig:motPoly},~\ref{fig:motStik},
\mbox{and~\ref{fig:dplts}--B}. Because the motion was dynamically
simulated, the exaggerated arm motion also had subtle effects on other
aspects of the running motion.
The magnitude of the exaggeration in arm motion was controlled by a
normalized parameter,~$\lambda$. A value of $\lambda = 0$ gave a
magnification factor of $1\times$ so that the modified motion was
identical to that of the original data. Larger values of $\lambda$
correspond linearly to higher magnification factors, with $\lambda =1$
yielding a $1.5\times$ magnification of the shoulder rotation.
Twenty-four student volunteers who had not participated in the first
experiment were subjects for this second experiment. Testing
procedures and format were identical to those used in the first
experiment.
\subsection {Experiment Three: Additive Noise} \label{subsec-noise}
The format of this experiment was identical to that of the first two,
except for the manner in which the running motion was modified. For
this experiment, time-varying noise was added to the joint angles for
the waist, shoulders, and neck. The noise was generated using a
sinusoidal wave generator\cite{Schlick:1995:WGCG} with frequency
varying randomly about that of the runner's gait at approximately 3~Hz.
The amplitude of the additive noise was controlled by a normalized
parameter, $\lambda$, as in the torso rotation test. A value of
$\lambda = 0$ resulted in motion data that was identical to the
original data (zero noise amplitude). The maximum noise amplitude
used, given by $\lambda = 1$, produced a variation of $\pm0.15$~radians
about the original joint angles
(\mbox{Figs.~\ref{fig:motPoly} and \ref{fig:motStik}}). One
representative joint angle, rotation of the torso at the waist about
the transverse axis, is shown in
\mbox{Figs.~\ref{fig:showRot}--C} \mbox{and~\ref{fig:dplts}--C}.
Twenty-six student volunteers who had not participated in the previous
experiments were selected as subjects. Testing procedures were
identical to those used in the first and second experiments.
\section {Results} \label{sec-results}
To analyze the data from the experiments, we used the responses to
compute the Choice Theory sensitivity measure for each subject on each
test set. The sensitivity measure, $log(\alpha)$, is defined as
\begin{equation}
log(\alpha) = { log(H/(1-H)) - log(F/(1-F)) \over 2 } ,
\end{equation}
where $H$ is the fraction of pairs in a set that were {\em different}
and which the subject labeled correctly, and $F$ is the fraction of
pairs in a section that were {\em the same} and which the subject
labeled incorrectly\cite{Mcmillan:1991:DT}.
This measure is zero when the subject's responses are uncorrelated
with the correct responses to cause a~$50\%$~correct score, and
increases as response correlation improves, as illustrated in
Fig.~\ref{fig:svspc}.
Additionally, the measure is symmetric,
naturally invariant with respect to response bias, and suitable for
use as a distance metric\cite{Mcmillan:1991:DT}.
After sensitivity scores had been determined, a {\it post hoc}
selection criteria was used to build a subgroup of ``skilled''
subjects who had achieved a sensitivity score indicating performance
significantly better than chance with {\bf \it either} the polygonal
or the stick figure models. Significantly better than chance was
defined as at least $73\%$ correct, which corresponds to a sensitivity
score of $log(\alpha) \geq 1.0$. Analysis was computed both
for the group of all subjects and for the group of skilled subjects.
Sensitivity scores for each experiment averaged within subject groups
are shown in Fig.~\ref{fig:togbar}.
\figureSingleN{
\centerline{\epsfysize=2in \epsfbox{./PsImages/sensitivityVsCorrect.eps}}
\vskip -0.25in
\caption{
Plot of sensitivity score, $log(\alpha)$, versus fraction correct
at zero bias.
} \label{fig:svspc}
}
\figureSingleN{
\centerline{\epsfxsize= 0.5 \textwidth \epsfbox{./PsImages/togetherBar.eps}}
\vskip -0.25in
\caption{
Sensitivity scores by experiment averaged over subject groups.
Skilled subjects are those who achieved a sensitivity score of
$log(\alpha) \geq 1.0$ on either the polygonal or the stick
figure portion of the test. Note that sensitivity scores are
consistently higher with the polygonal model.
} \label{fig:togbar}
}
In Section~\ref{sec-intro}, we proposed three possible answers to the
question of whether the geometric model used for rendering affects a
viewer's perception of motion. The third possible answer implied that
subjects would achieve similar sensitivity measures when asked
identical questions about the motion of stick figure models or
polygonal models. To test this hypothesis, we computed the difference
in sensitivity for each subject:
\begin{equation}
\Delta log(\alpha) = log(\alpha_{poly})-log(\alpha_{stick}) .
\end{equation}
The results from the three tests are summarized in
Table~\ref{tab:results}. For the torso rotation test, the mean of the
difference in sensitivities across all subjects was $0.43$ with a
standard deviation of $0.77$. Student's $t$--test for paired
samples\cite{Press:1992:NR} shows this difference to be significant,
$p < 0.012$. For the group of skilled subjects, the mean rose to
$0.73$ while the standard deviation fell to $0.68$. The $t$--test for
paired samples shows this difference to be significant, $p < 0.001$.
For the dynamic arm motion test, the mean of the difference in
sensitivities across all subjects was $0.41$ with a standard deviation
of $0.59$, a difference significant at $p < 0.003$. For the group of
skilled subjects, the mean was $0.55$ and the standard deviation was
$0.59$, a difference significant at $p < 0.001$.
For the additive noise test, the mean of the difference in
sensitivities across all subjects was $0.74$ with a standard deviation
of $0.69$, a difference significant at $p < 0.001$. For the group of
skilled subjects, the mean was $0.72$ and the standard deviation was
$0.73$, a difference significant at $p < 0.001$.
\begin{table*}[tb]
\centerline{\small
\begin{tabular}{|l||r|r|c||r|r|c|}
\hline
\hline & \multicolumn{3}{c||}{All Subjects} & \multicolumn{3}{c|}{Skilled Subjects} \\
& Mean & Std. Dev. & Prob. Err. & Mean & Std. Dev. & Prob. Err. \\
\hline
\hline Torso Rotation & $0.43$ & $0.77$ & $p < 0.012$ & $0.73$ & $0.68$ & $p < 0.001$ \\
\hline Dynamic Arm & $0.41$ & $0.59$ & $p < 0.003$ & $0.55$ & $0.59$ & $p < 0.001$ \\
\hline Additive Noise & $0.74$ & $0.69$ & $p < 0.001$ & $0.72$ & $0.73$ & $p < 0.001$ \\
\hline
\hline
\end{tabular}
}
\caption{
Summary of results from the three experiments. Mean and standard
deviation are for $\Delta log(\alpha)$ by subject group.
Probability of error is calculated with Student's $t$--test for
paired samples. Positive values for mean $\Delta log(\alpha)$ in
all six test/group combinations indicate that subjects were able
to discriminate better with the polygonal model.
} \label{tab:results}
\end{table*}
Fig.~\ref{fig:histogramAll} shows histograms of the sensitivity
differences, $\Delta log(\alpha)$, for the three test conditions.
Positive values correspond to higher sensitivity for the set rendered
with the polygonal model.
Our results indicate that, for the three types of motion variation
tested, subjects were better able to discriminate motion variations
using the polygonal model than they were with the stick figure model.
This result holds to a high level of significance both for the
analyses computed on the group of all subjects and for the group of
skilled subjects, although the magnitude of the differences are, in
general, greater within the group of skilled subjects.
\section {Discussion} \label{sec-disc}
\figureSingleW{
\centerline{
{\epsfxsize=2.1in \epsfbox{./PsImages/histogramsTorsoII.eps}}
{\epsfxsize=2.1in \epsfbox{./PsImages/histogramsElbowsII.eps}}
{\epsfxsize=2.1in \epsfbox{./PsImages/histogramsNoiseII.eps}}
}
\caption{
Histogram of sensitivity differences for {\bf A.} the torso
rotation test, {\bf B.} the dynamic arm motion test, and {\bf C.}
the additive noise test. The upper graphs show the occurrence
frequency for sensitivity differences, $\Delta log(\alpha)$,
across all subjects. The bottom graphs show the data for subjects
who had a sensitivity of $log(\alpha) \geq 1.0$ on either
the polygonal or the stick figure portion of the test. Positive
values of the sensitivity difference indicate a higher sensitivity
to changes in the motion with the polygonal model. (Bucket size
$=0.5$.)
} \label{fig:histogramAll}
}
Although the differences in sensitivity measures show that our
subjects were more sensitive to motion changes when a polygonal model
was used for rendering, our results can not be generalized to
say that polygonal models are always better than stick figure
models for perceiving motions. Rather, the two types of geometric models
are distinctly different and, in the cases we tested, polygonal models
allowed better discrimination. There may be variations for which
the difference in sensitivity has the opposite sign, implying that
stick figures might be a better model for making fine
discriminations about that particular motion variation.
Our results, however, do show that stick figures and polygonal models
are not equivalent for tasks that require making fine discriminations
about motion. This observation implies that any useful comparison of
motion sequences requires that the same models and rendering methods
be used for each and might indicate that the community would benefit
from adopting a standard set of human models. In particular,
comparing motions of a stick figure model to those of a more complex
model may be meaningless because viewer sensitivities can differ
substantially. As a practical matter, animators may want to avoid
conducting preliminary tests only with stick figures or other simple
models because it is likely that viewers would have different
sensitivities to the more complex models that would be used in the
final rendering.
Considerable familiarity with the motion appears to make differences
in the geometric models less significant. For example, when the
authors of this paper took the tests, they answered nearly all
questions correctly. Of course, the authors were not included among
the subjects whose data are reported above. If a larger subject pool
showed that subjects who were very familiar with particular animated
motions showed equal sensitivity to the two models, then we would have
evidence that using stick figures for preliminary {\it pencil tests}
of motion sequences will provide good information about the motion.
The subject, in this case the animator, is very familiar with the
motion and may be able to make subtle observations independent of the
geometric models used for rendering.
Our results do not conflict with the findings discussed in
Section~\ref{sec-background}. Previous studies have found that
subjects were able to use a variety of models to make judgments about
human motion, these studies did not address how the subject's
proficiency might be affected. As can be seen from
Fig.~\ref{fig:togbar}, the subjects we tested were able to make
distinctions using both polygonal and stick figure models; however
they were better able to make these distinctions when viewing motion
rendered with the polygonal model.
For the two models used in these experiments, the more complex model
was also more human-like but that may not always be the case. Complex
but abstract models may be useful for making particular features of the
motion visible in some applications. For example, crash test dummies
have markings for the center of mass of each body segment and other
visualization techniques such as force vectors have been
used successfully in biomechanics research.
We used simulation combined with kinematic modifications to generate
the motion for these studies because it allowed us to control the
variations explicitly. Motion capture data would be an interesting
source for this kind of study because it more closely matches human
motion. However, even two consecutive captures of an actor performing
a simple task will have significant differences because of the
variability of human motion. Capturing a set of consecutive motions
with a controlled variation for sensitivity tests would be difficult
because of this variability.
The three techniques used to modify the motion were chosen both for
their relevance to current animation techniques and for their
perceptual significance. We chose torso rotation because previous
studies have shown that the motion of the torso provides important
cues for gender determination and subject
recognition\cite{Cutting:1978:abifgp}. The kinematic modification
used for the torso rotation test is also similar to the modifications
an animator might make when keyframing motion or adjusting motion
capture data. Similarly, the adjustments of the desired shoulder
joint angles used in the dynamic arm motion test are typical of the
adjustments that an animator might make to a dynamic simulation in
order to change the style of the resulting motion. Finally, noise is
found in naturally occurring motions, and additive noise
generators have been used to synthesize natural and appealing human
motion\cite{Perlin:1995:RTR}.
A potential problem with the experimental design used in this study is
that the test must be of an appropriate difficulty. If the test is
too difficult, then subject responses will be guesses
regardless of which model is presented. Conversely, if the test is
too easy, then all subject responses will be correct. In
either case, the data gathered will not be useful. We can increase or
decrease the difficulty of a test by changing the spacing of the
$\lambda$ values for the trials or the amount of information given to
the subjects about the alterations to the motion. Unfortunately, it
can be difficult to devise a test sequence of appropriate difficulty.
This problem could be overcome by using tests that adaptively adjust
difficulty level by selecting subsequent questions based on past
responses. Alternatively, selection criteria can be used to cull
subjects whose responses are not significantly correlated with the
test stimuli.
While our assessment that the polygonal models allow greater
sensitivity holds irrespective of culling, it is interesting to note
how selection based on performance criteria does affect the data. As
can be seen from the average scores shown in Fig.~\ref{fig:togbar},
subjects who took the torso rotation test achieved lower scores than
did those who took the additive noise test, probably because the torso
rotation test was more difficult. Comparing the results of the torso
rotation test before and after culling shows that the mean of $\Delta
log(\alpha)$ as well as the shape of the histograms in
Fig.~\ref{fig:histogramAll}.A were notably different between the
group of all subjects and the group of skilled subjects. For the
easier, additive noise test, the selection criteria has essentially no
effect. Moreover, the effect of the selection criteria on the torso
rotation data appears to make it more closely resemble the data from
the additive noise test, thereby supporting the notion that lowering
the difficulty of the test and selecting subjects based on performance
criteria are approximately equivalent.
Although we did not formally measure the subjects' perceptions of how
well they did on the test, it appeared that their perceptions did not
always match their performance. Several subjects were certain that
they had scored higher on the section with the stick figure model when
in fact they had a higher sensitivity to motion changes with the
polygonal model.
To create the animation sequences for these tests, we altered only the
motion and the geometric models used; all other aspects of the
rendering were held constant. It would be interesting to explore
whether, and how, other aspects of the rendering affect the perception
of motion as well as whether these results hold for behaviors other than
running. For example, we have informally observed that the motion
of the simulated runner appears more natural when the tracking camera
has a constant velocity rather than one that matches the periodic
accelerations of the runner's center of mass. When the camera motion
matches the acceleration of the center of mass exactly, the running
motion appears jerky. More sophisticated models that incorporate
clothing and skin may help to smooth out rapid accelerations of the
limbs and make the motion appear more natural. Motion blur probably
plays a similar role. Textured ground planes and shadows help to
determine motion of the feet with respect to the ground and may
provide important clues about the details of the motion.
If we had enough psychophysical results to build a model of how people
perceive motion, we could optimize the rendering of animated sequences
by emphasizing those factors that would make the greatest differences
in how a viewer perceives the sequence either consciously or
unconsciously. This approach of using
results from the psychophysical literature to refine rendering
techniques has already been used successfully for still
images\cite{Ferwerda:1996:MVARIS,Kawai:1993:RGBR,Teo:1994:PID}.
\section*{Acknowledgments} \label{sec-ack}
The authors would like to thank Jacquelyn Gray, John Pani, Neff
Walker, and the reviewers for their valuable comments. This project
was supported in part by NSF NYI Grant No. IRI-9457621, Mitsubishi
Electric Research Laboratory, and a Packard Fellowship. An earlier
version of this work, reporting preliminary results, appeared in {\it
The Conference Proceedings of Graphics Interface '97}.
|
\section{Introduction}
The $T\bar{T}$ deformation of two dimensional quantum field theories provides a concrete set-up to study non-local effects in quantum field theory, in particular those which might arise from coupling the theory to gravity. Due to some remarkable properties of the $T\overline{T}$ operator found by Zamolodchikov \cite{Zamolodchikov:2004ce}, it turns out that the spectrum of energy eigenvalues of the deformed theory on the cylinder (i.e., when the spatial slice is a circle) can be solved exactly, given the undeformed spectrum. This spectrum shows some tantalizing properties which are reminiscent of string theory or theories with a UV completion, despite the operator being irrelevant \cite{Cavaglia:2016oda, Callebaut:2019omt, Smirnov:2016lqw, Dubovsky:2018bmo, Frolov:2019nrr, Sfondrini:2019smd}. For instance, with a particular sign of the deformation, the spectral density of the theory develops a Hagedorn growth of states. On the other hand, for the opposite sign of the coupling, the energies exactly match with the gravitational quasi-local energies of black holes in $AdS_3$ with a radial cutoff on the asymptotic region \cite{McGough:2016lol, Kraus:2018xrn, Guica:2019nzm}. This latter feature is particularly interesting because getting rid of the asymptotic region in AdS/CFT would be a very promising starting point in moving towards quantum gravity beyond asymptotically AdS spaces \cite{Gorbenko:2018oov}.
In the past few years, much effort has gone into understanding various apsects of $T\overline{T}$ deformed quantum field theories, such as the spectrum on the circle and its complexification, sphere and torus partition functions \cite{Datta:2018thy, Aharony:2018bad, Caputa:2019pam, Mazenc:2019cfg}, the holographic aspect of the $T\overline{T}$ deformation, correlation functions on the Euclidean plane \cite{Kraus:2018xrn, Aharony:2018vux, Cardy:2019qao} and higher-dimensional generalization \cite{Taylor:2018xcy, Hartman:2018tkw, AitorsLastHepth}. Furthermore, a particularly interesting direction is the study of the entanglement structure of states in these (non-local) theories \cite{Donnelly:2018bef, Lewkowycz:2019xse, Murdia:2019fax, Banerjee:2019ewu, Asrat:2020uib}. However, it would be fair to say that beyond the deformed energy spectrum and partition functions, many of these aspects are not fully understood. In 0+1 dimensions, i.e., in $T\overline{T}$ deformed quantum mechanics \cite{Gross:2019ach, Gross:2019uxi, Iliesiu:2020zld}\footnote{See also \cite{Stanford:2020qhm} for an interesting alternative proposal for finite cutoff JT gravity.}, the deformed spectrum of the theory is all one really needs, as this entirely fixes the correlation functions of the deformed theory. However, in 1+1 dimensions, this is not true -- along with the energy eigenvalues, the energy eigenstates of the theory also change under the $T\overline{T}$ deformation, something which is clearly important to keep track of when we study observables such as correlation functions or entanglement entropy. Furthermore, for the holographic sign of the deformation, the flow of eigenstates is intimately tied with the idea of the ``surface-state correspondence'' proposed in \cite{Miyaji:2015yva, Miyaji:2015fia} (see also \cite{Nomura:2018kji}), which was at least in part inspired by the analogy between AdS/CFT and tensor-networks (see, for instance, \cite{Swingle:2009bg, Swingle:2012wq, Nozaki:2012zj, Pastawski:2015qua, Czech:2015kbp, Hayden:2016cfa, Bao:2018pvs}). Our central objective here will be to study the flow of energy eigenstates under the $T\overline{T}$ deformation, and the effect this has on the flow of correlations functions. We hope that our results will also shed some light on other issues such as entanglement entropy, surface-state correspondence/tensor networks in AdS/CFT, etc.
\subsection*{Summary and outline}
We will focus primarly on the flow of energy eigenstates, operators and correlation functions in a $T\overline{T}$ deformed quantum field theory in Lorentzian signature. Motivated by the formula for the deformed energy spectrum, plus the results on $T\overline{T}$ deformation in $0+1$ dimensions \cite{Gross:2019ach,Gross:2019uxi}, we take as our starting point a definition of the $T\overline{T}$ deformed theory from a Hamiltonian point of view, namely that the Hamiltonian $H_\l$ and momentum $P$ of the deformed theory change under the flow as
\begin{eqnarray} \label{def}
\partial_\l H_{\l} = \int dx_1\;\mathcal{O}_{T\overline{T}}^{(\l)}(x_0,x_1),\quad \quad \partial_\l P = 0,
\end{eqnarray}
with $\l$ the deformation parameter. The superscript $\l$ on the $T\overline{T}$ operator is meant to indicate that the stress tensor is that of the theory at $\l$. With this definition, the translation symmetries of the original theory are maintained along the flow. Classically, this definition is equivalent to the definition in terms of flow of the action proposed by Smirnov and Zamolodchikov in \cite{Smirnov:2016lqw}, but quantum mechanically there could be differences arising from operator ordering related counter-terms. At any rate, we will take the definition \eqref{def} as our starting point. We will later show that this definition of the $T\overline{T}$ deformation in Lorentzian signature is consistent with the other known results, such as, for instance, the deformed $S$-matrix \cite{Cavaglia:2016oda, Dubovsky:2017cnj}
Given this definition, we begin our analysis in section \ref{sec:rewrite} with the following crucial observation: the spatial integral of the $T\overline{T}$ operator can always be written as a sum of two terms
\begin{eqnarray}\label{rewrite}
\pa_{\l}H_\l = i\left[H_\l, \mathcal{X}^{(\l)}\right]+\mathcal{Y}^{(\l)},
\end{eqnarray}
where explicit expressions for $\mathcal{X}^{(\l)}$ and $\mathcal{Y}^{(\l)}$ are given in equation \eqref{X&Y}. The first of these terms is clearly a total-in-time derivative; as such it does not change the energy eigenvalues, but merely implements a \emph{canonical transformation} on phase space, or equivalently a \emph{Bogoliubov transformation} on the Hilbert space. A lattice version of this term was also found in \cite{Pozsgay:2019ekd}. On the other hand, the second term $\mathcal{Y}^{(\l)}$ turns out to be a manifestly factorized operator, i.e., a product of two spatial integrals of the stress tensor (see equation \eqref{X&Y}). This rewriting directly implies the known formula for the deformed energy spectrum of the theory \cite{Zamolodchikov:2004ce, Smirnov:2016lqw}, and also simplifies the analysis of eigenstates in what follows.
With this observation in hand, we compute various quantities as a function of $\l$, both on the plane and cylinder. The most basic ones are the energy eigenstates. Since translation symmetries remain unbroken under the flow, these states $\ket{E(\l),k}$ are labelled by the energy and momentum. In case of the spatial topology being a circle, the momentum is quantized in units of the circle length. Due to the $T\overline{T}$ deformation, the energy eigenstates start to mix and we give an explicit expression for the unitary matrix $U$ implementing that mixing in section \ref{sec:unitary}. This unitary $U$ depends on the deformed stress tensor and in section \ref{sec:Cauchyslice}, we rewrite it in terms of a kernel which involves a path integral over a fluctuating ``worldsheet'', which we dub the \emph{Cauchy string}.
We then turn to the question of correlation functions in section \ref{sec:operators}. On the plane, we consider correlators of two types of operators -- the first type are operators of the original seed theory, but time evolved with the deformed Hamiltonian. We obtain a flow equation for the correlation functions of this class of operators on the plane, which agrees with that of \cite{Cardy:2019qao} and can be physically interpreted in terms of a ``state-dependent diffeomorphism'' via the attachement of a stress tensor ``Wilson line''. The second type of operators are what we call \emph{dressed operators}. The definition of these operators is motivated by the simple rewriting of the spatial integral of the $T\overline{T}$ operator in equation \eqref{rewrite}. In particular, the $\mathcal{Y}^{(\l)}$ term drops out on the plane if we restrict attention to finite energy/near-vacuum states, and so the $T\overline{T}$ deformation on the plane acts as a pure canonical transformation in classical terms, or a Bogoliubov transformation quantum mechanically. With this in mind, the dressed operators are defined as the ``canonically transformed'' operators, $\widetilde{O}=U O U^{-1}$. These dressed operators have the property that they are causal, i.e. they commute with each other at spacelike separation, and additionally their correlation functions, the structure constants in their commutator algebra etc. are invariant along the flow. However, the dressed operators do not spacelike commute with the operators of the seed theory, i.e., they are non-local with respect to the original seed operator algebra. In particular, we can also construct a (conserved) dressed stress tensor (which we emphasize is different from the \emph{local} stress tensor) such that its correlation functions on the plane, its algebra etc. remain invariant under the flow. A deformed CFT on the plane therefore continues to have a conserved, traceless stress tensor which satisfies the same commutator algebra as in the undeformed CFT, albeit one which is non-local with respect to the seed operators. As an example, we give an explicit expression for the dressed operators in the classical $T\overline{T}$ deformed free, scalar field theory. On the cylinder, the situation with correlation functions is much more complicated and we do not have a complete picture for the flow of operators/correlation functions. Nevertheless, for \emph{dressed} operators, we are able to write the deformed correlation functions as an integral transform of the original correlators, just as in 1d $T\overline{T}$ \cite{Gross:2019uxi}.
In section \ref{sec:further}, we briefly discuss how the expected CDD factor in the flat space S-matrix of $T\overline{T}$ deformed theories arises from our analysis. We then give a 2+1 dimensional gravitational viewpoint on the unitary $U$, reminiscent in spirit and form of the gravitational kernels which have appeared previously in \cite{Freidel:2008sh,Mazenc:2019cfg, Tolley:2019nmm}. Finally, we also propose a tensor network interpretation of our results in the context of AdS/CFT. We end with some remarks on future directions in section \ref{sec:discussion}.
\section{Energy eigenstates and their flow}
The $T\overline{T}$ deformation is a one-parameter deformation of a quantum field theory, which is often defined from a Lagrangian perspective as a flow of the Lagrangian density of the theory:
\begin{eqnarray}
\pa_{\l}\mathcal{L} =-\mathcal{O}_{T\overline{T}}^{(\l)}= -\varepsilon^{ab}\varepsilon^{cd}T^{(\l)}_{ac}T^{(\l)}_{cd},
\end{eqnarray}
where $T^{(\l)}_{ab}$ is the stress tensor of the theory at the flow parameter $\l$. Since the stress tensor can itself be constructed from the Lagrangian density, say by the Noether procedure, this defines a self-contained flow equation for the classical Lagrangian density of the field theory. Quantum mechanically, the common approach is to use the integral of this deformed Lagrangian density as the action inside the Feynman path integral, and this gives a definition for the partition function, generating functional of correlation functions etc. In this paper, we will take a Hamiltonian perspective on the $T\overline{T}$ deformation, i.e. we will define it via a flow of the Hamiltonian of the theory:
\begin{eqnarray}\label{HamDef}
\pa_{\l}H_{\l} = \int dy_1\,\varepsilon^{ab}\varepsilon^{cd}T^{(\l)}_{ac}(y_0,y_1)T^{(\l)}_{bd}(y_0,y_1),
\end{eqnarray}
where we have written this operator on the Cauchy slice at some time $y_0$, with $y_1$ being the spatial coordinate. Note that this was already used in the derivation of the deformed energy spectrum in \cite{Zamolodchikov:2004ce, Smirnov:2016lqw}. Classically, the two definitions are entirely equivalent (see Appendix \ref{App:A}). Quantum mechanically, the two may differ by operator-ordering related counterterms. At any rate, we will take equation \eqref{HamDef} as our starting point, and use it to construct energy eigenstates and correlation functions along the flow.
\subsection{Rewriting the $T\overline{T}$ operator}\label{sec:rewrite}
We can write the deformation of the Hamiltonian in a somewhat more illuminating way by using the properties of the $T\overline{T}$ operator. We will employ a variant of the Green function method explained in \cite{Cardy:2019qao} for this purpose. We begin by trivially rewriting the spatial integral of the $T\overline{T}$ operator in equation \eqref{HamDef} as a double integral at equal times by inserting a spatial delta function:
\begin{eqnarray} \label{Op1}
\int dy_1 \mathcal{O}_{T\overline{T}}(y_0,y_1) = \int dy_1 dw_1 \e^{ab}\e^{cd}\d(y_1-w_1)T^{(\l)}_{ac}(y_0,y_1)T^{(\l)}_{bd}(y_0,w_1).
\end{eqnarray}
Here the spatial slice can either be compact (in which case we have a circle of length $L$) or non-compact, and correspondingly the Lorentzian spacetime is either a cylinder or a plane. We now rewrite the spatial delta function in terms of the Green function for the spatial derivative, defined as
\begin{eqnarray}
\partial_{y_1}G(y_1-w_1) = \delta(y_1 - w_1) - \mu,
\end{eqnarray}
where the constant $\mu = 0$ when the spatial slice is non-compact, while for a compact spatial slice we have $\mu = 1/L$ (corresponding to the subtraction of the zero mode of the derivative operator). Explicitly, this Green function is given by
\begin{eqnarray}
G(x) = \frac{1}{2}{\rm sgn}(x)
\end{eqnarray}
in the non-compact case (i.e., when $x\in \mathbb{R}$), and
\begin{eqnarray}\label{Gsum}
G(x) = \sum_{n \in \mathbb{Z}, n\neq 0}\frac{e^{i\frac{2\pi nx}{L}}}{2\pi i n}= \frac{1}{2}\mathrm{sgn}(x)-\frac{x}{L}
\end{eqnarray}
in the compact case (i.e., when $x\in [-L/2,L/2]$ with perodic boundary conditions). Replacing the delta function in \eqref{Op1} in terms of the Green function, we find
\begin{align}
\int dy_1 \mathcal{O}_{T\overline{T}}^{(\l)}(y_0,y_1) &= - \int dy_1 dw_1 \e^{ab}\Big(\partial_{w_1}G(y_1-w_1) - \mu \Big)T^{(\l)}_{0a}(y_0,y_1)T^{(\l)}_{1b}(y_0,w_1)\nonumber\\
&- \int dy_1 dw_1 \e^{ab}\Big(\partial_{y_1}G(y_1-w_1) + \mu \Big)T^{(\l)}_{1a}(y_0,y_1)T^{(\l)}_{0b}(y_0,w_1).
\end{align}
Note that we can regulate the Green function $G(y_1-w_1)$ by requiring it to drop to zero sufficiently fast in the coincident limit $|y_1 - w_1| \ll \epsilon$ for some short distance cutoff $\epsilon$, where the stress tensors are approaching a coincident limit. Alternatively, one could regulate $G$ by truncating the sum in \eqref{Gsum} at some large $|n| = N_{\rm max}$. Upon a partial integration,\footnote{In the non-compact case, we should keep track of the boundary terms. Instead, here we will work with the cylinder and to get to the plane, take the limit $L\to \infty$ in the end. } we can rewrite this as
\begin{align}
\int dy_1 \mathcal{O}_{T\overline{T}}^{(\l)}(y_0,y_1) &= \int dy_1 dw_1 \e^{ab}G(y_1-w_1) T^{(\l)}_{0a}(y_0,y_1)\pa_{w_1}T^{(\l)}_{1b}(y_0,w_1)\nonumber\\
& +\int dy_1 dw_1 \e^{ab}G(y_1-w_1) \partial_{y_1} T^{(\l)}_{1a}(y_0,y_1)T^{(\l)}_{0b}(y_0,w_1)\nonumber\\
& + \mu\int dy_1 dw_1 \e^{ab}\e^{cd} T^{(\l)}_{ac}(y_0,y_1)T^{(\l)}_{bd}(y_0,w_1).
\end{align}
Now we can use conservation of the stress tensor, together with the fact that $H$ generates time translations, to finally rewrite this in the following form:
\begin{eqnarray}\label{spaceIntOTT}
\pa_{\l}H_\l=\int dy_1 \mathcal{O}_{T\overline{T}}^{(\l)}(y_0,y_1) = i\left[H_\l,\mathcal{X}^{(\l)}(y_0)\right] + \mathcal{Y}^{(\l)}(y_0),
\end{eqnarray}
where $\mathcal{X}$ and $\mathcal{Y}$ are given by the following bi-local integrals\footnote{We also note that $\mathcal{X}^{(\l)}$ can also be further rewritten as $\mathcal{X}^{(\l)} = i\left[H_{\l},\mathcal{W}^{(\l)}\right]$, where $$\mathcal{W}^{(\l)}= \int dx_1 dy_1 G_{\text{Lap.}}(y_1-w_1)T^{(\l)}_{00}(0,y_1)T^{(\l)}_{00}(0,w_1), $$ and $G_{\text{Lap.}}$ is the Green function for the Laplacian on the circle/line.}:
\begin{align}\label{X&Y}
\mathcal{X}^{(\l)}(y_0) &= \int dy_1 dw_1 \e^{ab}G(y_1-w_1) T^{(\l)}_{0a}(y_0,y_1)T^{(\l)}_{0b}(y_0,w_1),\\
\mathcal{Y}^{(\l)}(y_0) &= \mu\, \e^{ab}\e^{cd} \mathbf{P}_{ac}(y_0)\mathbf{P}_{bd}(y_0)\nonumber\\
&= \mu\, \left(\{H, \int dy_1 \Theta(y_0,y_1)\} + 2(H^2-P^2)\right).
\end{align}
Here we have used the following notation:
$$\mathbf{P}_{ab}(y_0) = \int dy_1 T^{(\l)}_{ab}(y_0,y_1),\;\;\Theta = {{T^{(\l)}}^a}_a.$$
Equation \eqref{spaceIntOTT} is the main formula we will utilize repeatedly in the following sections.
Note that the first term in \eqref{spaceIntOTT} can be removed by performing a \emph{canonical transformation}. For instance, in the classical theory, this term is of the form $\left\{H,\mathcal{X}\right\}_{PB}$, where the subscript PB stands for Poisson brackets. In classical mechanics, such a deformation is generated by a canonical transformation, with the generating function being $\mathcal{X}$.\footnote{In the language of symplectic geometry, this term arises from a symplectomorphism on phase space, i.e., a diffeomorphism which preserves the symplectic form.} Note however that this generating function $\mathcal{X}$ is not local in space, but rather a bi-local integral. As we will discuss below, the first term in \eqref{spaceIntOTT} thus merely has the effect of ``dressing'' the fundamental degrees of freedom, while leaving their energies unaffected (see section \ref{sec:operators}). The $\mathcal{Y}$ term, on the other hand, which is written entirely in terms of spatial integrals of the energy momentum tensor, does change the energy levels of the theory.
\subsection{Energy eigenvalues and eigenstates}
\label{sec:unitary}
With the simplified form of the spatial integral of the $T\overline{T}$ operator, \eqref{spaceIntOTT}, we proceed to study the flow of the energy eigenstates under the $T\overline{T}$ deformation. The flow of energy eigenvalues is already well-understood \cite{Zamolodchikov:2004ce, Smirnov:2016lqw}, but we begin by reviewing it briefly. Let us denote the set of deformed energy eigenstates by $\{\ket{n_{\l}}\}$ and the undeformed ones by $\{\ket{n_0}\}$. These states are also simultaneous eigenstates of the momentum operator, with the momentum eigenvalue constant along the flow. We will assume, without loss of too much generality, that for a given initial energy $E_n^{(0)}$ and momentum $k_n$, there is either no degeneracy, or that the degeneracy does not split along the $T\overline{T}$ flow, so we can use non-degenerate perturbation theory. If the degeneracy splits, then we instead need to use degenerate perturbation theory to begin with, but then after that point we can repeat our argument below. In the case of a 2d CFT as the initial theory, there are indeed degeneracies in the energy spectrum, but as was noted in \cite{LeFloch:2019wlf}, in the situation where these degeneracies arise due to other (commuting) charges, such as the Korteweg-de Vries charges, they do not split along the $T\overline{T}$ flow and so our arguments below apply. With this assumption, recall that under a deformation in the Hamiltonian $\pa_{\l}H_{\l}$, the energies get deformed as
\begin{eqnarray}
\pa_{\l} E_n(\l) = \langle n_{\l} |\pa_{\l}H_{\l} | n_{\l}\rangle,
\end{eqnarray}
which from equation \eqref{spaceIntOTT}, we can rewrite as
\begin{eqnarray}
\pa_{\l} E_n(\l) =i \langle n_{\l} |\left[H_{\l},\mathcal{X}^{(\l)}\right]| n_{\l}\rangle+\mu\, \e^{ab}\e^{cd}\langle n_{\l} | \mathbf{P}_{ac}\mathbf{P}_{bd}| n_{\l}\rangle .
\end{eqnarray}
The first term above drops out, and the second term, upon using $\mathbf{P}_{00}=H$ and $\mathbf{P}_{01}=P$ gives
\begin{eqnarray}
\pa_{\l}E_n = \frac{2}{L}E_n\int dy_1\langle n_{\l}|T_{11}(0,y_1)|n_{\l}\rangle - \frac{2}{L}k_n^2,
\end{eqnarray}
where $k_n$ is the momentum eigenvalue of the state $|n\rangle$. Finally, using (see Appendix \ref{App:B})
\begin{eqnarray}
\langle n_{\l}|T_{11}(0,y_1)|n_{\l}\rangle = -\pa_LE_n,
\end{eqnarray}
we arrive at the following differential equation:
\begin{eqnarray} \label{Burger}
\pa_{\l}E_n = -2E_n\pa_LE_n - \frac{2}{L}k_n^2.
\end{eqnarray}
This is the Burger's equation for the flow of energy eigenvalues which was derived in \cite{Zamolodchikov:2004ce, Smirnov:2016lqw}.
The solutions to \eqref{Burger} are well-known:
\begin{eqnarray}
E_n(\l) = \frac{L}{4\l}\left(1 - \sqrt{1-8\frac{\l E_n^{(0)}}{L} + 16 \frac{k_n^2 \l^2}{L^2}}\right).
\end{eqnarray}
Let us now turn to the flow of energy eigenstates. A standard result from non-degenerate perturbation theory gives
\begin{eqnarray}
\partial_{\l} \ket{n_{\l}} = \sum_{m\neq n} \frac{\braket{m_{\l}| \partial_{\l} H_{\l}|n_{\l}}}{E_\l^n - E_{\l}^m} \ket{m_{\l}}.
\end{eqnarray}
We simplify this expression replacing the denominator by an integral,
\begin{eqnarray} \label{integral}
\frac{1}{E^{n}_{\l} - E^m_{\l}+i\epsilon} = -i \int_{0}^\infty ds\;e^{i s(E_{\l}^n - E_{\l}^m + i\epsilon)},
\end{eqnarray}
with $\epsilon >0$, which is required to make the integral converge, for any state $\ket{n_\l}$ other than the vacuum.\footnote{For the vacuum, we could give $s$ a small imaginary part, but this does not work for general excited states.}
Furthermore, using $O(s) = e^{i s H}O(0)e^{-is H}$, we find
\begin{eqnarray}\label{flowU1}
\partial_{\l} \ket{n_{\l}} = -\sum_{m\neq n} i \int_{0}^{\infty} ds\;e^{-\epsilon s} \ket{m_{\l}}\braket{m_{\l}| \partial_{\l}H_{\l}(-s)|n_{\l}}.
\end{eqnarray}
At this stage, we will need to assume completeness of the $\{|m_{\l}\rangle\}$ basis of states. On the plane, or on the cylinder with $\l<0$ (assuming the ground state energy satisfies $E_0^{(0)}\geq 0$), we expect this to be true. However, on the cylinder with the holographic sign $\l > 0$, or in the situation that $\l<0$ but some of the low-lying states in the undeformed spectrum have negative energy, there is a subtlety -- in this case some of the energy eigenvalues become complex along the flow. This also clearly poses a problem for the convergence of the integral in equation \eqref{integral}. It is not clear whether one must discard the corresponding states or not, but if one does discard them, then we would need to ensure that $\pa_{\l}H_{\l}$ does not mix between the real and complex energy states. In what follows, we will simply restrict to the plane with either sign of $\l$, and the cylinder with $\l<0$ (assuming the ground state energy satisfies $E_0^{(0)}\geq 0$) to avoid the complexification of energies.
So going back to \eqref{flowU1}, using the completeness of the $\ket{m_{\l}}$ basis together with the previous assumption that the degeneracy of states does not change along the flo
, we get
\begin{eqnarray}\label{eqU}
\partial_{\l}\ket{n_{\l}} = -i \int_{0}^\infty ds\; e^{-\epsilon s}\partial_{\l}H_{\l}(-s) \ket{n_{\l}} + i \int_{0}^\infty ds \,e^{-\epsilon s}\braket{n_{\l}| \partial_{\l}H_{\l}(-s)|n_{\l}}\ket{n_{\l}}.
\end{eqnarray}
The above differential equation can be solved by making the following ansatz for the state $|n\rangle_{\l}$:
\begin{eqnarray}\label{ansatz}
|n\rangle_{\l} = e^{i\theta_n(\l)}U(\l)|n\rangle_0,
\end{eqnarray}
where $U$ is a unitary operator, and we have pulled out an eigenstate-dependent phase from it. In terms of this ansatz, equation \eqref{eqU} then translates to
\begin{eqnarray}\label{flowU}
\partial_{\l}U = - i \int_{-\infty}^0 ds \; e^{\epsilon s} e^{is H_{\l}}\partial_{\l} H_{\l} e^{-isH_{\l}} U,\;\;\pa_{\l}\theta_n = \frac{1}{\epsilon}\pa_{\l}E_n,
\end{eqnarray}
with formal solution given by,
\begin{eqnarray}\label{solU}
U = \mathcal{P} \exp\left( -i \int_0^{\l} d\l' \int_{-\infty}^0 ds\,e^{\epsilon s} \partial_{\l'} H_{\l'}(s) \right),\;\;\theta_n(\l)=\frac{1}{\epsilon}(E_n(\l)-E_n(0)).
\end{eqnarray}
Finally, using
$ \partial_\l H = \int d\theta\,\mathcal{O}_{T\overline{T}}$,
the operator $U$ in \eqref{solU} can be rewritten as
\begin{eqnarray} \label{Unitary}
U = \mathcal{P} \exp\left(-i \int_0^{\l} d\l' \int_{M_-}e^{\epsilon s} \mathcal{O}_{T\overline{T}}\right),
\end{eqnarray}
where $M_- = \mathbb{R}_- \times \Sigma$, with $\Sigma = \mathbb{R}$ or $S^1$. Note that if we try to naively take \eqref{solU} to be true even in the cases where the energy spectrum complexifies, then the $e^{i\theta_n}$ factor would either diverge or decay. The form of $U$ we have obtained in \eqref{Unitary} is rather formal, but we can get some further intuition in two ways. Firstly, by performing some manipulations using equation \eqref{spaceIntOTT}, the above $U$ can be re-written in terms of a kernel, which can be interpreted as the Cauchy slice becoming ``dynamical'', with the dynamics controlled by a string worldsheet action. We will present this in the next subsection. Secondly, one can also use the random metric approach of \cite{Cardy:2018sdv} where one interprets the $T\overline{T}$ deformation as coupling the seed theory to a random metric. This leads to an effective, three dimensional gravitational kernel for the unitary $U$ (similar in spirit to \cite{Dubovsky:2018bmo, Mazenc:2019cfg, Freidel:2008sh}). We will defer this 3d approach to section \ref{sec:further}.
\subsection{A kernel for $U$}\label{sec:Cauchyslice}
Going back to equation \eqref{spaceIntOTT}, the unitary operator $U$ can now be expressed in terms of the bi-local operators $\mathcal{X}$ and $\mathcal{Y}$ as
\begin{eqnarray} \label{Unitary2}
U = \mathcal{P}\exp\left[-i\int_0^{\lambda} d\lambda'\,\left(\mathcal{X}^{(\lambda')}(0)+\mu\int_{-\infty}^0ds\,e^{\epsilon s}\e^{ab}\e^{cd}\mathbf{P}^{(\lambda')}_{ac}(s)\mathbf{P}^{(\lambda')}_{bd}(s)\right) \right].
\end{eqnarray}
Note that the $\mathcal{X}$ term entirely localizes on the $s=0$ spatial slice.\footnote{We have taken the $\epsilon \to 0$ limit in the $\mathcal{X}$ term and dropped an $O(\epsilon)$ term resulting from integration by parts.} The second term proportional to $\mu$ is more complicated and involves operators at finite time, but at least on the plane, this term drops out. At any rate, this expression for the unitary $U$ makes it fairly easy to write a flow equation for correlation functions in the $T\overline{T}$ flowed CFT, as we will show in section \ref{sec:operators} below. Note that equation \eqref{Unitary2} is strikingly reminiscent of tensor networks \cite{Swingle:2009bg, Swingle:2012wq, Nozaki:2012zj, Pastawski:2015qua, Czech:2015kbp, Hayden:2016cfa, Bao:2018pvs} and the surface-state correspondence \cite{Miyaji:2015yva, Miyaji:2015fia} in the context of AdS/CFT, at least on the plane ($\mu=0$); we will return to this point later.
We can also rewrite this expression in terms of a path-integral kernel involving a ``string worldsheet'' as follows (see figure \ref{fig:WS}). We first break up the path-ordered exponential into infinitesimal exponentials:
\begin{eqnarray}
U = \lim_{\delta \l\to 0}\prod_{k=0}^NU_k,\;\;U_k=\exp\left[-i\delta \lambda \int_{M_-}e^{\epsilon s}O_{T\bar{T}}(\lambda_k =k\delta \l)\right],
\end{eqnarray}
where $N= \lambda/\delta \l$. Now using equation \eqref{Unitary2}, each of these infinitesimal unitaries can be written as
\begin{eqnarray}\label{inter}
U_k = \exp\left[-i\delta \lambda\,\mathcal{X}^{(\lambda_k)}(0)-i\delta \lambda\mu\left(\left\{H_{\l^k},\int_{M_-}\Theta^{(\l_k)}\right\}-\frac{2}{\epsilon}(P^2-H_{\l}^2)\right) \right],
\end{eqnarray}
where we have rewritten $T_{11}$ in terms of the trace of the stress tensor $\Theta$. Next, we rewrite this as
\begin{align}
U_k &= \int \left[D\xi_k(\sigma)DQ_kD\phi_k\right]\exp\Big[i\delta \l S[\xi_k,Q_k,\phi_k]- i\delta \l\oint d\sigma\,\xi_k^a(\sigma)T^{(\lambda_k)}_{0a}(0,\sigma)\nonumber\\
&-i\delta \l \left(Q^0_k H + Q^1_k P\right) -i\delta \l \phi_k \int_{M_-}\Theta^{(\l_k)}\Big],
\end{align}
where
\begin{eqnarray}\label{actionSk}
S[\xi_k,Q_k,\phi_k]= \frac{1}{4}\oint d\sigma\,\epsilon_{ab}\xi_k^a(\sigma)\pa_{\sigma}\xi_k^b(\sigma)-\frac{\epsilon}{8\mu}(Q^1_k)^2 - \frac{1}{2\mu\epsilon}\phi_k^2+\frac{1}{2\mu}\phi_kQ^0_k,
\end{eqnarray}
For each $k$th infinitesimal piece we have introduced a vector valued Hubbard-Stratanovich (HS) field $\xi_k^a(\sigma)$ which only depends on the spatial coordinate, a vector valued HS field $Q^a$ and a scalar HS field $\phi$ both of which are spacetime independent. We can combine $Q^a$ and $\xi^a(\sigma)$ into one field, with $Q^a$ being the zero mode and $\xi^a$ being the remaining non-zero modes, whose spatial integral vanishes. In fact, it is more convenient to define a field $X^a(\lambda,\sigma)$ such that
\begin{eqnarray}
\pa_{\l}X^a(\lambda,\sigma) = Q^a(\l)+\xi^a(\l,\sigma)
\end{eqnarray}
Now sending $\delta \l \to 0$, we can rewrite the full unitary $U$ as a path integral over the fields $X^a$ and $\phi$:
\begin{eqnarray}\label{UwigglyCauchy}
U = \int \frac{[DX D\phi]}{\mathcal{N}}e^{i(S+S_{\rm reg})}\mathcal{P}\exp\left[-i\int_0^{\l} d\l'\left(\oint d\sigma\pa_{\l'}X^a\,T^{(\l')}_{0a}(\sigma)+\phi(\l')\int_{M_-}\Theta^{(\l')}\right)\right]
\end{eqnarray}
where the action is given by
\begin{eqnarray}
S[X,\phi] = \frac{1}{4}\int_0^{\l} d\l'\left(\oint d\sigma\,\varepsilon_{ab}\pa_{\l'}X^a\pa_{\sigma}\pa_{\l'}X^b + 2\phi(\l')\oint \pa_{\l'}X^0 \right).
\end{eqnarray}
and the term $S_{\rm reg}$ regularizes the zero mode integrals:
\begin{eqnarray}
S_{\rm reg} = - \frac{1}{2}\int_0^\l d\l' \left(\frac{\epsilon \mu}{2}\oint d\sigma\oint d\sigma' \pa_{\l'}X^1(\sigma)\pa_{\l'}X^1(\sigma') + \frac{1}{\mu\epsilon}\phi(\l')^2\right).
\end{eqnarray}
We can interpret the $X^a$ field in terms of an effective ``Cauchy string'' (see figure \ref{fig:WS}).
\begin{figure}[t]
\centering
\includegraphics[height=5.5cm]{WS.pdf}
\caption{We can interpret the unitary $U$ as making the Cauchy slice a dynamical surface parametrised by $X^a(\l,\s)$.}
\label{fig:WS}
\end{figure}
The coordinate $\sigma$ is an intrinsic coordinate along the string, and $\lambda$ is an emergent Euclidean ``time'' direction, parametrizing the $T\overline{T}$ flow. $X^a(\l,\sigma)$ is then a map of the Cauchy string worldsheet to the target space, which is either $\mathbb{R}^2$ or $\mathbb{R}\times S^1$. Therefore, we may interpret the unitary $U$ as making the Cauchy slice in the CFT a dynamical object, in a manner of speaking. From the tensor network perspective mentioned above, we seem to have a superposition of tensor networks, at least on the plane. The interpretation of the $\phi$ field is not clear to us at this point, but it roughly seems to be a dilaton-like field implementing a rescaling of the cylinder.
\section{Flow of operators and correlation functions}\label{sec:operators}
In the previous section, we have shown how the energy eigenstates change under the flow triggered by the $T\overline{T}$ operator. In particular, we found an explicit form of a unitary operator $U$ that rotates these states amongst each other. Next, we would like to know how correlation functions change under the flow (pertubatively in $\l$ such correlator have been computed, for instance, see \cite{Kraus:2018xrn}\footnote{See also \cite{He:2020udl, He:2019vzf}} for a perturbative approach). This requires knowing how operators flow.
There are several different approaches one could consider for the flow of operators/correlation functions. Here, we consider two type of operators:
\noindent{\it (i)} The first type of operators, which we will call \emph{undeformed operators}, are those obtained from time evolution of the operators of the undeformed theory. More precisely, we consider some constant time Cauchy slice, say at $t=0$, and consider the undeformed operators $O(0,x)$ of the seed theory on this Cauchy slice. Operators at a time separation away from the Cauchy slice are of course defined in the usual way via
\begin{eqnarray}
O^{(\l)}(t,x)= e^{itH_{\l}} O(0,x) e^{-itH_{\l}},
\end{eqnarray}
and since the Hamiltonian of the theory is changing along the flow, these finite time operators will also change, but only via their dependence on $H_{\l}$. The one exception to this is the stress tensor -- since the Hamiltonian is $H_{\l}=\int dx T^{(\l)}_{00}(0,x)$, we are forced to let $T^{(\l)}_{\mu\nu}(0,x)$ change explicitly along the flow. At least classically, this explicit flow of $T^{(\l)}_{\mu\nu}(0,x)$ can be obtained via Noether's procedure from the flow of the Lagrangian density of the theory.
\noindent{\it (ii)} The second type of operators we will consider are what we will call \emph{dressed operators}, where we explicitly flow the operators on the initial time slice. This flow is motivated by the observation that the $T\overline{T}$ deformation on the plane can be removed by a canonical/Bogoliubov transformation. The operators at finite time are then again defined in the usual way via time evolution.
\subsection{On the plane}
For simplicity of presentation, we first consider the case of the theory on the plane and then on the cylinder. Again, we mention here that our discussion below applies only to finite energy states. If the energy of the state under consideration is not finite, but has a finite energy density, the $T\bar{T}$ deformation can change the energy density and the flow is morally similar to the one on the cylinder.
\subsubsection*{Undeformed operators}
We will first consider correlation functions of the undeformed operators defined above. Let us consider the following correlation function:
\begin{eqnarray}
C(\{t_i,x_i\})=\langle n_{\l} | O^{(\l)}(t_1,x_1)\cdots O^{(\l)}(t_n,x_p)| n_\l \rangle,
\end{eqnarray}
where $|n_{\l}\rangle$ is an energy eigenstate with energy $E_n$. We can derive a flow equation for this correlation function as follows: we first insert complete sets of energy eigenstates between the operators:
\begin{eqnarray} \label{Cf2}
C=\sum_{n_1,\cdots,n_{p-1}} e^{it_1(E_n-E_{n_1})+\cdots+ it_p(E_{n_{p-1}}-E_n)}\langle n_{\l} | O(0,x_1)|n_{1,\l}\rangle \cdots \langle n_{p-1,\l}|O(0,x_p)| n_\l \rangle.
\end{eqnarray}
Now we can use the fact that the energy eigenvalues on the plane are $\l$-independent, and so also are the operators $O(0,x_i)$ on the initial time slice, as per our choice. Therefore, the only $\l$-dependence in the correlation function comes from the energy eigenstates, which satisfy the following flow equation:
\begin{eqnarray}
\pa_{\l}|n_{\l}\rangle &=& -i\int_{-\infty}^0 ds\,e^{\epsilon s}\pa_{\l}H_{\l}(s)|n_{\l}\rangle \nonumber\\
&=&-i \int_{-\infty}^0 ds\,e^{\epsilon s}e^{isH_{\l}}i\left[H_{\l},\mathcal{X}^{(\l)}\right]e^{-isH_{\l}} |n_{\l}\rangle\nonumber\\
&=& -i\int_{-\infty}^0 ds\,e^{\epsilon s}\pa_s \left(e^{isH_{\l}}\mathcal{X}^{(\l)}e^{-isH_{\l}}\right) |n_{\l}\rangle=-i\mathcal{X}^{(\l)}|n_{\l}\rangle.
\end{eqnarray}
Note that we have dropped the $\mathcal{Y}$ term above, assuming that it is suppressed in the $L\to \infty$ limit at finite energy. Since we are primarily interested in vacuum correlation functions, we expect this to be a good assumption. Therefore, taking a $\l$ derivative of the correlation function \eqref{Cf2} gives
\begin{eqnarray} \label{Cf3}
\pa_{\l}C = i\sum_{i=1}^p \langle n_{\l}|O^{(\l)}(t_1,x_1)\cdots \left[\mathcal{X}^{(\l)}(t_i),O^{(\l)}(t_i,x_i)\right]\cdots O^{(\l)}(t_p,x_p)|n_{\l}\rangle,
\end{eqnarray}
where we have defined
\begin{eqnarray}
\mathcal{X}^{(\l)}(t)=\frac{1}{2}\int dy \int dw\; G(y-w)\varepsilon^{ab}T^{(\l)}_{0a}(t,y)T^{(\l)}_{0b}(t,w).
\end{eqnarray}
Note that on general grounds the commutator can be simplified,
\begin{eqnarray} \label{Cf4}
\left[\mathcal{X}^{(\l)}(t_i),O^{(\l)}(t_i,x_i)\right]=\int dy\, G(y-x_i)\varepsilon^{ab}T_{0a}(t_i,y)\pa^{(x_i)}_bO^{(\l)}(t_i,x_i)+\cdots,
\end{eqnarray}
where $\cdots$ denotes a theory-dependent, local operator, which, if we like, we can absorb via a local redefinition of the operators $O^{(\l)}$. Equations \eqref{Cf3} and \eqref{Cf4} agree with the flow equation for correlation functions derived recently by Cardy in \cite{Cardy:2019qao} using Euclidean path integral methods, up to the local operator re-definitions mentioned above. As suggested in \cite{Cardy:2019qao}, we can figuratively think of the effect of the $T\overline{T}$ deformation on correlation functions as implementing a ``state-dependent diffeomorphism'' via the attachment of a stress tensor ``Wilson line'' to the operators. Despite the non-locality of this ``Wilson line'', we emphasize that that since the operators on the initial time slice are those of the undeformed theory, their equal-time commutators at separate points will continue to vanish inside correlation functions. Furthermore, since the deformation preserves Lorentz invariance on the plane, commutators of more general spacelike separated operators will also continue to vanish. The non-local Wilson line attachment in the flow equation obscures the above causal properties of these correlation functions, nevertheless we expect their analytic structure to still be controlled by causality.
Equation \eqref{Cf4} is a bit formal, because we need to address the $UV$ divergencies which appear in the limit when the two operators becomes co-incident. As discussed previously, we can regulate these divergences by introducing a short distance cutoff $\epsilon$ in the Green function $G$, such that it drops to zero when $|y-x| \ll \epsilon$. Fortunately, these UV divergences were addressed in the analysis of Cardy in \cite{Cardy:2019qao}, where it was shown that the RHS of \eqref{Cf4} only has a logarithmic divergence in $\epsilon$ :
\begin{eqnarray}\label{div}
\int dy\, G(y-x_i)\varepsilon^{ab}T_{0a}(t_i,y)\pa^{(x_i)}_bO^{(\l)}(t_i,x_i)+ = -\log(|\epsilon| ) \nabla_{i}^2 O^{(\l)}(t_i,x_i) + \text{finite}.
\end{eqnarray}
Crucially, note that the divergence in the flow equation is proportional to a local operator, and thus corresponds to a cutoff-dependent, local redefinition of the operator at every step along the flow. In other words, we should locally redefine the operators $O^{(\l)}$ at every step along the flow in order to cancel the above divergence.
Equation \eqref{Cf3} gets slightly modified if one of the operators in the correlation function is the stress tensor. In this case we need to account for the explicit change in the stress tensor on the initial time slice along the flow. As mentioned previously, this explicit change in the stress tensor can be obtained, at least classically, from Noether's procedure:
\begin{eqnarray} \label{Noether}
\pa_{\l}T^{(\l)}_{ij}(\phi,\dot\phi)= \pa_{i}\phi\frac{\delta}{\delta \pa^{j}\phi} \left(\varepsilon^{ab}\varepsilon^{cd}T^{(\l)}_{ac}T^{(\l)}_{bd}\right)-\eta_{ij}\varepsilon^{ab}\varepsilon^{cd}T^{(\l)}_{ac}T^{(\l)}_{bd} + \cdots,
\end{eqnarray}
where $\phi$ denotes the elementary fields in the action and $\cdots$ denote potential improvement terms which may be required to make the stress tensor symmetric. An additional subtlety is that the above stress tensor is written in terms of $\phi$ and its time derivatives, but the operator written in terms of the canonical variables $(\phi,\pi)$ will have an additional contribution of the form $(\pa_{\l}\dot\phi) \frac{\delta}{\delta \pi}T_{ij}^{(\l)}$ coming from the change in the relation between $\pi$ and $\dot\phi$. All these contributions to correlation functions appear to be theory dependent.
\subsubsection*{Dressed operators}
Now we come to the second type of operators of interest to us, which we will call \emph{dressed} operators and will only be considered in detail on the plane for reasons that will be clear momentarily. To motivate the definition of these dressed operators, we go back to equation \eqref{spaceIntOTT}, which implies that the spatial integral of the $T\overline{T}$ operator on the plane (i.e., at $\mu=0$) is given by
\begin{align}\label{CanT}
\pa_{\l}H_{\l} = i\left[H_{\l}, \mathcal{X}^{(\l)}\right],\quad \mathcal{X}^{(\l)} =\frac{1}{2}\int dx_1 \int dy_1\,G(x_1-y_1)\varepsilon^{ab}T^{(\l)}_{0a}(0,x_1)T^{(\l)}_{0b}(0,y_1),
\end{align}
where we again emphasize that we have dropped the $\mathcal{Y}$ term above in the $L\to \infty$ limit, assuming that we are working at finite energy.
It is helpful to first look at the classical analog of equation \eqref{CanT}, which is
\begin{eqnarray}\label{CanT2}
\pa_{\l}H_{\l} = \left\{H_{\l},\mathcal{X}^{(\l)}\right\}_{PB},
\end{eqnarray}
where the subscript $PB$ stands for Poisson brackets. It is clear that such a deformation of the Hamiltonian can be removed by a canonical transformation, generated by $\mathcal{X}$. In more detail, say that the theory at $\lambda$ is naturally written in terms of some canonical degrees of freedom $(\phi_I^{\l}, \pi_J^{\l})$ satisfying
\begin{eqnarray}\label{PB}
\left\{\phi_I^{\l}, \pi^{\l}_J\right\}_{PB} = \delta_{IJ},
\end{eqnarray}
where the $I,J$ are meant to be generalized indices, including the spatial dependence of these fields. Then deforming the Hamiltonian, as in \eqref{CanT2}, is equivalent to keeping the Hamiltonian function unchanged but deforming the phase space coordinates as
\begin{eqnarray}
\pa_{\l}\phi^I_{\l} = -\left\{\mathcal{X}^{(\l)},\phi^I_{\l}\right\}_{PB},\;\;\pa_{\l}\pi^I_{\l} = -\left\{\mathcal{X}^{(\l)},\pi^I_{\l}\right\}_{PB}.
\end{eqnarray}
This flow of phase space coordinates is a canonical transformation/symplectic diffeomorphism, i.e. it preserves the Poisson brackets in \eqref{PB}. Thus, classically the $T\overline{T}$ deformation on the plane and at finite energy merely has the effect of implementing a $\l$-dependent canonical transformation along the flow. Quantum mechanically, we can replace the Poisson brackets above with commutators, and then it becomes evident that the flow simply implements a unitary rotation on phase space which preserves the canonical commutation relations, or in other words, a \emph{Bogoluibov transformation}.
This motivates us to define the \emph{dressed operators} $\widetilde{O}$ on the initial time slice via the following flow equation:
\begin{eqnarray} \label{DO}
\pa_{\l}\widetilde{O}^{(\l)}=-i\left[\mathcal{X}^{(\l)}, \widetilde{O}^{(\l)}\right].
\end{eqnarray}
This flow is rather formal, since we have not discussed $UV$ divergencies, but we will see that inside correlation function these operators do make sense. We can recast \eqref{DO} in the form
$$D_{\l}\widetilde{O}^{(\l)} \equiv \pa_{\l}\widetilde{O}^{(\l)}+i\left[\mathcal{X}^{(\l)}, \widetilde{O}^{(\l)}\right] = 0,$$
where we may think of the derivative $D_{\l}$ defined above as a \emph{covariant derivative}. From this point of view, the dressed operators are covariantly constant along the flow. The flow equation has a simple solution:
\begin{eqnarray} \label{FlowSol}
\widetilde{O}^{(\l)}(0,x) = U\,O(0,x)\,U^{-1}, \;\; U = \mathcal{P}\,e^{-i\int_0^{\l}d\l' \mathcal{X}^{(\l'})},
\end{eqnarray}
where the unitary $U$ is the same operator we considered in the previous section. From equation \eqref{CanT}, it follows that dressed operators at time $t$ are also related to the operators of the seed theory similarly, i.e.,
\begin{eqnarray}
\widetilde{O}^{(\l)}(t,x) = U\,O(t,x)\,U^{-1}, \;\; O(t,x)=e^{itH_0}O(0,x)e^{-itH_0}.
\end{eqnarray}
The \emph{dressed} operators are thus non-local, since $U$ is non-local.
Note that the dressed operators satisfy the same commutation relations as the seed operators; in particular, dressed operators commute with other dressed operators at spacelike separation. It should perhaps be emphasized that the ``dressing'' $\mathcal{X}^{(\l)}$ is non-local, and so a dressed operator will \emph{not} necessarily commute with an undeformed operator at a spacelike separated point. Nevertheless, the dressed operators do respect causality in that they commute with other dressed operators at spacelike separation, notwithstanding the non-locality of the dressing. Further, since energy eigenvalues on the plane do not flow and eigenstates flow by the action of the same unitary $U$, correlation functions of dressed operators are $\l$-independent:
\begin{eqnarray}
\pa_{\l}\widetilde{C}(\{t_i,x_i\})=\pa_{\l} \langle n_{\l}| \widetilde{O}^{(\l)}(t_1,x_1)\cdots \widetilde{O}^{(\l)}(t_p,x_p)| n_{\l}\rangle=0.
\end{eqnarray}
Thus, the dressed operators are a canonical choice of operators along the flow in terms of which the theory appears completely undeformed.
It is important to stress here that while the classical version of the flow equation \eqref{DO} (with Poisson brackets in place of commutators) is perfectly well defined, the quantum version is somewhat formal, since it suffers from the coincident divergences discussed around equation \eqref{div}. Indeed, as discussed there, the right hand side of \eqref{DO} has a local, logarithmic divergence. Thus, if we compute the correlation functions of these dressed operators in the original, undeformed vacuum state, then these will be divergent. However, the vacuum state of the theory also flows and the correlation functions in the flowed vacuum are indeed finite and unchanged. We emphasize that the sole reason this construction works is that the classical version of the deformation is a canonical transformation, i.e., a redefinition of the phase space coordinates. This is no longer the case on the cylinder, or even for finite energy density states on the plane, and in those situations, there is no natural way to define such dressed operators. \footnote{Alternatively, one can subtract off the $UV$ divergence along the flow as proposed in \cite{Cardy:2019qao}. The flow of the operator would then be defined as
\begin{eqnarray}
D_{\l}\widehat{O}_{ij}^{(\l)}(x) = - \log |\mu \varepsilon| \nabla_x^2 \widehat{O}_{ij}^{(\l)}(x),
\end{eqnarray}
with $\mu$ a renormalisation scale. Correlation functions of these operators in the undeformed state are finite, but are divergent in the deformed state as mentioned in the main text. Our perspective is therefore a little different than \cite{Cardy:2019qao} as we define correlation functions of dressed operators in the deformed states. This is a natural way to define them, because of the $T\bar{T}$ deformation is a canonical/Bogoliubov transformation on the plane.
}
Nevertheless, having said that, it remains curious why these operators (on the plane) only make sense inside correlation functions and at present we do not have a full understanding of it. Furthermore, this also touches upon the question whether two canonically related classical theories give the same quantum theory or not. It would be interesting to study this aspect of the $T\bar{T}$ deformation more.
We can also define a \emph{dressed} stress tensor $\widetilde{T}_{ij}$ in the same way as any other operator:
\begin{eqnarray}\label{flowT}
D_{\l}\widetilde{T}^{(\l)}_{\mu\nu}=0.
\end{eqnarray}
This is \emph{not} the same as the original stress tensor of the theory which was discussed in the previous section (see equation \eqref{Noether}). The dressed stress tensor is not local with respect to the undeformed operators, however it is local (i.e., microcausal) with respect to dressed operators. Furthermore, it is conserved and its spatial integrals give the expected energy-momentum charges. To show conservation, it is enough to show that if the dressed stress tensor is conserved at $\l$, then the dressed stress tensor at $\l+d\l$ will also be conserved. To this end, consider the conservation equation and
|
take a $\l$ derivative, replacing spacetime derivatives with commutators:
\begin{eqnarray}
\pa_{\l}\left( \pa^{\mu}\widetilde{T}^{(\l)}_{\mu\nu}\right)=\partial_\l \left( -i \left[H_{\l},\widetilde{T}^{(\l)}_{0\nu}(x)\right]+ i \left[P,\widetilde{T}^{(\l)}_{1\nu}(x)\right] \right).
\end{eqnarray}
Bringing the $\l$ derivative inside the commutators and using \eqref{spaceIntOTT}, we can write this as
\begin{eqnarray}
\pa_{\l}\left( \pa^{\mu}\widetilde{T}^{(\l)}_{\mu\nu}\right)=\left[\left[H_{\l},\mathcal{X}^{(\l)}\right],\widetilde{T}^{(\l)}_{0\nu}(x)\right]-i \left[H_{\l},\partial_\l \widetilde{T}^{(\l)}_{0\nu}(x)\right] + i \left[P,\partial_\l \widetilde{T}^{(\l)}_{1\nu}(x)\right].
\end{eqnarray}
The double commutator can be simplified using the Jacobi identity and after a little algebra, using conservation of $\widetilde{T}^{(\l)}_{\mu\nu}(x)$, we find
\begin{eqnarray}\label{finalflowT1}
\pa_{\l}\left( \pa^{\mu}\widetilde{T}^{(\l)}_{\mu\nu}\right)=\partial^{\mu} \left(D_{\l}\widetilde{T}^{(\l)}_{\mu\nu}(x)\right)-i\left[\left[P,\mathcal{X}^{(\l)}\right],\widetilde{T}^{(\l)}_{1\nu}\right]=0,
\end{eqnarray}
where we have used the fact that the dressed stress tensor is covariantly constant, by definition, and that $\left[P,\mathcal{X}^{(\l)}\right]=0$. Finally, since the dressed stress tensor matches onto the conserved stress tensor of the seed theory at $\l=0$, we conclude that it is conserved everywhere along the flow. Next, the dressed energy and momentum operators obtained from the dressed stress tensor:
\begin{eqnarray}
\widetilde{H}_{\l} = \int dx_1 \widetilde{T}^{(\l)}_{00}(0,x_1),\;\;\widetilde{P}_{\l} = \int dx_1 \widetilde{T}^{(\l)}_{01}(0,x_1),
\end{eqnarray}
satisfy the following flow equations
\begin{eqnarray}
\pa_{\l}\widetilde{H}_{\l} = \int dx_1\pa_{\l}\widetilde{T}^{(\l)}_{00}(0,x_1)=- i\int dx_1\left[\mathcal{X}^{(\l)},\widetilde{T}^{(\l)}_{00}(0,x_1)\right]
=- i\left[\mathcal{X}^{(\l)},\widetilde{H}_{\l}\right],
\end{eqnarray}
\begin{eqnarray}
\pa_{\l}\widetilde{P}_{\l} = \int dx_1\pa_{\l}\widetilde{T}^{(\l)}_{01}(0,x_1)=- i\int dx_1\left[\mathcal{X}^{(\l)},\widetilde{T}^{(\l)}_{01}(0,x_1)\right]
=- i\left[\mathcal{X}^{(\l)},\widetilde{P}_{\l}\right].
\end{eqnarray}
These first order flow equations for $\widetilde{H}_\lambda$ and $\widetilde{P}_\l$ are the same as their untilded counterparts and since they have the same $\l = 0$ limit, the tilded and untilded charges are the same. Note however that the dressed and undressed stress-tensor are still different and the equality of the charges merely states that they are related through improvement terms, albleit non-local ones. Also notice that even though the flow of the tilded stress tensor is formally divergent, the flow of the charges is not, because the divergence comes with a Laplacian. Its spatial part drops out because of the spatial integral and the temporal part gives a time derivative of the commutators $[\widetilde{H}_\l , \widetilde{H}_\l]$ and $[\widetilde{P}_\l , \widetilde{H}_\l]$, which thus also vanishes.
Thus, the energy and momentum operators obtained from the dressed stress tensor are the correct energy and momentum operators of the deformed theory.
Finally, if the seed theory is a conformal field theory, then the stress tensor of the seed theory is expected to satisfy an algebra of the form:
\begin{eqnarray}\label{Talgebra}
\left[T_{\mu\nu}(x),T_{\rho\sigma}(x')\right] = f_{\mu\nu\rho\s}^{\a\b}(x-x')T_{\a\b}(x) + \g_{\mu\nu\rho\s}(x-x'),
\end{eqnarray}
where $f_{\mu\nu\rho\s}^{\a\b}$ are the structure constants and $\g_{\mu\nu\rho\s}$ the central terms. Either by using the flow equation, or by using equation \eqref{FlowSol}, it is straightforward to show that the dressed stress tensor $\widetilde{T}_{ij}^{(\l)}$ also satisfies the same algebra, with $\l$-independent structure constants and central terms. In particular this has the interesting consequence that the dressed stress tensor behaves like the stress tensor of the seed conformal field theory, with the central charge equal to that of the seed theory, i.e. the Schwinger terms are equivalent.
To be a bit more explicit, let us consider the seed theory to be a 2d CFT. This theory has, amongst the usual Lorentz and special conformal currents, a dilatation current $j_{\mu}^D = T_{\mu\nu} x^\nu$. In the deformed theory this current is simply,
\begin{eqnarray}
\widetilde{j}_{\mu}^D = \widetilde{T}_{\mu\nu}^{(\l)} x^\nu,
\end{eqnarray}
and the charge $\widetilde{D}$ is the spatial integral of $\widetilde{j}_0^D$. Note, however, that this current is \emph{non}-local Commuting this charge (at equal time) with a dressed operator $\widetilde{O}^{(\l)}(x)$ it will have the same eigenvalue, i.e. conformal dimension $\D$, as in the undeformed theory. This can also be seen from the fact that the correlators of dressed operators do not flow. An interesting question is whether the global conformal group lifts to a full Virasoro symmetry. In these \emph{non-local} CFTs this is far from obvious and we will discuss this further in section \ref{sec:discussion}. Again here we mention that although these results are true classically, in the quantum theory there are UV divergencies that need to be dealt with. However, there are two pieces of evidence why this might not be a such a big issue. \footnote{It would be interesting and of great value to understand this better as there could be subtleties at the quantum level.} First, inside correlation functions of the deformed state these cancel and we end up with well-defined Ward identities. Second, the flow of the conserved charge is again finite by the same argument as given above about the conserved charges associated to translations in space and time.
Finally, one might wonder whether it is possible to define a new flow where at every step one adds to the Hamiltonian the $T\overline{T}$ operator made out of the dressed stress tensor. It is easy to check that in this case, the generating functional $\widetilde{\mathcal{X}}$ is $\l$-independent, because
\begin{eqnarray}
\pa_{\l}\widetilde{\mathcal{X}}^{(\l)} =-i \left[\widetilde{\mathcal{X}}^{(\l)},\widetilde{\mathcal{X}}^{(\l)}\right]=0,
\end{eqnarray}
and so such a deformation would be equivalent to the ``one-shot'' deformation where we turn on $\l$ times the $T\overline{T}$ operator of the seed theory.
\subsubsection*{Example: Classical, free scalar field}
Let us apply the discussion above to a simple example. Let the seed theory be a free, massless scalar field theory on the plane:
\begin{eqnarray}
\mathcal{L}^{(0)} = \frac{1}{2}\left(\dot\phi^2 -\phi'^2\right),
\end{eqnarray}
where $\dot \phi = \pa_t\phi$ and $\phi'=\pa_x\phi$. Classically, the deformed action corresponding to this seed theory was calculated in \cite{Cavaglia:2016oda}, and is given by the Nambu-Goto action:
\begin{eqnarray}\label{deformedLNG}
\mathcal{L}^{(\l)} = \frac{1}{4\l}\left(-1+\sqrt{1+4\l\left(\dot\phi^2 -\phi'^2\right)}\right).
\end{eqnarray}
The canonical momentum conjugate to $\phi$ is given by
\begin{eqnarray}
\pi = \frac{\delta \mathcal{L}^{(\l)}}{\delta \dot \phi}=\frac{\dot\phi}{\sqrt{1+4\l\left(\dot\phi^2 -\phi'^2\right)}},
\end{eqnarray}
from which we can easily obtain $\dot\phi$ as a function of $\pi$. In the Hamiltonian perspective, the canonical variables $(\phi,\pi)$ on an initial time slice (say, $t=0$) are to be regarded as $\l$-independent field variables, while $\dot\phi^{(\l)}(\phi,\pi)$ is $\l$-dependent. We will often suppress the explicit $\l$-dependence of $\dot\phi$, but the reader should bear this in mind. The Hamiltonian is given by
\begin{eqnarray}
H_\l = \int dx\,h_{\l}(x),\;\;\;h_{\l}=\frac{1}{4\l}\left(1-\sqrt{\left(1-4\l \pi^2 \right)\left(1-4\l \phi'^2\right)}\right).
\end{eqnarray}
Note that the Hamiltonian density at finite $\l$ can be rewritten in terms of that of the seed theory as
\begin{eqnarray}\label{DefHam}
h_{\l}(x) = \frac{1}{4\l}\left(1-\sqrt{1+16\l^2 p_0^2 -8\l h_0}\right),
\end{eqnarray}
where $h_0=\frac{1}{2}(\pi^2+\phi'^2)$ and $p_0=\pi\phi'$ are the energy and momentum density of the seed theory. The (canonical) stress tensor can be obtained using Noether's procedure:
\begin{eqnarray}
T^{(\l)}_{ij} = -\pa_i\phi \frac{\delta \mathcal{L}^{(\l)}}{\delta \pa^j\phi}+\eta_{ij}\mathcal{L}^{(\l)}.
\end{eqnarray}
Applying this to the action \eqref{deformedLNG}, we find
\begin{eqnarray}
T^{(\l)}_{00} = \frac{1}{4\l}\left(1-\sqrt{\left(1-4\l \pi^2 \right)\left(1-4\l \phi'^2\right)}\right)=h_{\l}, \quad T^{(\l)}_{01} = \pi \phi'= p_0,
\end{eqnarray}
and
\begin{eqnarray}
T^{(\l)}_{11} = \frac{1}{4\l}\left\{-1+4\l \phi'^2\sqrt{\frac{1-4\l\pi^2}{1-4\l \phi'^2}}+\sqrt{\frac{1-4\l\phi'^2}{1-4\l \pi^2}}\right\},
\end{eqnarray}
where we observe that the momentum density $p_{\l}(x)$ at finite $\l$ is actually $\l$-independent at $t=0$. One can readily check that this stress tensor satisfies the flow equation $\partial_\l T_{00} = \e^{ab}\e^{cd}T_{ac}T_{bd}$. From here, we can compute the generator of the canonical transformation:
\begin{eqnarray}\label{Xscalar}
\mathcal{X}^{(\l)}=\int dx dy\,\mathrm{sgn}(x-y)h_{\l}(x)p_0(y).
\end{eqnarray}
If we have some observable $\mathcal{O}(\phi,\pi)$ in the seed theory, then the corresponding dressed observable $\widetilde{\mathcal{O}}^{(\l)}(\phi,\pi)$ can be obtained by solving the following flow equation
\begin{eqnarray}\label{ObsFlow}
\pa_{\l}\widetilde{\mathcal{O}}^{(\l)}(\phi,\pi)= -\left\{\mathcal{X}^{(\l)},\widetilde{\mathcal{O}}^{(\l)}(\phi,\pi)\right\}_{PB}.
\end{eqnarray}
This equation may look complicated because of the $\l$-dependence in $\mathcal{X}^{(\l)}$, but a closer look at equations \eqref{Xscalar} and \eqref{DefHam} reveals that we can transform this into a $\l$-independent flow by defining the new variables (assuming, for convenience, $\l>0$):
\begin{eqnarray} \label{DimLess0}
x= \sqrt{\l}\,\widehat{x},\;\;\phi(x) = \widehat{\phi}(\widehat{x}),\;\;\pi(x)=\frac{1}{\sqrt{\l}}\widehat{\pi}(\widehat{x}).
\end{eqnarray}
Note that this change of phase space coordinates is also a canonical transformation, i.e., it preserves the Poisson brackets. Thus, we can rewrite equation \eqref{ObsFlow} in these new variables as
\begin{eqnarray}\label{ObsFlow2}
\l\pa_{\l}\widetilde{\mathcal{O}}^{(\l)}(\widehat{\phi},\widehat{\pi})= -\left\{\widehat{\mathcal{X}},\widetilde{\mathcal{O}}^{(\l)}(\widehat{\phi},\widehat{\pi})\right\}_{PB},
\end{eqnarray}
where we have defined the new $\l$-independent generator $\widehat{\mathcal{X}}$ as
\begin{eqnarray}\label{G1}
\widehat{\mathcal{X}}=\mathcal{D}+\mathcal{K},
\end{eqnarray}
where we have defined
\begin{eqnarray} \label{G2}
\mathcal{D}= \int dx\,x \widehat{T}_{01}(x),\quad
\mathcal{K} = \frac{1}{2}\int dx\,dy\,\mathrm{sgn}(x-y)\varepsilon^{ab}\widehat{T}_{0a}(x)\widehat{T}_{0b}(y),
\end{eqnarray}
and the hatted stress tensor is defined in terms of $\widehat{\phi}$ and $\widehat{\pi}$:
\begin{eqnarray}
\widehat{T}_{00} = \frac{1}{4}\left(1-\sqrt{\left(1-4 \widehat{\pi}^2 \right)\left(1-4 \widehat{\phi}'^2\right)}\right),\;\;\; \widehat{T}_{01} = \widehat{\pi} \widehat{\phi}',
\end{eqnarray}
and does not depdent explicitly on $\l$ anymore.
Thus, in these dimensionless variables, the flow equation for the dressed observables becomes $\l$-independent. We can also rewrite equation \eqref{ObsFlow2} in terms of a $\l$-independent vector field $\mathcal{V}$ on phase space:
\begin{eqnarray}
\l\pa_{\l}\widetilde{\mathcal{O}}^{(\l)}(\widehat{\phi},\widehat{\pi}) = -\int dx \left[\mathcal{V}^{\widehat{\pi}}\frac{\delta}{\delta \widehat{\pi}(x)}+\mathcal{V}^{\widehat{\phi}}\frac{\delta}{\delta \widehat{\phi}(x)}\right]\widetilde{\mathcal{O}}^{(\l)}(\widehat{\phi},\widehat{\pi}),
\end{eqnarray}
where $\mathcal{V}^{\widehat{\pi}}=\frac{\delta \widehat{\mathcal{X}}}{\delta \widehat{\phi}}$ and $\mathcal{V}^{\widehat{\phi}}=-\frac{\delta \widehat{\mathcal{X}}}{\delta \widehat{\pi}}$. The vector field $\mathcal{V}$, which, in the language of symplectic geometry is the Hamiltonian vector field dual to the generating function $\widehat{\mathcal{X}}$, entirely encodes the flow of the dressed observables. At any rate, the key point is that $\mathcal{V}$ is $\l$-independent, and so we can formally integrate this flow:
\begin{eqnarray}
\widetilde{\mathcal{O}}^{(\l)}(\widehat{\phi},\widehat{\pi})= e^{\log(\frac{\l_0}{\l})\,\int dx \left[\mathcal{V}^{\widehat{\pi}}(x)\frac{\delta}{\delta \widehat{\pi}(x)}+\mathcal{V}^{\widehat{\phi}}(x)\frac{\delta}{\delta \widehat{\phi}(x)}\right]}\widetilde{\mathcal{O}}^{(\l_0)}(\widehat{\phi},\widehat{\pi}).
\end{eqnarray}
This gives an explicit, albeit formal, construction of the classically dressed observables in this theory. Above, we saw that the flow equation for the dressed observables could be expressed in terms of a $\l$-independent flow. Although we have only shown this in the special example of the classical, free scalar field, we expect this phenomenon to be generally true of all $T\overline{T}$ deformed CFTs on the plane. If so, the path-ordering in the unitary $U$ can be removed very generally for CFTs on the plane, by repeating the same argument above. Furthermore, equation \eqref{ObsFlow2} seems to fit nicely within the circle of ideas involving tensor networks (especially the MERA) and the surface state correspondence in AdS/CFT, if we interpret the operator $\mathcal{K}$ above as a ``disentangler''. We will return to this point in the next section.
\subsection{On the cylinder}\label{sec:cylinderCorr}
In contrast with the plane, we do not have a complete picture of how operators/correlation functions behave on the cylinder. We present some preliminary results below.
\subsubsection*{Undeformed operators}
We can define the undeformed operators on the cylinder in the same way as we did for the plane -- we take the operators on an initial time slice to be those of the seed theory (except for the stress tensor), and then operators at a time separation away are defined by time evolution with the deformed Hamiltonian. Even so, correlation functions on the cylinder are much more complicated because both energy eigenvalues and eigenstates change along the flow. For simplicity, let us consider a two-point function of two scalar operators in the vacuum:
\begin{eqnarray}
G_\l(t,x) = \langle 0_\l|O^{(\l)}(t,x)O^{(\l)}(0,0)|0_{\l}\rangle.
\end{eqnarray}
By inserting a complete set of energy eigenstates of the deformed theory, this correlator can be rewritten as
\begin{eqnarray}
G_{\l}(t,x) = \sum_{n} |\braket{0_\l|O(0,0)|n_{\l}}|^2 e^{-i t \Delta E_n(\l)} e^{-ik_n x},
\end{eqnarray}
with $\Delta E_n = (E_n-E_0)$ is the energy relative to the ground state energy in the deformed theory. Analogously to the Euclidean computation of the finite temperature partition function \cite{Dubovsky:2018bmo, Hashimoto:2019wct}, we rewrite the exponential factors using an integral transform,
\begin{eqnarray} \label{Kernel}
e^{-i t \Delta E_n(\l) - i k_n x} = \int d^2 x'\, K_{\l}(t,x;t',x') e^{-i t' \Delta E_n(0) - i k_n x'}.
\end{eqnarray}
We can obtain the kernel $K_{\l}$ by a suitable Wick rotation of the contour of integration from the Euclidean formula in \cite{Dubovsky:2018bmo, Hashimoto:2019wct}:
\begin{eqnarray}\label{KernelExplicit}
K_\l(t,x;t',x') = -\frac{t L}{8\pi \l} \frac{1}{t'^2}\exp\left(\frac{L}{8i\l t'}\left( -(t-t')^2 + (x-x')^2 \right) \right)
\end{eqnarray}
The integration region in \eqref{Kernel} for $x'$ is the full real line, whereas for $t'$ it lies on the positive real axis. With this kernel, we can write the deformed correlator as an integral transform of the undeformed one,
\begin{eqnarray}
G_{\l}(t,x) = \int d^2 x\,K_{\l}(t,x;t',x')\widehat{G}(t',x'),
\end{eqnarray}
with
\begin{eqnarray}
\widehat{G}(t',x') = \langle 0_0 | e^{it'H_0} U^{-1}O(0,x')U e^{-it'H_0}U^{-1}O(0,0)U|0_0\rangle.
\end{eqnarray}
\subsubsection*{Dressed operators}
Given that the deformation on the cylinder is not a pure canonical transformation, it is not immediately clear how we should define dressed operators. We will provisionally\footnote{It would be worthwhile to see whether this definition makes sense in the full quantum theory} define them as a generalization of \eqref{FlowSol} in the plane case:
\begin{eqnarray}
\widetilde{O}^{(\l)}(0,x) = UO(0,x)U^{-1},\;\;U=\mathcal{P}\,e^{-i\int_0^{\l}d\l'\left(\mathcal{X}^{(\l')}+\mu\int_{-\infty}^0 ds e^{\varepsilon s}\mathcal{Y}(s)\right)},
\end{eqnarray}
or in terms of a flow equation, we have
\begin{eqnarray} \label{DressedOp}
\pa_{\l}\widetilde{O}^{(\l)}(0,x)=-i\left[\mathcal{X}^{(\l)},\widetilde{O}^{(\l)}(0,x)\right]-i\mu \int_{-\infty}^0ds\,e^{\varepsilon s}\left[\mathcal{Y}(s),\widetilde{O}^{(\l)}(0,x)\right],
\end{eqnarray}
where recall that $\mathcal{Y}=\varepsilon^{ac}\varepsilon^{bd}\mathbf{P}_{ab}(s)\mathbf{P}_{cd}(s)$, with $\mathbf{P}_{ab}(s) = \oint dx T^{(\l)}_{ab}(s,x)$. Operators at finite time can be obtained by time evolution with the deformed Hamiltonian. The dressing in the cylinder case is substantially more complicated because of the presence of the term proportional to $\mu$. Correlation functions of these operators are, nevertheless, simpler; for instance the two-point function is given by
\begin{eqnarray}
\widetilde{G}_{\l}(t,x) = \int d^2 x\,K_{\l}(t,x;t',x')G_0(t',x'),\;\;G_0(t',x') = \langle 0_0 | O(t,x')O(0,0)|0_0\rangle,
\end{eqnarray}
where $G_0$ is the two-point function in the original seed theory and $K_\l$ given in \eqref{KernelExplicit}. Given the difficulties in computing the unitary matrix $U$ and the flow of the stress tensor needed to compute the deformed matrix elements, the deformed correlator of dressed operators is remarkably simple and does not suffer from these difficulties, which partly justifies their definition. Unlike the plane case, however, correlation functions of dressed operators do flow on the cylinder -- they are merely smeared versions of the seed correlation functions, with the smearing function $K_{\l}$. This can be thought of as the two dimensional version of the prescription put forward in\cite{Gross:2019ach, Gross:2019uxi} for computing deformed correlation functions in quantum mechanics. A slightly different point of view can be obtained through a differential equation for the deformed correlator, again inspired from the one for the torus partition function \cite{Aharony:2018bad, Datta:2018thy}. The change in the energy levels then follows from the differential equation. It is straighforward to check that the appropriate differential operator acting on $\widetilde{G}_{\l}$ is
\begin{eqnarray} \label{diffeqG}
\frac{iL}{2}\partial_\l \widetilde{G}_\l(t,x) = \left[t(\partial_x^2 - \partial_t^2-E_0^2) - 2\l\left(\partial_t - \frac{1}{t}\right) \partial_\l\right] \widetilde{G}_\l(t,x).
\end{eqnarray}
This point of view has the advantange, that we do not need to worry about the existence of a kernel and analytic continuation. From here we can actually also see the smearing. For instance, consider small $\l$, then the only term on the LHS that is going to contribute is the Laplacian on 2d Minkowski space. The differential equation then looks like a diffusion equation with $\l$ playing the role of an additional fictitious time, and the diffusion constant $D\sim t/L$.
From this differential equation we can actually learn some more. Consider for instance chiral correlators, say $\widetilde{G}_{\l}(x_+)$, then the differential equation for that correlator becomes,
\begin{eqnarray}
\frac{iL}{2}\partial_\l \widetilde{G}_\l(x_+) = \left[ -\l \partial_+ \partial_\l + \frac{4\l}{x_+ - x_-}\partial_\l \right]\widetilde{G}_{\l}(x_+),
\end{eqnarray}
whose solution is the undeformed chiral correlator $G_0(x_+)$, since the other solution depends on $x_-$. We thus see that not only the energy eigenvalues of states with $E = k$ do not flow, also chiral correlators are independent of $\l$.
Thusfar we have only considered correlators of scalar operators. For the stress tensor we expect the flow of correlation functions to be much more complicated. To calculate, for instance, the entanglement entropy of a region on the circle using twist operators such correlation functions and their flows would be required. We leave the study of these computations to future work and discuss them briefly in the discussion section.
We would also like to define a dressed stress tensor. However, naively defining the dressed stress tensor in the same way as in \eqref{DressedOp} is not enough; we want to ensure that the dressed stress tensor is conserved and that its spatial integrals reproduce the energy and momentum operators. One can check that a naive definition of the dressed stress tensor following \eqref{DressedOp} violates the conservation condition. However, we can deduce the appropriate flow for the stress tensor by studying the conservation equation. Following the same steps leading to equation \eqref{finalflowT1} in the plane case, we get on the cylinder:
\begin{eqnarray}\label{finalflowT2}
\pa_{\l}\left( \pa^{\mu}\widetilde{T}^{(\l)}_{\mu\nu}\right)=\partial^{\mu} \left(D_{\l}\widetilde{T}^{(\l)}_{\mu\nu}(x)\right)-i\left[\mathcal{Y},\widetilde{T}^{(\l)}_{1\nu}\right],
\end{eqnarray}
where $D_{\l}=\partial_\l + i[\mathcal{X}^{(\l)},\,\cdot\,]$ is the same covariant derivative defined previously, and recall that $\mathcal{Y}=\mu \varepsilon^{ab}\varepsilon^{cd}\mathbf{P}_{ac}\mathbf{P}_{bd}$. Therefore, conservation of the dressed stress tensor implies:
\begin{eqnarray}
\partial^{\mu} D_{\l}\widetilde{T}^{(\l)}_{\mu\nu}(x) = i[\mathcal{Y},\widetilde{T}^{(\l)}_{0\nu}(x)].
\end{eqnarray}
From here, it is possible to extract the flow equation for the deformed stress tensor. The final expressions are a bit complicated, so we will present them in Appendix \ref{App:CylinderT}. Note, however, that this flow equation for the dressed stress tensor is different from that of the other operators we guessed in equation \eqref{DressedOp}, and this implies that the commutation relations of the dressed stress tensor with itself and with the other dressed operators will not be preserved along the flow. In particular, we have not checked whether the dressed stress tensor satisfies microcausality (i.e., whether it commutes at spacelike separation with the other dressed operators). It would be nice to understand the causality structure of these dressed operators, or to see if one can define a fully causal set of dressed operators; we leave this to future work.
\section{Further developments}
\label{sec:further}
\subsection{$S$-matrix}
So far we have discussed (arguably) the most important players in a field theory: the operators, spectrum and correlation functions. By knowing how these objects change under the $T\overline{T}$ flow, we know, in principle, everything there is to know about the deformed theory. In this section, we will consider the $S$-matrix on the plane. This quantity has been discussed extensively \cite{Caselle:2013dra, Dubovsky:2012wk, Dubovsky:2013ira} and here we give yet another derivation from our perspective.
Let us start with the $T\overline{T}$ deformed theory on the plane at some value of the coupling $\l$. We wish to ask how the S-matrix of the theory changes when we flow from $\l \to \l + \delta \l$. To set up a scattering process, we need to define in and out states at the asymptotic past and future. In the undeformed theory, such states where constructed using insertions of particle creation and annihilation operators at the past and future null infinities. As a result of the $T\overline{T}$ deformation, these operators will now get dressed in the same way as was discussed in section \ref{sec:operators}, i.e., $a_{p^i} \to U a_{p_i} U^{-1}$. At any rate, the momenta of these particles will be taken as an input for the S-matrix computation. We then deform $\l \to \l + \delta \l$, and ask how the S-matrix changes under this deformation. This is given by
\begin{eqnarray}
S_{\l+\delta \l} =\lim_{t\to \infty}\, _{\text{out}}\langle p'_1,\cdots, p'_{m}| \mathcal{T}e^{-i\delta \l\int_{-t}^tdt'\pa_{\l}H(t')} | p_{1},\cdots p_{n}\rangle_{\text{in}}.
\end{eqnarray}
Using the fact that $\pa_{\l} H = i\left[H, \mathcal{X}^{(\l)}\right]$, i.e. $\pa_{\l}H$ is a total time-derivative, we learn that the deformation only gives rise to boundary terms at asymptotic infinity:
\begin{eqnarray}
S_{\l+\delta \l} =\lim_{t\to \infty}\, _{\text{out}}\langle p'_1,\cdots, p'_{m}| e^{-i\delta \l \mathcal{X}^{(\l)}(t)}e^{i\delta \l \mathcal{X}^{(\l)}(-t)} | p_{1},\cdots p_{n}\rangle_{\text{in}}.
\end{eqnarray}
We can conveniently rewrite the contribution at past asymptotic infinity by introducing a Hubbard-Stratanovich field:
\begin{eqnarray}
e^{i\delta \l \mathcal{X}^{(\l)}(-t)} | p_{1},\cdots p_{m}\rangle_{\text{in}} = \int [D\xi] e^{-2i\delta \l\int du\,\left(\varepsilon_{ab}\xi^a(u)\pa_{u}\xi^b(u) + \xi^a T^{(\l)}_{0a}(-t,u)\right)}| p_{1},\cdots p_{n}\rangle_{\text{in}},
\end{eqnarray}
where $u$ is a coordinate along the asymptotic spatial slice which approaches past infinity in the limit $t\to \infty$. There is a similar term coming for future infinity as well. If we now take the action of $T_{0a}$ on the in state to be given by $T_{0a}(u) = \sum_{i=1}^n p_a^i \delta(u_i - u)$ to represent the $n$-particle in state, and similarly account for the term from future infinity, we precisely land on the gravitational dressing proposed in \cite{Dubovsky:2017cnj} and therefore the $S$-matrix,
\begin{eqnarray}\label{Smatrix}
S_{\l+\delta \l}(\{p^i\}) = e^{-i\frac{\delta \l}{2}\sum_{i<j}\e_{ab}p_a^ip_b^j}S_\l(\{p^i\})
\end{eqnarray}
where we have collectively denoted all the in and out momenta by $\{p^i\}$ in this last formula. Since the momenta are $\l$-independent, we
|
us (D.M.) would like to thank T.
|
Dahm
for useful discussions and numerical help.
|
\section{Introduction}\fontsize{9}{12}\selectfont
Quantum dots (QDs) are known for their tunable and strongly energy-dependent electron transport properties, which result in a nonlinear response to an applied electrical bias $V_{SD}$. Nonlinear conductance due to the Coulomb blockade \cite{VanHouten1992} is perhaps the most well known example of such nonlinear behavior. It is also well established that the energy-dependent electron transport properties of QDs strongly influence their thermoelectric behavior \cite{Beenakker1992, Staring1993}, which has made them attractive model systems for fundamental studies of quantum thermoelectric effects \cite{Humphrey2002, Edwards1993, ODwyer2006, Esposito2009, Nakpathomkun2010, Jordan2013, Zianni2009}. Nonlinear response to an applied thermal bias $\Delta T$, in particular, has been theoretically investigated in various mesoscopic systems, including resonant tunneling structures \cite{WANG2006, Snchez2013}, multi-terminal quantum conductors \cite{Snchez2013, Meair2013, Whitney2013} and Kondo-correlated devices \cite{Boese2001, Azema2012}. For QDs, one can expect that the quasi-discrete resonance energy spectrum of a QD alone should lead to nonlinear thermoelectric response \cite{Nakpath2010, Svensson2013}. This behavior was explored in detail by Sierra and Sanchez who predicted a strongly nonlinear regime behavior in QDs when $\Delta T$ is about an order of magnitude larger than the background temperature $T_0$ \cite{Sierra2014}.
In experiments, a nonlinear thermovoltage as a function of thermal bias $\Delta T$ has been observed in semiconductor QDs \cite{Staring1993, Svensson2013, Pogosov2006, Hoffmann2009} and in molecular junctions \cite{Reddy2007}. Most recent studies using a tunable thermal bias have shown a strongly nonlinear thermovoltage and thermocurrent in semiconductor nanowire QDs that could not be fully explained by the energy-dependence of the QD resonance energy spectrum alone, and was attributed to a renormalization of resonance energies as a function of heating \cite{Svensson2013}.
The key experimental challenge in the observation of nonlinear thermoelectric behavior in QDs is the ability to apply a tunable and large enough thermal bias $\Delta T$ across a nanoscale object without significant overall heating of the device. The latter can prevent the ability to perform low-temperature experiments, and makes it difficult to distinguish temperature-dependent transport effects from the true nonlinear response to the thermal bias $\Delta T$.
Here, we report measurements of a strongly nonlinear thermocurrent as a function of $\Delta T$ across a QD that is defined by two InP segments within an InAs nanowire. To a large extent the measurements presented here were enabled by a recently developed heater architecture that allows local and electrically non-invasive thermal biasing of a nanowire \cite{Gluschke2014}. This architecture enables tuning of $\Delta T$ over a wide range by applying a relatively small heating power, thus minimizing the parasitic heating effects. We also use theoretical calculations based on Master equations to demonstrate that the experimentally measured thermocurrent can be fully understood from the QD resonance energy spectrum, and is consistent with the previously presented theory in Ref.\hspace{1mm}\cite{Sierra2014}.
\section{Experiment}
\subsection{Device Fabrication}
The device consists of a heterostructured InAs/InP nanowire with a 60 nm diameter (see Fig.\hspace{0.5mm}\ref{fig:1}a) that was grown by chemical beam epitaxy seeded by a gold particle \cite{Froberg2008, Persson2007}. Based on transmission electron microscopy (TEM) analyses of $11$ nanowires from the same growth, the InAs/InP nanowire (starting from the seed particle) consists of a $350\smallpm70$ nm InAs segment, followed by a $17\smallpm1.5$ nm long InAs QD defined by two, $4\smallpm3$ nm thick, InP segments, and a second InAs segment of $265\smallpm60$ nm in length. The remaining nanowire, which is not used in the device, consists of a $25$ nm InP plug incorporated for growth reasons and another InAs segment.
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Fig1.pdf}
\caption{
(a) Transmission electron microscope image of a nanowire nominally identical to the one used in our thermoelectric device. (b) Device schematic with circuitry diagram for the $I_{th}$ measurement setup. The source and drain contacts in yellow, top-heaters in orange, InAs/InP nanowire in green, quantum dot in light green. The heater over the drain lead is unused. (c) Stability diagram of the InAs quantum dot. Magnitude of differential conductivity, $g=dI/dV_{SD}$, in log$_{10}$-scale as a function of back-gate bias, $V_G$, and source drain bias, $V_{SD}$.
}\label{fig:1}
\end{figure}
The nanowire is contacted to metallic source and drain contacts, as illustrated in Fig.\hspace{0.5mm}\ref{fig:1}b. Electrically isolated metallic top-heaters pass over the source and drain contacts enabling local dissipation of Joule heat directly on top of the contacts; ensuring heat transfer to the nanowire. Only the heater on top of the source contact was used in the experiments presented here. The device fabrication followed the process developed by Gluschke et al \cite{Gluschke2014}. In brief, electron-beam lithography (EBL) was used to define a pair of source and drain contacts centered around the QD and separated by $300$ nm. A dilute sulfur passivation is performed before source and drain contacts are deposited on the nanowire \cite{Suyatin2007}. A $10$ nm thick layer of HfO$_2$ was deposited via atomic layer deposition to insulate the metallic contacts from the overlying heaters, which were aligned and exposed in a second EBL step. Both the contacts and the heaters were deposited thermally with a metal stack of $25$ nm Ni and $75$ nm Au for the contacts and $25$ nm Ni and 125 nm Au for the heaters. The heater layer was thicker to ensure continuity as the heater steps onto the contact region. The entire device rests on $100$ nm of thermally grown SiO$_2$, allowing the underlying doped Si substrate to be used as a global back gate.
\subsection{Electrical Characterization}
Measurements were conducted in a cryostat in which the estimated electron temperature in the device, $T_0$, was below $1$ K without heating. Bias spectroscopy of the device was carried out using a Stanford Research SRS-830 lock-in amplifier. The voltage from the oscillation output was reduced using a $1:20000$ voltage divider circuit to provide a stable AC source-drain bias amplitude $dV_{SD}=25$ \textmu V $\ll k_B T_0/e$ ($k_B$ - Boltzmann constant, $e$ - elementary charge). To measure the differential conductance $g=dI/dV_{SD}$ as a function of a DC source-drain bias $V_{SD}$, the differential current amplitude, $dI$, was measured in response to $dV_{SD}$, while adding the AC and DC source-drain bias components in a summing box.
To measure Coulomb oscillations (Fig.\hspace{0.5mm}\ref{fig:2}a), a source-drain current, $I_{SD}$, was measured in DC mode using Yokogawa 7651 voltage source to bias the source lead at $100$ \textmu V and a SR570 current preamplifier with $1$ M\textOmega $\mbox{ }$ input impedance.
The set-up used for thermoelectric characterization of the QD nanowire device is shown in Fig.\hspace{0.5mm}\ref{fig:1}b. A thermal bias, $\Delta T$, was applied by running a current $I_H$ through the heater on top of the source contact using a Yokogawa 7651 DC voltage source. The dissipated Joule heat mostly heats the underlying source contact, but is expected to also create a fractional temperature rise in the drain contact \cite{Gluschke2014}. The resulting thermocurrent through the QD nanowire device, $I_{th}$, was amplified via the SR570 current preamplifier.
\begin{figure*}[ht]
\includegraphics[width=\textwidth]{Fig2.pdf}
\caption{
(a) Coulomb oscillations in source-drain current $I_{SD}$ as a function of back-gate voltage $V_G$, with the source potential set to $100$ \textmu V. (b) Thermocurrent, $I_{th}$, as a function of back-gate bias for different heater currents $I_H\!=\!(0, 0.35, 0.70, 1.06, 1.41, 3.17 \mbox{ mA})$. (c) Thermocurrent (color) as a function of back-gate voltage, $V_G$, and heating current $I_H$. Arrows along the top correspond to $V_G$ values for traces in (d) as indicated by their color. (d) Thermocurrent as a function of heating current $I_H$ for different $V_G$ values $(-0.165, -0.154, -0.141, -0.131, -0.115, -0.101, -0.085\mbox{ V})$ taken from data in (c).
}\label{fig:2}
\end{figure*}
\subsection{Experimental Results and Discussion}
The QD's stability diagram, measured as a function of the source-drain voltage, $V_{SD}$, and a back-gate voltage, $V_G$, is shown in Fig.\hspace{0.5mm}\ref{fig:1}c. The dark diamond-like regions represent bias conditions at which the conductivity is suppressed due to Coulomb blockade. From the bias spectroscopy data we estimate a charging energy $E_C$ of $4.0\smallpm0.2$ meV, which is a measure of electron-electron interaction strength in the QD. We also determine the value of the coupling constant $\alpha_G=0.042\smallpm0.04$, which characterizes the capacitive coupling strength between the QD and the back-gate electrode.
Figure \ref{fig:2}b shows $I_{th}$ as a function of $V_G$. The data confirms that our device's thermoelectric response is typical for QDs \cite{Beenakker1992, Svensson2013, Svensson2012} where $I_{th}$ goes to zero and changes direction at those $V_G$ values where the Coulomb peaks in Fig.\hspace{0.5mm}\ref{fig:2}a are centered. The locations of these thermocurrent zeros do not depend on the heating current, as can be seen in Fig.\hspace{0.5mm}\ref{fig:2}c, which shows $I_{th}$ as a function of $V_G$ and $I_H$. This independence of the $I_{th}$ zeros from $I_H$ is in contrast to previous studies \cite{Svensson2013}, where the nonlinear behavior of $I_{th}$ was strongly influenced by a heating dependent renormalization (shift) of the resonance energies of the QD. The stability of the resonances in the present study is attributed to the benefits of the top-heater architecture where a higher $\Delta T$ can be applied with much less overall background heating of the device \cite{Gluschke2014}.
The core observation of our experiments is the strongly nonlinear behavior of the thermocurrent as a function of $\Delta T$. This nonlinearity is clearly apparent in Fig.\hspace{0.5mm}\ref{fig:2}d where several back-gate voltage traces, taken from the data in the Fig.\hspace{0.5mm}\ref{fig:2}c, are plotted as a function of $I_H$.
Several key features can be identified in the observed nonlinear behavior of $I_{th}$, all of which can be understood in terms of the QD's resonance energy spectrum at different thermal biases. In the following we base our discussion on Ref.\hspace{0.5mm}\cite{Sierra2014} and use phenomenological sketches of a QD resonance spectrum and Fermi-Dirac distributions in the leads to illustrate how the increase in $\Delta T$ can lead to nonlinear effects (Fig.\hspace{0.5mm}\ref{fig:3}). The currents $I_{\varepsilon 1}$ and $I_{\varepsilon 2}$ in Fig.\hspace{0.5mm}\ref{fig:3}b combine to give the overall thermocurrent $I_{th}$ through the QD.
First, we observe that the $I_H$ at which $I_{th}$ starts to rapidly increase depends on $V_G$ (Fig.\hspace{0.5mm}\ref{fig:2}d). As shown in sketch A in Fig.\hspace{0.5mm}\ref{fig:3}a, this behavior can be understood based on the energy of the QD resonances, $\varepsilon_1$ and $\varepsilon_2$. Until the temperature on the hot side reaches a
|
certain value, there is no net current because the electronic states at energies $\varepsilon_1$ and $\varepsilon_2$ in both leads are equally occupied - either completely full or completely empty. This is reflected in point A in Fig.\hspace{0.5mm}\ref{fig:3}b.
The second interesting experimental feature in Fig.\hspace{0.5mm}\ref{fig:2}d is the nonlinear increase of $I_{th}$, as a function of thermal bias. Sketch B in Fig.\hspace{0.5mm}\ref{fig:3}a illustrates how increased heating on the source side leads to a misbalance of the electronic state occupancy in the leads at $\varepsilon_1$. This misbalance leads to a net current as indicated by an arrow in the sketch and by point B in Fig.\hspace{0.5mm}\ref{fig:3}b. Thus, the origin of the nonlinear increase in $I_{th}$ is the nonlinear change of the electronic state occupancy in the leads due to heating.
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Fig3.pdf}
\caption{
(a) Schematic representation of electron distribution in source (red) and drain (blue) leads when the thermal bias is (A) $k_B \Delta T_H/E_C = 0.02$, (B) $0.1$ and (C) $0.3$. Current direction through resonances of a quantum dot is indicated with arrows. Electron energy increases up the vertical axis. (b) Simulated thermocurrent as a function of thermal bias for the back-gate voltage $e\alpha_G V_G/E_C = 0.24$ (black). Brown curves are thermocurrent contributions through each resonance of the quantum dot. See Fig.\hspace{0.5mm}\ref{fig:4} for simulation parameters and Sec.\hspace{0.5mm}\ref{sec:3} for a detailed description.
}\label{fig:3}
\end{figure}
Finally, $I_{th}$ tends to decrease at higher $I_H$. Ref. \cite{Sierra2014} predicts such behavior due to an increasing backflow of electrons at large thermal bias values $(\Delta T/T\geq 10)$. We believe that the same is true for $I_{th}$ in our experiment, except we expect that we also parasitically heat the drain lead when aiming for high $\Delta T$. Sketch C in Fig.\hspace{0.5mm}\ref{fig:3}a illustrates that the major current contribution, $I_{\varepsilon 1}$, is still provided by the electron transport through $\varepsilon_1$, however, the thermally excited electrons on the source side also leak back through $\varepsilon_2$, thus contributing to the decrease in $I_{th}$. We note that any decrease of the current through $\varepsilon_1$ in the sketch is, in fact, caused by the overall increase in temperature; e.g. slight heating of the drain. However, the backflow of electrons through $\varepsilon_2$ is caused purely by the thermal bias.
\section{Theory}\label{sec:3}
\subsection{Model Description}
We model electron transport through the InAs/InP nanowire by considering a QD which is tunnel-coupled to two electron reservoirs (source and drain leads). Following the experimental setup showed in Fig.\hspace{0.5mm}\ref{fig:1}b the QD is considered in series with a resistive load $R$ to model the input impedance of the current preamplifier. The source and drain leads are characterized by their electrochemical potentials, $u_S = E_F - eV_S$ and $u_D = E_F - eV_D$, where $E_F$ is Fermi energy, and their temperatures, $T_S$ and $T_D$. Electrons in the leads are assumed to occupy states according to the Fermi-Dirac distribution $f_r (E) = \{ 1 + \exp \left[ (E-u_r) / (k_B T_r) \right] \}^{-1}$ and the density of states in the leads is assumed to be a constant. The QD is capacitively coupled to the leads with capacitances $C_S$ and $C_D$, and to the global back-gate with a capacitance $C_G$, giving rise to a charging energy $E_C = e^2 / ( C_S + C_D + C_G )$. In order to model resonance energies we consider a QD in which adding the $N^{th}$ electron changes its state from $i$ to $f$ and that has an electrochemical potential of the form
\begin{equation*}
\mu_{fi}=\epsilon_{fi}+(N-1)E_C- \!\!\!\!\sum\limits_{r=G,S,D} \!\!\!\!\alpha_r V_r.
\end{equation*}
Here $\epsilon_{fi}$ is energy of the single-electron orbital in which the electron is added and $\alpha_r = C_r/( C_S + C_D + C_G )$ are dimensionless coupling constants. We label the probability of the $f^{th}$ state to be occupied $p_f$. Steady-state probabilities for each state occupancy can be represented by a vector $\mathbf{P}$ and are found using the Master equation for a stationary case
\begin{equation*}
\mathbf{W P}=\mathbf{0}.
\end{equation*}
Here $\mathbf{W}$ is a matrix with elements $W_{fi}$ given by
\begin{equation*}
W_{fi} = \begin{cases}
\sum\limits_{r=S,D}\!\!\left\{\Gamma_{fi}^{r,in}f_r(\mu_{fi})+
\Gamma_{fi}^{r,out}\left[1-f_r(\mu_{fi})\right]\right\} \text{,\hspace{2.5mm}if $i\neq f$}\\
-\sum\limits_m W_{mf}\text{,\hspace{4.65cm}if $i = f$}
\end{cases}
\end{equation*}
where $\mathbf{\Gamma^{S,in}}$, $\mathbf{\Gamma^{D,in}}$, $\mathbf{\Gamma^{S,out}}$ and $\mathbf{\Gamma^{D,out}}$ are matrices containing tunnel rates for single electron tunneling in or out of the QD, involving source or drain leads. Here non-diagonal matrix elements $W_{fi}$ express physical rates at which the QD changes its state from $i$ to $f$. Probability normalization requires that the sum of all occupancy probabilities pf must be $1$.
The current $I_{SD}$ through the QD is then found by adding up current contributions from all possible QD states given the calculated steady state occupancies $p_f$
\begin{equation*}
I_{SD}=-e\sum\limits_{i,f}p_f \{\Gamma_{fi}^{S,in}f_S(\mu_{fi})-\Gamma_{fi}^{S,out}\left[1-f_S(\mu_{fi})\right]\}.
\end{equation*}
In order to calculate the current $I_{SD}$ through the circuit with the QD and the load $R$ in series, a bias value on the drain side $V_D$ is calculated self-consistently using the Ohms law $V_D=I_{SD}R$.
For the purpose of comparing with our experimental results it is sufficient to consider a QD with only one single electron orbital, in which $N$ can take values 0, 1 or 2. Including electron spin this gives four possible QD states $i,f=\{0, \uparrow, \downarrow, \uparrow\downarrow\}$. In this case, the phenomenological resonance energies $\varepsilon_1$ and $\varepsilon_2$ discussed in the experimental section (Fig.\hspace{0.5mm}\ref{fig:3}) thus correspond to the electrochemical potentials $\mu_{\sigma 0}=\varepsilon_1$ and $\mu_{\uparrow\downarrow\sigma}=\varepsilon_2$, with $\sigma=$ $\uparrow,\downarrow$. For qualitative comparison with experiment we consider the tunnel-barriers to be identical and characterized by a constant tunnel rate $\Gamma$.
\subsection{Simulation Results}\label{sec:3.2}
We now calculate the thermocurrent as a function of temperature in source and drain leads. Since in our experiment the source lead is heated, we label the source temperature $T_S=T_H=T_0+\Delta T_H$ and the drain temperature $T_D=T_C=T_0+\Delta T_C$. In simulations the base temperature $T_0$ is chosen such that $k_B T_0/E_C=0.01$, which is close to the experimental value. Because in the experiments the drain lead is also expected to be somewhat heated we assume $\Delta T_C=\Delta T_H/3$. The ratio between $\Delta T_H$ and $\Delta T_C$ is chosen to obtain a qualitative agreement with the experimental data, but the precise value is not important for the discussed physics.
\begin{figure}[h]
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{
(a) Simulated thermocurrent as a function of back-gate voltage for different thermal biases $k_B \Delta T_H/E_C\!\!=\!\!(0, 0.04, 0.08, 0.12, 0.16, 0.32)$. (b) Simulated thermocurrent as a function of the thermal bias for several back-gate voltage values $e\alpha_G V_G/E_C\!\!=\!\!(0.11,0.24,0.37,0.50,0.63,0.76,0.89)$. (c) Simulated thermocurrent (color) as a function of both, back-gate voltage and thermal bias. Other parameters: $\Gamma=5$ GHz, $R=1$ M\textOmega, $T_0=0.01E_C$.
}\label{fig:4}
\end{figure}
In Fig.\hspace{0.5mm}\ref{fig:4} we sum up our thermocurrent simulation results. Thermocurrent as a function of the back-gate voltage for different thermal bias values is shown in Fig.\hspace{0.5mm}\ref{fig:4}a (compare with the corresponding experimental data in Fig. \ref{fig:2}b). Similarly, we plot the simulated thermocurrent as a function of the thermal bias for different back-gate voltage values in Fig. \ref{fig:4}b. The dimensionless range of thermal bias shown is chosen based on the similarity to Fig.\hspace{0.5mm}\ref{fig:2}d. Finally, the color plot in Fig.\hspace{0.5mm}\ref{fig:4}c is produced using the ranges of the electrochemical potential and the thermal bias used in Figs.\hspace{0.5mm}\ref{fig:4}a and b, and closely matches the experimental result shown in Fig.\hspace{0.5mm}\ref{fig:2}c.
According to our simulations, the source-drain bias $V_{SD}$ that develops across the QD due to the series load at peak currents is estimated to be below $\raisebox{.3\height}{\scalebox{.7}{ $\pm$ }} 0.04$ $E_C/e$ and therefore does not significantly influence the behavior of the thermocurrent. Note that it is very challenging to measure the temperature in the leads leading up to the QD directly and this was not attempted in the experiment. However, given the qualitative agreement between the experimental thermocurrent data in Fig.\hspace{0.5mm}\ref{fig:2} and the simulated thermocurrent in Fig.\hspace{0.5mm}\ref{fig:4}, one can conclude that the relation between $I_H$ and $\Delta T$ must be close to linear. Moreover, the agreement also suggests that $1$ mA of $I_H$ gives rise to a thermal bias $\Delta T$ of several Kelvin between the source and drain leads.
\section{Conclusions}
In summary, we have reported measurements of a strongly nonlinear thermocurrent in a QD. By comparing our measurements to simulation results, we show that the nonlinear behavior can be fully explained in terms of the QD's energy-dependent transport properties \cite{Sierra2014}. This is in contrast to earlier experiments \cite{Svensson2013} where this behavior was masked by effects that can also be explained by the overall heating of the device. Our results were enabled by use of a novel heating technique \cite{Gluschke2014} that allows the application of very large $\Delta T$ across a nanoscale device with minimal overall heating of the sample space, even at low temperatures. The ability demonstrated here opens a wide range of quantum thermoelectric experiments in mesoscopic systems.
\acknowledgments
We thank Sebastian Lehmann for the TEM image in Fig.\hspace{0.5mm}\ref{fig:1}a. This work was supported by the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7-People-2013-ITN) under REA Grant agreement no. 608153, by the Swedish Energy Agency (Project P38331-1), by the Swedish Research Council (Project 621-2012-5122) and by NanoLund.
\begin{comment}
\end{comment}
\pagebreak
|
\section{Introduction}
Many applications in modern robotics require a rich perception of the environment. The automatic detection of known objects, for instance, may help in scenarios like quality control, automated assembly, pick-and-place, bin-picking and so on. The specific task of \emph{object detection}, as many others in computer vision, has witnessed dramatic advances in recent years due to the introduction of deep learning methods capable of recognizing many different object categories in real time \cite{redmon2016yolo9000,huang2016speed,Lin_2017_ICCV}. Yet, these state-of-the-art methods mandate training on large datasets of images annotated with bounding boxes surrounding the objects of interest. The process of manually annotating the training images is time-consuming, tedious and prone to errors that may often lead to flimsy datasets or noisy annotations, especially when these are performed by non-professional users. Although a number of high-quality, large, multi-category training datasets are publicly available \cite{lin2014microsoft,Rennie2016,Calli2017}, they usually concern general classes (such e.g. person, car, cat, etc.) that may not suit the needs of a specific -- industrial -- task. In particular, robotic applications typically require detection of a relatively small set of specific object instances in cluttered and heavily occluded scenes captured from many different viewpoints, and with the sought objects possibly changing frequently overtime. In these settings, the richness and fickleness of the training dataset play a fundamental role to any deep learning solution. Indeed, handling a new object may easily require thousands of annotated images, which translates into tons of man-hours.
As at such large scale manual annotation turns out impractical and often inaccurate, we propose a user-friendly approach that allows to gather effortlessly and almost automatically huge datasets of annotated images in order to train state-of-the-art 2D Object Detectors based on deep learning like \cite{redmon2016yolo9000,huang2016speed,Lin_2017_ICCV}, or even 3D Pose Estimators like \cite{Kehl_2017_ICCV}, \cite{xiang2017posecnn}, \cite{rad2017bb8}.
We start by acquiring a sequence of 2D images alongside with the camera pose in each frame. Based on this input information, our method deploys Augmented Reality techniques to enable the user to create easily the manual annotations for the first frame (or few ones) and then can deliver automatically accurate annotations for all the other frames of the sequence without any further human intervention. Hence, we dub our proposal \textbf{ARS{}: Augmented Reality Semiautomatic-labeling{}}. It is worth pointing out that the camera poses required by our method may be obtained in different manners, such as, e.g., by a monocular SLAM algorithm like \cite{mur2017orb,engel2014lsd} or a motion capture system. Yet, as in this paper we mainly address robotic applications, we propose leveraging on a camera directly mounted on a robotic arm in an \emph{eye-on-hand} configuration in order to gather tracked image sequences with high tracking accuracy, as also shown, \textit{e.g. }, in \cite{Zeng2016,de2016robotfusion,Mitash}. We expect such an approach to become increasingly more practical and affordable with the advent of lightweight, but precise, \textit{collaborative robots}, allowing the dataset creation directly on the application scenario, reducing in this way the gap, with regard to data distribution, between training and real conditions.
We rely only on a plain 2D camera, which is cheap and does not set forth any restriction as long as objects are visible whereas a Stereo or RGB-D camera would have hindered flexibility, e.g. due to constraints on the minimum object-camera distance (and thus object size) or the inability to sense poorly textured or black surfaces. We developed a publicly available ROS package
\footnote{\url{https://github.com/m4nh/ars}}
implementing all the tools described in this paper, which can be used to realize the ARS{} labelling pipeline starting from a video sequence with associated camera poses. Furthermore, we distribute all the datasets used throughout the experiments (see \autoref{sec:experimental}), which enables reproducibility of the experimental results.
\section{Related work}\label{sec:related}
A popular approach to speed-up creation of training datasets consists in the use of synthetic images rendered \cite{mayer2016large,Ros2016,movshovitz2016useful,Carlucci2016} or even grabbed from realistic videogames \cite{Richter2016,Johnson-Roberson2016}. These techniques can deliver countless perfectly annotated images with human effort/time spent only to build synthetic scenes.
However, obtaining a large dataset of photo realistic images usually comes at a cost as it may require hours of highly specialized human work to construct suitable synthetic environments plus many hours of computation on high-performance graphical hardware for rendering. In some practical settings, useful synthetic objects may be available beforehand in the form of CAD models although, indeed quite often, the textures may either be missing or look quite diverse with respect to the appearance of the actual objects.
Moreover, it is well known \cite{movshovitz2016useful,Carlucci2016} that training deep neural network by synthetic images does not yield satisfactory performance upon testing on real data due to the inherent difference between the ideal and real images, an issue often referred to as \emph{domain gap} and calling for specific \emph{domain adaptation} techniques, such as, e.g., fine-tuning the network by -- fewer -- manually annotated real images.
Recent works \cite{shrivastava2016learning,zhang2018fully,tzeng2017adversarial,bousmalis2017using} focus on developing ad-hoc adaptation techniques to close the performance gap between training and test distribution. Unfortunately the performance achievable are still quite far from those obtainable training on real data or fine tuning on few annotated samples.
Alternatively, to ameliorate the domain shift, \cite{Georgakis2017} proposes an hybrid approach whereby an object detection system is trained on rendered views of synthetic 3D objects superimposed on real images; however, the blend between synthetic and real is far from perfect such that an additional fine-tuning on a real dataset is still needed.
Differently, in this paper we propose a methodology to ease and speed-up the acquisition of large labeled datasets of real images which may be acquired directly in the deployment scenario, thereby avoiding any gap between the training and test domains.
Several approaches tailored to dataset creation have been proposed within the robotic research community: \cite{Kendrick2017} proposes a system suitable for 3D face annotation, \cite{Zeng2016} proposes a semi-automatic technique to acquire a training dataset for object segmentation and \cite{Mitash} extends the idea to support object detection by leveraging on physical simulation to create realistic object arrangements. All this proposals require depth information from the sensor, and for the last two, realistic 3D models (\textit{e.g. } textured CAD models). A similar solution is proposed in \cite{nguyen2016robust} and \cite{wong2015smartannotator} where an environment is reconstructed by means of an RGB-D sensor and the labeling procedure is performed on the resulting 3D model. Conversely, our approach does not need any clue about he 3D shape of the object nor does it require depth information at training or test time. Moreover, our approach is the first technique usable with very small objects (as shown in \cite{de2018integration}).
\section{Method description}\label{sec:method_description}
Given a set of images, each equipped with the 6-DoF pose of the camera, and knowing the pose of the observed objects \textit{w.r.t.} each vantage point, it is possible to project in each image some simplified representation of these items through augmented reality in order to generate automatically annotations (e.g. 2D bounding boxes, class labels, etc.) useful to train machine learning models. \autoref{sec:input_dataset} describes formally the input data required by our proposed ARS{} labeling pipeline, which, as depicted in \autoref{fig:teaser}, can be summarized in the following main steps:
\begin{enumerate}
\setcounter{enumi}{-1}
\item Scene Setup (\textit{i.e. } Arrange the objects randomly);\vspace{-0.0cm}
\item Outline virtual boxes around the target objects;\vspace{-0.0cm}
\item Scan the environment by a tracked camera;\vspace{-0.0cm}
\item Refine virtual boxes by visual analysis of the scan;\vspace{-0.0cm}
\item Generate automatically a training dataset.
\end{enumerate}
\noindent ARS{} can be used to generate a dataset starting from scratch (\textit{i.e. } following all the steps 0,1,2,3,4) or by exploiting archived material (\textit{i.e. } following only steps 3 and 4, assuming the availability of an off-line camera tracker algorithm, like \cite{mur2017orb}, to be applied to a recorded video sequence).
The reminder of this section will describe in detail all the important steps of the labeling pipeline: \autoref{sec:input_dataset} deals with the input data, \autoref{sec:augmente_pen} presents a way to define virtual boxes by an Augmented Reality Pen directly interacting with the physical environment, \autoref{sec:pose_refinement} addresses refining (or create a posteriori) virtual boxes around the objects and, finally, \autoref{sec:gen_training_data} describes the procedure used to generate annotated images. In \autoref{sec:notation}, we introduce the notation adopted throughout the rest of the paper.
\begin{figure*}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.31\textwidth]{images/arp/arp-eps-converted-to.pdf}&\includegraphics[width=0.31\textwidth]{images/arp/arp_with_squared-eps-converted-to.pdf}&\includegraphics[width=0.31\textwidth]{images/arp/arp_with_spherical-eps-converted-to.pdf}\\
\textbf{(a)} ARP &
\textbf{(b)} Drawing on geometrical object &
\textbf{(c)} Drawing on rounded object \\
\end{tabular}
\caption{(a) The \emph{Augmented Reality Pen} (ARP) used to draw virtual boxes. The pen features several Augmented Reality Markers with a known pose $^{tip}\mathbf T_{mk_i}$ \textit{w.r.t.} the tip. (b) By tracking the tip position we can easily draw a virtual box around a target object by touching its edges. (c) Conversely, its not simple to draw a virtual box around a rounded object.}
\label{fig:arp}
\end{figure*}
\subsection{Notation}\label{sec:notation}
We denote as $^{A}\mathbf{T}_{B} \in \mathbb R^{4 \times 4}$ a 3D reference frame (briefly RF) $B$ expressed in the base $A$. So $^{0}\mathbf{T}_{cam}$ represents the RF linked to the camera in the $zero$ reference frame (\textit{i.e. } the \textit{world} RF). $m_i$ denotes a generic image and $b = \{ x_b,y_b,w_b,h_b, c_b \}$ a square region (box) therein, with $(x_b,y_b)$ the coordinates of the center of $b$, $w_b$ and $h_b$ the \emph{width} and \emph{height}, respectively, and, optionally, $c_b \in \mathbb Z^{+}$ the \emph{class} of the object contained in $b$.
\subsection{The input data}\label{sec:input_dataset}
The input data to our labelling pipeline consist of two separate sets $\mathbb{F}$ and $\mathbb{I}$:
\begin{equation}
\begin{gathered}
\mathbb{F} = \{ F_i = \{^{0}C_i , m_i \} , i \in [0,...,n] \} \\
\mathbb{I} = \{ V}\newcommand{\bboxset}{\mathcal{B}_j=\{^{0}\mathbf T_j , s_j, c_j \} , j \in [0,...,k] \}
\end{gathered}
\end{equation}\label{eq:dataset}
\noindent $\mathbb{F}$ represent the acquired images (\emph{frames}) $F_i=\{^{0}C_i , m_i \}$ with $^{0}C_i$ being the camera matrix for image $m_i$. Each $^{0}C_i$ can be expressed as the multiplication of the \emph{intrinsics} matrix $\hat{A} \in \mathbb R^{4\times4}$ encoding camera specific parameters and the \emph{extrinsics} matrix:
\begin{equation}
^{0}\mathbf T_{cam_i} =\small{ \begin{bmatrix}
\mathbf R_{cam_i} & \mathbf{t}_{cam_i} \\
0 & 1
\end{bmatrix}}\in \mathbb R^{4\times4}
\end{equation}
\noindent encoding camera orientation ($\mathbf R_{cam_i} \in \mathbb R^{3\times3}$) and position ($\mathbf{t}_{cam_i} \in \mathbb R^{3}$) with respect to the \textit{world} frame.
\noindent $\mathbb{F}$ can be obtained using any method to track the camera movement, for example: a motion capture system, a SLAM based method like \cite{mur2017orb} or a camera mounted on a robotic arm in an \textit{eye-on-hand} configuration.
The $\mathbb{I}$ set, instead, corresponds to a collection of $k$ object instances present in the scenes. Each instance can be thought of as a 3D Virtual box, as shown in \autoref{fig:arp}(b-c), and can be expressed as a tuple $V}\newcommand{\bboxset}{\mathcal{B}_j=\{^{0}\mathbf T_j , \mathbf{s}_j, c_j \}$ with $^{0}\mathbf T_j$ the 6-DoF pose of the instance in the \textit{world} reference frame, $\mathbf{s}_j \in \mathbb R^{3}$ the box dimensions and $c_j \in \mathbb Z^{+}$ its \emph{class}. In the following we will cover different methodologies to collect a suitable $\mathbb{I}$ set.
\subsection{Online Labeling by the Augmented Reality Pen}\label{sec:augmente_pen}
To create $\mathbb{I}$ quickly and effortlessly, we developed the 3D printed artifact pictured in \autoref{fig:arp}, which resembles a pen covered with Augmented Reality Markers (in short: Markers \cite{munoz2012aruco}); we will refer to this device as to the \emph{ARP} (Augmented Reality Pen). This tool can be used for labeling \emph{online} object instances by interacting directly with the environment.
Each Marker on the ARP has a known pose $^{tip}\mathbf T_{mk_i}$ \textit{w.r.t.} the tip of the pen (the placement is CAD-driven). Using the OpenCV Marker Detector\footnote{\url{https://opencv.org/}} and the calibrated camera parameters we can estimate the pose of each of the markers, and consequently of the tip, \textit{w.r.t.} the camera $^{cam}\mathbf T_{mk_i}$, then : $\small ^{cam}\mathbf T_{tip} = ^{cam}\mathbf T_{mk_i} \cdot (^{tip}\mathbf T_{mk_i})^{-1}$.
To achieve proper vertices estimation, the ARP is tracked exploiting multiple markers simultaneously, each of which contributes to refine the estimated tip position:
by averaging those given by all the visible markers we can obtain a more accurate estimation of the real position.
Accuracy therefore depends very much on the type of marker used, and has therefore not been dealt over in detail in this paper. As an indication, using this type of squared marker results in an error on the pose estimation proportional to the angle of inclination of the marker itself.
A more detailed explanation of this approach, along with an accuracy analysis, is described in \cite{jiawei2010three} and \cite{wu2017dodecapen}.
In particular, in \cite{wu2017dodecapen} a dodecahedron (instead of our parallelepiped-shaped pen) is used to reduce the angle of view of visible markers, thus reducing the estimation error.
As shown in \autoref{fig:arp}(b), as the ARP can be tracked while being used to -- virtually -- draw a box around an object, those 3D points can then be used to construct the corresponding tuple $V}\newcommand{\bboxset}{\mathcal{B}_j = \{^{0}\mathbf T_j , s_j, c_j \}$. Our approach requires only four specific points $p_0,p_1,p_2,p_3$ placed as shown in \autoref{fig:arp}(b). Then, from those spatial positions we can obtain the corresponding components of $V}\newcommand{\bboxset}{\mathcal{B}_j$ as:
\begin{equation}\label{eq:virtual_object_buildings}
\begin{gathered}
^{cam}\mathbf T_j =
\begin{bmatrix}
\mathbf{v}_x & \mathbf{v}_z \times \mathbf{v}_x & \mathbf{v}_z & \mathbf{p_0} \\
0 & 0 & 0 & 1
\end{bmatrix}
\\
\\
\mathbf{v}_z =\frac{• \mathbf{p}_1 - \mathbf{p}_0}{\norm{ \mathbf{p}_1 - \mathbf{p}_0}} ,
\mathbf{v}_x =\frac{• \mathbf{p}_2 - \mathbf{p}_1}{\norm{ \mathbf{p}_2 - \mathbf{p}_1}}
\\
\\
s_j =
\begin{bmatrix} s_x \\ s_y \\ s_z \end{bmatrix} =
\begin{bmatrix}
\norm{ \mathbf{p}_2 - \mathbf{p}_1} \\
\norm{ \mathbf{p}_3 - \mathbf{p}_2} \\
\norm{ \mathbf{p}_1 - \mathbf{p}_0}
\end{bmatrix}
\end{gathered}
\end{equation}
\noindent The class $c_j$ should instead be specified by the user. The
\autoref{eq:virtual_object_buildings} specifies the RF of the virtual object $^{cam}\mathbf T_j $ referred to the camera RF; however, it can be easily transformed into the \textit{world} RF by knowledge of the current camera pose $^{0}\mathbf T_j = {^{0}}\mathbf T_{cam} \cdot {^{cam}}\mathbf T_j $.
By exploiting the ARP, a tight 3D bounding box can be easily created around box-like objects, such as in the case of electromechanical components, see \autoref{fig:arp}(b). However, this method will not work properly in case of arbitrarily shaped objects, such as the fruits depicted in \autoref{fig:arp}(c). To overcome this limitation, an off-line procedure to sketch $V_j$ by labeling a pair of frames picturing the same object from different vantage points has been developed, as reported in the following.
\subsection{Offline Labeling}\label{sec:pose_refinement}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{images/refinement/refinement_screens-eps-converted-to.pdf}
\caption{ Four random \emph{frames} acquired during the creation of our new dataset: the first row refers to the Industrial{} dataset and displays virtual boxes drawn by the ARP; the second row shows samples from the Fruits{} dataset with annotations created off-line by the technique described in \autoref{sec:pose_refinement}.}
\label{fig:refinement_screens}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{images/frustum_intersection/frustum_intersection-eps-converted-to.pdf}
\caption{Graphical representation of the Visual Hull procedure used to build a virtual object from multiple 2D images.}
\label{fig:frustum}
\end{figure}
In case of object with non-boxed shape, an offline procedure has been developed to increase the annotation accuracy with respect to what can be achieved by the ARP method previously described. The Fruits Dataset has been built by exploiting this off-line approach with good results, as shown in \autoref{fig:refinement_screens}. The offline procedure is based on a graphic interface through which the user can manually draw suitable masks around at least two different views of the object directly on the image frame. The complete procedure is detailed in the following.
As shown in \autoref{fig:refinement_screens}, upon acquisition of $\mathbb{F}$ and $\mathbb{I}$ it is quite straightforward to display frames from $\mathbb{F}$ with superimposed a 2D re-projection of the object instances defined in $\mathbb{I}$: each virtual object $V}\newcommand{\bboxset}{\mathcal{B}_j$ can be represented as a list of 3D points corresponding, in this specific case, to the eight vertices of the box. By arranging the vertices as columns of the matrix $^{0}P_{V_j} \in \mathbb R^{4 \times 8}$ ($4$ rows are a result of the homogeneous coordinates conversion) and converting them in the $i_{\text{th}}$ camera RF, $^{cam}P_{V_j}=^{cam_i}\mathbf T_{0} \cdot {^{0}}P_{V_j}$, we can simply perform a 3D-to-2D re projection through:
\begin{equation}
\begin{bmatrix}
\lambda H_{V_j} & 1
\end{bmatrix}^{\intercal} = \hat{A} \cdot {^{cam}}P_{V_j}
\end{equation}
\noindent where $H_{V_j}$ is the set of corresponding 2D points and $\lambda$ the scale factor.
This procedure is quite general and can be applied to any set of points (\textit{e.g. } instead of virtual boxes we could have used virtual squares made by only 4 points if we are dealing with planar objects or arbitrary complex polygons for arbitrary shaped objects).
\autoref{fig:refinement_screens} shows many examples of reprojected virtual boxes $H_{V_j}$ each one being the 2D re projection of the virtual box $V_j$.
However, as shown in \autoref{fig:refinement_screens}(top-right), the $H_{V_j}$ produced with the ARP tool can sometimes not be highly accurate due to several nuisances (\textit{e.g. } the user hand-shake); therefore, we included in our software GUI a graphical tool that shows in augmented reality both the object and its bounding box from multiple points of view, as if the user was immersed in a 3D environment (see the third frame of \autoref{fig:teaser}), by offering a browsable \textit{Mixed Reality} scene \cite{milgram1994taxonomy}.
This tool allows also to manually edit the bounding boxes, in order to correct the position, orientation and size of the virtual objects. Moreover, through the same GUI also a novel technique can be exploited to annotate the fruit dataset, in order to estimate the position of an object by analyzing two tracked frames (i.e. two rgb images of which I know the exact camera pose) without any initial guess.
The aforementioned method consists simply in estimating a 3D virtual box (or a more complex 3D shape) by manually drawing its 2D re-projection on at least two frames, as depicted in \autoref{fig:frustum}.
Considering a pair of frames $F_1,F_2$ we can draw two 2D masks $k_1,k_2$ around the appointed object (the red apple in this case); knowing the two camera matrices $^{0}C_1,^{0}C_2$ we can compute two view frustums which shall intersect -- likely -- in the center of mass of the real object producing what is generally called \emph{Visual Hull} (VH) \cite{baker2005shape}. The pose of the new $V_j$ (\textit{i.e. } $^{0}\mathbf T_j$) for the translational part can be computed easily using the center of mass of the VH, while for the rotational part we can choose the canonical identity matrix $I \in \mathbb R^{3 \times 3}$ as a starting point. Finally, to complete the new $V_j$, we use the minimum bounding box algorithm, over the produced VH, to compute its dimensions $s_j$. It is important to note that a VH tends to be equal to the real 3D shape of the target object with the increase of the input frames labeled with the abovementioned technique, however the coarse $V_j$ created by this pipeline can be always manually refined using the already described interactive procedure.
\subsection{Genaration of the Training Data}\label{sec:gen_training_data}
The final step of the ARS{} labeling pipeline, once both $\mathbb{F}, \mathbb{I}$ have been acquired and refined, is the creation of a training dataset suitable for modern machine learning models.
In this work, as stated in the introduction, we mainly address the object detection task, which we will consider as an explanatory use case to show how to generate a training set. Our goal is to create a dataset consisting of images annotated with labeled 2D bounding boxes, $b_j$, surrounding each of the visible target objects. This information can be attained straightforwardly by simply reprojecting in each frame of $\mathbb{F}$ the 3D virtual objects in $\mathbb{I}$, \textit{i.e. } $H_{V_j}$, and then computing a function to produce a squared bounding box $b_j$ from it. A graphical representation of the process is depicted in \autoref{fig:refinement_screens} (bottom-right), where it is clear that $b_j = \tau(H_{V_j})$ is obtained through function $\tau(\cdot)$ (\textit{i.e. } \textit{the minimum 2D bounding box}), which indeed may be replaced by any custom function to obtain different kinds of labeled data (\textit{e.g. } \textit{the convex hull}).
\section{Experimental evaluation}
\label{sec:experimental}
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.45\columnwidth]{images/manual_00049_reduced.png}&\includegraphics[width=0.45\columnwidth
|
]{images/auto_00049_reduced.png}\\
\textbf{(a)} Manual Industrial & \textbf{(b)} Auto Industrial
\end{tabular}
\caption{Samples from the dataset used for our experiments. The green rectangles display the annotated bounding boxes, the white text over each box is the class of that box. (a) Electromechanical components dataset annotated manually, (b) Electromechanical components dataset with auto-generated labels.}
\label{fig:dataset}
\end{figure}
To validate the ARS{} pipeline we performed three sets of tests on two novel datasets that we are going to introduce in \autoref{ssec:datasets}. In \autoref{ssec:annotation}, we compare our automatic labeling procedure against a manual one. In \autoref{ssec:objDetTest}, we show how our datasets can be used to train CNN based object detectors. In \autoref{sec:slam_comparison}, we will compare monocular SLAM versus the use of a Robot for tracking the camera pose. Finally, in \autoref{ssec:histo} we introduce a new and interesting way of analyzing datasets, made possible only by our labeling approach.
Some qualitative results are shown in \autoref{fig:live_detections} as in the supplementary material, which also features a live demo of the ARS{} labeling procedure from scratch.
\subsection{Datasets and evaluation metrics}
\label{ssec:datasets}
To validate our proposal we choose as test beds two different detection tasks: one concerning recognition of 7 types of electromechanical components (\emph{Industrial{}}), the other with 5 classes of fruits (\emph{Fruits{}}); some samples from the two datasets are depicted in \autoref{fig:live_detections}. The first task concerns \textit{instance detection}, as the items display low intra and inter class variability: the same components are seen across all the images from different vantage points, with different components looking remarkably similar. This kind of instance detection is used in the WIRES project
(described in \cite{wires2018})
to implement a \textit{quality control} and \textit{automated assembly} systems for switchgear. The second task, instead, concerns \textit{object detection}, as items show high intra and inter class variability: each fruit is quite different from the others, yet also fruits belonging to the same class can show quite different appearances, e.g. in our acquisition we have two kinds of apples and pears showing different peel colors. The produced fruits object detector can be used for a simple \textit{pick\&place} manipulation application. The first dataset is composed of 9 acquisition ($\sim36000$ frames), the second by 8 shorter ones ($\sim7500$ frames).
Both datasets were built by means of a camera mounted
on an industrial manipulator, a COMAU Smart Six, with a position repeatability lesser than $0.05mm$. Using the industrial manipulator to move the camera we achieved a nearly perfect camera tracking, by computing the 6-DoF sensor pose through the robot kinematics, as to establish an upper bound for the labeling performance of ARS{}.
We will show in \autoref{sec:slam_comparison} how for less constrained applications, such us mobile robotics or home service robots, ARS{} could rely on a classical monocular SLAM pipeline for tracking the camera (although sacrificing some accuracy).
We chose one sequence from Industrial{} dataset and two from the Fruits{} dataset, manually annotating them, to be used as test sets for the trained object detectors, we will refer to them as Industrial{}\_Test and Fruits{}\_Test, respectively. The other sequences are randomly rearranged to create sets with increasing number of samples, each splitted in 80\% train and 20\% validation. We will refer to each one of such set as: \texttt{$\langle$dataset name$\rangle$\_$\langle$number of samples$\rangle$}, \emph{e.g.} Industrial{}\_1000 identifies 1000 sample from the training sequences of Industrial{} that will be split into 800 training samples and 200 for validation.
All datasets have been automatically annotated with ARS{} and for further tests we enriched Industrial{}\_1000 with manual annotation as well. We will use a "\_M" suffix for a manually annotated dataset (e.g. Industrial{}\_1000\_M) and "\_A" suffix for annotation using ARS{} (e.g. Industrial{}\_1000\_A).
\autoref{fig:teaser} shows a graphic comparison of the different efforts in human work hours needed to manually annotate a dataset (growing linearly with the number of required images) vs using ARS{} (constant once sequences are acquired and virtual boxes are created). For reference, factoring out the common acquisition time, the manual annotation of 1000 frames took us slightly more than 10 hours, while using ARS{} we were able to annotate all the 9 sequence of the Industrial{} dataset in less than an hour ($\sim35000$ frames), with a gain of factor $\sim 450$.
To measure the detector performance we will use the standard object recognition metrics defined for the PASCAL VOC challenge \cite{Everingham10}. Given a prediction $b_j^p$ and the corresponding ground truth box $b_i$, we consider $b_j$ correct if they have the same class and ${IOU(b_j^p,b_i)>IOU_{th}}$ with $IOU(\cdot)$ intersection over union of the boxes and $IOU_{th}$ a threshold parameters. Given the set of correct predictions we can measure: \emph{Precision},\emph{Recall} and \emph{average intersection over union for correct predictions (avgIOU)}. Usually a detector produces quite a lot of $b_j^p$ each one associated with a certain confidence value $t_j^p \in [0,1]$, by thresholding the minimum confidence allowed we can tune the behaviour of the system. We represent the global performance with different confidence threshold using Precision/Recall curves and, concisely, with the \textbf{mean average precision}($mAP$), defined as the approximation of the area under the precision recall curve.
\subsection{Annotation Study}
\label{ssec:annotation}
In this section we want to test if the annotation obtained by ARS{} resemble what a human annotator would do. To verify this, we have compared the two sets of annotations, manual and auto-generated, by considering the first as the output of an ideal detector, while the second as ground truth annotations.
Using $\small\text{IOU}_{th}=0.3$ we obtain the following performance: $\small\text{Precision}=98.49\%$, $\small\text{Recall}=95.02\%$ and $\text{avgIOU}=0.7$, \textit{i.e. } comparing manual to automatic annotation the first set has fewer annotations (5\% less Recall) and there exist some class misalignment (1.5\% less Precision).
To gain more insights on those differences, we visually examined the boxes obtained from the two sets of annotations and found out that the missing $1.5\%$ Precision can be mostly explained by class mistakes made by the human annotator during the labelling process, while ARS{} cannot assign any wrong label by construction. The $5\%$ missing recall is instead due to situations like that depicted in \autoref{fig:dataset}(a-b), where the visible portion of an object (the bottom object \emph{Cls\_1}) is too small to allow the human annotator to recognize it.
Finally, the relatively low $avgIOU$ highlights a key difference between ARS{} and a human annotator, the former always produces a box large enough to enclose the whole object as side effect of the re-projection of the virtual 3D box, while the latter usually encloses only the visible portion of the object (See the \autoref{fig:refinement_screens} (top-right) image where a virtual box is drawn also where the object is occluded). As a result, the manual and auto annotations does not always have matching shapes, especially in cluttered environment, as it can be observed in \autoref{fig:dataset} (a-b).
Nevertheless, as we will prove in the following paragraphs, the dataset labelled with ARS{} can effectively be used to train and validate any machine learning based object detector obtaining performance comparable with a manually annotated dataset of the same size, while also enabling an effortless training with many more images so as to create quite very robust object detectors.
\subsection{Object Detector Test}
\label{ssec:objDetTest}
\begin{figure*}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{images/plots/yolo_indust_trainingsize-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{images/plots/ssd_indust_trainingsize-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{images/plots/yolo_vs_ssd_indust-eps-converted-to.pdf} \\
\textbf{(a)} YOLO & \textbf{(b)} SSD & \textbf{(c)} Comparison \\
\end{tabular}
\caption{Precision/Recall curves for the two type of detector trained on different subsets of the Industrial{} dataset. (a) and (b) report the results for YOLO and SSD respectively; (c) instead displays a comparison between them.}
\label{fig:industrial_results}
\end{figure*}
\begin{figure*}
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.3\textwidth]{images/plots/yolo_fruits_trainingsize_manual-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{images/plots/ssd_fruits_trainingsize_manual-eps-converted-to.pdf} &
\includegraphics[width=0.3\textwidth]{images/plots/ssd_yolo_fruits_singleclass-eps-converted-to.pdf}
\\
\textbf{(a)} YOLO & \textbf{(b)} SSD & \textbf{(c)} Class specific Comparison \\
\end{tabular}
\caption{Precision/Recall curves for the two type of detector trained on different subsets of the Fruits{} dataset. (a) and (b) report the results for YOLO and SSD respectively; (c) instead displays a comparison between SSD and YOLO over the best class (\textit{Banana}) and the worst class (\textit{Pear}) of the Fruits{} dataset.}
\label{fig:fruitResult}
\end{figure*}
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{\textbf{\emph{mAP}} (\text{th}=0.5)}& \multicolumn{2}{c|}{\textbf{\emph{avgIOU}}}\\
\hline
\textbf{Training set}& YOLO&SSD& YOLO&SSD\\
\hline
Industrial{}\_1000\_M&0.589&0.619&0.7479&\textbf{0.795}\\
Industrial{}\_1000\_A&0.731&0.562&0.719&0.728\\
Industrial{}\_3000\_A&0.799&0.809&0.713&0.720\\
Industrial{}\_5000\_A&0.828&0.831&0.705&0.729\\
Industrial{}\_15000\_A&0.834&\textbf{0.851}&0.709&0.732\\
\hline
\end{tabular}%
}
\captionsetup{size=small,skip=0.333\baselineskip}
\small
\caption{Mean average precision (\emph{mAP}) and average intersection over union (\emph{avgIOU}) on the Industrial{}\_Test+ for YOLO and SSD trained using 5 different training sets with increasing number of images. Best result highlighted in bold.}
\label{tab:elettroresult}
\end{table}
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{%
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{\textbf{\emph{mAP}} (\text{th}=0.5)}& \multicolumn{2}{c|}{\textbf{\emph{mAP}} (\text{th}=0.3)}
& \multicolumn{2}{c|}{\textbf{\emph{avgIOU}}}\\
\hline
\textbf{Training set}& YOLO&SSD& YOLO&SSD& YOLO&SSD\\
\hline
Fruits{}\_2500\_A&0.438&0.468&0.895&0.888&0.710&0.744 \\
Fruits{}\_5000\_A&0.440&0.465&0.894&0.889&0.6818&0.749 \\
Fruits{}\_7500\_A&0.469&\textbf{0.504}&0.902&\textbf{0.904}&0.734&\textbf{0.756} \\
\hline
\end{tabular}%
}
\captionsetup{size=small,skip=0.333\baselineskip}
\small
\caption{Mean average precision (\emph{mAP}), at two different $IOU_{th}$, and average intersection over union (\emph{avgIOU}) on Fruits{}\_Test for YOLO and SSD trained using 3 different training sets with increasing number of images. "A" suffix marks training sets with annotations produced by ARS{}, best result highlighted in bold.}
\label{tab:fruitresult}
\end{table}
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{images/plots/yolo_orbslam-eps-converted-to.pdf}
\caption{Precision/Recall curves for the YOLO detector trained on a subset of the Industrial{} dataset where the camera is also tracked with the \textit{Orb-Slam-v2} algorithm\cite{mur2017orb}.}
\label{fig:slam_results}
\end{figure}
Once assessed that annotations obtained by ARS{} are comparable with manual ones, we test how effective is an object detector system trained on them. We choose as detectors YOLO \cite{redmon2016yolo9000} and SSD \cite{huang2016speed}, using for both the original author's implementation and public pre-trained networks as initial weights.
The first tests concern the Industrial{} dataset where we defined a test bench of 126 images randomly picked from Industrial{}\_Test plus 20 external smartphone pictures, all carefully manually annotated. We will call Industrial{}\_Test+ this dataset.
Thanks to the fast labelling obtained by ARS{} we were able to actually measure the performance boost related to the training set size using four different sets with increasing number of images, respectively Industrial{}\_1000, 3000, 5000 and 15000. The samples for Industrial{}\_1000 have both automatic and manual annotations, as to test if and how much the final detection performances change according to the label used.
Given the five different training sets, we trained both detectors on them for 100000 steps with \texttt{batch\_size=24} using the default hyperparameters recommended by the authors. The results are ten slightly different detectors that we tested on Industrial{}\_Test+, we reported in \autoref{fig:industrial_results} the obtained precision/recall curves and in \autoref{tab:elettroresult} the \emph{mAP} and \emph{avgIOU}.
As expected, for both detectors the performance increases proportionally to the size of the training set used, \textit{e.g. } +0.23 mAP gain between SSD trained on 1000 images and the best performing one trained on 15000; vouching the need for a method to ease and speed up the creation of huge training dataset.
Inspecting the \emph{avgIOU} obtained by the detectors we can see how the best performing methods are, unsurprisingly, the two trained on the manually annotated Industrial{}\_1000\_M. It is due to the fact the the testing images are manually annotated, so a manual label is better suited for an IOU score.
We repeated similar experiments on the Fruits{} dataset: we annotated all the eight sequences using ARS{}, then we produced three different training and validation sets with increasing number of samples and sequences, respectively Fruits{}\_2500 (2 sequences), 5000 (4 sequences) and 7500 (6 sequences).
As stated above, the remaining two sequences are used as the test bench for the detector creating a manual labelled test set of 1000 images (refereed as Fruits{}\_Test).
We tested the six different resulting detectors on Fruits{}\_Test and report the result in \autoref{fig:fruitResult} and in \autoref{tab:fruitresult}. Once again all the performance indexes increase alongside with the size of the training set, with best absolute performance obtained by SDD using the Fruits{}\_7500 dataset.
\autoref{fig:fruitResult}(c) reports an intra-detector comparison between YOLO and SSD on the best and the worst class.
\subsection{Comparision between visual SLAM and robot-based camera tracking}\label{sec:slam_comparison}
Aiming at the comparison between the ARS approach and what can be achieved through different kinds of camera trackers, the state-of-the-art monocular SLAM described in \cite{mur2017orb} has been exploited to track the camera on the acquired datasets, and the results are compared with the ones obtained through robot based camera tracking previously described.
To this end, we use all the original Industrial training sequences to feed the visual SLAM algorithm, obtaining in this way a new version of the frame set in which the camera poses are estimated by tracking the visual features instead of measuring it through the robot.
Unfortunately, state of the art SLAM system featuring a pose optimization graph, can produce camera poses only for a subset of frames (\textit{i.e. } the keyframes) much smaller than the original set.
In this specific case, the experimental compression ratio is about $1:30$, which results in reducing the dataset dimension to about 1000 images (starting from the 36000 of the Industrial dataset). We refer with Industrial 1000 SLAM to the direct outcome of the visual SLAM algorithm, while with Industrial 1000 Robot we refer to the corresponding frames in which the camera pose is estimated through the robot kinematics.
Moreover, due to the known scaling factor problem of a monocular SLAM, the estimated camera poses may be slightly different from the real one, generating a non-perfect match with the original objects and, as a consequence, misaligned annotations.
However, by using ARS we manually corrected them with the procedure described in \autoref{sec:pose_refinement} producing an additional training set called Industrial 1000 SLAM Corrected. In Figure 8 we plot precision-recall curves obtained by YOLO trained on the three training sets. The performance of original SLAM procedure is worse than the corrected counterpart that is indeed comparable to the Robot version.
\begin{figure}
\centering
\includegraphics[width=1\columnwidth]{images/live_detections/live_detections-eps-converted-to.pdf}
\caption{Detectors output (all correct) computed on images acquired with a usb webcam. The detector is able to distinguish \emph{apple}s and \emph{pear}s only looking at their lower side. }
\label{fig:live_detections}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=0.9\textwidth]{images/coverage_density/coverage_density_test.pdf}
\caption{The Viewpoint Coverage{} computed for object of class 0 for the Industrial{}\_3000/5000/15000 datasets. The ColorBar maps the color with the number of views voting for corresponding polar bin. }
\label{fig:viewpoint_coverage}
\end{figure*}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.6\textwidth]{images/coverage_comparison/coverage_comparison.png} &
\includegraphics[width=0.25\textwidth]{images/coverage_comparison/coverage_density_plot-eps-converted-to.pdf}
\\ \textbf{(a)} & \textbf{(b)}
\end{tabular}
\caption{(a) Shows the VC over Industrial{}\_3000 and Industrial{}\_5000 with the latter filtered removing half the viewpoints. (b) Depicts the performance of an object detector trained on the two dataset, highlighting the importance of a full coverage at training time.}
\label{fig:viewpoint}
\end{figure*}
\subsection{Viewpoint Coverage{}}
\label{ssec:histo}
All the results reported so far show that ARS{} is fast and effective for the dataset creation, however, it has one additional useful side effect: for each image in the dataset we know the position of the camera with respect to each object in the scene. Thus, for each virtual box $V}\newcommand{\bboxset}{\mathcal{B}_j=\{^{0}\mathbf T_j , s_j, c_j \}$ we can compute the position of the camera \textit{w.r.t.} that object at the $i_{\text{th}}$ frame $^{j}\mathbf T_{cam_i}$. We can express the position of the camera in the object RF in polar coordinates $^{j}\mathbf{p}_{cam_i}=^{j}(r,\theta,\phi)_{cam_i}$ (\textit{i.e. } \emph{radial}, \emph{azimuthal}, \emph{polar}) and build a 2D histogram by aggregating $(\theta,\phi)$ into bins and counting the number of frames which contribute to that viewpoint. We dubbed this histogram the \emph{Viewpoints Coverage} (VC) of an object in the training dataset. \autoref{fig:viewpoint_coverage} shows as heat map the histograms for object class $0$ on three of the Industrial{} training sets presented above, with hotter colors corresponding to higher coverage, \textit{i.e. } to more frames acquired from that viewpoint. On the middle histogram of \autoref{fig:viewpoint_coverage} we highlighted with a dashed circle a region with low score, \textit{i.e. } a viewpoint poorly covered in the training set and thus a potential flaw in the final detector when watching the object from that vantage point in a test image. Therefore the VC representation may be used during the creation of the dataset to guide the user (or a Robot), \textit{e.g. } suggesting how to acquire new sequences carrying out better trajectories.
To highlight the importance of having as much object coverage as possible we define two dataset Industrial{}\_3000/360$^{\circ}$ (featuring only 3000 images but covering all viewpoints) and Industrial{}\_5000/180$^{\circ}$ (featuring 5000 images but only covering half of the possible viewpoints) considering only object class $0$, with \autoref{fig:viewpoint}(a) depicting the corresponding VCs. We use those as training set for YOLO and report in \autoref{fig:viewpoint}(b) the resulting precision/recall curves for that object class. As expected, even featuring 2000 images more, Industrial{}\_5000/180$^{\circ}$ perform much worse due to having seen only a limited set of object appearances. In conclusion, with the Viewpoint Coverage analysis we demonstrate that methods like this are crucial to control the distribution of training data , which is much more relevant than the size of an uncontrolled dataset. In addition, in case of ARS deployed in a robotic scenario, the VC metrics could guide the robot to perform optimal trajectories to maximize the coverage of the viewpoints autonomously, or, in case of impossible trajectories, inform the user that there is a need to rearrange objects in the scene.
\section{Conclusion and Future Work}
In this paper, we demonstrated how, by using robotic vision (i.e. robotics at the service of vision and vice versa), it is possible to create systems, such as the one here presented, that use the robot itself to learn. By exploiting the dexterity of the robot, a large number of viewpoints are automatically generated for each object (as depicted in \autoref{fig:viewpoint_coverage}), to allow the neural networks to generalize well distilling a knowledge of them. Moreover, the possibility to generate self-annotated images without human intervention allows to effortlessly collect countless environmental variations (such as light or working table color changes) thus generating robust visual perception even in unstructured environments.
By the proposed approach, two novel datasets are effortlessly created, one on electromechanical components (industrial scenario) and one on fruits (daily-living scenario). From these datasets, two state-of-the-art object detectors based on convolutional neural networks, such as YOLO and SSD, are trained robustly. The proposed approach based on ARS allows to annotate 9 sequences of about 35000 frames in less than one hour, that compared to conventional manual annotation of 1000 frames that takes us slightly more than 10 hours, results in a gain factor of about 450 considering both the time saved and the dataset dimension. From the point of view of performance in the object detection, both the precision and recall is increased by about 15\% with respect to manual labelling. The proposed approach allows to embed into robots novel and more performing perception and learning capabilities at the expense of a very limited human intervention. All the software generated to implement the proposed approach is available as a ROS package in a public repository alongside with the novel annotated datasets \footnote{\url{https://github.com/m4nh/ars}}.
As a future extension of the proposed method, we will exploit the knowledge of the 6-DOF pose of objects with respect to the camera provided by the robot in each viewpoint to train more complex systems than a simple 2D detector. To this end, the approaches adopted to estimate the 3D position and orientation of objects from single 2D images reported in literature, see for example \cite{Kehl_2017_ICCV,rad2017bb8,sundermeyer2018implicit}, will be trained with the output provided by the ARS pipeline. This approach will enable the realization of a fully automated self-learning BinPicking system.
Another ARS{} extension that we plan to explore concerns the use of out of the box augmented reality toolkit offered by nowadays mobile platforms
(\textit{e.g. } ARKit for iOS and ARCore for Android) in order to track the camera pose while acquiring video sequences. This extension would allow quick and easy creation of a training dataset by an off-the-shelf mobile device such as a smartphone or a tablet.
\bibliographystyle{spmpsci}
|
\section{Introduction}
Elliptic PDEs arise in a vast number of applications in scientific computing. A significant class of these involve the Laplace operator, which appears not only in potential calculations but also in, for example, Stokes and Navier-Stokes problems~\cite[Chapters 5 and 7]{elman2005}, electron density computations~\cite[Part II]{martin2004} and reaction-convection-diffusion equations~\cite[Part IV]{hundsdorfer2003}. Consequently, the rapid solution of PDEs involving the Laplace operator is of wide interest.
Although many successful numerical methods for such PDEs exist, changing computer architectures necessitate new paradigms for computing and the development of new algorithms. Computer architectures of the future will favor algorithms with high concurrency, high arithmetic intensity (Flop/Byte), asynchronicity, and data locality. This trend is manifest on GPUs and co-processors, where some algorithms are accelerated much less than others on the class of architectures that can be extended to extreme scale. There is always a balance between algorithmic efficiency in a convergence sense, and how well an algorithm scales on parallel architectures. This balance is shifting towards increased parallelism, even at the cost of increasing computation. There is therefore a strong incentive to choose/modify the algorithms that promise the best use of future architectures. Since the processor frequency has plateaued for the last decade, Moore's law holds continued promise only for those who are willing to make algorithmic changes.
Among the scientific applications ripe for reconsideration, those governed by elliptic PDEs will be among the most challenging. A common solution strategy for such systems is to discretize the partial differential equations by low-order finite element, finite volume or finite difference methods and then solve the resulting large, sparse linear system. However, elliptic systems are global in nature, and this conflicts with the requirements of future architectures, \textit{e.g.} high concurrency, asynchronicity, and data locality. The linear solver must enable the transfer of information from one end of the domain to the other, either through successive local communications (as in many iterative methods), or a direct global communication (as in direct solvers with global recurrences and Krylov methods with global reductions). In either case, avoiding synchronization and reducing communication are the main challenges. There has been considerable effort in this direction in the dense linear algebra community~\cite{demmel2008}. The directed-acyclic-graph-based technology developed in such efforts could be combined with iterative algorithms of optimal complexity for solving elliptic PDEs at extreme scale.
Scalable algorithms for solving elliptic PDEs tend to have a hierarchical structure, as in multigrid methods~\cite{trottenberg2001}, fast multipole methods (FMM)~\cite{greengard1987}, and $\mathcal{H}$-matrices~\cite{hackbusch1999}. This structure is crucial, not only for achieving optimal arithmetic complexity, but also for minimizing data movement. For example, a 3-D FFT requires $\mathcal{O}(\sqrt{P})$ communication for the transpose between pencil-shaped subdomains on $P$ processes \cite{czechowski2012}, whereas these hierarchical methods require $\mathcal{O}(\log P)$ communication \cite{lashuk2012}. This $\mathcal{O}(\log P)$ communication complexity is likely to be optimal for elliptic problems, since an appropriately coarsened form of a local representation must somehow arrive to all other parts of the domain for the elliptic equation to converge. In other words, an elliptic problem for which the solution is desired everywhere cannot have a communication complexity of $\mathcal{O}(1)$. However, a disadvantage of these hierarchical methods is that for certain problems we see slow convergence, or even divergence.
Krylov subspace methods provide another popular alternative to direct methods for general operators, although we note that methods such as Chebyshev semi-iteration can require even less communication when information about the spectrum of the coefficient matrix is known~\cite[Section 10.1.5]{golub1996}, \cite{golub1961}. Among the best known Krylov methods are the conjugate gradient method~\cite{hestenes1952}, MINRES~\cite{paige1975} and GMRES~\cite{saad1986}, although a multitude of Krylov solvers are available in popular scalable solver libraries. The great advantage of these solvers is their robustness -- for any consistent linear system there exists a Krylov method that will converge, in exact arithmetic, for sufficiently many iterations. However, the convergence rate of Krylov methods typically deteriorates as the discretization of an elliptic PDE is refined.
Mesh-independent convergence for Krylov methods applied to systems from elliptic PDEs can often be recovered by preconditioning. Among the best performing preconditioners are the optimal hierarchical methods or, for multiphysics problems such as Stokes and Navier-Stokes equations, block preconditioners with these methods as components. By combining these hierarchical methods and Krylov subspace solvers we get the benefits of both approaches and obtain a linear solver that is fast but robust. These hierarchical methods all have multiple parameters for controlling the precision of the solution and are able to trade-off accuracy for speed, which is a useful feature for a preconditioner. Furthermore, in analogy to geometric multigrid and algebraic multigrid, $\mathcal{H}^2$-matrices can be thought of as an algebraic generalization of what FMMs do geometrically. There are advantages and disadvantages to using algebraic and geometric methods, and both have their place as preconditioners.
There has been some recent work on algebraic multigrids (AMG) in anticipation of the future hardware constraints mentioned above. Gahvari \textit{et al.} developed a performance model for AMG and tested it on various HPC systems -- Intrepid, Jaguar, Hera, Zeus, and Atlas~\cite{gahvari2011}. They found that network distance and contention were both substantial performance bottlenecks for AMG. Adams presents a low-memory matrix-free full multigrid (FMG) with a full approximation storage (FAS)~\cite{adams2012}. He revives an idea from the 1970s~\cite{brandt1977}, which processes the multigrid algorithm vertically, and improves data locality and asynchronicity. Baker \textit{et al.} compared the scalability of different smoothers -- hybrid Gauss-Seidel, $l_1$ Gauss-Seidel, and Chebyshev polynomial, and showed that $l_1$ Gauss-Seidel and Chebychev smoothers scale much better~\cite{baker2012}. There is continuous effort in the multigrid community to adapt the algorithm according to future hardware constraints, and it is likely that multigrid will evolve to remain competitive.
On the other hand, performing a hierarchical low rank approximation (HLRA) of the off-diagonal blocks of a matrix leads to a whole new variety of $\mathcal{O}(N)$ solvers or preconditioners. HLRA based methods include FMM~\cite{greengard1987}, $\mathcal{H}$-matrices~\cite{hackbusch1999}, hierarchically semi-separable matrices (multifrontal methods) \cite{chandrasekaran2006}, and recursive skeletonization~\cite{ho2012}. These techniques can be applied to a dense matrix or the fill-in during a sparse direct solve, thus enabling an $\mathcal{O}(N)$ matrix-vector multiplication of a $N\times N$ dense matrix or an $\mathcal{O}(N)$ direct solve of a $N\times N$ sparse matrix to within a specified accuracy. These HLRA based methods require a decaying kernel which yields a block low-rank structure. The distinguishing features of the variants come in the way the low rank approximation is constructed -- rank-revealing LU~\cite{pan2000}, rank-revealing QR~\cite{gu1996}, pivoted QR~\cite{kong2011}, truncated SVD~\cite{grasedyck2003}, randomized SVD~\cite{liberty2007}, adaptive cross approximation~\cite{rjasanow2002}, hybrid cross approximation~\cite{borm2005}, and Chebychev interpolation~\cite{dutt1996} are all possibilities. Multipole/local expansions in the FMM constitute another way to construct the low rank approximations. In fact, many of the original developers of FMM are now working on these algebraic variants~\cite{greengard2009}.
Literature on the HLRA based methods mentioned above mainly focuses on the error convergence of the low rank approximation and there is little investigation of the parallel scalability or direct comparison against multigrid. An exception is the work by Grasedyck \textit{et al.}~\cite{grasedyck2008}, where their $\mathcal{H}$-LU preconditioner is compared with BoomerAMG, Pardiso, MUMPS, UMFPACK, SuperLU, and Spooles. However, their executions are serial, and show that their $\mathcal{H}$-matrix code is not yet competitive with these other highly optimized libraries.
In the present work, we consider the Laplace and Stokes equations and devise highly scalable preconditioners for these problems. Our Poisson preconditioner is based on a boundary element method in which matrix-vector multiplies are performed using FMM; the result is an $\mathcal{O}(N)$ preconditioner that is scalable. For the Stokes problem, we apply a block diagonal preconditioner, in which our Poisson preconditioner is combined with a simple diagonal matrix. Such FMM based preconditioners were first proposed by Sambavaram \textit{et al.} \cite{sambavaram2003}. Such methods lacked practical motivation when flops were expensive, since they turn a sparse matrix into a dense matrix of the same size before hierarchically grouping the off-diagonal blocks. But in a world of cheap flops, the notion of a ``compute-bound preconditioner" sounds more attractive. In the present work, we perform scalability benchmarks and compare the time-to-solution with state-of-the-art multigrid methods such as BoomerAMG in a high performance computing environment.
The rest of the manuscript is organized as follows. In Section \ref{sec:poisson} we present the model problems and in Section \ref{sec:krylov} we give an overview of Krylov subspace methods and preconditioning. The basis of our preconditioner is a boundary element method that is discussed in Section \ref{sec:bem} and the FMM, the essential kernel that makes our method efficient and scalable, is described in Section \ref{sec:fmm}. Our numerical results in Section \ref{sec:results} examine the convergence rates of FMM and multigrid for small Poisson and Stokes problems. Then, in Section \ref{sec:performance} we scale up the Poisson problem tests and perform strong scalability runs, where we compare the time-to-solution against BoomerAMG \cite{henson2002} on up to 1024 cores. Our conclusions are given in Section \ref{sec:conc}.
\section{Model problems} \label{sec:poisson}
In this section we introduce the Poisson and Stokes model problems we
wish to solve and describe properties of the linear systems that result
from their discretization. We focus on low-order finite elements
but note that discretization by low-order finite difference or finite volume
methods give linear systems with similar properties.
\subsection{Poisson model problem}
The model Poisson problem we wish to solve is of the form
\begin{subequations} \label{e:p}
\begin{alignat}{3}
-\nabla^2u & = f && \text{ in } & \Omega,\label{e:pint}\\
u & = g && \text{ on } & \Gamma,\label{e:pbound}
\end{alignat}
\end{subequations}
where $\Omega \in \mathbb{R}^d$, $d = 2,3$ is a bounded connected domain with piecewise smooth boundary $\Gamma$, $f$ is a forcing term and $g$ defines the Dirichlet boundary condition.
Discretization of~\eqref{e:p} by finite elements or finite differences leads to a large, sparse linear system of the form
\begin{equation} \label{e:axb}
A\bv{x} = \bv{b},
\end{equation}
where $A\in\R^{N\times N}$ is the stiffness matrix and $\bv{b}\in\R^N$ contains the forcing and boundary data. The matrix $A$ is symmetric positive definite and its eigenvalues depend on the mesh size, which we denote by $h$, as is typical of discretizations of elliptic PDEs. In particular, the condition number $\kappa = \lambda_{max}(A)/\lambda_{min}(A)$, the ratio of the largest and smallest eigenvalues of $A$, grows as $O(h^{-2})$ (see, for example,~\cite[Section 1.6]{elman2005}).
\subsection{Stokes model problem}
Incompressible Stokes problems are important when modelling viscous flows and for solving Navier-Stokes equations by operator splitting methods~\cite[Section 2.1]{benzi2005}. The equations governing the velocity $\bv{u}\in\mathbb{R}^d$, $d=2,3$, and pressure $p\in\mathbb{R}$ of a Stokes fluid in a bounded connected domain $\Omega$ with piecewise smooth boundary $\Gamma$ are \cite{benzi2005},~\cite{elman2005}:
\begin{subequations}\label{e:stokes}
\begin{alignat}{3}
-\nabla^2{\bv{u}} + \nabla p &= {0} && \text{ in } & \Omega,\label{e:stokes1} \\
\nabla\cdot{\bv{u}} &=0 && \text{ in } & \Omega, \label{e:stokes2}\\
{\bv{u}} &= {\bv{w}} && \text{ on } & \Gamma.\label{e:stokesbc}
\end{alignat}
\end{subequations}
Discretizing~\eqref{e:stokes} by a stabilized\footnote{Although we treat only stabilized discretizations here, stable discretizations are no more difficult to precondition and are discussed in detail in Elman \emph{et al.}~\cite[Chapter 6]{elman2005}.} finite element or finite difference approximation leads to the symmetric saddle point system
\begin{equation}\label{e:sp}
\underbrace{
\begin{bmatrix}
A & B^T\\
B & -C
\end{bmatrix}}_{\mathcal A}
\begin{bmatrix}
\bv{u}\\ \bv{p}
\end{bmatrix}
=
\begin{bmatrix}
\bv{f} \\ \bv{g}
\end{bmatrix},
\end{equation}
where ${A}\in\R^{N\times N}$ is the vector-Laplacian, a block diagonal matrix with blocks equal to the stiffness matrix from~\eqref{e:axb}, ${B}\in\R^{M\times N}$ is the discrete divergence matrix, $C\inR^{M\times M}$ is the symmetric positive definite pressure mass matrix and $\bv{f}\in\R^N$ and $\bv{g}\in\R^M$ contain the Dirichlet boundary data.
The matrix $\mathcal A$ is symmetric indefinite
and the presence of the stiffness matrix means that the condition number of $\mathcal{A}$ increases as the mesh is refined. However, as we will see in the next section, the key ingredient in a preconditioner for $\mathcal A$ that mitigates this mesh dependence is a good preconditioner for the Poisson problem. This allows us to use our preconditioner for the Poisson problem in this more complicated fluid dynamics problem as well.
\section{Iterative solvers and preconditioning}\label{sec:krylov}
\subsection{Krylov Subspace Methods}
Large, sparse systems of the form~\eqref{e:axb} are often solved by Krylov subspace methods. We focus here on two Krylov methods: the conjugate gradient method (CG)~\cite{hestenes1952} for systems with symmetric positive definite coefficient matrices and MINRES~\cite{paige1975} for systems with symmetric indefinite matrices. For implementation and convergence details, we refer the reader to the books by Greenbaum~\cite{greenbaum1997} and Saad~\cite{saad2003}.
The convergence of these Krylov subspace methods depends on the spectrum of the coefficient matrix which for the Poisson and Stokes problems, as well as other elliptic PDEs, deteriorates as the mesh is refined. This dependence is removed by preconditioning. In the case of the Poisson problem~\eqref{e:axb}, we can conceptually think of solving the equivalent linear system $P^{-1}A\boldsymbol{x} = P^{-1}\boldsymbol{b}$ (left preconditioning), or $AP^{-1}\boldsymbol{y} = \bv{b}$, with $P^{-1}\boldsymbol{y} = \bv{x}$ (right preconditioning), for some $P^{-1}\in\R^{N\times N}$, and analogously for the Stokes equations~\eqref{e:sp}. However, when the coefficient matrix is symmetric, we would like to preserve this property when preconditioning; this can be achieved by using a symmetric positive definite preconditioner~\cite[Chapters 2 and 6]{elman2005}. We also note that in practice we never need $P^{-1}$ explicitly but only the action of this matrix on a vector. This enables us to use matrix-free approaches such as multigrid or the fast multipole method.
Many preconditioners for the Poisson problem reduce the number of iterations, with geometric and algebraic multigrid among the most effective strategies~\cite{elman2005},~\cite{trottenberg2001}. However, to achieve a lower time-to-solution than can by obtained for the original system, it is also necessary to choose a preconditioner that can be cheaply applied at each iteration. Both geometric and algebraic multigrid methods are $O(N)$, and therefore exhibit good performance on machines and problems for which computation is expensive. However, stresses arise in parallel applications as discussed in the introduction.
For Stokes problems we consider the block diagonal preconditioner
\begin{equation}\label{e:sppre}
{\mathcal P} =
\begin{bmatrix}
P_A & 0\\
0 & P_S
\end{bmatrix},
\end{equation}
where $P_A\in\R^{N\times N}$ and $P_S\inR^{M\times M}$ are symmetric positive definite matrices.
The advantage of this preconditioner is that there is no coupling between the blocks,
so $\mathcal P$ is scalable provided the blocks $P_A$ and $P_S$ are.
Appropriate choices for $P_A$ and $P_S$ have been well studied and it is known that mesh-independent convergence
of MINRES can be recovered when $P_A$ is spectrally equivalent to $A$ in~\eqref{e:sp} and $P_S$ is spectrally equivalent to
the pressure mass matrix $Q\inR^{M\times M}$~\cite{cahouet1988},~\cite[Chapter 6]{elman2005}. These spectral equivalence requirements
imply that the eigenvalues of $P_A^{-1}A$ and $P_S^{-1}Q$ are bounded in an interval on the positive real line independently of the
mesh width $h$.
It typically suffices to use the diagonal of $Q$~\cite[Chapter 6]{elman2005}, \cite{wathen1987} or a few steps of Chebyshev semi-iteration~\cite{wathen2009} for $P_S$. Moreover, the diagonal matrix is extremely parallelizable.
Thus, the key to obtaining a good preconditioner for $\mathcal A$ is to approximate the vector Laplacian effectively. This is typically the most computationally intensive part of the preconditioning process, since in most cases $M \ll N$.
\subsection{The FMM-BEM preconditioner}
In this paper we propose an alternative preconditioner for Poisson and Stokes problems that heavily utilizes the fast multipole method (FMM). The FMM is $\mathcal{O}(N)$ with compute intensive inner kernels. It has a hierarchical data structure that allows asynchronous communication and execution. These features make the FMM a promising preconditioner for large scale problems on future computer architectures. We show that this preconditioner improves the convergence of these Krylov subspace methods, and is effectively parallelized on today's highly distributed architectures.
The FMM in its original form relies on free-space Green's functions and is able to solve problems with free-field boundary conditions. In Section \ref{sec:bem} the FMM preconditioner is extended to Dirichlet, Neumann or Robin boundary conditions for arbitrary geometries by coupling it with a boundary element method (BEM). Our approach uses the FMM as a \textit{preconditioner} inside a \textit{sparse} matrix solver and the BEM solve is \textit{inside} the preconditioner. Numerous previous studies use FMM for the matrix-vector multiplication inside the Krylov solver for the dense matrix arising from the boundary element discretization. In the present method we are calculating problems with non-zero sources in the volume, and the FMM is used to calculate the volume-to-volume contribution. This means we are performing the action of an $N\times N$ dense matrix-vector multiplication, where $N$ is the number of points in the volume (not the boundary). Additionally, as discussed in Section~\ref{subsec:varcoeff}, it is possible to extend the boundary element method to problems with variable diffusion coefficients, particularly since low accuracy solves are often sufficient in preconditioning.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{flow_chart.pdf}
\caption{Flow chart of the FMM-BEM preconditioner within the conjugate gradient method.}
\label{f:flow_chart}
\end{figure}
Figure \ref{f:flow_chart} shows the flow of calculation of our FMM-BEM preconditioner within the conjugate gradient method; its role in other Krylov solvers is similar. The FMM is used to approximate the matrix-vector multiplication of $A^{-1}$ within the preconditioner. The BEM solver adapts the FMM to finitely applied boundary conditions. During each step of the iteration, the $u$ vector from the previous iteration is used to determine $\partial u/\partial n$ at the boundary from~\eqref{e:ubound}, then~\eqref{e:uinter} is used to compute the new $u$ in the domain $\Omega$.
\section{Boundary Element Method}\label{sec:bem}
\subsection{Formulation}
We use a standard Galerkin boundary element method \cite{sauter2011} with volume contributions to solve the Poisson equation. A brief description of the formulation is given here. Applying Green's third identity to~\eqref{e:pint} gives
\begin{equation}
\int_\Gamma u\frac{\partial G}{\partial n}d\Gamma-\int_\Gamma\frac{\partial u}{\partial n}Gd\Gamma-\int_\Omega u(\nabla^2G)d\Omega=\int_\Omega fGd\Omega,
\label{e:green}
\end{equation}
where $G$ is the Green's function of the Laplace operator, $\frac{\partial}{\partial n}$ is the derivative in the outward normal direction, and $\Gamma$ is the boundary. Following the definition of the Green's function $\nabla^2G=-\delta$, the third term in~\eqref{e:green} becomes
\begin{equation}
-\int_\Omega u(\nabla^2G)d\Omega=\int_\Omega u\delta d\Omega=
\begin{cases}
\frac{1}{2}u\ \text{on}\ \partial\Omega, \\
u\quad \text{in}\ \Omega.
\end{cases}
\end{equation}
Therefore, we may solve the constant coefficient inhomogeneous Poisson problem by solving the following set of equations
\begin{align}
\int_\Gamma\frac{\partial u}{\partial n}Gd\Gamma&=\int_\Gamma u\left(\frac{1}{2}\delta+\frac{\partial G}{\partial n}\right)d\Gamma-\int_\Omega fGd\Omega\quad \text{on}\ \partial\Omega, \label{e:ubound} \\
u&=\int_\Gamma\frac{\partial u}{\partial n}Gd\Gamma-\int_\Gamma u\frac{\partial G}{\partial n}d\Gamma+\int_\Omega fGd\Omega\quad \text{in}\ \Omega. \label{e:uinter}
\end{align}
As an example, consider the case where Dirichlet boundary conditions are prescribed on $\partial\Omega$. The unknowns are $\partial u/\partial n$ on $\Gamma$ and $u$ in $\Omega\backslash \Gamma$, where~\eqref{e:ubound} solves for the former and~\eqref{e:uinter} can be used to determine the latter. For Neumann boundary conditions one can simply switch the two boundary integral terms in~\eqref{e:ubound} and solve for $u$ instead of $\partial u/\partial n$. In either case, we obtain both $u$ and $\partial u/\partial n$ at each point on the boundary, then calculate~\eqref{e:uinter} to obtain $u$ at the internal points. The last term in~\eqref{e:uinter} takes up most of the calculation time since it is a volume integral for every point in the volume, whereas other terms are either for every point on the boundary or are boundary integrals.
\subsection{Singular integrals}
The Laplace Green's function in 2-D
\begin{equation}
G=-\frac{1}{2\pi}\log r
\end{equation}
is singular. Therefore, the integrals involving $G$ or $\partial G/\partial n$ in~\eqref{e:ubound} and~\eqref{e:uinter} are singular integrals. As we will mention in the following subsection, these singular integral are discretized into piecewise integrals, which are evaluated using Gauss-Legendre quadratures with special treatment for the singular piecewise integral. For boundary integrals in~\eqref{e:ubound} and~\eqref{e:uinter}, analytical expressions exist for the piecewise integral. However, for the volume integral an analytical expression does not exist. For this reason, we used a smoothed Green's function of the form
\begin{equation}
G=-\frac{1}{2\pi}\log(\sqrt{r^2+\epsilon^2})
\end{equation}
where $\epsilon$ is a small number that changes with the grid resolution.
\subsection{Discretization}
The integrals in equations~\eqref{e:ubound} and~\eqref{e:uinter} are discretized in a similar fashion to finite element methods. In the following description of the discretization process, we will use the term on the left hand side in~\eqref{e:ubound} as an example. The first step is to break the global integral into a discrete sum of piecewise local integrals over each element
\begin{equation}
\int_\Gamma\frac{\partial u}{\partial n}Gd\Gamma\approx\sum_{j=1}^{N_\Gamma}\int_{\Gamma_j}\frac{\partial u_j}{\partial n}Gd\Gamma_j,
\end{equation}
where $N_\Gamma$ is the number of boundary nodes. These piecewise integrals are performed by using quadratures over the basis functions \cite{sauter2011}. In the present case, we use constant elements so there are no nodal points at the corners of the square domain for the tests in Sections \ref{sec:results} and \ref{sec:performance}. By applying this discretization technique to all terms in~\eqref{e:ubound} we obtain
\begin{equation*}
N_\Gamma
\left\{
\phantom{
\begin{bmatrix}
\ddots\\
&G_{ij}\\
&&\ddots
\end{bmatrix}
}
\right.
\hspace{-24mm}
\overbrace{
\begin{bmatrix}
\ddots\\
&G_{ij}\\
&&\ddots
\end{bmatrix}
}^{N_\Gamma}
\underbrace{
\begin{bmatrix}
\vdots\\
\frac{\partial u_j}{\partial n}\\
\vdots
\end{bmatrix}
}_\text{unknown}
=
\overbrace{
\begin{bmatrix}
\hspace{-18mm}\ddots\\
\frac{1}{2}\delta_{ij}+\frac{\partial G_{ij}}{\partial n}\\
\hspace{18mm}\ddots
\end{bmatrix}
}^{N_\Gamma}
\begin{bmatrix}
\vdots\\
u_j\\
\vdots
\end{bmatrix}
-
\overbrace{
\begin{bmatrix}
\ddots\\
&G_{ij}\\
&&\ddots
\end{bmatrix}
}^{N_\Omega}
\begin{bmatrix}
\vdots\\
f_j\\
\vdots
\end{bmatrix},
\end{equation*}
where $N_\Omega$ is the number of internal nodes. All values on the right hand side are known, and $\partial u/\partial n$ at the boundary is determined by solving the linear system. Similarly, we apply the discretization to~\eqref{e:uinter} to have
\begin{equation*}
\small
N_\Omega
\left\{
\begin{bmatrix}
\vdots\\
u_i\\
\vdots
\end{bmatrix}
\right.
=
\overbrace{
\begin{bmatrix}
\hspace{-18mm}\ddots\\
\frac{\partial G_{ij}}{\partial n}\\
\hspace{18mm}\ddots
\end{bmatrix}
}^{N_\Gamma}
\begin{bmatrix}
\vdots\\
u_j\\
\vdots
\end{bmatrix}
-
\overbrace{
\begin{bmatrix}
\ddots\\
&G_{ij}\\
&&\ddots
\end{bmatrix}
}^{N_\Gamma}
\begin{bmatrix}
\vdots\\
\frac{\partial u_j}{\partial n}\\
\vdots
\end{bmatrix}
+
\overbrace{
\begin{bmatrix}
\ddots\\
&G_{ij}\\
&&\ddots
\end{bmatrix}
}^{N_\Omega}
\begin{bmatrix}
\vdots\\
f_j\\
\vdots
\end{bmatrix}.
\end{equation*}
At this point, all values on the right hand side are known so one can perform three matrix-vector multiplications to obtain $u$ at the internal nodes, and the solution to the original Poisson equation~\eqref{e:pint}. The third term on the right hand side involves an $N_\Omega\times N_\Omega$ matrix, and is the dominant part of the computational load. This matrix-vector multiplication can be approximated in $\mathcal{O}(N)$ time by using the FMM described in Section ~\ref{sec:fmm}. We also use the FMM to accelerate all other matrix-vector multiplications.
\subsection{Variable coefficient problems}\label{subsec:varcoeff}
A natural question that arises is how to extend the boundary element
method, which is the basis of our preconditioner, to problems with
variable diffusion coefficients, i.e., problems of the type
\begin{equation}\label{e:varcoeff}
\nabla\cdot (a(\bx) \nabla u(\bx)) = f(\bx),
\end{equation}
where $\bx\in\mathbb{R}^d$, $d = 2,3$ and $a\in\mathbb{R}$.
Several strategies for extending boundary element methods to
problems with variable diffusion coefficients have been proposed
(see, for example, the thesis of Brunton~\cite[Chapter 3]{brunton1996}).
Additionally, in this preconditioner setting we may not need to capture the variation
in the diffusion coefficient to a high degree of accuracy;
for a similar discussion in the context of additive
Schwarz preconditioners see, for example, Graham et al.~\cite{graham2007}.
Although analytic fundamental solutions can sometimes be found for problems with
variable diffusion
(see, e.g., Cheng~\cite{cheng1984} and Clements~\cite{clements1980}),
in most cases numerical techniques are employed.
One popular method is to introduce a number of subdomains,
on each of which the diffusion coefficient is approximated by a constant
function~\cite{langer2007,wardle1978}.
A second option is to split the differential operator into a part for which a fundamental
solution exists and another which becomes part of the source term.
Specifically, starting from~\eqref{e:varcoeff},
a similar approach to that described in Banerjee~\cite{banerjee1979}
and Cheng~\cite{cheng1984} leads to
$$
\int_\Gamma a u \frac{\partial G}{\partial n}d\Gamma
- \int_\Gamma a \frac{\partial u}{\partial n} G
- \int_\Omega u \nabla a\cdot \nabla G d\Omega - \int_\Omega au\nabla^2G d\Omega
= \int_\Omega fG d\Omega,
$$
where again $G$ is the standard fundamental solution for the Laplace operator, i.e., not
the fundamental solution for~\eqref{e:varcoeff}.
We can then proceed as described above for~\eqref{e:green}.
\section{Fast Multipole Method} \label{sec:fmm}
\subsection{Introduction to FMM}
The last term in Eq.~\eqref{e:uinter} when discretized, has the form
\begin{equation}
u_i=\sum_{j=1}^{N_\Omega}f_jG_{ij}.
\label{e:volume_integral}
\end{equation}
where $i=1,2,...,N_\Omega$. If we calculate this equation directly, it will require $\mathcal{O}(N^2)$ operations. In Figure \ref{f:fmm_schamatic}, we show a schematic of how the fast multipole method is able to calculate this in $\mathcal{O}(N)$ operations. Figures \ref{f:direct_interaction} and \ref{f:fmm_interaction} show how the source particles (red) interact with the target particles (blue) for the direct method and FMM, respectively. In the direct method, all source particles interact with all target particles directly. In the FMM, the source particles are first converted to multipole expansions using the P2M (particle to multipole) kernel. Figure \ref{f:fmm_flow} shows the corresponding geometric view of the hierarchical domain decomposition of the particle distribution. Then, multipole expansions are aggregated into larger groups using the M2M (multipole to multipole) kernel. Following that, the multipole expansions are translated to local expansions between well-separated cells using the M2L (multipole to local) kernel. Both Figures \ref{f:fmm_interaction} and \ref{f:fmm_flow} show that the larger cells interact if they are significantly far away, and smaller cells may interact with slightly closer cells. The direct neighbors between the smallest cells are calculated using the P2P (particle to particle) kernel, which is equivalent to the direct method between a selected group of particles. Then, the local expansions of the larger cells are translated to smaller cells using the L2L (local to local) kernel. Finally, the local expansions at the smallest cells are translated into the potential on each particle using the L2P (local to particle) kernel. The mathematical formulae for these kernels will be given in Section \ref{sec:expansions}.
\begin{figure}
\centering
\subfigure[Direct method]{\includegraphics[width=0.45\textwidth]{direct_interaction.pdf}\label{f:direct_interaction}}
\subfigure[Fast Multipole Method]{\includegraphics[width=0.43\textwidth]{fmm_interaction.pdf}\label{f:fmm_interaction}}
\subfigure[Flow of data in FMM]{\includegraphics[width=0.9\textwidth]{fmm_flow.pdf}\label{f:fmm_flow}}
\caption{Schematic of Fast Multipole Method. (a) shows the interactions for a $\mathcal{O}(N^2)$ direct method. (b) shows the interactions for the $\mathcal{O}(N)$ FMM, describing the type of interaction between elements in the tree data structure. (c) shows the same FMM kernels as in (b), but from a geometric point of view of the hierarchical domain decomposition.}
\label{f:fmm_schamatic}
\end{figure}
In order to perform the FMM calculation mentioned above, one must first decompose
|
the domain in a hierarchical manner. It is common to use an octree in 3-D and quad-tree in 2-D, where the domain is split by its geometrical centerline. The splitting is performed recursively until the number of particles per cell reaches a prescribed threshold. The splitting is usually performed adaptively, so that the densely populated areas result in a deeper branching of the tree. A common requirement in FMMs is that these cells must be isotropic (cubes or squares and not rectangles), since they are used as units for measuring the well-separatedness as shown in Figure \ref{f:fmm_flow} during the M2L interaction. However, our FMM does not use the size of cells to measure the distance between them and allows the cells to be of any shape as long as they can be hierarchically grouped into a tree structure. Once the tree structure is constructed, it is trivial to find parent-child relationships between the cells/particles. This relation is all that is necessary for performing P2M, M2M, L2L, and L2P kernels. However, for the M2L and P2P kernels one must identify a group of well-separated and neighboring cells, respectively. We will describe an efficient method for finding well-separated cells in the following subsection.
\subsection{Dual Tree Traversal}\label{sec:ddt}
The simplest method for finding well-separated pairs of cells in the FMM is to ``loop over all target cells and find their parent's neighbor's children that are non-neighbors", as shown by Greengard and Rokhlin~\cite{greengard1987}. A scheme that permits the interaction of cells at different levels for an adaptive tree was introduced by Carrier \textit{et al.}~\cite{carrier1988}. This scheme is used in many modern FMM codes, and is sometimes called the UVWX-list \cite{lashuk2012}. Another scheme to find well-separated pair of cells is to ``simultaneously traverse the target and source tree while applying a multipole acceptance criterion", as shown by Warren and Salmon~\cite{warren1995}. Teng~\cite{Teng1998} showed that this dual tree traversal can produce interaction pairs that are almost identical to the adaptive interaction list by Carrier \textit{et al.}~\cite{carrier1988}. A concise explanation and optimized implementation of the dual tree traversal is provided by Dehnen~\cite{dehnen2002}.
The dual tree traversal has many favorable properties compared to the explicit construction of interaction lists. First of all, the definition of well-separatedness can be defined quite flexibly. For example, if one were to construct explicit interaction lists by extending the definition of neighbors from $3\times3\times3$ to $5\times5\times5$ using the traditional scheme, the M2L list size will increase rapidly from $6^3-3^3=189$ to $10^3-5^3=875$ in 3-D, which is never faster for any number of expansions. On the other hand, the dual tree traversal can adjust the definition of neighbors much more flexibly and the equivalent interaction list always has a spherical shape. (We say ``equivalent interaction list" because there is no explicit interaction list construction in the dual tree traversal.) The cells no longer need to be cubic, since the cells themselves are not used to measure the proximity of cells. The cells can be any shape or size -- even something like a hierarchical K-means. Of course, the explicit interaction list construction can be modified to include more flexibility, too~\cite{gumerov2008}. However, the resulting code becomes much more complicated than the dual tree traversal, which is literally a few lines of code. \footnote{https://bitbucket.org/rioyokota/exafmm-dev} This simplicity is a large advantage on its own. Furthermore, the parallel version of the dual tree traversal simply traverses the local tree for the target with the local essential tree \cite{warren1992} for the sources, so the serial dual tree traversal code can be used once the local essential tree is assembled.
A possible (but unlikely) limitation of dual tree traversals is the loss of explicit parallelism -- it has no loops. It would not be possible to simply use an OpenMP ``parallel for" directive to parallelize the dual tree traversal. In contrast, the traditional schemes always have an outer loop over the target cells, which can be easily parallelized and dynamically load balanced with OpenMP directives. However, this is not an issue since task based parallelization tools such as Intel Thread Building Blocks (TBB) can be used to parallelize the dual tree traversal. With the help of these tools, tasks are spawned as the tree is traversed and dispatched to idle threads dynamically. In doing so, we not only assure load-balance but also data-locality, so it may actually end up being a superior solution than parallelizing ``for loops" with OpenMP, especially on NUMA architectures.
Considering the advantages mentioned above, we have decided to use the dual tree traversal in our current work. This allows us to perform low accuracy optimizations by adjusting the multipole acceptance criterion without increasing the order of expansions too much, which is the secret to our speed \cite{yokota2013a}. These low accuracy optimizations can give the FMM a performance boost when used as a preconditioner.
\subsection{Multipole Expansions} \label{sec:expansions}
For the 2-D Laplace equation, the free space Green's function has the form
\begin{equation}
G_{ij}=\frac{1}{2\pi}\log\left(\frac{1}{r_{ij}}\right),
\end{equation}
where $r_{ij}=|\mathbf{x}_i-\mathbf{x}_j|$ is the distance between point $i$ and point $j$. By using complex numbers to represent the two-dimensional coordinates $z=x+\iota y$, Eq.~\eqref{e:volume_integral} can be written as
\begin{equation}
u_i=\sum_{j=1}^{N_\Omega}\frac{f_j}{2\pi}\Re\left\{-\log(z_{ij})\right\},
\end{equation}
where $\Re(z)$ represents the real part of $z$. Figure \ref{f:vectors} shows the decomposition of vector $\mathbf{x}_{ij}$ into five parts, $\mathbf{x}_{ij}=\mathbf{x}_{i\lambda}+\mathbf{x}_{\lambda\Lambda}+\mathbf{x}_{\Lambda M}+\mathbf{x}_{M\mu}+\mathbf{x}_{\mu j}$, where $\lambda$ and $\Lambda$ are the center of local expansions and $\mu$ and $M$ are the center of multipole expansions. The lower case is used for the smaller cells and upper case is used for the larger cells. When assuming the relation $|\mathbf{x}_{\Lambda M}|>|\mathbf{x}_{i\lambda}+\mathbf{x}_{\lambda\Lambda}|+|\mathbf{x}_{M\mu}+\mathbf{x}_{\mu j}|$ the following FMM approximations are valid \cite{carrier1988}. We denote the $n$th order multipole expansion coefficient at $\mathbf{x}$ as $M_n(\mathbf{x})$, and the $n$th order local expansion coefficient as $L_n(\mathbf{x})$, where $n=0,1,...,p-1$ for a $p$th order truncation of the series.
\begin{figure}
\centering
\includegraphics[width = 0.8\textwidth]{vectors.pdf}
\caption{Decomposition of the distance vector $\mathbf{x}_{ij}=\mathbf{x}_i-\mathbf{x}_j$ into five parts, that correspond to the five stages P2M, M2M, M2L, L2L, and L2P in the FMM.}
\label{f:vectors}
\end{figure}
\begin{enumerate}
\item P2M from particle at $\mathbf{x}_j$ to multipole expansion at $\mathbf{x}_{\mu}$,
\begin{eqnarray}
M_0(\mathbf{x}_{\mu})&=&\sum_{j=1}^{N}f_j,\\
M_n(\mathbf{x}_{\mu})&=&\sum_{j=1}^{N}\frac{-f_j(-z_{\mu j})^n}{n}\hspace{20mm} n=\{1,2,...,p-1\}.
\end{eqnarray}
\item M2M from multipole expansion at $\mathbf{x}_\mu$ to multipole expansion at $\mathbf{x}_M$,
\begin{eqnarray}
M_0(\mathbf{x}_M)&=&M_0(\mathbf{x}_\mu),\\
M_n(\mathbf{x}_M)&=&-M_0(\mathbf{x}_\mu)\frac{(-z_{M\mu})^n}{n}+\sum_{k=1}^nM_k(\mathbf{x}_\mu)(-z_{M\mu})^{n-k}{{n-1}\choose{k-1}}.
\end{eqnarray}
\item M2L from multipole expansion at $\mathbf{x}_M$ to local expansion at $\mathbf{x}_\Lambda$,
\begin{eqnarray}
L_0(\mathbf{x}_\Lambda)&\approx&M_0(\mathbf{x}_M)\log(z_{\Lambda M})+\sum_{k=1}^{p-1}\frac{M_k(\mathbf{x}_M)}{z_{\Lambda M}^k},\\
L_n(\mathbf{x}_\Lambda)&\approx&-\frac{M_0(\mathbf{x}_M)}{(-z_{\Lambda M})^nn}+\sum_{k=1}^{p-1}\frac{(-1)^nM_k(\mathbf{x}_M)}{z_{\Lambda M}^{n+k}}{{n+k-1}\choose{k-1}}.
\end{eqnarray}
\item L2L from local expansion at $\mathbf{x}_\Lambda$ to local expansion at $\mathbf{x}_\lambda$,
\begin{equation}
L_n(\mathbf{x}_\lambda)\approx\sum_{k=n}^{p-1}L_k(\mathbf{x}_\Lambda)z_{\lambda\Lambda}^{k-n}{{k}\choose{n}}.
\end{equation}
\item L2P from local expansion at $\mathbf{x}_\lambda$ to particle at $\mathbf{x}_i$,
\begin{equation}
u_i\approx\Re\left(\sum_{n=0}^{p-1}L_n(\mathbf{x}_{\lambda})z_{i\lambda}^n\right).
\end{equation}
\end{enumerate}
For the P2M, M2M, and M2L kernels, the first term requires special treatment. The expansions are truncated at order $p$, so the accuracy of the FMM can be controlled by adjusting $p$. When recurrence relations are used to calculate the powers of $z$ and the combinations they can be calculated at the cost of one multiplication per inner loop ($k$ loop) iteration. In our implementation, we do not construct any matrices during the calculation of these kernels. The P2P kernel is vectorized with the use of SIMD intrinsics, and the $\log()$ function is calculated using a polynomial fit for $\log_2(x)/(x-1)$ using SIMD.
\section{Numerical results} \label{sec:results}
In this section we demonstrate the potential of the FMM-based preconditioner by applying it to a number of test problems and comparing it with standard preconditioners.
The primary aim is to assess the effectiveness of the preconditioner at reducing the number of Krylov subspace iterations that are required for convergence to a given tolerance. Additionally, we seek to ascertain whether mesh independence is achieved. We defer reporting on performance to Section \ref{sec:performance}. Accordingly, we choose problems that are small enough to enable solution by Matlab.
Our Poisson problems are all two dimensional and include examples with homogeneous and inhomogeneous Dirichlet boundary conditions. We additionally present a two-dimensional Stokes flow problem and show that, as predicted, combining the FMM-based Poisson preconditioner with a block diagonal matrix gives an effective preconditioner for the saddle point problem~\eqref{e:sp}.
Throughout, our stopping criterion is a decrease in the relative residual of six orders of magnitude. If such a decrease is not achieved after $maxit$ iterations the computations are terminated; this is denoted by `---' in the tables. This maximum number of iterations is stated for each problem below. For all problems and preconditioners the initial iterate is zero.
\subsection{The Poisson equation} \label{subsec:poisson}
We first test our preconditioner on three two-dimensional Poisson problems with a constant diffusion coefficient on the domain on $[-1,1]^2$. We discretize the problems by $Q_1$ finite elements using IFISS~\cite{elman2007},~\cite{ifiss}, with default settings. Our fast multipole preconditioner is compared with the incomplete Cholesky (IC) factorization~\cite{meijerink1977} with zero fill implemented in Matlab and the algebraic multigrid (AMG) and geometric multigrid (GMG) methods in IFISS. Within the GMG preconditioner we select point-damped Jacobi as a smoother instead of the default ILU, which is less amenable to parallelization. Otherwise, default settings for both multigrid methods are used. For all preconditioners, $maxit=20$ and we apply preconditioned conjugate gradients.
Our first example is the first reference problem in Elman \emph{et al.}~\cite[Section 1.1]{elman2005} for which
$$
\nabla^2 u = 1 \text{ in } \Omega = [-1,1]^2, \ u = 0 \text{ on } \Gamma.
$$
Table~\ref{t:p1} lists the preconditioned CG iterations for each preconditioner applied. The FMM preconditioner, as well as GMG and AMG appear to give mesh-independent convergence, although the incomplete Cholesky factorization does not.
\begin{table}
\centering
\begin{tabular}{crrrr}
\hline
$h$ & GMG & AMG & FMM & IC\\
\hline
$2^{-4}$ & 6 & 5 & 5 & 10 \\
$2^{-5}$ & 6 & 6 & 5 & 18 \\
$2^{-6}$ & 7 & 6 & 5 & --- \\
$2^{-7}$ & 7 & 6 & 6 & --- \\
$2^{-8}$ & 7 & 6 & 6 & --- \\
\hline
\end{tabular}
\caption{Preconditioned CG iterations for the relative residual to reduce by six orders of magnitude for the problem with $-\nabla^2 u = 1$ and homogeneous boundary conditions.}
\label{t:p1}
\end{table}
In Table~\ref{t:p1eig} we plot the eigenvalues of the FMM preconditioned stiffness matrix for $h = 2^{-4}, 2^{-5} \text{ and } 2^{-6}$. It is clear that the smallest eigenvalue of $A$ decreases as the mesh is refined; this is particularly problematic for Krylov subspace methods, since small eigenvalues can significantly hamper convergence. However, the eigenvalues of the FMM-preconditioned matrix are bounded away from the origin in a small interval that does not increase in size as the mesh is refined. This hints at spectral equivalence between the FMM-based preconditioner and the stiffness matrix. The condition number appears to be bounded, which is unsurprising given the mesh-independent convergence observed.
\begin{table}
\centering
\begin{tabular}{crrrrrr}
\hline
$h$ & $\lambda_{min}(A)$ & $\lambda_{max}(A)$ & $\kappa(A)$ & $\lambda_{min}(P^{-1}A)$ & $\lambda_{max}(P^{-1}A)$ & $\kappa(P^{-1}A)$\\
\hline
$2^{-4}$ & 0.076& 3.94 & 52 & 0.73 & 1.29 & 1.77\\
$2^{-5}$ & 0.019 & 3.99 & 207 & 0.72 & 1.29 & 1.79\\
$2^{-6}$ & 0.005 & 4.00 & 830 & 0.72 & 1.30 & 1.80\\
\hline
\end{tabular}
\caption{Smallest ($\lambda_{min}$) and largest ($\lambda_{max}$) eigenvalues and condition number ($\kappa$) of the stiffness matrix $A$ and FMM-preconditioned matrix $P^{-1}A$ for the problem with $-\nabla^2 u = 1$ and homogeneous boundary conditions.
\label{t:p1eig}
\end{table}
Our second example is the third reference problem from Elman \emph{et al.}~\cite[Section 1.1]{elman2005} posed on $[-1,1]^2$ which is characterized by inhomogeneous Dirichlet boundary conditions and the analytic solution
$$u(x,y) = \frac{2(1+y)}{(3+x)^2 + (1+y)^2}.$$
From Table~\ref{t:p2} we find that, similarly to the previous problem, the FMM preconditioner and both multigrid preconditioners are mesh independent but the Cholesky preconditioner is not. The FMM preconditioner is also competitive with the multigrid methods. Thus, on systems on which applying the FMM preconditioner is significantly faster than applying the multigrid preconditioners, we will achieve a faster time-to-solution with the former. We note that the eigenvalues and condition numbers obtained for the FMM preconditioned stiffness matrix are the same as those computed for the previous example.
\begin{table}
\centering
\begin{tabular}{crrrr}
\hline
$h$ & GMG & AMG & FMM & IC\\
\hline
$2^{-4}$ & 5 & 5 & 5 & 11 \\
$2^{-5}$ & 5 & 5 & 5 & 19 \\
$2^{-6}$ & 5 & 5 & 5 & --- \\
$2^{-7}$ & 5 & 5 & 5 & --- \\
$2^{-8}$ & 5 & 5 & 5 & --- \\
\hline
\end{tabular}
\caption{Preconditioned CG iterations for the relative residual to reduce by six orders of magnitude for the problem with $-\nabla^2 u = 0$ and inhomogeneous boundary conditions.}
\label{t:p2}
\end{table}
The final problem we consider in this section is the Poisson problem with solution
$$u(x,y) = x^2 + y^2$$
on $[-1,1]^2$, which has forcing term $f\equiv -4$ in the domain and inhomogeneous Dirichlet boundary conditions.The convergence results for this problem, given in Table~\ref{t:p3}, are similar to those for the previous problems. They show that the FMM preconditioner gives mesh independent convergence and is competitive with AMG and GMG. We also obtain the same eigenvalue results as for the previous examples.
\begin{table}
\centering
\begin{tabular}{crrrr}
\hline
$h$ & GMG & AMG & FMM & IC\\
\hline
$2^{-4}$ & 5 & 5 & 5 & 10 \\
$2^{-5}$ & 5 & 5 & 5 & 18 \\
$2^{-6}$ & 5 & 5 & 5 & --- \\
$2^{-7}$ & 5 & 5 & 5 & --- \\
$2^{-8}$ & 5 & 5 & 5 & --- \\
\hline
\end{tabular}
\caption{Preconditioned CG iterations for the relative residual to reduce by six orders of magnitude for the problem with $-\nabla^2 u = -4$ and inhomogeneous boundary conditions.}
\label{t:p3}
\end{table}
\begin{figure}
\centering
\includegraphics[width = 0.9\textwidth,height=10cm]{convergence.pdf}
\caption{Convergence rate of the FMM preconditioner with different precision, plotted along with algebraic multigrid, geometric multigrid, and incomplete Cholesky preconditioners. The $\epsilon$ represents the precision of the FMM, where $\epsilon=10^{-6}$ corresponds to six significant digits of accuracy.}
\label{f:convergence}
\end{figure}
\subsection{Effect of FMM precision on convergence}
For the results shown above, the FMM precision was set to preserve six significant digits. However, the FMM can be accelerated further by trading precision for speed. Since we are using the FMM as a preconditioner, the accuracy requirements are somewhat lower than that of general applications of FMM. Although this balance between the accuracy and speed of FMM is a critical factor for evaluating the usefulness of FMM as a preconditioner, the relation between the FMM precision and convergence rate has not been studied previously.
In Figure \ref{f:convergence} the relative residual at each CG iteration is plotted against the number of iterations for FMM, AMG, GMG, and IC. The problem is the same as in Table \ref{t:p1}. Three cases of FMM are used with six, four, and two significant digits of accuracy, respectively. The $\epsilon=10^{-6}$ case corresponds to the condition for the tests in Tables \ref{t:p1}--\ref{t:p3}. Decreasing the FMM accuracy to four digits has little effect during the first few iterations, but slows down the convergence near the end. Decreasing the FMM accuracy further to two digits slows down the convergence somewhat, but is still much better than the incomplete Cholesky.
Increasing the precision of the FMM past six digits did not result in any noticeable improvement because truncation error begins to dominate. We are preconditioning a matrix resulting from a FEM discretization by using a integral equation with Green's function kernels. Each has its own error, below which algebraic error need not be reduced. We show in Figure \ref{f:discretization} the convergence of spatial discretization error for the FEM and BEM approaches. We use the same reference problem as in Table \ref{t:p1}, which has an analytical solution. The discretization error is measured by taking the relative $L^2$ norm of the difference between the analytical solution and the individual numerical solutions. We see that the FEM is second order and BEM is first order. The five different values of $\Delta x$ correspond to $h=\{2^{-4},2^{-5},2^{-6},2^{-7},2^{-8}\}$, which were used in the previous experiments. For the current range of grid spacing, the discrepancies between the FEM and BEM truncation error is in the range of $10^{-3}$ to $10^{-4}$.
\begin{figure}
\centering
\includegraphics[width = 0.8\textwidth]{discretization.pdf}
\caption{Convergence of spatial discretization error for the FEM and BEM. The relative $L^2$ norm of the difference between the analytical solution is plotted against the grid spacing $\Delta x$.}
\label{f:discretization}
\end{figure}
\subsection{Stokes problem}
Finally, we examine convergence for a two-dimensional Stokes flow. The leaky cavity problem~\cite[Example 5.1.3]{elman2005} on $[-1,1]$ is discretized by $Q_1-P_0$ elements in Matlab using IFISS with default settings. As described in Section~\ref{sec:krylov}, by combining a stiffness matrix preconditioner $P_A$ with the diagonal of the pressure mass matrix $P_S$, an effective preconditioner~\eqref{e:sppre} for the saddle point system~\eqref{e:sp} is obtained. Here, we are interested in using the FMM preconditioner for $P_A$, and we compare its performance with AMG and GMG. We do not consider the incomplete Cholesky factorization of $A$ because of its poor performance on the stiffness matrix (see Tables~\ref{t:p1}, \ref{t:p2} and~\ref{t:p3}). We set $maxit=50$ and apply preconditioned MINRES to the saddle point system.
As for the Poisson problem, the FMM-based preconditioner provides a mesh-independent preconditioner that is comparable to algebraic and geometric multigrid. Although three or four more iterations are required by the FMM preconditioner than the AMG preconditioner, if each iteration is faster the time-to-solution may well be lower.
\begin{table}
\centering
\begin{tabular}{crrr}
\hline
$h$ & GMG & AMG & FMM\\
\hline
$2^{-4}$ & 37 & 37 & 41 \\
$2^{-5}$ & 39 & 39 & 43 \\
$2^{-6}$ & 39 & 41 & 43 \\
$2^{-7}$ & 39 & 39 & 43 \\
$2^{-8}$ & 38 & 38 & 41\\
\hline
\end{tabular}
\caption{Preconditioned MINRES iterations for the relative residual to reduce by six orders of magnitude for the Stokes problem.}
\label{t:s1}
\end{table}
\section{Performance analysis}\label{sec:performance}
In this section we evaluate the performance of the FMM-based preconditioner by comparing its time-to-solution to an algebraic multigrid code BoomerAMG. We have implemented our FMM-preconditioner into PETSc~\cite{petsc-user-ref}~,\cite{petsc-web-page} via PetIGA~\cite{PetIGA}. PetIGA is a software layer that sits on top of PETSc that facilitates NURBS-based Galerkin finite element analysis. For our present analysis, we simply use PetIGA to reproduce the same finite element discretization as the tests shown in Section \ref{sec:results}, but in a high performance computing environment. We select the first problem in Section \ref{subsec:poisson} with $-\nabla^2u=1$ and homogeneous Dirichlet boundary conditions for the following performance evaluation.
All codes that were used for the current study are made publicly available. A branch of PetIGA that includes the FMM preconditioner is hosted on bitbucket. \footnote{https://bitbucket.org/rioyokota/petiga-fmm} The 2-D extension of exaFMM, which we call from PetIGA is also hosted on bitbucket. \footnote{https://bitbucket.org/rioyokota/exafmm2d}
All calculations were performed on the TACC Stampede system without using the coprocessors. Stampede has 6400 nodes, each with
two Xeon E5-2680 processors and one Intel Xeon Phi SE10P coprocessor and 32GB of memory. We used the Intel compiler (version 13.1.0.146) and configured PETSc with ``\texttt{COPTFLAGS=-O3 FOPTFLAGS=-O3 --with-clanguage=cxx\\ --with-mpi-dir=/opt/apps/intel13/impi/4.1.0.030/intel64/\\ --download-f-blas-lapack --download-hypre --download-metis\\ --download-parmetis --download-superlu\_dist --with-debugging=0}".
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{complexity.pdf}
\caption{Time-to-solution for different problem sizes of the FMM and AMG preconditioners on a single core of a Xeon E5-2680.}
\label{fig:complexity}
\end{figure}
\subsection{Serial results}
We first evaluate the serial performance of our method using the same two-dimensional Poisson problem used in Section \ref{sec:poisson}. We confirmed that the iteration counts shown in Table \ref{t:p1} did not change for the PETSc version of our code. Then, we measured the time-to-solution for different problem sizes. Since the domain size is $[-1,1]$, the grid spacing of $h=\{2^{-4},2^{-5},2^{-6},2^{-7},2^{-8}\}$ in Section \ref{sec:results} correspond to a grid size of $N=\{32^2,64^2,128^2,256^2,512^2\}$. In the PETSc version, the time-to-solution improves significantly so we tested for larger problem sizes of $N=\{64^2,128^2,256^2,512^2,1024^2,2048^2,4096^2\}$.
The time-to-solution is plotted against the problem size $N$ in Figure \ref{fig:complexity}. Since we are using PETSc, it is trivial to change the preconditioner to AMG by passing the option ``\texttt{--pc\_type hypre}" during runtime. Therefore, the time-to-solution of BoomerAMG is shown as a reference in the same figure. For BoomerAMG we compared different relaxation, coarsening, and interpolation methods and found that ``\texttt{-pc\_hypre\_boomeramg\_relax\_type\_all backward-SOR/Jacobi -pc\_hypre\_boomeramg\_coarsen\_type modifiedRuge-Stueben -pc\_hypre\_bommeramg\_interp\_type classical}" gives the best performance.
Both FMM and AMG runs are serial, where we used a single MPI process and a single thread. The majority of the time goes into the setup of the preconditioner ``PCSetUp" and the actual preconditioning ``PCApply", so only these events are shown in the legend. The ``PCSetUp" is called only once for the entire run, while ``PCApply" is called every iteration. For the present runs, both FMM and AMG required six iterations for the relative residual to drop six digits, so all runs are calling ``PCApply" six times. The order of expansion for the FMM is set to $p=6$ and $\theta=0.4$, which gives about six significant digits of accuracy. With this accuracy for the FMM, we are still able to converge in six iterations. The P2P kernel in the FMM code is performed in single precision using SIMD intrinsics, but this does not prevent us from reaching the required accuracy of six significant digits because we use Kahan's summation technique \cite{kahan1965} for the reduction.
By taking a closer look at Figure \ref{fig:complexity}, one can see that both the FMM and AMG show $\mathcal{O}(N)$ asymptotic behavior. The FMM seems to have a slower preconditioning time, but a much faster setup time compared to AMG. The FMM also has a constant overhead which becomes evident when $N$ is small. In summary, the time-to-solution of the FMM is approximately an order of magnitude larger than that of AMG for the serial runs. This is consistent with our intuition that FMM is not the preconditioner of choice for solving small problems on a single core. We will show in the following section that the FMM becomes competitive when scalability comes into the picture.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{strong_scaling.pdf}
\caption{Strong scaling of the FMM and AMG preconditioners.}
\label{fig:strong_scaling}
\end{figure}
\subsection{Parallel results}
Using the same Poisson problem, we now compare the performance of FMM and AMG for parallel runs on Stampede. We also compare with a sparse direct solver MUMPS by invoking at runtime ``\texttt{-ksp\_type\ preonly -pc\_type lu -pc\_factor\_mat\_solver\_package mumps}".
The strong scaling of FMM, AMG, and MUMPS are shown in Figure \ref{fig:strong_scaling}. We use the largest grid size in the previous runs $N=4096^2$. Stampede has 16 cores per node so all runs first parallelize among these cores and then among the nodes after the 16 cores are filled. The FMM strong scales quite well up to 1024 cores, while the parallel efficiency of AMG starts to decrease after 128 cores. The sparse direct solver has a much larger time-to-solution even on a single core, and is much less scalable than the other two hierarchical preconditioners. For this particular Poisson problem on this particular machine using this particular FMM code we see an advantage over BoomerAMG past 512 cores.
\subsection{Extension to 3-D}
The results presented in this article are only for two dimensional problems. A natural question that arises is whether the extension to 3-D is straightforward, and whether FMM will still be competitive as a preconditioner or not. Our results showed that a dominant part of the calculation time for the FMM preconditioner is the ``PCApply" stage, which is the dual tree traversal for calculation of M2L and P2P kernels. For 3-D kernels, the M2L operation is much more complicated so the calculation time of the FMM will increase, even for the same number of unknowns $N$.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{2dvs3d.pdf}
\caption{Calculation time of 2-D and 3-D FMM for the same problem size.}
\label{f:2dvs3d}
\end{figure}
Figure \ref{f:2dvs3d} shows the calculation time of our 2-D FMM and 3-D FMM, both for the Laplace kernel with four significant digits of accuracy on a single core of a Xeon E5-2680, 2.7 GHz CPU. The problem size $N$ varies from $10^5$ to $10^7$. We see that the 3-D FMM is about an order of magnitude slower than the 2-D FMM for the same problem size. The curse of dimensionality exists for FMMs as well, but it is not clear how competitive our 3-D FMM preconditioner will be until we perform a direct comparison against other preconditioners in 3-D. The logical next step is to actually perform these 3-D comparisons.
\section{Conclusions}\label{sec:conc}
The Fast Multipole Method, originally developed as a free-standing solver, can be effectively combined with Krylov iteration as a scalable and highly performant preconditioner for traditional low-order finite discretizations of elliptic boundary value problems. In model problems it performs similarly to algebraic multigrid in convergence rate, while excelling in scalings where AMG becomes memory bandwidth-bound locally and/or synchronization-bound globally. Additional algorithmic development and additional testing of implementations on emerging architectures are necessary to more fully define the niche in which FMM is the preconditioner of choice. Two of the extensions required relative to our current results are to coefficient variability and in the order of discretization of the boundary integral technique. No preconditioner considered in isolation can address the fundamental architectural challenges of Krylov methods for sparse linear systems, which are being simultaneously adapted to less synchronization tolerant computational environments, but it is important to address the bottlenecks of preconditioning this most popular class of solvers by making a wide variety of tunable preconditioners available and better integrating them into the overall solver. Fast multipole-based preconditioners are demonstrably ready to play an important role in the migration of sparse iterative solvers to the exascale.
\noindent
{\bf Acknowledgements}
The authors would like to acknowledge the open source software packages that made this work possible:
PETSc~\cite{petsc-user-ref,petsc-web-page}, PetIGA~\cite{PetIGA}
and IFISS~\cite{elman2007,ifiss}.
We thank Dave Hewett, David May, Andy Wathen, and Ulrike Yang for helpful discussions
and comments and are indebted to Nathan Collier for his help with the PetIGA
framework. We thank Lorena Barba for the support in developing ExaFMM.
This publication was based on work supported in part by Award No KUK-C1-013-04 ,
made by King Abdullah University of Science and Technology (KAUST). This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575.
\bibliographystyle{abbrv}
|
\section*{Introduction}\label{S:intro}
Let $\Omega$ be a domain in ${\mathbb C}^n$. If $\alpha$ is a $\bar \partial$-closed $(0,1)$-form on $\Omega$, consider
solutions to the Cauchy-Riemann system $\bar \partial v=\alpha$. A fundamental problem is to determine whether particular solutions
satisfying norms estimates exist for various norms. Such solutions lead to construction of non-trivial holomorphic functions
on $\Omega$.
Solving $\bar \partial$ with estimates depends both on the geometry of $\Omega$ and the norms considered. In this paper
results on two classes of domains are established -- product domains, especially with non-smooth factors, and the Hartogs triangle -- with estimates in $L^p$ norms,
$1 \le p<\infty$. Obtaining $L^p$ estimates for $\bar \partial$ on the Hartogs triangle motivated our investigation and is achieved in Theorem \ref{HarLp'}. However this result is
proved by transferring the $\bar \partial$ problem to a 2-dimensional product domain, so $L^p$ estimates for $\bar \partial$ on product spaces are established first. The results on product domains are
new and of independent interest. That such estimates were not previously established is unusual, given the success of integral formulas on domains with more complicated
geometry. The study of $\bar \partial$ on product spaces accounts for most of the paper's length.
A successful method for obtaining non-$L^2$ estimates on $\bar \partial$ starts by establishing integral representation formulas with holomorphic kernels
for forms.\footnote{Or almost holomorphic kernels; see \cite{Range13} for a result in this direction.}
This approach was inaugurated by \cite{Henkin69} and \cite{GrauertLieb}
on strongly pseudo convex domains and was intensely pursued in the two decades after \cite{Henkin69}, \cite{GrauertLieb}; see \cite{Kerzman71}, \cite{Lieb70}, and \cite{Ovrelid}
for some early foundational results. There are many significant results in this direction, too numerous to survey; see \cite{Range86} and \cite{HenkinLeiterer} for references to
the main results prior to 1985.
Integral formulas follow from a general procedure, the Cauchy-Fantappi\' e method, once a {\it generating form} is constructed; see \cite{Range86}, \cite{RangeSiu}, and \cite{LanSte13}.
However Cauchy-Fantappi\' e integral formulas have almost exclusively been derived for domains with smooth boundary (plus additional, restrictive geometric conditions) because Stokes' theorem is freely applied during the construction.
Two notable exceptions are \cite{RangeSiu}, on strongly pseudoconvex domains with piecewise smooth boundary, and \cite{Henkin71}, on analytic polyhedra.
For a product domain, the boundary is not smooth nor strongly pseudoconvex away from its boundary singularities. So while many techniques used below are well-known, modifications of the ``standard recipe'' are also required to establish our integral formulas. The first goal is to obtain the abstract integral formula \eqref{T1} on a 2-dimensional product domain with smoothly bounded factors; the derivation crucially uses an
idea from \cite{RangeSiu}. A formula for products with higher-dimensional factors is also obtained, see Remark \ref{R:productSolution}.
The 2-dimensional formula is then converted into an explicit solution operator using the Cauchy generating form, when the data is sufficiently smooth:
\begin{proposition}\label{I:main} Suppose $D_1, D_2\subset{\mathbb C}$ are domains with $C^1$ boundary. If $f\in C^1_{0,1}\left(\overline{D_1\times D_2}\right)$ satisfies $\bar \partial f=0$, the function
\begin{equation}
\label{I:derivativeT1}
\begin{split}
T(f)
&=\frac{-1}{2 \pi i}\int_{D_2}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2+\frac{-1}{2 \pi i}\int_{D_1}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\\
&+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{{\mathscr D}(f)(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2
\end{split}
\end{equation}
solves $\bar \partial (Tf) =f$. In \eqref{I:derivativeT1}, ${\mathscr D} f =\frac{\partial f_1}{\partial \bar z_2}=\frac{\partial f_2}{\partial \bar z_1}$.
\end{proposition}
This is proved as Proposition \ref{P:strong} below. Even in the case of the bidisc, i.e., when $D_j={\mathbb D}=\left\{z\in{\mathbb C}: |z|<1\right\}$,
Proposition \ref{I:main} is new. Since the kernels in the integrands are Cauchy kernels, or iterated Cauchy kernels, mapping properties of the operator $T$ are easy to derive.
Additionally, a re-expression of \eqref{I:derivativeT1} -- see Remark \ref{T1forDxA} -- corrects a minor error in a formula displayed on page 212 of \cite{Henkin71}, given without proof, and reproduced in \cite{FornaessLeeZhang}. (The error is inconsequential for the estimates proved in \cite{FornaessLeeZhang}.)
The solution operator \eqref{I:derivativeT1} is extended to non-$C^1$ bounded domains -- including ${\mathbb D}\times{\mathbb D}^*$, where ${\mathbb D}^*=\left\{ 0 <|z| <1\right\}$ is the punctured disc -- and to forms not necessarily smooth
up to the boundary in Section \ref{SS:punctured}. The $L^p$ boundedness of the integral operators is also proved in Section \ref{LpofT}. In contrast to results on strongly pseudoconvex domains, non-standard $L^p$ boundedness of the data is needed to obtain an $L^p$ bound on the solution. Define the norm $\|f\|_{{\mathcal B}}:=\|f_1\|_{L^p(D_1\times D_2)}+\|f_2\|_{L^p(D_1\times D_2)}+\|{\mathscr D}(f)\|_{L^p(D_1\times D_2)}$ on $(0,1)$-forms $f=f_1d\bar z_1+ f_2 d\bar z_2$, see Definition \ref{D:banach}.
The main $L^p$ result on product domains, Theorem \ref{Lpwithdbarf}, says the ordinary $L^p$ norm of $Tf$ is dominated by $\|f\|_{{\mathcal B}}$.
In Section \ref{S:necessity}, the condition $f\in L^p\left(D_1\times D_2\right)$ alone is shown not to be sufficient to conclude $Tf\in L^p$. More dramatically, the example there shows that $Tf$ can fail to {\it exist} for
$f\in L^1\left(D_1\times D_2\right)$, or more generally for $f\in L^p$, $1\leq p<2$. This contrasts sharply with results on the Henkin-Cauchy-Fantappi\' e operator known on strongly pseudoconvex domains \cite{Kerzman71}, \cite{Ovrelid},
some finite type domains \cite{ChaNagSte}, \cite{FefKohn88}, and even some infinite type domains \cite{HaKhaRai}. The contrast is interesting and should be understood more fully. The observation in Section \ref{S:necessity}
merely inaugurates this new phenomena; finding actual necessary conditions on $f$ that follow from $Tf$ existing or satisfying $L^p$ estimates remains open. Such conditions, beyond $f\in L^p\left(D_1\times D_2\right)$, obviously have consequences when using the estimates in application. To be clear: our computations in Section \ref{S:necessity} are made only on the Henkin solution operator.
The main previous result on $\bar \partial$-estimates for the bidisc are the $L^\infty$ estimates in \cite{FornaessLeeZhang}. There is a point connecting \cite{FornaessLeeZhang}, the undetermined necessary conditions mentioned above, and the older literature on the Henkin solution. Norm control of derivatives of the data $f$ is assumed in \cite{FornaessLeeZhang}, though somewhat obliquely. In that paper the estimate
$\|Tf\|_{L^\infty({\mathbb D}\times{\mathbb D})} \leq\, C\|f\|_{L^\infty({\mathbb D}\times{\mathbb D})}$ is proved, but only under the assumption that $f\in C^1_{0,1}\left(\overline{{\mathbb D}\times{\mathbb D}}\right)$. Thus the question posed by Kerzman in 1971 -- does there exist a solution operator satisfying $\|Sf\|_{L^\infty}\leq C \|f\|_{L^\infty}$ for {\it all} $\bar \partial$-closed forms in $L^\infty$, \cite{Kerzman71} remark on pages 311--312 -- is still unresolved on the bidisc.
Finally in Section \ref{S:Hartogs}, the
biholomorphism between the Hartogs triangle ${\mathbb H}$ and ${\mathbb D}\times{\mathbb D}^*$ transfers the $\bar \partial$ problem on ${\mathbb H}$ to a $\bar \partial$ problem on ${\mathbb D}\times{\mathbb D}^*$ with different data. $L^p$ estimates for a solution operator on ${\mathbb H}$ are then inferred from those on ${\mathbb D}\times{\mathbb D}^*$, cf. Theorem \ref{HarLp'}. There are previous results about solving $\bar \partial$ with estimates in H\" older spaces
$C^{k,\alpha}\left({\mathbb H}\right)$, including the degenerate case $L^\infty\left({\mathbb H}\right)$. See \cite{ChaumatChollet91}, \cite{ChaumatChollet93}, and \cite{MaMichel}. In these papers, a reduction of extending the $\bar \partial$ data to supersets of ${\mathbb H}$ is allowed by the H\" older norms and used essentially. This reduction does not occur for $L^p$ data.
Our analysis has a surprising consequence: solution operators for $\bar \partial$ on ${\mathbb H}$ exist that are better behaved than the canonical solution operator in terms of $L^p$ boundedness. Recent results, \cite{EdhMcN16}, \cite{ChaZey16}, and \cite{Chen17}
show the Bergman projection on ${\mathbb H}$ is only $L^p$ bounded for $p\in\left(\frac 43, 4\right)$. The canonical solution operator inherits this limited $L^p$ boundedness. However our solution operator for $\bar \partial$ on ${\mathbb H}$ is $L^p$ bounded for all $1\leq p<\infty$, at least on a subclass of forms. See Sections \ref{SS:example} and \ref{SS:extra} for details. The only other situation we know where the canonical solution operator is demonstrably not the best solution operator for $\bar \partial$ is the Diederich-Fornaess worm domain $W$, for estimates in the $C^k\left(\overline{W}\right)$ scale of norms, c.f. \cite{Christ96} and \cite{Kohn73}.
The authors thank Dror Varolin for an insightful comment about section \ref{SS:product}. The authors are also grateful to the anonymous referee, who pointed out an error in an earlier version of Section \ref{S:necessity} and suggested several expositional improvements.
\section{The Cauchy-Fantappi\'{e} Formalism}\label{cfformalism}
In this section the Cauchy-Fantappi\'{e} formalism is reviewed and applied to product spaces. For more information, see \cite{Range86,RangeSiu}.
\subsection{Domain with $C^1$ boundary}
\label{C1domain}
Let $D$ be a domain in ${\mathbb C}^n$ with $C^1$ boundary $bD$. Let $U$ be an open neighborhood of $bD\times\bar D$ and $U^*=\{(\zeta,z)\in U | \zeta\ne z\}$.
\begin{definition}\label{D:generateC1domain}
A generating form $w$ on $bD\times D$ is a $C_{1,0}^1$-form in $\zeta$ and a $C^{\infty}$ function in $z$,
\[
w(\zeta,z)=\sum_{l=1}^n w^l(\zeta,z)\,d\zeta_l,
\]
with the following property
\[
\langle w(\zeta,z),\zeta-z\rangle = \sum_{l=1}^n w^l(\zeta,z)(\zeta_l-z_l)=1\qquad\text{for}\,\,\,(\zeta,z)\in U^*.
\]
\end{definition}
When applying a differential operator to forms depending on multiple sets of independent variables (like $w$), subscripts will be used to indicate which variables are differentiated. For instance,
$\bar \partial_z w(\zeta, z) =\sum_{k,l=1} \frac{\partial w^l}{\partial\bar z_k} d\bar z_k\wedge d\zeta_l$, while $\partial_\zeta w(\zeta, z) =\sum_{k,l=1} \frac{\partial w^l}{\partial\zeta_k} d\zeta_k\wedge d\zeta_l$.
The same convention is used on functions.
The universal form
\[
w_0(\zeta,z)=\frac{\partial_{\zeta} (|\zeta-z|^2)}{|\zeta-z|^2}=\frac{\sum_{l=1}^n(\bar\zeta_l-\bar z_l)\,d\zeta_l}{|\zeta-z|^2}
\]
is called the Bochner-Martinelli generating form. This form satisfies Definition \ref{D:generateC1domain} for any domain $D$. Let $\mu\in I=[0,1]$. If $w$ is a generating form on $bD\times D$, define the homotopy between $w$ and $w_0$ by
\begin{equation}\label{E:homotopy}
\hat w(\zeta,z,\mu)=\mu w(\zeta,z)+(1-\mu) w_0(\zeta,z).
\end{equation}
Note that for each fixed $\mu\in I$, the form $\hat w$ also satisfies Definition \ref{D:generateC1domain}.
A piece of notation simplifies writing formulas below: let $\bar \partial_{\zeta,\mu}=\bar \partial_{\zeta}+d_{\mu}$.
\begin{definition}
\label{CFkernel}
The Cauchy-Fantappi\'{e} kernel of order $q$ generated by $\hat w$ is
\begin{equation}\label{E:CFkernel1}
\Omega_q(\hat w)=\frac{(-1)^{q(q-1)/2}}{(2\pi i)^n}\left( \begin{array}{c} n-1 \\ q \end{array} \right) \hat w \wedge \left( \bar \partial_{\zeta,\mu} \hat w \right)^{n-q-1} \wedge \left( \bar \partial_{z} \hat w \right)^q
\end{equation}
for $0 \le q \le n-1$, and $0$ otherwise ($q=-1$ and $q=n$).
\end{definition}
\smallskip
\begin{remark}\label{R:1}
The kernel $\Omega_q(w)$ associated to an arbitrary generating form $w$ is defined in the same manner:
\begin{equation}\label{E:CFkernel2}
\Omega_q(w)=\frac{(-1)^{q(q-1)/2}}{(2\pi i)^n}\left( \begin{array}{c} n-1 \\ q \end{array} \right) w \wedge \left( \bar \partial_{\zeta} w \right)^{n-q-1} \wedge \left( \bar \partial_{z} w \right)^q
\end{equation}
Note that $\bar\partial_{\zeta,\mu}$ has been replaced by $\bar\partial_{\zeta}$. Moreover, if $q\ge1$ and $w$ is holomorphic in $z$, $\Omega_q(w)=0$ because of the final factor in \eqref{E:CFkernel2}.
\end{remark}
\smallskip
\begin{remark}\label{R:2}
The kernel $\Omega_q(w_0)$ is also denoted $K_q$ and called the Bochner-Martinelli-Koppelman kernel, following \cite{Range86}. For $D\subset\subset{\mathbb C}^n$ with piecewise $C^1$ boundary, the Bochner-Martinelli-Koppelman representation for $f\in C_{0,q}^1(\bar D)$ is
\begin{equation}\label{E:BMK}
f(z)=\int_{bD}f\wedge K_q(\cdot,z)-\int_D\bar\partial f\wedge K_q(\cdot,z)-\bar \partial_z\int_Df\wedge K_{q-1}(\cdot,z)
\end{equation}
where $0\le q\le n$. It holds that $\int_Df\wedge K_{q-1}(\cdot,z)\in C_{0,q-1}^1(D)$. See \cite[Chap. IV, Theorem 1.10]{Range86} for proofs of these facts.
\end{remark}
A more general representation formula than \eqref{E:BMK} uses the following ingredient:
\begin{definition}\label{D:solutionop}
Let $D\subset\subset{\mathbb C}^n$ be a domain with $C^1$ boundary and $w$ a generating form on $bD\times D$.
For $1 \le q \le n$, define the integral operator
\[
T_{q}^{w}: C_{0,q}(\bar D) \to C_{0,q-1}(D)
\]
by
\[
T_{q}^{w}(f)=\int_{bD \times I}f \wedge \Omega_{q-1}(\hat w)-\int_D f \wedge K_{q-1}.
\]
Set $T_0^w=T_{n+1}^w \equiv 0$.
\end{definition}
The following theorem is proved in \cite[Chap. IV, Theorem 3.6]{Range86}.
\begin{theorem}
Let $D\subset\subset{\mathbb C}^n$ be a domain with $C^1$ boundary and $w$ a generating form on $bD\times D$.
For $0 \le q \le n$ and $f \in C_{0,q}^1(\bar D)$,
\begin{equation}\label{E:SolutionOp1}
f=\int_{bD}f \wedge \Omega_q(w) + \bar \partial T_q^w(f) + T_{q+1}^w(\bar \partial f)\qquad\text{on}\,\,\, D.
\end{equation}
Moreover, for $k=0,1,2,\dots,\infty$, if $f \in C_{0,q}^k(D) \cap C_{0,q}(\bar D)$ then $T_q^w(f) \in C_{0,q-1}^k(D)$.\\
\end{theorem}
\begin{remark}\label{R:3}
Suppose $q\geq1$. If the generating form $w$ is holomorphic in $z$ and $f \in C_{0,q}^1(\bar D)$ is $\bar \partial$-closed, \eqref{E:SolutionOp1} implies $u=T_{q}^w(f)$ solves
\[
\bar \partial u=f,
\]
since $\Omega_q(w)=0$ as noted in Remark \ref{R:1}
\end{remark}
\subsection{Product domains}\label{SS:product} An idea from \cite{RangeSiu} is used to construct a generating form on a product domain from known generating forms on the factors. In \cite{RangeSiu}, only domains with piecewise smooth boundaries that are {\it strongly pseudoconvex} away from boundary singularities are considered. However, strong pseudoconvexity is only used to build the integral kernels on
smooth pieces of the boundary, not to piece the kernels together to get a solution operator for $\bar \partial$. This latter idea is what we extract.
Let $D=D_1 \times D_2\subset{\mathbb C}^n$, where $D_1\subset{\mathbb C}^{n_1}$ and $D_2\subset{\mathbb C}^{n_2}$ are domains with $C^1$ boundary. Let $S_1=bD_1\times\bar D_2$ and $S_2=\bar D_1\times bD_2$.
\begin{definition}
\label{D:generateproductdomain}
For $j=1,2$, a generating form $w_j(\zeta,z)$ on $S_j\times D$ is a $(1,0)$-form in $\zeta$
\[
w_j(\zeta,z)=\sum_{l=1}^n w_j^l(\zeta,z)\,d\zeta_l
\]
with the following properties
\begin{enumerate}
\item for each fixed $\zeta\in S_j$, $w_j^l(\zeta,\cdot)$ is $C^{\infty}$ in $D$,
\item for each fixed $z\in D$, $w_j^l(\cdot,z)$ is $C^1$ in a neighborhood $U_j^z$ of $S_j$, and
\item for each $z\in D$
\begin{equation}
\label{geniden}
\langle w_j(\zeta,z),\zeta-z\rangle = \sum_{l=1}^n w_j^l(\zeta,z)(\zeta_l-z_l)=1\qquad \text{for all} \,\,\,\zeta\in U_j^{z}.
\end{equation}
\end{enumerate}
\end{definition}
\begin{remark}\label{R:generating}
The forms in Definition \ref{D:generateproductdomain} are generating for only part of $bD$, namely $S_j$.
To connect this with the previous definition, suppose $\tilde w_j$ is a generating form on $bD_j\times D_j$ as in Definition \ref{D:generateC1domain}, for $j=1,2$. Define
\[
w_1(\zeta,z)=\sum_{l=1}^{n_1}\tilde w_1^l(\zeta^1,z^1)\,d\zeta^1_l
\]
and
\[
w_2(\zeta,z)=\sum_{l=1}^{n_2}\tilde w_2^{l}(\zeta^2,z^2)\,d\zeta^2_l
\]
where $\zeta=\left(\zeta^1,\zeta^2\right)\in{\mathbb C}^{n_1}\times{\mathbb C}^{n_2}$ and $z=\left(z^1,z^2\right)\in{\mathbb C}^{n_1}\times{\mathbb C}^{n_2}$. Note that $w_1, w_2$ are independent of $\left(\zeta^2, z^2\right)$, $\left(\zeta^1, z^1\right)$ respectively.
Elementary algebra shows that $w_1$ and $w_2$ are generating forms on $S_1\times D$ and $S_2\times D$, respectively, as given by Definition \ref{D:generateproductdomain}.
\end{remark}
Following \cite{RangeSiu}, let
\[
\Delta=\{\lambda=(\lambda_0,\lambda_1,\lambda_2)\in{\mathbb R}^{3}\,|\,\lambda_0,\lambda_1,\lambda_2\ge0,\lambda_0+\lambda_1+\lambda_2=1\},
\]
\[
\Delta_{0}=\{\lambda\in\Delta\,|\,\lambda_1=\lambda_2=0\},
\]
\[
\Delta_{01}=\{\lambda\in\Delta\,|\,\lambda_2=0\},
\]
\[
\Delta_{02}=\{\lambda\in\Delta\,|\,\lambda_1=0\}.
\]
As before, let
\[
w_0(\zeta,z)=\frac{\partial_{\zeta} (|\zeta-z|^2)}{|\zeta-z|^2}=\frac{\sum_{l=1}^n(\bar\zeta_l-\bar z_l)\,d\zeta_l}{|\zeta-z|^2}.
\]
Consider the partial convex combination of $w_0$, $w_1$, and $w_2$
\begin{equation}
\label{cvxcmb}
W(\zeta,z,\lambda)=\sum_{j=0}^{2}\lambda_j w_j(\zeta,z),
\end{equation}
defined only on the following sets
\begin{enumerate}
\item $(\zeta,z,\lambda)\in\bar D\times D\times\Delta_0$ with $\zeta\neq z$,
\item $(\zeta,z,\lambda)\in S_1\times D\times\Delta_{01}$,
\item $(\zeta,z,\lambda)\in S_2\times D\times\Delta_{02}$,
\item and $(\zeta,z,\lambda)\in (bD_1\times bD_2)\times D\times\Delta$.
\end{enumerate}
\smallskip
\begin{remark}
For $\zeta$ fixed, the form $W$ in \eqref{cvxcmb} is $C^{\infty}$ for $z\in D$. For $z\in D$ fixed, $W$ is $C^1$ in $\zeta$ and satisfies \eqref{geniden} in the corresponding neighborhood depending on $z$. Also, the form $W$ is differentiable in $\lambda$ in the interiors of $\Delta$, $\Delta_{01}$, and $\Delta_{02}$.
\end{remark}
When derivatives with respect to the vector $\lambda$ are written, the meaning is that derivatives with respect to $\lambda_0, \lambda_1, \lambda_2$ are taken and the results are added. Notationally
\[
\bar \partial_{\zeta,\lambda}=\bar \partial_{\zeta}+d_{\lambda_0}+d_{\lambda_1}+d_{\lambda_2}.
\]
Similar to Definition \ref{CFkernel}, a kernel is associated to the form $W$ in \eqref{cvxcmb}:
\begin{definition}
The Cauchy-Fantappi\'{e} kernel of order $q$ generated by the form $W$ in \eqref{cvxcmb} is defined
\[
\Omega_q(W)=\frac{(-1)^{q(q-1)/2}}{(2\pi i)^n}\left( \begin{array}{c} n-1 \\ q \end{array} \right) W \wedge \left( \bar \partial_{\zeta,\lambda} W \right)^{n-q-1} \wedge \left( \bar \partial_{z} W \right)^q
\]
for $0 \le q \le n-1$, and $0$ otherwise ($q=-1$ and $q=n$).
\end{definition}
\begin{remark}\label{R:CF}
If $q\ge1$ and if for $j=1,2$ $w_j$ is holomorphic in $z$ for $\zeta$ fixed, then $\Omega_q(W)=0$ on the set where $\lambda_0=0$. This follows since $\lambda_0=0$ implies that $W$ defined by \eqref{cvxcmb} is holomorphic in $z$.
On the other hand, note that none of the sets (1)-(4) in \eqref{cvxcmb} allow $\lambda_0=0$. Nevertheless, $\Omega_q(w)=0$ on this set is needed in order to show that the operator in Definition \ref{D:solution} below is solution operator for $\bar \partial$; for a proof see \cite[\S(2.5)]{RangeSiu}.
\end{remark}
Parallel to \S\ref{C1domain}, an integral operator associated to the form $W$ in \eqref{cvxcmb} is defined.
\begin{definition}\label{D:solution}
For $1 \le q \le n$, define the integral operator
\begin{align*}
T_{q}^{W}: C_{0,q}(\bar D) \to C_{0,q-1}(D)
\end{align*}
by
\begin{align*}
T_{q}^{W}(f)=-\int_{bD_1\times bD_2\times \Delta}f\wedge\Omega_{q-1}(W)+\int_{S_1\times\Delta_{01}}f\wedge\Omega_{q-1}(W)&+\int_{S_2\times\Delta_{02}}f\wedge\Omega_{q-1}(W)\\ &-\int_{D\times\Delta_0} f \wedge\Omega_{q-1}(W).
\end{align*}
Set $T_0^W=T_{n+1}^W \equiv 0$.
\end{definition}
\begin{remark}\label{R:productSolution}
For $1\le q\le n$, if $w_j$ is holomorphic in $z$ for $j=1,2$, then $T_q^W$ is a solution operator to the $\bar \partial$-equations; i.e. $u=T^W_q(f)$ solves
\[
\bar \partial u=f
\]
when $\bar\partial f=0$. This follows from Stokes' theorem, but non-trivially as the different dimensional facets of the simplex $\Delta$ must be handled. As for Remark \ref{R:CF}, a detailed proof is given in \cite[\S(2.5)]{RangeSiu}.
\end{remark}
\begin{remark}
Since $\lambda_0+\lambda_1=1$ on $\Delta_{01}$, $d\lambda_0=-d\lambda_1$ on this set. By change of variables, it follows that
\[
\int_{S_1\times\Delta_{01}}f\wedge\Omega_{q-1}(W)=\int_{S_1\times I}f\wedge\Omega_{q-1}(\hat w_1),
\]
where $\hat w_1$ is the homotopic form as in \S\ref{C1domain} and $\mu\in I$. Similarly,
\[
\int_{S_2\times\Delta_{02}}f\wedge\Omega_{q-1}(W)=\int_{S_2\times I}f\wedge\Omega_{q-1}(\hat w_2).
\]
Moreover, since $\lambda_0=1$ on $\Delta_0$, $w=w_0$ on this singleton. Thus
\begin{align*}
T_{q}^{W}(f)=-\int_{bD_1\times bD_2\times \Delta}f&\wedge\Omega_{q-1}(W)+\int_{bD_1\times D_2 \times I}f \wedge \Omega_{q-1}(\hat w_1)\\&+\int_{D_1\times bD_2\times I}f\wedge\Omega_{q-1}(\hat w_2) -\int_D f \wedge K_{q-1}
\end{align*}
for $f\in C_{0,q}(\bar D)$ and $1\le q\le n$.
\end{remark}
\begin{remark}
Of particular importance here, when $D_1$ and $D_2$ are 1-dimensional the first integral in the displayed equation above vanishes. I.e.,
\begin{equation*}
\int_{bD_1\times bD_2\times \Delta}f\wedge\Omega_{0}(W)=0.
\end{equation*}
This follows since the degree of the form (with respect to the integration variable) in the integrand must equal the dimension of the set over which it is integrated, otherwise the integral is 0. This argument will be called
{\it dimension-degree counting} when used below. When $D=D_1\times D_2$ is 2-dimensional, the operator $T_1^W$ reduces to
\begin{equation}
\label{T1}
T_{1}^{W}(f)=\int_{bD_1\times D_2 \times I}f \wedge \Omega_{0}(\hat w_1)+\int_{D_1\times bD_2\times I}f\wedge\Omega_{0}(\hat w_2)-\int_D f \wedge K_{0}
\end{equation}
for $f\in C_{0,1}^1(\bar D)$, where
\[
\Omega_0(\hat w_j)=\frac{1}{(2\pi i)^2}w_j\wedge w_0\wedge d\mu\qquad\text{for}\,\,\,j=1,2
\]
and
\[
K_0=\frac{1}{(2\pi i)^2}\frac{(\bar\zeta_1-\bar z_1)\,d\zeta_1\wedge d\bar \zeta_2 \wedge d\zeta_2+(\bar \zeta_2-\bar z_2)\,d\zeta_2\wedge d\bar \zeta_1\wedge d\zeta_1}{|\zeta-z|^4}.
\]
The form on the operator $T^W_1$ given by \eqref{T1} is the starting point for the computations in the next section.
By dimension-degree counting, it is also easy to see
\[
T_2^{W}(f)=-\int_Df\wedge K_1\qquad\text{for}\,\,\,f\in C_{0,2}^1(\bar D),
\]
where
\[
K_1=\frac{1}{(2\pi i)^2}\frac{(\bar\zeta_1-\bar z_1)\,d\zeta_2 \wedge d\zeta_1\wedge d\bar z_2+(\bar \zeta_2-\bar z_2)\,d\zeta_1\wedge d\zeta_2\wedge d\bar z_1}{|\zeta-z|^4}.
\]
\end{remark}
\section{The $\bar \partial$-equation on product spaces}
For a two-dimensional product domain, the right hand side of \eqref{T1} can be written as explicit integral operators.
This is now derived for arbitrary bounded domains $D_1, D_2\subset{\mathbb C}^1$ with
$C^1$ boundary, using the Cauchy generating form $w$.
\subsection{The product space $D_1\times D_2\subset{\mathbb C}^2$}\label{SS:product}
Definition \ref{D:generateC1domain} shows the Cauchy kernel
\[
w=\frac{d\zeta}{\zeta-z}
\]
is a generating form for any domain $\Omega\subset{\mathbb C}^1$. This form is holomorphic in $z$, away from $z=\zeta$. Set
\[
w_j=\frac{d\zeta_j}{\zeta_j-z_j}\qquad\text{for}\,\,\,j=1,2,
\]
the Cauchy kernels on the two domains $D_j$, $j=1,2$. Note that Remark \ref{R:generating} shows that $w_1, w_2$ give generating forms on $S_1\times\left(D_1\times D_2\right)$ and $S_2\times\left(D_1\times D_2\right)$ respectively.
For the rest of this section $w_j $ will refer to the Cauchy forms above, and $\hat w_j$ is defined via \eqref{E:homotopy} relative to these particular $w_j$.
Direct computation from \eqref{E:CFkernel1} gives
\[
\Omega_0(\hat w_1)=\frac{1}{(2 \pi i)^2}\frac{(\bar \zeta_2 - \bar z_2)\,d\zeta_1\wedge d\zeta_2\wedge d\mu}{(\zeta_1-z_1)|\zeta-z|^2}
\]
and
\[
\Omega_0(\hat w_2)=\frac{1}{(2 \pi i)^2}\frac{(\bar \zeta_1 - \bar z_1)\,d\zeta_2\wedge d\zeta_1\wedge d\mu}{(\zeta_2-z_2)|\zeta-z|^2}.
\]
Let $f\in C_{1,0}^1(\bar D)$ and write $f=f_1d\bar \zeta_1+f_2d\bar\zeta_2$. The first term on the right hand side of \eqref{T1} becomes
\begin{equation*}
\begin{split}
\int_{bD_1\times D_2 \times I}f \wedge \Omega_{0}(\hat w_1)
&=\int_{bD_1\times D_2 \times I}f\wedge\frac{1}{(2 \pi i)^2}\frac{(\bar \zeta_2 - \bar z_2)\,d\zeta_1\wedge d\zeta_2\wedge d\mu}{(\zeta_1-z_1)|\zeta-z|^2}\\
&=\frac{1}{(2 \pi i)^2}\int_{bD_1\times D_2 \times I}\frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)\,d\bar \zeta_2 \wedge d\zeta_1\wedge d\zeta_2\wedge d\mu}{(\zeta_1-z_1)|\zeta-z|^2}\\
&=\frac{1}{(2 \pi i)^2}\int_{bD_1\times D_2}\frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)\,d\bar \zeta_2 \wedge d\zeta_1\wedge d\zeta_2}{(\zeta_1-z_1)|\zeta-z|^2}.
\end{split}
\end{equation*}
The second equality follows from dimension-degree counting.
Now focus on the integration in $\zeta_1$ and apply Stokes' theorem to $\zeta_1$ on $D_1\setminus D(z_1;\tau)$, where $D(z_1;\tau)$ is the disk centered at $z_1$ with radius $\tau$. This yields
\begin{equation*}
\begin{split}
\int_{bD_1\times D_2 \times I}f \wedge \Omega_{0}(\hat w_1)
&=\frac{1}{(2 \pi i)^2}\int_{D_2} \Big[\int_{D_1\setminus D(z_1;\tau)} \frac{\partial}{\partial \bar \zeta_1} \left( \frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)}{(z_1-\zeta_1)|\zeta-z|^2} \right) d\bar \zeta_1 \wedge d\zeta_1\\
&+ \int_{|\zeta_1-z_1|=\tau} \frac{f_2\cdot(\bar \zeta_2-\bar z_2)}{(z_1-\zeta_1)|\zeta-z|^2}d\zeta_1 \Big]d\bar \zeta_2 \wedge d \zeta_2\\
=\frac{1}{(2 \pi i)^2}&\int_{D_2} \Big[\int_{D_1\setminus D(z_1;\tau)} \left( \frac{\partial f_2}{\partial \bar \zeta_1} \cdot \frac{ \bar \zeta_2 - \bar z_2}{(z_1-\zeta_1)|\zeta-z|^2} + \frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)}{|\zeta-z|^4} \right) d\bar \zeta_1 \wedge d\zeta_1\\
&~+ \int_{|\zeta_1-z_1|=\tau} \frac{f_2\cdot(\bar \zeta_2-\bar z_2)}{(z_1-\zeta_1)|\zeta-z|^2}d\zeta_1 \Big]d\bar \zeta_2 \wedge d \zeta_2.
\end{split}
\end{equation*}
Letting $\tau\to 0^+$ gives
\begin{equation*}
\lim_{\tau\to0^+}\int_{D_1\setminus D(z_1;\tau)} \frac{\partial f_2}{\partial \bar \zeta_1} \cdot \frac{ \bar \zeta_2 - \bar z_2}{(z_1-\zeta_1)|\zeta-z|^2} d\bar \zeta_1 \wedge d\zeta_1=\int_{D_1} \frac{\partial f_2}{\partial \bar \zeta_1} \cdot \frac{ \bar \zeta_2 - \bar z_2}{(z_1-\zeta_1)|\zeta-z|^2} d\bar \zeta_1 \wedge d\zeta_1
\end{equation*}
\begin{equation*}
\lim_{\tau\to0^+}\int_{D_1\setminus D(z_1;\tau)}\frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)}{|\zeta-z|^4} d\bar \zeta_1 \wedge d\zeta_1=\int_{D_1}\frac{f_2 \cdot (\bar \zeta_2 - \bar z_2)}{|\zeta-z|^4} d\bar \zeta_1 \wedge d\zeta_1,
\end{equation*}
and
\begin{equation*}
\lim_{\tau\to0^+}\int_{|\zeta_1-z_1|=\tau} \frac{f_2\cdot(\bar \zeta_2-\bar z_2)}{(z_1-\zeta_1)|\zeta-z|^2}d\zeta_1=-2\pi i \frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}.
\end{equation*}
Substituting these terms in the previous equation, the first term on the right hand side of \eqref{T1} can be written
\begin{equation*}
\begin{split}
\int_{bD_1\times D_2 \times I}f \wedge \Omega_{0}(\hat w_1)
&=\frac{-1}{2 \pi i}\int_{D_2}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2 \\&+\frac{1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{f_2\cdot(\bar \zeta_2 - \bar z_2)}{|\zeta-z|^4} d\bar \zeta_1 \wedge d\zeta_1\wedge d \bar \zeta_2\wedge d \zeta_2 \\
&+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{\partial f_2}{\partial \bar \zeta_1}\cdot\frac{\bar \zeta_2-\bar z_2}{(\zeta_1-z_1)|\zeta-z|^2}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{split}
\end{equation*}
Similarly, the second term on the right hand side in \eqref{T1} can be written
\begin{equation*}
\begin{split}
\int_{D_1\times bD_2\times I}f\wedge\Omega_{0}(\hat w_2)
&=\frac{-1}{2 \pi i}\int_{D_1}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\\&+\frac{1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{f_1\cdot(\bar \zeta_1 - \bar z_1)}{|\zeta-z|^4} d\bar \zeta_1 \wedge d\zeta_1\wedge d \bar \zeta_2\wedge d \zeta_2\\
&+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{\partial f_1}{\partial \bar \zeta_2}\cdot\frac{\bar \zeta_1-\bar z_1}{(\zeta_2-z_2)|\zeta-z|^2}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{split}
\end{equation*}
Note that the last term on the right hand side of \eqref{T1} is
\begin{equation*}
\int_{D_1\times D_2} f \wedge K_{0}=\frac{1}{(2 \pi i)^2}\int_{D_1\times D_2}\frac{f_1\cdot(\bar \zeta_1 - \bar z_1)+f_2\cdot(\bar \zeta_2-\bar z_2)}{|\zeta-z|^4} d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{equation*}
Hence, formula \eqref{T1} can be expressed
\begin{equation}
\label{4termT1}
\begin{split}
T^W_1(f)
&=\int_{bD_1\times D_2 \times I}f \wedge \Omega_{0}(\hat w_1)+\int_{D_1\times bD_2\times I}f\wedge\Omega_{0}(\hat w_2)-\int_{D_1\times D_2} f \wedge K_{0}\\
&=\frac{-1}{2 \pi i}\int_{D_2}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2+\frac{-1}{2 \pi i}\int_{D_1}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\\
&~+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{\partial f_2}{\partial \bar \zeta_1}\cdot\frac{\bar \zeta_2-\bar z_2}{(\zeta_1-z_1)|\zeta-z|^2}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2\\
&~+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{\partial f_1}{\partial \bar \zeta_2}\cdot\frac{\bar \zeta_1-\bar z_1}{(\zeta_2-z_2)|\zeta-z|^2}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{split}
\end{equation}
\begin{definition}\label{D:D} If $f=f_1d\bar z_1+f_2d\bar z_2$ is a $(0,1)$-form, let $${\mathscr D} f= \frac 12\left( \frac{\partial f_1}{\partial\bar z_2} + \frac{\partial f_2}{\partial\bar z_1}\right),$$
with derivatives taken in the distributional sense.
If $\bar\partial f=0$, note ${\mathscr D} f =\frac{\partial f_1}{\partial \bar z_2}=\frac{\partial f_2}{\partial \bar z_1}$.
\end{definition}
Using Definition \ref{D:D}, combine the last two terms in \eqref{4termT1}. The following expression for a strong solution operator on $D_1\times D_2$ is obtained:
\begin{proposition}\label{P:strong} Suppose $D_1, D_2\subset{\mathbb C}$ are domains with $C^1$ boundary. If $f\in C^1_{0,1}\left(\overline{D_1\times D_2}\right)$ satisfies $\bar \partial f=0$, define
\begin{equation}
\label{derivativeT1}
\begin{split}
T(f)
&=\frac{-1}{2 \pi i}\int_{D_2}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2+\frac{-1}{2 \pi i}\int_{D_1}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\\
&~+\frac{-1}{(2 \pi i)^2}\int_{D_1 \times D_2}\frac{{\mathscr D}(f)(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{split}
\end{equation}
Then $\bar \partial (Tf) =f$.
\end{proposition}
\begin{remark}
\label{T1forDxA}
Consider the third term on the right hand side of \eqref{4termT1}. If the idea from \cite{FornaessLeeZhang, Henkin71} is followed and Stokes' theorem is applied in the $\zeta_2$ variable, this term becomes
\begin{align*}
\int_{D_1\times D_2}\frac{\partial f_1}{\partial \bar \zeta_2} \cdot \frac{ \bar \zeta_2 - \bar z_2}{(\zeta_1-z_1)|\zeta-z|^2} d\bar \zeta_1 \wedge d\zeta_1\wedge d \bar \zeta_2 \wedge d\zeta_2
&=\int_{D_1} \Big[ \int_{bD_2} \frac{f_1\cdot(\bar \zeta_2-\bar z_2)}{(\zeta_1-z_1)|\zeta-z|^2} d\zeta_2\\
&~-\int_{D_2} \frac{f_1\cdot(\bar \zeta_1-\bar z_1)}{|\zeta-z|^4}d\bar \zeta_2\wedge d\zeta_2 \Big] d\bar \zeta_1\wedge d\zeta_1.
\end{align*}
Similarly, the last term in \eqref{4termT1} can be rewritten as
\begin{align*}
\int_{D_1\times D_2}\frac{\partial f_2}{\partial \bar \zeta_1} \cdot \frac{ \bar \zeta_1 - \bar z_1}{(\zeta_2-z_2)|\zeta-z|^2} d\bar \zeta_1 \wedge d\zeta_1\wedge d \bar \zeta_2 \wedge d\zeta_2
&=\int_{D_2} \Big[ \int_{bD_1} \frac{f_2\cdot(\bar \zeta_1-\bar z_1)}{(\zeta_2-z_2)|\zeta-z|^2} d\zeta_1\\
&-\int_{D_1} \frac{f_2\cdot(\bar \zeta_2-\bar z_2)}{|\zeta-z|^4}d\bar \zeta_1\wedge d\zeta_1 \Big] d\bar \zeta_2\wedge d\zeta_2.
\end{align*}
Thus an alternative expression for the operator $T=T_1^W$ is
\begin{equation*}
\begin{split}
T(f)
&=\frac{-1}{2 \pi i}\int_{D_2}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2-\frac{1}{(2 \pi i)^2}\int_{D_1\times bD_2}\frac{f_1\cdot(\bar \zeta_2-\bar z_2)}{(\zeta_1-z_1)|\zeta-z|^2} d\bar \zeta_1\wedge d\zeta_1\wedge d\zeta_2\\
&~+\frac{-1}{2 \pi i}\int_{D_1}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1-\frac{1}{(2 \pi i)^2}\int_{bD_1\times D_2}\frac{f_2\cdot(\bar \zeta_1-\bar z_1)}{(\zeta_2-z_2)|\zeta-z|^2} d\zeta_1\wedge d \bar \zeta_2 \wedge d\zeta_2\\
&+\frac{1}{(2 \pi i)^2}\int_{D_1\times D_2}\frac{f_1\cdot(\bar \zeta_1 - \bar z_1)+f_2\cdot(\bar \zeta_2-\bar z_2)}{|\zeta-z|^4} d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2.
\end{split}
\end{equation*}
This formula corrects a small error in \cite{Henkin71, FornaessLeeZhang}, where a different constant appears before the last term.
\end{remark}
\section{The $L^p$ estimate of the solution operator}
\label{LpofT}
For the rest of the paper, $T$ denotes the operator defined by \eqref{derivativeT1}.
\subsection{The $L^p$ estimate of $T$}
As a integral operator, $T$ is first shown to be well-defined and bounded between particular Banach spaces. The following lemma is used.
\begin{lemma}
\label{lem135}
Let $D_1, D_2\subset{\mathbb C}$ be bounded domains and $1\le p<\infty$. If $g\in L^p(D_1\times D_2)$, the functions
\[
\frac{-1}{2 \pi i}\int_{D_1}\frac{g(\zeta_1, z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\quad\text{and}\quad\frac{-1}{2 \pi i}\int_{D_2}\frac{g(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2
\]
belong to $L^p(D_1\times D_2)$. Their $L^p$-norms are bounded by $C\|g\|_{L^p(D_1\times D_2)}$, for a constant $C>0$ independent of $g$.
\end{lemma}
\begin{proof} The symmetry of the functions show it suffices to prove the result for either one; consider the second function.
Let $g^{z_1}(\zeta_2)=g(z_1,\zeta_2)\chi_{D_2}(\zeta_2)$, where $\chi_{D_2}$ is the characteristic function over $D_2$. Since $g\in L^p(D_1\times D_2)$, $g^{z_1}\in L^p({\mathbb C})$.
Let $B=B(0;R)$ be the disk centered at $0$ of radius $R$ in ${\mathbb C}$ and $h(\zeta)=\frac{1}{|\zeta|}\chi_{B}(\zeta)$, where $\chi_{B}$ is the characteristic function over $B$ and $R$ is sufficiently large (say $R>\text{diam}(D_2)$). Then $h\in L^1({\mathbb C})$.
By Young's inequality, $g^{z_1}*h\in L^p({\mathbb C})$. Note that, for any $z_2,\zeta_2\in D_2$, $|z_2-\zeta_2|\le\text{diam}(D_2)<R$, so $z_2-\zeta_2\in B$. Therefore,
\begin{align*}
\int_{D_2}\left|\int_{D_2}\frac{g(z_1,\zeta_2)\,dA(\zeta_2)}{\zeta_2-z_2}\right|^p\,dA(z_2)
&\le\int_{{\mathbb C}}\left(\int_{{\mathbb C}}\frac{|g^{z_1}(\zeta_2)|\chi_B(\zeta_2-z_2)\,dA(\zeta_2)}{|\zeta_2-z_2|}\right)^p\,dA(z_2)\\
&\le C\int_{D_2}|g(z_1,z_2)|^p\,dA(z_2).
\end{align*}
The conclusion follows by integrating this inequality in $z_1$ over $D_1$.
\end{proof}
Lemma \ref{lem135} applies to the first two terms on the right hand side of \eqref{derivativeT1}. Thus, the $L^p$-norms of these terms are bounded by $C\|f_2\|_{L^p(D_1\times D_2)}$ and $C\|f_1\|_{L^p(D_1\times D_2)}$ respectively, provided $f_1,f_2\in L^p(D_1\times D_2)$. Additionally, if ${\mathscr D}(f)\in L^p(D_1\times D_2)$, two applications of Lemma \ref{lem135} show the last term in \eqref{derivativeT1} is in $L^p(D_1\times D_2)$, with $L^p$-norm bounded by $C\|{\mathscr D}(f)\|_{L^p(D_1\times D_2)}$.
\begin{definition}\label{D:banach} For $1\le p<\infty$, define the Banach space of $(0,1)$-forms on $D_1\times D_2$
\[
{\mathcal B}=\left\{f=f_1d\bar z_1+f_2d\bar z_2~|~f_1,f_2\in L^p\,\text{and}\, {\mathscr D}(f)\in L^p\right\}
\]
with norm
$\|f\|_{{\mathcal B}}:=\|f_1\|_{L^p(D_1\times D_2)}+\|f_2\|_{L^p(D_1\times D_2)}+\|{\mathscr D}(f)\|_{L^p(D_1\times D_2)}$.
\end{definition}
The argument above Definition \ref{D:banach} proves
\begin{lemma}
\label{lem24}
The operator $T$ given by \eqref{derivativeT1} maps ${\mathcal B}$ to $L^p(D_1\times D_2)$ boundedly.
\end{lemma}
\subsection{Passage to a weak solution}\label{SS:punctured}
Lemma \ref{lem24} shows that $T:{\mathcal B}\to L^p(D_1\times D_2)$ is well-defined. By \eqref{derivativeT1}, $T$ is a strong solution operator to $\bar \partial(Tf)=f$ if $T$ is restricted to $f\in C^1_{0,1}\left(\overline{D_1\times D_2}\right)$ and $\bar \partial f=0$. In this section $Tf$ is shown to be a weak solution to $\bar \partial(Tf)=f$, if $\bar \partial f=0$ weakly and $f\in{\mathcal B}$ , by a limit argument.
\begin{theorem}
\label{Lpwithdbarf}
For $j=1,2$, let $D_j\in{\mathbb C}$ be bounded domains with $C^1$ boundary. Let $f$ be a $(0,1)$-form that is $\bar \partial$-closed in the weak sense on $D_1\times D_2$. For $1\le p<\infty$, assume that $f\in{\mathcal B}$.
Then $u=Tf$, defined by \eqref{derivativeT1}, is a weak solution to the equation $\bar \partial u=f$ on $D_1\times D_2$ and satisfies the estimate
\[
\|T(f)\|_{L^p(D_1\times D_2)}\le C \|f\|_{\mathcal B}
\]
for a constant $C>0$ independent of $f$.
\end{theorem}
\begin{proof}
For $j=1,2$, let $\rho_j$ be a defining function for the domain $D_j$. Let $D_j^{\delta}=\{z\in{\mathbb C}\,|\,\rho_j(z)<-\delta\}$ for $\delta>0$ sufficiently small. Denote by $T^{\delta}$ the operator in \eqref{derivativeT1} with $D_j$ replaced by $D_j^{\delta}$. Then
\[
\bar \partial T^{\delta}(f)=f\qquad\text{on\,\,}D_1^{\delta}\times D_2^{\delta}\qquad\text{for\,\,}\bar \partial\text{-closed\,\,}(0,1)\text{-form\,\,}f\in C^{1}(\overline{D_1^{\delta}\times D_2^{\delta}}).
\]
Let $1\le p<\infty$. For $f\in{\mathcal B}$, the standard mollifier argument (see for example \cite[Chap. 5.3, Theorem 2]{Evans98}) gives a sequence $\{f^{\varepsilon_j}\}\subset C^{1}(\overline{D_1^{\delta}\times D_2^{\delta}})$, so that
\[
\bar \partial f^{\varepsilon_j}=0,
\]
\[
f^{\varepsilon_j}\to f\,\,\,\,\text{and}\,\,\,\,{\mathscr D}(f^{\varepsilon_j})\to {\mathscr D}(f)\,\,\,\,\text{in}\,\,L^p(D_1^{\delta}\times D_2^{\delta})\,\,\,\,\text{as}\,\,\varepsilon_j\to0.
\]
Thus
\[
\bar \partial T^{\delta}(f^{\varepsilon_j})=f^{\varepsilon_j}\qquad\text{on\,\,}D_1^{\delta}\times D_2^{\delta}
\]
in the strong sense.
On the other hand, replacing $D_1\times D_2$ by $D_1^{\delta}\times D_2^{\delta}$ in Lemma \ref{lem24} and denoting the Banach space on $D_1^{\delta}\times D_2^{\delta}$ by ${\mathcal B}^{\delta}$, it follows that $T^{\delta}$ is bounded from ${\mathcal B}^{\delta}$ to $L^p(D_1^{\delta}\times D_2^{\delta})$. So for $f\in{\mathcal B}$, $\lim T^{\delta}(f^{\varepsilon_j})=T^{\delta}(f)$ in $L^p(D_1^{\delta}\times D_2^{\delta})$ as $\varepsilon_j\to0$. Hence $T^{\delta}$ weakly solves the $\bar \partial$-equation on $D_1^{\delta}\times D_2^{\delta}$.
Next, for each $f\in{\mathcal B}$, extend $T^{\delta}(f)$ to a function on $D_1\times D_2$ by setting it equal $0$ outside $D_1^{\delta}\times D_2^{\delta}$. Consider $\|T(f)-T^{\delta}(f)\|_{L^p(D_1\times D_2)}$.
Note that $f_2$ can be replaced by $f_2\cdot\chi_{D_2\setminus D_2^{\delta}}(\zeta_2)$ in Lemma \ref{lem135}, where $\chi_{D_2\setminus D_2^{\delta}}$ is the characteristic function over $D_2\setminus D_2^{\delta}$. Thus, the $L^p$-norm in the conclusion in Lemma \ref{lem135} is be bounded by $C\|f_2\|_{L^p(D_1\times(D_2\setminus D_2^{\delta}))}$, which tends to $0$ as $\delta\to0^+$. A similar argument holds for Lemma \ref{lem24}. Therefore, $\lim T^{\delta}(f)=T(f)$ in $L^p(D_1\times D_2)$ as $\delta\to0^+$. This argument shows that the limit $T(f)$ is unique, and is independent of the defining functions for $D_1$ and $D_2$ used.
To show $T$ weakly solves the $\bar \partial$-equation on $D_1\times D_2$, argue as follows. For any $\phi\in C_c^{\infty}(D_1\times D_2)$, there is a $\delta_0>0$ so that $\text{supp}\phi\subset D_1^{\delta_0}\times D_2^{\delta_0}$. Let $K=D_1^{\delta_0}\times D_2^{\delta_0}$. Then
\[
(\bar \partial T(f),\phi)_{D_1\times D_2}=(\bar \partial T(f),\phi)_K=(T(f),\bar\partial^*\phi)_K=\lim_{\delta\to0^+}(T^{\delta}(f),\bar\partial^*\phi)_K
\]
by $L^p(D_1\times D_2)$-norm convergence. Note that $T^{\delta}$ weakly solves the $\bar \partial$-equation on $D_1^{\delta}\times D_2^{\delta}$. Thus for $0<\delta<\delta_0$,
\[
\lim_{\delta\to0^+}(T^{\delta}(f),\bar\partial^*\phi)_K=\lim_{\delta\to0^+}(\bar \partial T^{\delta}(f),\phi)_K=(f,\phi)_K=(f,\phi)_{D_1\times D_2}.
\]
\end{proof}
Now let $D_1\times D_2={\mathbb D}\times A$, where $A=A(0;1,\delta)=\{z\in{\mathbb C}\,|\,\delta<|z|<1\}$. Theorem \ref{Lpwithdbarf} directly applies to ${\mathbb D}\times A$. However the proof of Theorem \ref{Lpwithdbarf} also applies, allowing the limit $\delta\to0^+$ to be taken. This yields the following result.
\begin{corollary}\label{C:mainProduct}
Let $f=f_1d\bar z_1+f_2d\bar z_2$ be a $(0,1)$-form that is $\bar\partial$-closed in the weak sense on ${\mathbb D}\times{\mathbb D}^*$. For $1\le p<\infty$, assume that $f\in {\mathcal B}$.
Then $Tf$ defined
\begin{equation}
\label{T1forDxD*}
\begin{split}
T(f)
&=\frac{-1}{2 \pi i}\int_{{\mathbb D}^*}\frac{f_2(z_1,\zeta_2)}{\zeta_2-z_2}d\bar \zeta_2\wedge d\zeta_2+\frac{-1}{2 \pi i}\int_{{\mathbb D}}\frac{f_1(\zeta_1,z_2)}{\zeta_1-z_1}d\bar \zeta_1\wedge d\zeta_1\\
&~+\frac{-1}{(2 \pi i)^2}\int_{{\mathbb D}\times{\mathbb D}^*}\frac{{\mathscr D}(f)(\zeta_1,\zeta_2)}{(\zeta_1-z_1)(\zeta_2-z_2)}\,d\bar \zeta_1\wedge d\zeta_1\wedge d\bar \zeta_2\wedge d\zeta_2
\end{split}
\end{equation}
is well-defined, weakly solves $\bar\partial(Tf)=f$ on ${\mathbb D}\times{\mathbb D}^*$, and satisfies the estimate
\[
\|T(f)\|_{L^p({\mathbb D}\times {\mathbb D}^*)}\le C \|f\|_{\mathcal B}
\]
for a constant $C>0$ independent of $f$.
\end{corollary}
\section{$f\in L^p$ is not sufficient for existence of $Tf$}\label{S:necessity}
The assumption $\left\|{\mathscr D}(f)\right\|_p<\infty$ is part of the hypotheses in Theorem \ref{Lpwithdbarf} and Corollary \ref{C:mainProduct} via the condition $f\in{\mathcal B}$. Under this hypothesis and $f\in L^p(D_1\times D_2)$,
these results imply $Tf$ exists and belongs to $L^p$.
In this section we show that only assuming $f\in L^p(D_1\times D_2)$ is not enough to conclude that $Tf\in L^p(D_1\times D_2)$ for $1\leq p <2$ in general. Thus on product domains, estimates on the data beyond $f\in L^p$
are generally needed for the Henkin solution to belong to $L^p$, unlike the situation for the Henkin solution on strongly pseudoconvex domains, \cite{Kerzman71, Ovrelid}.
Consider the $\bar \partial$-equation on the bidisc ${\mathbb D}^2$. For each $k=1,2,\dots$, let $f^k=f^k_1d\bar z_1+f^k_2d\bar z_2$ on ${\mathbb D}^2$, where
\[
\begin{array}{ccc}
\displaystyle f^k_1(z_1,z_2)=\bar z_1^{k-1}z^k_1 \bar z_2^kz_2^k & \text{and} & \displaystyle f^k_2(z_1,z_2)=\bar z_1^kz_1^k \bar z_2^{k-1}z_2^k.
\end{array}
\]
Note each $f
|
^k$ is $\bar\partial$-closed. Moreover, direct computation shows
\begin{align}\label{E:fL1}
\|f_1^k\|_{L^1({\mathbb D}^2)}=\int_{{\mathbb D}^2}|\bar z_1^{k-1}z^k_1 \bar z_2^kz_2^k|\,dV(z)&=O\left(\frac{1}{k^2}\right) \notag\\
\|f_2^k\|_{L^1({\mathbb D}^2)}=\int_{{\mathbb D}^2}|\bar z_1^kz_1^k \bar z_2^{k-1}z_2^k|\,dV(z)&=O\left(\frac{1}{k^2}\right).
\end{align}
An elementary calculation will be used to compute $T(f^k)$.
\begin{lemma}
\label{Cauchytrans}
For $z\in{\mathbb D}$ and $k\in{\mathbb Z}^+$,
\[
\int_{{\mathbb D}}\frac{\bar\zeta^{k-1}\zeta^k\,d\bar\zeta\wedge d\zeta}{\zeta-z}=\frac{2\pi i}{k}(1-\bar z^k z^k).
\]
\end{lemma}
\begin{proof} This follows from the generalized Cauchy Integral formula. Details are provided for completeness.
Let $\omega=\frac{1}{k}\cdot\frac{\bar\zeta^k\zeta^k}{\zeta-z}\,d\zeta\text{ on }{\mathbb D}\setminus B$,
where $B=B(z;\varepsilon)$ is a disk centered at $z$ of radius $\varepsilon$ sufficiently small so that $B\subset{\mathbb D}$. By Stokes' theorem,
$\int_{{\mathbb D}\setminus B}d\omega=\int_{b{\mathbb D}}\omega-\int_{bB}\omega$. Since
$d\omega=(\partial+\bar \partial)\omega=\frac{\bar\zeta^{k-1}\zeta^k}{\zeta-z}\,d\bar\zeta\wedge d\zeta$,
it follows
\begin{align*}
\int_{{\mathbb D}}\frac{\bar\zeta^{k-1}\zeta^k\,d\bar\zeta\wedge d\zeta}{\zeta-z}
=\lim_{\varepsilon\to0^+}\int_{{\mathbb D}\setminus B}d\omega
=\lim_{\varepsilon\to0^+}\left(\int_{b{\mathbb D}}\frac{1}{k}\cdot\frac{\bar\zeta^k\zeta^k}{\zeta-z}\,d\zeta-\int_{bB}\frac{1}{k}\cdot\frac{\bar\zeta^k\zeta^k}{\zeta-z}\,d\zeta\right).
\end{align*}
By the Cauchy integral formula, the first term on the right hand side is
\[
\int_{b{\mathbb D}}\frac{1}{k}\cdot\frac{\bar\zeta^k\zeta^k}{\zeta-z}\,d\zeta=\frac{1}{k}\int_{b{\mathbb D}}\frac{d\zeta}{\zeta-z}=\frac{2\pi i}{k}.
\]
Writing $\zeta=z+\varepsilon e^{i\theta}$ on $bB$, the second term becomes
\[
\lim_{\varepsilon\to0^+}\int_{bB}\frac{1}{k}\cdot\frac{\bar\zeta^k\zeta^k}{\zeta-z}\,d\zeta=\lim_{\varepsilon\to0^+}\int_0^{2\pi}\frac{1}{k}\cdot\frac{|z+\varepsilon e^{i\theta}|^{2k}\varepsilon e^{i\theta}i\,d\theta}{\varepsilon e^{i\theta}}=\frac{2\pi i}{k}|z|^{2k}.
\]
The conclusion follows by combining these terms.
\end{proof}
Compute $T(f^k)$ using the explicit expression $\eqref{derivativeT1}$. For the first term,
\begin{align*}
\frac{-1}{2\pi i}\int_{{\mathbb D}}\frac{f^k_2(z_1,\zeta_2)\,d\bar\zeta_2\wedge d\zeta_2}{\zeta_2-z_2}
&=\frac{-1}{2\pi i}\int_{{\mathbb D}}\frac{\bar z_1^k z_1^k \bar \zeta_2^{k-1} \zeta_2^k\, d\bar \zeta_2\wedge d\zeta_2}{\zeta_2-z_2}
=\frac{-1}{2\pi i}\cdot\bar z_1^k z_1^k\cdot\int_{{\mathbb D}}\frac{\bar\zeta_2^{k-1}\zeta_2^k\,d\bar\zeta_2\wedge d\zeta_2}{\zeta_2-z_2}\\
&=\frac{-1}{2\pi i}\cdot\bar z_1^k z_1^k\cdot\frac{2\pi i}{k}(1-\bar z_2^k z_2^k) =\frac{1}{k}|z_1z_2|^{2k}-\frac{1}{k}|z_1|^{2k}.
\end{align*}
The third equality follows from Lemma \ref{Cauchytrans}. Similarly, the second term on the right hand side in \eqref{derivativeT1} is
\begin{equation*}
\frac{-1}{2\pi i}\int_{{\mathbb D}}\frac{f^k_1(\zeta_1,z_2)\,d\bar\zeta_1\wedge d\zeta_1}{\zeta_1-z_1}=\frac{1}{k}|z_1z_2|^{2k}-\frac{1}{k}|z_2|^{2k}.
\end{equation*}
For the last term in \eqref{derivativeT1}, separate the variables in the integral and apply Lemma \ref{Cauchytrans} twice to get
\begin{align*}
\frac{-1}{(2\pi i)^2}\int_{{\mathbb D}^2}\frac{{\mathscr D}(f^k)(\zeta_1,\zeta_2)\,d\bar\zeta_1\wedge d\zeta_1\wedge d\bar\zeta_2\wedge d\zeta_2}{(\zeta_1-z_1)(\zeta_2-z_2)}
&=\frac{-1}{(2\pi i)^2}\int_{{\mathbb D}^2}\frac{k\bar \zeta_1^{k-1}\zeta_1^k \bar \zeta_2^{k-1}\zeta_2^k\,d\bar\zeta_1\wedge d\zeta_1\wedge d\bar\zeta_2\wedge d\zeta_2}{(\zeta_1-z_1)(\zeta_2-z_2)}\\
&=\frac{-k}{(2\pi i)^2}\int_{{\mathbb D}}\frac{\bar\zeta_1^{k-1}\zeta_1^k\,d\bar\zeta_1\wedge d\zeta_1}{\zeta_1-z_1}\int_{{\mathbb D}}\frac{\bar\zeta_2^{k-1}\zeta_2^k\,d\bar\zeta_2\wedge d\zeta_2}{\zeta_2-z_2}\\
&=\frac{-k}{(2\pi i)^2}\cdot\frac{2\pi i}{k}(1-\bar z_1^k z_1^k)\cdot\frac{2\pi i}{k}(1-\bar z_2^k z_2^k)\\
&=-\frac{1}{k}+\frac{1}{k}|z_1|^{2k}+\frac{1}{k}|z_2|^{2k}-\frac{1}{k}|z_1z_2|^{2k}.
\end{align*}
Therefore,
\begin{equation}\label{E:TfnotL1}
T(f^k)=\frac{1}{k}|z_1z_2|^{2k}-\frac{1}{k}.
\end{equation}
Now define, for each $L\in{\mathbb Z}^+$, $g^L=g^L_1d\bar z_1+g^L_2d\bar z_2$ with $g^L_1=\sum_{k=1}^L f^k_1$ and $g^L_2=\sum_{k=1}^L f^k_2$. Clearly each $\bar \partial g^L=0$. Since $\sum \frac 1{k^2}<\infty$, \eqref{E:fL1} implies $g^L\in L^1_{0,1}\left({\mathbb D}^2\right)$ with $L^1$ norm bounded independent of $L$.
However \eqref{E:TfnotL1} implies
\begin{align*}
T\left(g^L\right)(z_1,z_2)= \sum_{k=1}^L T\left(f^k\right)
=\frac{|z_1z_2|\left(1-|z_1z_2|^{2L}\right)}{1-|z_1z_2|^2}-\sum_{k=1}^L\frac{1}{k}.
\end{align*}
If $K\subset {\mathbb D}^2$ is a compact set and $(z_1,z_2)\in K$, the last expression tends to $-\infty$ as $L\to\infty$, by divergence of the harmonic series.
Thus there does not exist a constant $C$ such that $\left\|T\left(g^L\right)\right\|_{L^1}\leq C\left\|g^L\right\|_{L^1}$ for all $L$.
\begin{remark}
Taking the full sums, $g=g_1d\bar z_1+g_2d\bar z_2$ with $g_1=\sum_{k=1}^\infty f^k_1$ and $g_2=\sum_{k=1}^\infty f^k_2$, gives an example where $Tg$ does not even exist. In this case, \eqref{E:fL1} still shows $g\in L^1_{0,1}\left({\mathbb D}^2\right)$, while the analogue of the above computation yields
$$T(g)(z_1,z_2)=-\ln(1-|z_1z_2|^2)-\sum_{k=1}^{\infty}\frac{1}{k}\equiv\infty.$$
\end{remark}
\begin{remark}
A careful inspection of the integrals shows that $g\in L^p\left({\mathbb D}^2\right)$ for $1\leq p <2$; details are left to the interested reader. Thus for the $L^1$ problem, ``over-prescribing'' integrability by requiring $g\in L^p$ for $p<2$ is still not sufficient to guarantee $Tg\in L^1$.
\end{remark}
\section{Non-canonical Solution}
The solution $u=T(f)$ in \eqref{derivativeT1} is compared with the $L^2$-minimal solution $u_{\text{can}}$ on $D_1\times D_2$. The first observation is that $u=T(f)\neq u_{\text{can}}$ on ${\mathbb D}^2$.
Let $h\in C^1(\overline{{\mathbb D}})$ be a holomorphic function on ${\mathbb D}$ and let
\begin{equation}
\label{noncansol}
f=z_1^kh(z_2)d\bar z_1
\end{equation}
for some positive integer $k$. It is easily checked that $\bar \partial f=0$. Since $f_2=0$, \eqref{derivativeT1} becomes
\begin{equation*}
u=T(f)=\frac{-1}{2\pi i}\int_{{\mathbb D}}\frac{f_1(\zeta_1,z_2)\,d\bar\zeta_1\wedge d\zeta_1}{\zeta_1-z_1}=\frac{-1}{2\pi i}\cdot h(z_2)\int_{{\mathbb D}}\frac{\zeta_1^k\,d\bar\zeta_1\wedge d\zeta_1}{\zeta_1-z_1}.
\end{equation*}
Let $\omega=\zeta_1^k\bar\zeta_1/(\zeta_1-z_1)$ and apply Stokes theorem to the integral
$\int_{{\mathbb D}\setminus B}d\omega$,
where $B=B(z_1,\varepsilon)$ is the disk centered at $z_1$ of radius $\varepsilon$ for some $\varepsilon>0$ sufficiently small. It follows that
\begin{equation}
\label{T1example}
\frac{1}{2\pi i}\int_{{\mathbb D}}\frac{\zeta_1^k\,d\bar\zeta_1\wedge d\zeta_1}{\zeta_1-z_1}=\lim_{\varepsilon\to0^+}\left(\frac{1}{2\pi i}\int_{b{\mathbb D}}\frac{\zeta_1^k\bar\zeta_1\,d\zeta_1}{\zeta_1-z_1}-\frac{1}{2\pi i}\int_{bB}\frac{\zeta_1^k\bar\zeta_1\,d\zeta_1}{\zeta_1-z_1}\right).
\end{equation}
By the Cauchy integral formula, the first term in \eqref{T1example} is $z_1^{k-1}$. For the second term of \eqref{T1example}, write $\zeta_1=z_1+\varepsilon e^{i\theta}$ and note
\[
\lim_{\varepsilon\to0^+}\frac{1}{2\pi i}\int_{bB}\frac{\zeta_1^k\bar\zeta_1\,d\zeta_1}{\zeta_1-z_1}=z_1^k\bar z_1.
\]
Thus, $u=h(z_2)\left(z_1^k\bar z_1-z_1^{k-1}\right)$.
But the $L^2$-minimal solution of $\bar\partial u=f$ is
\begin{equation*}
u_{\text{can}}=h(z_2)\left(z_1^k\bar z_1-\frac{k}{k+1}z_1^{k-1}\right),
\end{equation*}
since it is easy to verify
\[
\langle u_{\text{can}}\,,\,z_1^mz_2^n\rangle=\int_{{\mathbb D}\times{\mathbb D}}z_1^k\bar z_1h(z_2)\bar z_1^m\bar z_2^n\,dV(z)-\frac{k}{k+1}\int_{{\mathbb D}\times{\mathbb D}}z_1^{k-1}h(z_2)\bar z_1^m\bar z_2^n\,dV(z)=0
\]
for all integers $m,n\ge0$.
\begin{remark}
Let $D_1$ and $D_2$ be bounded simply connected planar domains. Then $D_1\times D_2$ is biholomorphic to ${\mathbb D}^2$ under a mapping $\psi_1\otimes\psi_2$, where $\psi_j$ is the biholomorphism from $D_j$ to ${\mathbb D}$ for $j=1,2$. Now consider the form with the same expression as in \eqref{noncansol} and let $h\equiv 1$, that is
$f=z_1^kd\bar z_1$
on $D_1\times D_2$, for some positive integer $k$. The solution operator $T$ gives the same solution
$u=z_1^k\bar z_1-z_1^{k-1}$
as above.
However, to obtain the $L^2$-minimal solution, one must transfer the orthonormal basis on $A^2({\mathbb D}^2)$ to one on $A^2(D_1\times D_2)$. Thus the expression of $u_{\text{can}}$ necessarily involves $\psi'_1$ and $\psi'_2$, unlike the expression of $u=T(f)$.
\end{remark}
\section{The Hartogs Triangle}\label{S:Hartogs}
In this section, consider the $\bar \partial$-equation on the Hartogs triangle ${\mathbb H}$:
\begin{equation*}
{\mathbb H}=\left\{(z_1, z_2)\in{\mathbb C}^2: \left|z_1\right| <\left|z_2\right| <1\right\}.
\end{equation*}
The first step is to transfer the equation on ${\mathbb H}$ to the product space ${\mathbb D}\times{\mathbb D}^*$.
\subsection{Transform the $\bar \partial$-equation}
Let
\[
\phi: {\mathbb H}\to{\mathbb D}\times{\mathbb D}^*
\]
\[
\phi(z_1,z_2)=(z_1/z_2,z_2)=(w_1,w_2)
\]
be the usual biholomorphism. Consider
\begin{equation}
\label{dbaronh}
\bar \partial_zv=\alpha=\alpha_1\,d\bar z_1+\alpha_2\,d\bar z_2
\end{equation}
on ${\mathbb H}$, where $\alpha$ is $\bar \partial$-closed. Using the chain rule
\[
\left\{
\begin{array}{r}
d\bar z_1=\frac{\partial\bar z_1}{\partial \bar w_1}\,d\bar w_1+\frac{\partial\bar z_1}{\partial \bar w_2}\,d\bar w_2 \\
d\bar z_2=\frac{\partial\bar z_2}{\partial \bar w_1}\,d\bar w_1+\frac{\partial\bar z_2}{\partial \bar w_2}\,d\bar w_2
\end{array}
\right.,
\]
it follows that equation \eqref{dbaronh} is equivalent to
\begin{equation}
\label{dbarondd}
\begin{split}
\bar \partial_w u
&=\Big(\tilde \alpha_1\frac{\partial \bar z_1}{\partial \bar w_1}+\tilde \alpha_2\frac{\partial \bar z_2}{\partial \bar w_1}\Big)\,d\bar w_1+\Big(\tilde \alpha_1\frac{\partial \bar z_1}{\partial \bar w_2}+\tilde \alpha_2\frac{\partial \bar z_2}{\partial \bar w_2}\Big)\,d\bar w_2\\
&=\bar w_2\cdot\tilde \alpha_1\,d\bar w_1+(\bar w_1\cdot\tilde \alpha_1+\tilde \alpha_2)\,d\bar w_2\\
&=f_1\,d\bar w_1+f_2\,d\bar w_2\\
&=f
\end{split}
\end{equation}
on ${\mathbb D}\times{\mathbb D}^*$, where $u=v\circ\phi^{-1}$ and $\tilde \alpha_j=\alpha_j\circ\phi^{-1}$ for $j=1,2$.
Note that
\[
\frac{\partial f_1}{\partial \bar w_2}=\frac{\partial }{\partial \bar w_2}(\bar w_2\cdot\tilde \alpha_1)=\tilde \alpha_1+\bar w_2\bar w_1\frac{\partial \alpha_1}{\partial \bar z_1}+\bar w_2\frac{\partial \alpha_1}{\partial \bar z_2}
\]
and
\[
\frac{\partial f_2}{\partial \bar w_1}=\frac{\partial }{\partial \bar w_1}(\bar w_1\cdot\tilde \alpha_1+\tilde \alpha_2)=\tilde \alpha_1+\bar w_1\bar w_2\frac{\partial \alpha_1}{\partial \bar z_1}+\bar w_2\frac{\partial \alpha_2}{\partial \bar z_1}.
\]
Since $\alpha$ is $\bar \partial$-closed on ${\mathbb H}$, it follows that $f$ is $\bar \partial$-closed on ${\mathbb D}\times{\mathbb D}^*$.
\subsection{An $L^p$ assumption on ${\mathbb H}$}
Based on the transformation \eqref{dbarondd}, a vanishing condition on $\alpha_j$ at the origin,
\[
\|\alpha_j\|^p_{L^p_{-2}({\mathbb H})}=\int_{{\mathbb H}}|\alpha_j|^p|z_2|^{-2}\,dV(z)<\infty
\]
for $j=1,2$, implies that $f_1,f_2\in L^p({\mathbb D}\times{\mathbb D}^*)$. This follows since
\[
\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^p\,dV(w) \le \int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\,dV(w)=\|\alpha_1\|^p_{L^p_{-2}({\mathbb H})}
\]
and
\[
\int_{{\mathbb D}\times{\mathbb D}^*}|\bar w_1\cdot\tilde \alpha_1+\tilde \alpha_2|^p\,dV(w) \le C_p\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p + |\tilde \alpha_2|^p\,dV(w)=C_p\left(\|\alpha_1\|^p_{L^p_{-2}({\mathbb H})}+\|\alpha_2\|^p_{L^p_{-2}({\mathbb H})}\right).
\]
In addition, a vanishing condition on derivatives of $\alpha_1$ at the origin,
\[
\left\|\frac{\partial \alpha_1}{\partial \bar z_j}\right\|^p_{L^p_{-1}({\mathbb H})}=\int_{{\mathbb H}}\left|\frac{\partial \alpha_1}{\partial \bar z_j}\right|^p\cdot |z_2|^{-1}\,dV(z)<\infty
\]
for $j=1,2$, implies that ${\mathscr D} f \in L^p({\mathbb D}\times{\mathbb D}^*)$. This follows since for $1\le p<\infty$
\begin{align*}
\int_{{\mathbb D}\times{\mathbb D}^*}\left| \tilde \alpha_1+\bar w_1\bar w_2\frac{\partial \alpha_1}{\partial \bar z_1}+\bar w_2\frac{\partial \alpha_1}{\partial \bar z_2} \right|^p
&\le C_p\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p+|w_2|\cdot\left|\frac{\partial \alpha_1}{\partial \bar z_1}\right|^p+|w_2|\cdot \left|\frac{\partial \alpha_1}{\partial \bar z_2}\right|^p\\
\le C_p&\left(\|\alpha_1\|^p_{L^p_{-2}({\mathbb H})}+\left\|\frac{\partial \alpha_1}{\partial \bar z_1}\right\|^p_{L^p_{-1}({\mathbb H})}+\left\|\frac{\partial \alpha_1}{\partial \bar z_2}\right\|^p_{L^p_{-1}({\mathbb H})}\right).
\end{align*}
Thus, the following $L^p$ estimate for a solution of $\bar \partial$ on ${\mathbb H}$ holds.
\begin{theorem}
\label{HarLp'}
Let $v$, $\alpha$ be as in \eqref{dbaronh} and $u$, $f$ be as in \eqref{dbarondd}. Suppose $\alpha$ is $\bar \partial$-closed in the weak sense on ${\mathbb H}$. For $1\le p<\infty$, assume that
\begin{enumerate}
\item $\alpha_1,\alpha_2\in L^p_{-2}({\mathbb H})$,
\item $\partial \alpha_2/\partial \bar z_1=\partial \alpha_1/\partial \bar z_2\in L^p_{-1}({\mathbb H})$ and $\partial \alpha_1/\partial \bar z_1\in L^p_{-1}({\mathbb H})$.
\end{enumerate}
Then there is a weak solution $v=u\circ\phi=T(f)\circ\phi$, where $T$ is the solution operator in \eqref{T1forDxD*}, satisfying the $L^p$ estimate
\[
\|v\|_{L^p({\mathbb H})}\le C\left(\sum_{j=1}^2\|\alpha_j\|_{L^p_{-2}({\mathbb H})}+\sum_{j=1}^2\|\partial \alpha_1/\partial \bar z_j\|_{L^p_{-1}({\mathbb H})}\right)
\]
for some constant $C>0$ independent of $\alpha$.
\end{theorem}
\subsection{An example}\label{SS:example} The $L^p$ estimates of the solution $v$ given in Theorem \ref{HarLp'}, and the cannonical solution $v_{\text{can}}$ on ${\mathbb H}$ can be compared via the simple example
\[
\alpha=d\bar z_2.
\]
Verifying that $\alpha$ satisfies the conditions in Theorem \ref{HarLp'} is easy. Thus the solution $v$ in Theorem \ref{HarLp'} belongs to $L^p({\mathbb H})$ for $1\le p<\infty$.
On the other hand, we claim the $L^2$-minimal solution of $\bar \partial v= \alpha$ is
\[
v_{\text{can}}=\bar z_2-cz_2^{-1}
\]
for some nonzero constant $c$. Clearly $\bar \partial v_{\text{can}}=\alpha$. To see that $v_{\text{can}}$ is orthogonal to holomorphic functions on ${\mathbb H}$, it suffices to take its inner product with the orthogonal basis $\{z_1^nz_2^m\}$ on ${\mathbb H}$, for so-called allowable indices $(n,m)\in {\mathbb Z}^+\times{\mathbb Z}$. See Sections of \cite{EdhMcN17} for the definition of allowable indices and Section 5 of that paper of that paper for a proof that $\left\langle v_{\text{can}}, z^\alpha\right\rangle =0$ for all allowable exponents $\alpha$.
Note that $\alpha\in L^p({\mathbb H})$. On the other hand, the proof of Proposition 5.5 in \cite{EdhMcN17} implies that $v_{\text{can}}\notin L^p({\mathbb H})$ for $p\ge4$. Thus, $v$ behaves better than $v_{\text{can}}$ in terms of $L^p$ regularity.
At the operator level, it follows that the canonical solution operator for $\bar \partial$ on ${\mathbb H}$ doesn't map $L^p$ $\bar \partial$-closed $(0,1)$-form to $L^p$ functions for $p\ge4$. This is consistent with results on the Bergman projection on ${\mathbb H}$, see \cite{EdhMcN16}, \cite{ChaZey16}, and \cite{Chen17}.
\subsection{Extra condition}\label{SS:extra}
An extra condition on $\alpha$, namely
\begin{equation}
\label{f2=0}
\bar z_1\cdot\alpha_1+\bar z_2\cdot\alpha_2=0,
\end{equation}
and \eqref{dbarondd} shows that $f=f_1d\bar w_1$ on ${\mathbb D}\times{\mathbb D}^*$. By \eqref{T1forDxD*}, $T(f)$ only involves the second term. Thus a better $L^p$ estimate holds in this case:
\begin{theorem}
Let $v$, $\alpha$ be as in \eqref{dbaronh} and $u$, $f$ be as in \eqref{dbarondd}. Suppose $\alpha$ is $\bar \partial$-closed in the weak sense on ${\mathbb H}$ and satisfies \eqref{f2=0}. For $1\le p<\infty$, assume that $\alpha_1\in L^p({\mathbb H})$. Then there is a weak solution $v=u\circ\phi=T(f)\circ\phi$, where $T$ is the solution operator in \eqref{T1forDxD*}, satisfying the $L^p$ estimate
\[
\|v\|_{L^p({\mathbb H})}\le C\|\alpha_1\|_{L^p({\mathbb H})}
\]
for a constant $C>0$ independent of $\alpha$.
\end{theorem}
\begin{proof}
Starting with \eqref{T1forDxD*} and applying Lemma \ref{lem135}, it follows that
\begin{equation}
\label{lpboundednessondd}
\int_{{\mathbb D}\times{\mathbb D}^*}|T(f)|^p\,dV(w) \le C\int_{{\mathbb D}\times{\mathbb D}^*}|f_1|^p\,dV(w).
\end{equation}
(i) When $p \ge 2$, it holds that
\[
\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^p\,dV(w) \le \int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^2\,dV(w)=\int_{{\mathbb H}}|\alpha_1|^p\,dV(z).
\]
Since $f_1=\bar w_2\cdot\tilde \alpha_1$ and $v=u\circ\phi=T(f)\circ\phi$, by \eqref{lpboundednessondd}, it follows
\[
\int_{{\mathbb H}}|v|^p\,dV(z)=\int_{{\mathbb D}\times{\mathbb D}^*}|u|^p\cdot |w_2|^2\,dV(w)\le \int_{{\mathbb D}\times{\mathbb D}^*}|T(f)|^p\,dV(w) \le C\int_{{\mathbb H}}|\alpha_1|^p\,dV(z).
\]
(ii) When $1 \le p<2$, consider
$\bar \partial_w (u \cdot w_2)=w_2\cdot\bar \partial_w u=w_2\cdot f=w_2\cdot f_1\,d\bar w_1$.
By \eqref{T1forDxD*}, $T(w_2\cdot f)=w_2\cdot T(f)=w_2\cdot u$. Therefore, replacing $f$ by $w_2\cdot f$ in \eqref{lpboundednessondd},
\[
\int_{{\mathbb D}\times{\mathbb D}^*}|u\cdot w_2|^p\,dV(w) \le C\int_{{\mathbb D}\times{\mathbb D}^*}|f_1\cdot w_2|^p\,dV(w)=C\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^{2p}\,dV(w).
\]
Since $1 \le p<2$,
\[
\int_{{\mathbb H}}|v|^p\,dV(z)=\int_{{\mathbb D}\times{\mathbb D}^*}|u|^p\cdot |w_2|^2\,dV(w) \le \int_{{\mathbb D}\times{\mathbb D}^*}|u|^p\cdot|w_2|^p\,dV(w)
\]
and
\[
\int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^{2p}\,dV(w) \le \int_{{\mathbb D}\times{\mathbb D}^*}|\tilde \alpha_1|^p\cdot|w_2|^2\,dV(w)=\int_{{\mathbb H}}|\alpha_1|^p\,dV(z).
\]
Hence,
\[
\int_{{\mathbb H}}|v|^p\,dV(z) \le C\int_{{\mathbb H}}|\alpha_1|^p\,dV(z).
\]
\end{proof}
\bibliographystyle{alpha}
|
\section{Introduction}
A popular solution of the strong CP problem involves extending the Standard Model by introducing a new particle, the axion. This pseudoscalar boson arises from the Peccei-Quinn (PQ) mechanism~\cite{Peccei:1977hh}, where an anomalous U(1)$_{\rm PQ}$ global symmetry is spontaneously broken~\cite{Weinberg:1977ma,Wilczek:1977pj}. When nonperturbative QCD corrections generate a potential, the axion relaxes to its CP-conserving minimum, solving the strong CP problem. Furthermore, for some mass ranges, the axion can be a good cold dark matter candidate~\cite{Preskill:1982cy,Abbott:1982af,Dine:1982ah}. The fact that two problems in the Standard Model are simultaneously addressed has led to an extensive experimental effort in searching for the axion.
An underlying assumption of the axion solution is that the U(1)$_{\rm PQ}$ global symmetry is well-preserved. However, this is in conflict with the expectation that global symmetries are explicitly broken by gravity~\cite{Banks:2010zn,Banks:1988yz}. In order not to misalign the minimum of the axion potential and reintroduce the strong CP problem, the global symmetry must then be preserved up to high-dimension terms in the effective Lagrangian~\cite{Holman:1992us,Kamionkowski:1992mf, Barr:1992qq, Ghigna:1992iv}. Using a slice of AdS$_5$~\cite{Randall:1999ee}, the axion-quality problem was recently addressed~\cite{Cox:2019rro} by delocalizing a bulk axion zero mode away from the UV brane, which sources explicit (gravitational) violations of the PQ symmetry. By the AdS/CFT correspondence~\cite{Maldacena:1997re}, this 5D geometric solution has a dual 4D interpretation where the U(1)$_{\rm PQ}$ symmetry is an accidental global symmetry of some underlying strong dynamics, such as that considered in \cite{Gavela:2018paw}.
The advantage of solving the axion-quality problem in a slice of AdS$_5$ is that one can use the 5D geometry to also explain the Standard Model fermion mass hierarchy.
In this work, we extend the model considered in Ref.~\cite{Cox:2019rro}, containing a PQ-charged, bulk complex scalar field, to also include bulk Standard Model fermions. The Higgs sector, consisting of two Higgs doublets, can either be localized on the UV boundary or propagate in the bulk. This 5D model is essentially the DFSZ model~\cite{Zhitnitsky:1980tq,Dine:1981rt} with the singlet scalar field, containing the axion, propagating in the bulk. The fact that the PQ symmetry is gauged in the bulk, is also in agreement with the expectation that only gauge symmetries are present in quantum gravity.
A feature of this 5D model is that the axion-fermion couplings are obtained while automatically addressing the fermion mass hierarchy and axion quality problem, unlike the original 4D DFSZ model. The bulk fermion profiles are controlled by order one 5D fermion mass parameters. Once these parameters are chosen to explain the Standard Model fermion mass hierarchy and mixings, they give predictions for the axion couplings to fermions. For a boundary-localized Higgs sector, only flavour-diagonal couplings are generated. This follows from the orthonormality of the bulk fermion profiles. However, when the Higgs sector propagates in the bulk, there is a non-trivial wavefunction overlap between the axion and the fermion profiles that gives off-diagonal fermion couplings.
The predictions for the off-diagonal couplings involving quarks and charged leptons are consistent with the current experimental limits~\cite{MartinCamalich:2020dfe,Calibbi:2020jvd}.
Assuming an axion decay constant $F_a \sim 10^9$\,GeV, the effective scale of the axion-fermion off-diagonal couplings is of order $10^{11}-10^{15}$\,GeV.
Furthermore, we also discuss the axion couplings to gluons and photons. In particular, using the known form of the 5D anomaly~\cite{ArkaniHamed:2001is,Hirayama:2003kk}, we derive the axion-gluon/photon couplings. These couplings can also be directly calculated from a Kaluza-Klein sum over 4D fermion modes, which provides a nontrivial check of our results. A 5D Chern-Simons term can also generate an axion coupling to gauge bosons, and we show how such interactions arise from integrating out bulk fermions, extending the calculation of Ref.~\cite{Witten:1996qb}.
The outline of our paper is as follows. In Section \ref{sec:5Daxionmodel} we review the 5D axion model~\cite{Cox:2019rro}, and then introduce bulk Standard Model fermion fields. There are two choices for the Higgs fields. UV boundary Higgs are first considered in Section \ref{sec:boundaryHiggs} where the axion-fermion couplings are shown to be flavour diagonal. Next, in Section \ref{bulkHiggsSection}, we consider the bulk Higgs case and derive the flavour-dependent, off-diagonal axion-fermion couplings for both a massless and massive axion in Section~\ref{sec:axionfermionBulk}. The axion-gluon/photon couplings from both the 5D anomaly and Chern-Simons term are discussed in Section~\ref{sec:axion-gluonphoton}. The concluding remarks are presented in Section~\ref{sec:conclusion}. The appendices contain further details of our calculations. In appendix~\ref{app:SMflavour} we present the approximations used in obtaining the Standard Model fermion masses and mixings for both the quark and lepton sector. Appendix~\ref{KKcomputationAppendix} contains a direct 4D calculation of the axion couplings to gauge bosons that uses the axion interactions with the fermionic KK modes. The boundary axion couplings can be related to a 5D Chern Simons interaction and this connection is presented in appendix~\ref{CStermAppendix}.
\section{The 5D Axion Model}
\label{sec:5Daxionmodel}
Consider a 5D $U(1)_{PQ}$ gauge theory in a slice of AdS$_5$. The metric is given by
\begin{equation}
ds^2=A^2(z)\left(dx^2+dz^2\right)\equiv g_{MN} dx^M dx^N\,,
\label{eq:AdSmetric}
\end{equation}
with coordinates $x^M=(x^\mu,z)$, and where $A(z)=1/(kz)$ with $k$ the AdS curvature scale. We denote the $U(1)_{PQ}$ gauge field by $V_M=(V_\mu,V_z)$ and introduce a complex scalar $\Phi=\eta\, e^{ia}$ with $PQ$ charge $X_\Phi=1$. The action is~\cite{Cox:2019rro}
\begin{align} \label{eq:5D_action}
S =~& 2\int^{z_{IR}}_{z_{UV}}d^5x\, \sqrt{-g} \left( -\frac{1}{4g_5^2}F^{MN}F_{MN} - \frac{1}{2} \big(\mathcal{D}^M\Phi\big)^\dagger\big(\mathcal{D}_M\Phi\big) - \frac{1}{2}m_\Phi^2\Phi^\dagger\Phi \right. \notag \\
&\left.-\frac{1}{2g_5^2\xi_{PQ}}\left( g^{\mu\nu}\partial_\mu V_\nu + \xi_{PQ} A^{-3}\partial_z\left(AV_z\right) - \xi_{PQ} g_5^2 X_\Phi \eta^2 a \right)^2 \right) \notag \\
&- \int d^4x\, \sqrt{-g_4} \, U(\Phi) \,,
\end{align}
where $\mathcal{D}_M=\partial_M -iX_\Phi V_M$, $g_5$ is the 5D gauge coupling and $\xi_{PQ}$ is a gauge-fixing parameter.
The scalar potentials on the UV and IR branes, located at $z=z_{UV}$ and $z=z_{IR}$ respectively, are taken to be
\begin{align}
U_{UV}(\Phi) &= b_{UV} k\, \Phi^\dagger\Phi \,, \label{eq:UV-potential} \\
U_{IR}(\Phi) &= \frac{\lambda_{IR}}{k^2}\left(\Phi^\dagger\Phi-k^3v_{IR}^2\right)^2 \,. \label{eq:IR-potential}
\end{align}
Neglecting the backreaction of the scalar field on the metric\footnote{This requires $|(\partial_z\eta)^2-m_\Phi^2\eta^2|\ll 12k^2M_5^3$, where the 5D Planck mass $M_5$ is related to the 4D Planck mass via $M_P^2 \simeq M_5^3/k$.}, the equation of motion for the scalar yields the background solution
\begin{equation} \label{eq:bulk-scalar}
\eta(z) = k^{3/2} \left( \lambda\, (kz)^{4-\Delta} + \sigma\, (kz)^{\Delta} \right) \,,
\end{equation}
where $\Delta>2$ is related to the bulk scalar mass via $m_\Phi^2=\Delta(\Delta-4)\,k^2$.
The real parameters $\sigma$ and $\lambda$ are determined by the boundary conditions:
\begin{align}
\sigma &= \sqrt{v_{IR}^2-\frac{\Delta}{2\lambda_{IR}}}\,(kz_{IR})^{-\Delta}\equiv \sigma_0\,(kz_{IR})^{-\Delta} \,, \\
\lambda &= \frac{\Delta-b_{UV}}{\Delta-4+b_{UV}}(kz_{UV})^{2\Delta-4}\sigma \,,
\end{align}
assuming $z_{UV} \gg z_{IR}$.
\subsection{Axion profile}
The action \eqref{eq:5D_action} leads to coupled equations of motion for the scalar degrees of freedom, $a(x^\mu,z)$ and $V_z(x^\mu,z)$.
These can be solved via the Kaluza-Klein (KK) expansion:
\begin{align} \label{eq:KK-expansion}
a(x^\mu,z) &= \sum_{n=0}^\infty f_{a}^n(z) a^n(x^\mu) \,, \\
V_z(x^\mu,z) &= \sum_{n=0}^\infty f_{V_z}^n(z) a^n(x^\mu) \,,
\end{align}
where the 4D modes $a^n(x^\mu)$ satisfy $\Box a^n =m_n^2 a^n$.
The axion is identified with the massless zero mode.
The solution for the axion profile was obtained in~\cite{Cox:2019rro}, and for $\lambda=0$ is approximately given by
\begin{align} \label{eq:axion-profile}
|
f_a^0(z) &\simeq \frac{z_{IR}}{\sigma_0} \sqrt{\Delta-1} \left(1 + \frac{g_5^2 k\sigma_0^2}{4\Delta(\Delta-1)}\left( \frac{(\Delta-1)^2}{2\Delta-1} + \frac{z^2}{z_{IR}^2}\left( \left(\frac{z}{z_{IR}}\right)^{2(\Delta-1)} - \Delta\right)\right) \right) \,, \notag \\
f_{V_z}^0(z) &\simeq \frac{-1}{2\sigma_0\sqrt{\Delta-1}} \frac{z}{z_{IR}}\left( g_5^2 k\sigma_0^2 \left(1-\left(\frac{z}{z_{IR}}\right)^{2(\Delta-1)}\right) \right) \,,
\end{align}
up to corrections of order $(g_5^2k\sigma_0^2/\Delta^2)^2$. Note that the exact profiles are used to obtain our numerical results in later sections. When PQ-violating terms are added on the UV boundary the above profiles are modified in the UV, with the expressions given in \cite{Cox:2019rro}.
\subsection{Bulk Standard Model fermions}
In addition to the bulk $U(1)_{PQ}$ there is also the Standard Model gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$. The bulk Standard Model gauge bosons have Neumann boundary conditions so that the massless zero modes are identified with the Standard Model gauge bosons (see Ref.~\cite{Gherghetta:2010cj}). Later, we will consider two possibilities for breaking the electroweak gauge symmetry.
The bulk Standard Model gauge group allows for the Standard Model fermions to be located in the bulk. The localization of the zero modes is then responsible for generating the fermion mass hierarchy and will also lead to flavour-dependent axion-fermion couplings. Denoting the 5D $SU(2)_L$ quark doublet field by $Q$ and the singlet fields by $U$, $D$, the bulk fermion action for the quark sector is given by~\cite{Gherghetta:2010cj,Gherghetta:2000qt}
\begin{align}
S_f = -2\int^{z_{IR}}_{z_{UV}}d^5x\, \sqrt{-g} \Bigg( &\frac{1}{2}\left( \bar{Q}_i \Gamma^M \mathcal{D}_M Q_i - (\mathcal{D}_M \bar{Q}_i) \Gamma^M Q_i \right) + M_{Q_i} \bar{Q}_i Q_i \notag \\
+ &\,\frac{1}{2}\left( \bar{U}_i \Gamma^M \mathcal{D}_M U_i - (\mathcal{D}_M \bar{U}_i) \Gamma^M U_i \right) + M_{U_i} \bar{U}_i U_i \notag \\
+ &\,\frac{1}{2}\left( \bar{D}_i \Gamma^M \mathcal{D}_M D_i - (\mathcal{D}_M \bar{D}_i) \Gamma^M D_i\right) + M_{D_i} \bar{D}_i D_i \Bigg) \,,
\end{align}
where $\Gamma^M = e^M_A \gamma^A = A(z)^{-1}(\gamma^\mu, \gamma^5)$, with $\gamma^5=((\mathbb{1},0),(0,-\mathbb{1}))$, and the fermions carry PQ charges $X_{Q,U,D}$. The 5D masses, $M_X \equiv c_X k$, determine the localization of the chiral zero modes, to be identified with the SM fermions, and $i$ is a flavour index. Decomposing the Dirac spinor $Q_i$ in terms of its Weyl components $Q_i=(Q_{iL}, Q_{iR})^T$, the equation of motion is
\begin{equation}
\gamma^\mu\partial_\mu Q_{iL(R)} \mp \partial_z Q_{iR(L)} + \frac{1}{z} \(c_{Q_i} \pm 2\) Q_{iR(L)} = 0 \,.
\end{equation}
To solve this equation, we perform the KK expansion,
\begin{equation}
Q_{iL(R)}(x^\mu,z) = \sum_{n=0}^\infty f_{Q_{iL(R)}}^n(z) Q_{iL(R)}^n(x^\mu) \,,
\label{KKexpansionBulkFermions}
\end{equation}
where $\slashed\partial Q_{iL(R)}^n = -m_n Q_{iR(L)}^n$, and similarly for $U$ and $D$. After imposing Dirichlet conditions $Q_{iR}=U_{iL}=D_{iL}=0$ on both boundaries, there are chiral zero modes with profiles
\begin{align}
\label{eq:fermionprofiles}
f_{Q_{iL}}^0(z) &= \mathcal{N}_{Q_i} (k z)^{2-c_{Q_i}} \,, \notag \\
f_{U_{iR}}^0(z) &= \mathcal{N}_{U_i} (k z)^{2+c_{U_i}} \,, \notag \\
f_{D_{iR}}^0(z) &= \mathcal{N}_{D_i} (k z)^{2+c_{D_i}} \,.
\end{align}
Normalising the 4D kinetic terms fixes the constants
\begin{equation}
\mathcal{N}_{X} =\sqrt{\frac{(1\mp 2c_X)k}{2((kz_{IR})^{1\mp 2c_X}-(kz_{UV})^{1\mp 2c_{X}})}}\,,
\end{equation}
where $-(+)$ refers to the left (right) handed profiles.
Similar expressions are obtained in the lepton sector.
\section{Boundary Higgs fields}
\label{sec:boundaryHiggs}
We first consider a setup with boundary-localized Higgs fields $H_{u,d}$ to construct a 5D model of the DFSZ axion~\cite{Zhitnitsky:1980tq,Dine:1981rt}. The Higgs doublet fields, which transform as $H_{u,d} \sim ({\bf 2}, \mp \frac{1}{2})$ under the $SU(2)_L\times U(1)_Y$ electroweak gauge group, are localized on the UV boundary. They are also charged under the $U(1)_{PQ}$ symmetry with charges $X_{H_u,H_d}$, such that $X_{H_u}+X_{H_d} +2 X_\Phi =0$. The most general scalar potential on the UV boundary is thus
\begin{align}
\label{eq:UVscalarpotential}
U_{UV}(\Phi,H_u, H_d) &= \lambda_u (|H_u|^2 - v_u^2)^2 + \lambda_d (|H_d|^2 - v_d^2)^2 + b_{UV} k |\Phi|^2 \nonumber\\
&+\, (a |H_u|^2 + b |H_d|^2) |\Phi|^2 + c( H_u H_d\Phi^2 + h.c.) \nonumber\\
&+\, d|H_u H_d|^2 + e |H_u^\dagger H_d|^2\,,
\end{align}
where $H_u H_d = \epsilon_{ij}H_u^i H_d^j$ with $\epsilon_{ij}$ the $SU(2)$ antisymmetric tensor.
To obtain the axion couplings, we first parametrise the scalar fields by
\begin{equation}
\label{eq:scalarVEVs}
H_u = \frac{v_u}{\sqrt{2}}
e^{i\frac{a_u(x)}{v_u}}
\begin{pmatrix} 1\\0\end{pmatrix}, \qquad H_d = \frac{v_{d}}{\sqrt{2}}
e^{i\frac{a_d(x)}{v_{d}}}
\begin{pmatrix} 0\\1\end{pmatrix}, \qquad \Phi = \eta(z) e^{i a(x,z)} \,,
\end{equation}
where we have ignored the radial components and the electromagnetically-charged NG bosons in $H_{u,d}$. The global 4D $U(1)_{PQ}$ symmetry is a remnant of the 5D local $U(1)_{PQ}$ symmetry and is realised by choosing the 5D gauge transformation parameter $\alpha(x,z) = \alpha_0 f_a^0(z)$, such that the axion zero mode transforms as $a^0(x) \rightarrow a^0(x) + \alpha_0$~\cite{Cox:2019rro}. The 4D PQ current can then be written as
\begin{equation}
J_\mu^{PQ} =
X_\Phi f_a^0(z_{UV})^{-1}{\partial_\mu}a^0 +
X_{H_u} H_u^\dagger i\overleftrightarrow{\partial_\mu} H_u +
X_{H_d} H_d^\dagger i\overleftrightarrow{\partial_\mu} H_d +
\ldots \,,
\end{equation}
where $H_i^\dagger\overleftrightarrow{\partial_\mu} H_i = \partial_\mu(H_i^\dagger) H_i - H_i^\dagger\partial_\mu H_i$.
The physical 4D axion, $a_4$, is then defined by using the Goldstone theorem $\langle 0| J_\mu^{PQ}|a_4\rangle = i F_a p_\mu$. This gives:
\begin{equation}
\label{eq:axiondefn}
F_a a_4(x) \equiv X_\Phi f_a^0(z_{UV})^{-1} a^0 + X_{H_u} v_u a_u + X_{H_d} v_d a_d\,,
\end{equation}
where
\begin{equation}
\sum_i X_i^2 v_i^2 = F_a^2\,,
\end{equation}
with $i=\Phi,H_{u,d}$ and $v_\Phi = f_a^0(z_{UV})^{-1}$. Since $v_{u,d}\ll v_\Phi$ we obtain that $F_a\simeq v_\Phi$.
Similarly, the 4D hypercharge current is given by
\begin{equation}
J_\mu^Y =
Y_u H_u^\dagger i\overleftrightarrow{\partial_\mu} H_u +
Y_d H_d^\dagger i\overleftrightarrow{\partial_\mu} H_d =
\frac{1}{2} \partial_\mu(v_u a_u-v_d a_d)\,,
\end{equation}
where $Y_{u,d} = \mp 1/2$ and $a_Z\propto v_u a_u - v_d a_d$ is the NG boson eaten by the $Z$ boson. Requiring orthogonality between the PQ and hypercharge currents, i.e. $\langle 0|J_\mu^Y|a_4\rangle = 0$, leads to
|
time. With tabbed browsing only a single page at a time is visible, and users have to switch between tabs to view the individual loaded pages. Naturally, multiple windows and tabbed browsing can be used in combination. Figure~\ref{fig:window-tab-usage} shows the behavior for the five longest participating users, Figure~\ref{fig:window-tab-usage-windows} for the usage of multiple windows, Figure~\ref{fig:window-tab-usage-tabs} for the usage of multiple tabs. In both figures the number on the x-axes are sorted refer to the same user in both figures (i.e., bars for each user are directly above each other). Figure~\ref{fig:window-tab-usage} shows that User 1 is browsing most of the
time with one browser window. Only $\sim$5\% of the time s/he is using at least two windows in parallel (and almost never three or more). Regarding User 1's usage of multple tabs, $\sim$78\% of the time s/he has has at least two parallel open tabs, $\sim$40\% of the time at least four parallel open tabs, $\sim$18\% of the time at least eight parallel open tabs, and $\sim$1\% of the time at least 16 parallel open tabs.
\begin{figure}[tb]
\centering
\subfigure[Multiple windows.]{
\includegraphics[width=0.65\textwidth]{window-overlap}
\label{fig:window-tab-usage-windows}
}
\subfigure[Multiple tabs.]{
\includegraphics[width=0.65\textwidth]{tab-overlap}
\label{fig:window-tab-usage-tabs}
}
\caption[]{Window and tabs usage of each user.}
\label{fig:window-tab-usage}
\end{figure}
The results show, even given the small sample size, that the behavior of users with respect to parallel browsing can vary significantly. For example, User 3 is the only one that often opens more than one browser window in parallel, User 1 does occasionally, the other users almost never. As some kind of extreme case, User 5 typically uses only one browser window at a time, but always open the same tabs loading the same pages at start-up (Firefox provides the feature to restore the previous browsing session at start-up). Considering both figures in combination, User 2 is almost never browsing different pages in parallel. User 3, on the other hand, regularly exploits both multiple browser windows and tabbed browsing. The other users typically browse several pages in parallel, but do this solely using tabbed browsing. The fourth case, i.e., that a user regularly opens multiple browser windows but restrains from opening multiple tabs does not occur in the current dataset.
\\
\\
\textbf{Browsing the Web is not everything.} As outlined in the introduction, we argue that online browsing is for many users no longer a dedicated task. For example, users can watch a video clip or listening to online radio while writing a document or are busy with something completely different (cleaning, cooking, etc.). The DOBBS add-on leverages two event mechanisms of the Firefox browser to explicitly quantify the time users do not actively browse the Web: a user's explicit time of inactivity, and the time a browser window is in the background, i.e., has not the focus among all applications (see Section~\ref{sec:dobbs}). Further, the DOBBS dataset also allows us identifying phases of inactivity implicitly by the prolonged absence of any new event. For example, Figure~\ref{fig:idle-vs-deactive-time3} shows the average ratios for (a) the explicitly observed inactive time of users, (b) the implicitly calculated inactive time of users (where a user is considered to be inactive after 1min without a new event)
, and (c) the explicitly observed background time of browser windows.
\begin{figure}[t]
\centering
\includegraphics[width=0.65\textwidth]{idle3-vs-deactive-time}
\caption{Comparison between the explicit and implicit inactive time, and the window background time for each user.}
\label{fig:idle-vs-deactive-time3}
\end{figure}
As anticipated, it is very common for users to suspend their on-going browsing session, indicated by three measures. Further, both measures are not necessary closely related. For example, the case that the average explicit idle time exceeds the average background time indicates that users stop browsing but do not switch to another application, but, e.g., just sitting back to watch a video clip. The opposite case, i.e., that the average background time is higher than the average explicit idle time is less intuitive. Two reasons account for that situation. Firstly, the idle time is measured across all parallel open browser windows. Only if a user is inactive in all windows the corresponding event is fired. The background time, on the other hand, is window-specific. Thus, a user having opened multiple browser windows but just using one is considered active, while background browser windows increase the average background time. And secondly, a user is considered active as soon as s/he moves the mouse curser over
the browser window, even if the window is in the background. Summing up, the explicit and implicit idle time, as well as background time provide different perspectives to look on the inactivity of users in the context of browsing which can be analyzed.
\\
\\
\textbf{What users are (really) interested in}.
In the context of the analysis of web server or search engine logs, the identification of popular websites or pages is a common task. Given the information provided by such logs, the popularity of sites or pages is typically derived from the number of visits. We argue, however, that this is a rather limited view, since the number of visits typically is not related the time users actually spent on a page. In contrast, DOBBS provides detailed information about (a) the \textit{loaded time}, i.e., the overall time that pages of a domain were loaded in the browser either in an active or in an background tab, (b) the \textit{display time}, i.e., the time pages of a domain were actually visible because they were in the active tab of a browser window that had the focus, and (c) the \textit{viewing time}, i.e., the part of the display time during the user was considered active with respect to the explicit active/inactive events. The viewing time may comprise multiple individual views, e.g., switches between tabs. To
illustrate this, Figure~\ref{fig:page-loaded-vs-focus} shows the distribution of the loaded time, the display tome, and the viewing time for a single browsing session of a user grouped by the domain of the individual web pages. The number above each bar represents the number of page loads of URLs with the same domain. Note that DOBBS allows the same analysis down to individual URLs; the aggregation over domains was chosen to simplify the representation.
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{page-loaded-vs-focused-vs-activefocused-domain-1086613761}
\caption{Distribution of times a domain has been loaded and actually in the active browser tab for a single session.}
\label{fig:page-loaded-vs-focus}
\end{figure}
It is easy to see that the measures (loaded time, display time, viewing time) as well as the number of revisits typically induce different rankings. Further, one can consider combining individual measures to derive new ones. For example, the ratio between viewing time and displaying time might represent a good indicator how ``absorbing'' a website or a web page is -- again, as Figure~\ref{fig:page-loaded-vs-focus} also indicates, this ratio would induce a different ranking. Having these different measures to quantify the popularity of a web page actually broadens the notion of popularity. Which measure to apply does eventually depend on the research questions motivating an analysis of the DOBBS dataset. For example, while advertisers might mainly be interested in the absolute viewing time, the frequency of visits is particularly interesting for web server administrators from a performance point of view.
\\
\\
\textbf{Following the footsteps of users.}
DOBBS not only allows to investigate how much time users spend on web pages but actually allows us to retrace each navigation step. This particularly refers to the usage of multiple tabs with one browser window. With page loads and tabs as the two dimensions to specify the browser usage, Figure~\ref{fig:graph-session-history} gives examples for the four different cases derived according to these two dimensions. Data points represent page loads (note that we do not consider duplicate page loads here, i.e., two data points may refer two the same URL.) Points on a horizontal line indicate new page loads in the same tab; diagonal lines represent new page loads in a new browser tab originating from the currently displayed tab. Figure~\ref{fig:graph-session-history} (top) shows the two cases where users do not use tabbed browsing: navigating from one page to another using the same tab, or simply open the browser for a single page load. In Figure~\ref{fig:graph-session-history} (middle), a user used individual tabs
for each page load. More specifically, the user opened four tabs directly after opening the browser window, and the loaded a page in each tab. Finally, Figure~\ref{fig:graph-session-history} (bottom) shows a session of a user who regularly opened new pages new tabs mostly (but not always) originating from the first tab.
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{single-tab}
\includegraphics[width=0.65\textwidth]{multiple-tab-single-load}
\includegraphics[width=0.65\textwidth]{multiple-tab-multiple-load}
\caption{Graphical visualization of examples for the different basic usages of tabbed browsing.}
\label{fig:graph-session-history}
\end{figure}
Figure~\ref{fig:graph-session-history} aims to illustrate how users use tabbed browsing to navigate between pages. However, alternative graph representations are conceivable. For example, the graph in Figure~\ref{fig:gephi-example} shows the same browsing session using a ``traditional'' representation, where the size of the nodes reflect the loaded times of the different pages, i.e., the times how long the pages were loaded in the active or an background tab. Note that this representations may obscure tabbed browsing. Naturally, the sizes of the nodes -- or optionally different shapes or colors -- can come from different measures, e.g., the displaying or viewing time, or combined measures (see above). Further, the edges can be distinguished, either using labels or colors, to, e.g., indicate if a user clicked a link or a bookmark, etc. Such visualizations may provide the basis for sophisticated graphical user interfaces with which users
|
can more easily and more intuitively overview and navigate through their
past browsing history.
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{gephi-414431039}
\caption{Graphical visualization of a single browsing session. The size of a node reflects the duration the user has spent on the corresponding web page.}
\label{fig:gephi-example}
\end{figure}
Using a graphical representation the browsing behavior can be depicted as a directed tree with the root being the startup of the browser. Particularly in the case of parallel browsing using multiple tabs, each used for one or more page loads, can result in very different types of trees. Quantifying these trees to, e.g., categorize different types of parallel browsing, requires appropriate measures. These can be straightforward measures such as the average number of page loads per tab or more sophisticated approaches applying graph-based measures such as the outdgree of nodes, the depth of the tree, the average shortest path from the root, etc. Table~\ref{tab:example-measures} lists the values as calculated for the same session shown in Figure~\ref{fig:graph-session-history} (bottom) for various measures. Again, the choice of the measure(s) will largely depend on the specific set of research questions to be answered through an analysis of the dataset.
\begin{table}
\centering
\begin{tabular}[htb]{|l|c|}
\hline
\textbf{Measure} & \textbf{Value} \\
\hline\hline
number of opened and used tabs & 21 \\
\hline
number of page loads & 50 \\
\hline
(number of tabs) / (number of page loads) & 0.42 \\
\hline
number of focus changes & 77 \\
\hline\hline
diameter of graph & 18 \\
\hline
average path length & 5.8 \\
\hline
maximum outdegree & 7 \\
\hline
modularity & 0.727 \\
\hline
\end{tabular}
\caption{Various charaterizing measures for a browsing session.}
\label{tab:example-measures}
\end{table}
Besides the number and sequence of page loads, DOBBS allows us also to analyze the source of each page load, i.e., whether the user clicked a link or a bookmark, or whether the user entered a new URL into the address bar. Figure~\ref{fig:page-load-sources-bw} exemplarily shows the distribution of different sources derived from the current DOBBS dataset. According to these numbers, the main cause for a page load is users clicking on a link, closely followed by entering a new URL. Note that the latter case also includes the case that a user is starting to type a URL and using the autocomplete feature of Firefox to select an already visited URL. The results also indicate that the history feature is of less importance. This is in line with previous studies (e.g.,~\cite{Dubroy10AStudyOfTabbed,Zhang11MeasuringWebPage,Huang12NoSearchResult}) showing that tabbed browsing significantly reduced the usage of the history buttons. Interestingly, bookmarks are very rarely used. Our explanation is that Firefox provides
various features that make the usage of bookmarks almost obsolete. This includes the autocomplete function for the address bar, and the option to restart the previous browsing session at startup of Firefox (i.e., all tabs with the last loaded web pages are opened at startup).
\begin{figure}
\centering
\includegraphics[width=0.65\textwidth]{page-load-sources-bw}
\caption{Distribution of causes for page loads.}
\label{fig:page-load-sources-bw}
\end{figure}
\section{Lessons Learned}
\label{sec:lessonslearned}
To the best of our knowledge, DOBBS represents a rather unique effort towards investigating the online browsing behavior of Web users. The granularity of the collected data goes far beyond the possibilities of conventional sources such as web server access logs or search engine transactions logs.
\\
\\
\textbf{Unsupervised experiments.}
Unlike most client-side studies that are conducted as some kind of supervised lab experiment, DOBBS is an open and unsupervised environment. Once a user has installed the add-on, there is no interference from any controlling entity. In fact, one of the most fundamental design decision was to make the add-on as unobtrusive as possible (cf. Section~\ref{sec:designdecisions}) to elicit the normal browsing behavior of users. However, this includes, in principle, that users can consciously manipulate the resulting logging data by behaving in a specific manner, e.g., by always leaving the same web page open when leaving the desk for a longer time. Thus, any analysis of the DOBBS dataset must be performed with a careful interpretation of the results. This particularly holds true when it comes to the identification of ``outliers'', i.e., browsing behavior that significantly differs from the average.
\\
\\
\textbf{Incomplete logging data.}
As outlined in Section~\ref{sec:limitations}, there are a few technical limitations that may cause incomplete logs. As a result, the DOBBS dataset inherently contains incomplete information. We have pointed out the types of logging data that could be absent. If an analysis of the dataset does not (heavily) depend on these types of missing logging data their absence can be ignored. If, however, missing data have potentially a significant effect on the results of an analysis, we also proposed alternative solutions: preprocessing steps to filter out affected data, or extrapolating the missing information using the available data. The comprehensiveness of the DOBBS dataset makes both approaches valid and applicable for most evaluation scenarios. Still, any alteration of the dataset needs to done in a careful fashion to ensure the correctness of the results.
\\
\\
\textbf{Dependencies between measure parameters.}
DOBBS collects a large variety of information all describing users' online browsing behavior, such as the times users are idle, the duration the tab containing a web page is in the foreground, or the duration the browser window has the focus among all open desktop applications. Despite the comprehensive set of measured parameters, the add-on cannot completely capture the exact behavior of users. This often leaves room for different interpretations of the logging data. For example, one can argue about whether if a page should be considered to be viewed by a user if a browser window did not have the focus (but was, however, not minimized) and/or the user was inactive during that time. Thus, any evaluation of the dataset should be preceded by a careful analysis of the alternative interpretations. Further, the final result should always be accompanied by the assumptions made for the evaluation.
\\
\\
\textbf{Spreading the word.}
The expected benefits of DOBBS naturally depend on the number of participants actively contributing to the dataset by installing the add-on. Motivating user to participate is, however, very challenging for several reasons. Firstly, the add-on is only available for Firefox and thus excluding any interested participants using different browsers. And this is unlikely to change since an adoption to other browser is difficult due to the extensive use of browser-specific event handling mechanisms. Secondly, users do not directly benefit from the add-on, it provides no added value to them. Plainly speaking, contributing to DOBBS is essentially an act of goodwill (apart from, e.g., academics that might be interested in analyzing the dataset for their own research). And thirdly, despite the anonymisation and application of encryption techniques, user might perceive privacy risks. To address this last issue our approach is to be as open and responsive as possible. To this end, DOBBS features its dedicated project
website containing all relevant information and providing the possibility to get in contact with the project team (even anonymously if necessary). Further, we not only make the dataset publicly available for download, but also make the add-on available as open source under the very open BSD license.\footnote{http://code.google.com/p/deri-dobbs/}
\section{Conclusions}
\label{sec:conclusions}
In this paper, we introduced DOBBS, our approach towards creating a comprehensive dataset capturing browsing behavior of online users. DOBBS provides a browser add-on that keeps track of the most relevant events: window events, e.g., the opening and closing of browser windows and tabs, session events, e.g., the duration of browsing sessions and the change in users' activities, and browsing events, e.g., the duration a web page was loaded and the duration the user has actually viewed the page. To avoid any impact on a user's browsing experience, the add-on runs silently in the background. The logging is done in a privacy-preserving manner, with users only identified by a randomly generated, non-retraceable identifier, and all sensitive data being encrypted. We also presented results based on the current dataset showcasing the potential benefits of the DOBBS dataset to gain deeper insights into users' browsing behavior.
The collected data yield some interesting results that we did not anticipate. Firstly, while parallel browsing is common, the way \textit{how} user conduct it (i.e., using multiple browser windows or tabs, and the number parallel open windows/tabs) can vary significantly. Secondly, and orthogonal to the actual approach to measure the time users are inactive, passive browsing occurs very frequently. Thirdly, to quantify the time a page has been loaded, displayed, and actually been viewed by a user, allows us to formulate new measures to quantify the popularity of a web page (apart from number and frequency of visits as the standard measure). And lastly, we have shown how we are able to very accurately retrace a user's browsing history. This knowledge, e.g., depcited as a browsing graph, enable new ways of describing users' browsing behavior. All these results provided by DOBBS go far beyond the capabilities of traditional sources such as web server access logs or search engine transaction logs.
DOBBS is a long-term effort. The collected data will be provided as a public dataset for research purposes on the project website (http://dobbs.deri.ie). Naturally, the value of this dataset increases with the number of participants and the length of their participation. We therefore would like to encourage every interested Internet user to download and install the browser add-on, thus contributing to DOBBS. For anyone interested in updates, the results, and the latest version of the dataset, we refer to the DOBBS project website. Besides providing the dataset and all project-relevant information, the sites also features a contact section allowing participants or interested users to leave comments or feedback, as well as to ask questions in an anonymous manner, i.e., without revealing any personal data such as their email addresses.
\vspace{-0.3cm}
\renewcommand{\baselinestretch}{0.97}
\bibliographystyle{abbrv}
|
\section{Introduction}
Decoherence and noisy dynamics in open quantum systems are major hurdles to the realization of working quantum computers and practical quantum communication devices \cite{Nielsen}.
Information encoded in a controlled quantum system will leak into its surrounding environment when there is coupling between the two systems, resulting in shorter qubit lifetimes and lower gate fidelities.
It is necessary to characterize these quantum dynamics in order to better understand the sources of system-environment coupling.
Characterization of the dynamical process can then be used to mitigate sources of noise and improve qubit coherence times.
\par
Given the density matrix $\rho$ for a $d$-dimensional system, the completely-positive quantum process
\begin{equation}
\label{eq:PM}
\mathcal{E}(\rho) = \sum_{a} {E_a \rho E_a^{\dagger}} = \sum_{m,n=0}^{d^2-1}{\chi_{mn} F_m \rho F_n^{\dagger}}
\end{equation}
may be described in terms of its Kraus operators $\{E_{a}\}$ or the Hermitian process matrix $\chi$ defined with respect to the operator basis $\{F_m\}$.
Experimental characterization of the process matrix $\chi$ provides a concrete representation of $\mathcal{E}$ that can be used to study and refine system behavior.
In standard quantum process tomography (SQPT), measurements characterizing the state $\mathcal{E}(\rho)$ are used to reconstruct the process matrix by inverting Eq.~(\ref{eq:PM}) over a complete set of input states \cite{Poyatos, Chuang_97, Nielsen, O_Brien, Bialczak, Kim}.
Ancilla-assisted process tomography (AAPT) performs a similar inversion using fewer input states by exploiting correlations between the principal system ({\bf P}) and an ancilla system ({\bf A}) isolated from non-trivial quantum process \cite{Altepeter}.
Such a composite process can be written as $\mathcal{E}_{\bf P} \otimes \mathds{1}_{\bf A} (\rho)$ where the subscripts indicate which processes occur on each subsystem.
\par
In contrast, direct-characterization of quantum dynamics (DCQD) avoids inverting Eq.~(\ref{eq:PM}) by measuring the process elements $\chi_{mn}$ directly \cite{Mohseni_PRL,Mohseni_PRA}.
DCQD techniques have recently been applied to characterize trapped ion \cite{Nigg_PRL} and hyper-entangled photon \cite{Graham_PRL} dynamics.
Like AAPT, the principal and ancilla subsystems are initially entangled in a probe state before being subjected to non-trivial and trivial quantum processes respectively.
Interestingly, DCQD probe states can be described as the codewords of a quantum error correction (QEC) code.
In this framework, a quantum process maps the joint system probe state either within or outside the codespace.
Processes mapping the probe state outside the original codespace are detected and characterized by their error syndrome, i.e., the measured eigenvalues of each QEC code generators.
Syndrome frequencies derived from an ensemble of stabilizer measurements are sufficient to directly characterize the underlying process matrix $\chi$.
\par
The DCQD framework shows that the mathematical tools developed for QEC can be leveraged for process characterization.
In particular, code design plays an important role in probing the process matrix \cite{Mohseni_PRL,Mohseni_PRA}.
Recent extensions to DCQD involve generalized characterization codes which also encode logical quantum information \cite{Omkar1,Omkar2}.
This offers the ability to characterize processes occurring during an arbitrary quantum computation.
Complimentary works, from a QEC perspective, have shown that syndrome data generated by error correction protocols can be used for noisy parameter estimation \cite{Combes2014,Fujiwara2014,Fowler}, with
recent experiments involving stabilizer QEC circuits over 9 and 4 qubits are prime candidates for these types of characterization methods \cite{Martinis_15,IBM_15}.
\par
Despite advances within the DCQD paradigm, a significant and persistent drawback in all existing schemes is the requirement that the ancilla system be perfectly noiseless.
This assumption is necessary for correctly interpreting the measured syndromes in the context of Eq.~(\ref{eq:PM}).
Noisy ancilla lead to spurious data that corrupts the process tomography and adds errors to the process matrix.
However, noise is certainly present in any realistic experiment and it is important to ask how QEC-based process characterization can be extended to include noisy ancilla.
\par
We address the use of noisy ancilla for process characterization by introducing a new class of quantum process codes that remove the requirement of noise-free ancilla.
Our approach is based on concatenated encoding of the ancilla system using a second quantum error detection code.
We show that by monitoring syndrome values of the composite code measurements of the principal system that have been corrupted by ancilla noise can be filtered out.
By removing measurements attributed to noisy ancilla, we generate a higher fidelity construction of the process matrix than possible with direct characterization alone.
We also examine the question of efficiency, which we define as a tradeoff between the syndromes collected and the accuracy of the process characterization.
\par
The remainder of the paper is organized as follows:
in Sec.~\ref{sec:CDCQD} we review notation for stabilizer QEC codes and outline the conventional DCQD procedure for constructing the process matrix before introducing a concatenated six-qubit code used to characterized the dynamics of a two-qubit principal system in the presence of full system noise.
This includes a discussion of how ancilla error detection is used to filter tomographic information prior to characterization.
In Sec.~\ref{sec:Monte-Carlo} we present a numerical case study of an amplitude damping channel on various codes with and without depolarizing noise affecting the ancilla subsystem.
We discuss results of our simulation, the degree to which the code faithfully characterizes dynamics on the principal system, and the probability that high weight errors, which can pass through our concatenated error filter thus corrupting the tomographic data, occur in Sec.~\ref{sec:analysis}.
Our conclusions and discussion appear in Sec.~\ref{sec:conclusion}.
\section{Characterization with Noisy Ancilla}
\label{sec:CDCQD}
An $[[n,k,d]]$ stabilizer code maps $k$ logical qubits onto $n$ qubits with a distance $d$ between distinct codewords \cite{Gottesman_97}.
Let $\mathcal{S} = \< g_1, \cdots, g_r \>$ denote an Abelian stabilizer group whose $r=n-k$ generators are drawn from the $n$-qubit Pauli group, i.e., $g_i \in \mathcal{P}_n$.
These stabilizer generators define a set of commuting observables called the syndrome that partitions the $n$-qubit Hilbert space into a set of mutually orthogonal subspaces, each encoding $k$ qubits.
The $i$-th subspace $\mathscr{H}_i$ corresponds to a syndrome eigenvalue $e_i$.
We will represent each syndrome as a string of classical bits such that $e_{ij} = 0$ or $1$, respectively, for the $+1$ or $-1$ eigenstate of the generator $g_j$.
In this notation, the logical codespace $\mathscr{H}_0$ corresponds to the trivial syndrome $e_0$ generated by the stabilizer group $\mathcal{S} = \{s \in \mathcal{P}_n : s \ket{\psi} = \ket{\psi}, \; \forall \ket{\psi} \in \mathscr{H}_0 \}$.
\par
Errors due to a set of operators $\mathds{E}$ are said to be correctable if the QEC condition
\begin{equation}
\label{eq:QECC}
\bra{i} E_a^\dagger E_b \ket{j} =C_{ab} \delta_{ij}
\end{equation}
is satisfied for all $E_{a},E_b \in \mathds{E}$, where $\ket{i},\ket{j}$ are orthonormal basis vectors spanning $\mathscr{H}_0$ and $C_{ab}$ is a Hermitian matrix.
In particular, a correctable error $E_a$ maps states in the codespace $\mathscr{H}_0$ to another codespace.
Upon measuring a syndrome value $e_i$, a state $\ket{\psi}$ is projected into the subspace $\mathscr{H}_i$.
For purposes of error correction, the syndrome dictates what recovery operation should be applied to return the state to the logical codespace $\mathscr{H}_0$.
\par
In the context of quantum process characterization the operator set $\mathds{E}$ represents a basis for the dynamical processes by which the encoded state evolves.
Instead of correcting $\mathds{E}$, the goal of process characterization is to unambiguously detect these operations.
As such, there is a significant difference between how QEC and process characterization are affected by undetectable errors.
In particular, operators commuting with the stabilizer group belong to the normalizer group $\mathcal{N}(\mathcal{S})$, where elements of the normalizer act as logical operators that map one codestate to another or stabilize the state (since $\mathcal{S} \in \mathcal{N}(\mathcal{S})$).
Elements of $\mathcal{N}(\mathcal{S})$ yield trivial syndromes and lead to logical errors because no recovery operation is applied.
For process characterization, however, undetectable stabilizer and normalizer operators lead to faulty tomographic data as they cannot be distinguished from the identity operation.
As seen below, concatenated ancilla codes ensure {\em all} weight one errors, across the entire system, are detectable via error syndromes distinct from $e_i$.
\par
DCQD involves partitioning the $n$-qubit system into a $n_{\bf P}$-qubit principal subsystem {\bf P} and a $n_{\bf A}$-qubit ancilla subsystem {\bf A}.
After the two subsystems are initially entangled, the principal system is subjected to dynamics $\mathcal{E}$ while the ancilla system has previously been assumed to remain isolated from any such process.
Syndrome measurements project the state into one of the $2^r$ subspaces $\mathscr{H}_i$ defined by the QEC code, and the relative frequency with which each syndrome is observed characterizes the process matrix $\chi$.
A schematic circuit expressing this partition for DCQD is shown in Fig.~\ref{fig:schematic}(a).
\subsection{Clean Ancilla DCQD}
\label{sec:cleandcqd}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=\columnwidth]{NewCharacterizationSchemeColor.pdf}
\caption{\label{fig:schematic}
Outline of a quantum circuit used to directly characterize the dynamics of a principal system {\bf P}.
The principal system is first entangled with the ancilla system {\bf A} before being subject to some noisy dynamics.
The process matrix is then constructed from an ensemble of stabilizer measurements (for details see Sec.~\ref{sec:cleandcqd} and Fig.~\ref{fig:DCQD_flowchart}).
Panel (a) illustrates a conventional DCQD circuit, where the {\bf P} and {\bf A} are maximally entangled in a Bell state $\ket{\Phi^+}$ and characterization assumes a noiseless ancilla.
Panel (b) illustrates a concatenated DCQD circuit, with an initial entangled state $\ket{0}$ (Eq.~\ref{eq:codeket}), which supports errors on the ancilla subsystem. }
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{DCQD_Flow_Simple.pdf}
\caption{\label{fig:DCQD_flowchart}
Schematic of the DCQD process.
Begin with the state $\rho = \ket{0}\bra{0}$ initialized in the codespace of $\mathcal{S}_1$.
Next, subject $\rho$ to some dynamics resulting in $\mathcal{E}(\rho)$.
A pre-processing operation $O_j= \{\mathds{I}, U_j, P_j\}$ is then applied just prior to the stabilizer generators yielding an error syndrome $e_i$.
An ensemble of syndrome measurements given the pre-processing $O_j$ is used to deduce part of the process matrix $\chi_{mn}(O_j)= \{\chi_{ii}, \text{Im} \chi_{Ji}, \text{Re} \chi_{Ji}\}$ as described in Eqs.~\ref{eq:syn_prob_diag},\ref{eq:syn_prob_coherence},\ref{eq:syn_prob_coherence_2}.}
\end{center}
\end{figure}
The probability for a state $\rho \equiv \ket{\psi}\bra{\psi}$ to be projectively measured into the subspace $\mathscr{H}_i$ with the error syndrome $e_i$ is
\begin{equation}
\label{eq:pi}
p_i=\text{Tr} \[ \Pi_i \rho \]
\end{equation}
where $\Pi_i$ is the projector onto $\mathscr{H}_i$.
We assume a one-dimensional codespace and write the projector into each orthogonal subspace in the {\em stabilizer basis} as $\Pi_i=\ket{i}\bra{i}$ where $\ket{i} = E_i \ket{0}$ for the unique operator $E_i \in \mathds{E}$.
(Our results will also hold when the syndrome subspaces are $j=2^k$ dimensional and the subspace projectors are $\Pi_i=\sum_j \ket{i_j}\bra{i_j}$.)
Let the encoded state of the system be initialized as $\rho_L=\ket{0}\bra{0}$, so that
before any quantum operation is applied the trivial syndrome $e_0$ occurs with probability $p_0=1$.
\par
After $\mathcal{E}$ acts on {\bf P}, the probability for each error syndrome becomes \cite{Mohseni_PRL,Omkar1}
\begin{eqnarray} \label{eq:syn_prob_diag}
p_i&=&\text{Tr}\[ \Pi_{i} \sum_{mn} \chi_{mn} F_m \rho_L F_n^\dagger \] \\ \nonumber
& = &\chi_{ii}
\end{eqnarray}
where we use the QEC condition in Eq.~(\ref{eq:QECC}) and perform the trace in the stabilizer basis.
Therefore, the quantum dynamical populations (diagonal elements) of $\chi$ are simply the relative frequencies with which each syndrome $e_i$ appears in an ensemble of stabilizer generators measurements.
\par
Off-diagonals of $\chi$ represent quantum dynamical coherence and are similarly measured by first applying the unitary operation $U_{j} = (\mathds{1} + i F_j) / \sqrt{2}$, where $F_j \in \mathds{E} $ is a member of the Pauli group with a trivial Pauli phase factor of $+1$. The corresponding probability for each syndrome measurement is then
\begin{eqnarray}
\label{eq:syn_prob_coherence}
p_i(U_j)&=& \text{Tr} \[ \Pi_i U_j \mathcal{E} (\rho) U_j^\dagger\] \\ \nonumber
& = & \frac{\chi_{ii} +\chi_{JJ}}{2} -\text{Im} \(\phi_J \chi_{Ji} \)
\end{eqnarray}
where $\phi_{J} \in \{\pm 1, \pm i\} $ is a Pauli phase factor which, along with $F_J$, depends on the indices $i,j$ according to $\phi_J F_J = F_i^\dagger F_j $ \cite{Omkar1}.
\par
Applying the unitary $U_j$ enables measurements to probe either the real or imaginary part of $\chi_{ij}$.
The complimentary part of a given element $\chi_{ij}$ is recovered by applying instead the projective measurement $P_{j \pm}=\mathds{1} \pm F_j$.
This corresponds to measuring an eigenvalue of $\pm1$ for the operator $F_j$.
In this case the syndrome probabilities become
\begin{eqnarray} \label{eq:syn_prob_coherence_2}
p_i(P_{j \pm})&=& \text{Tr} \[ \Pi_i P_{j \pm} \mathcal{E} (\rho) P_{j \pm}^\dagger\] \\ \nonumber
& = & \frac{\chi_{ii} +\chi_{JJ}}{2} \pm \text{Re}\( \phi_J \chi_{Ji}\)
\end{eqnarray}
for the $\pm 1$ eigenvalue.
\par
Equations (4)-(5) represent a system of linear equations that determine the elements $\chi_{ij}$ of the process $\mathcal{E}$.
Direct characterization can either be used to construct the complete process matrix or it may be applied partially to characterize only specific elements of $\chi$.
Partial characterization is especially useful when a priori knowledge about the quantum dynamics is available.
For example, partial dynamics can determine the relaxation times $T_1,T_2$ efficiently by only characterizing $\chi_{\mathds{1}Z}$ and $\chi_{XY}$ \cite{Mohseni_PRL}.
\subsection{$[[6,0,2]]$ Concatenated Ancilla Code and Error Filtering}
\begin{table}[t!]
\centering
\begin{ruledtabular}
\begin{tabular}{cccccc}
$i$ & $E_i$ & $e_i$ &$i$ & $E_i$ & $e_i$ \\[3pt]
\hline
$0 $ & $ \mathds{1}\mathds{1}$ & $ 0 0 0 0 0 0$ & $8 $ & $ XY$ & $ 0 0 0 1 1 1 $ \\
$1 $ & $ X \mathds{1}$ & $ 0 0 0 1 0 0$ & $9 $ & $ XZ$ & $ 0 0 0 1 1 0 $ \\
$2 $ & $ Y\mathds{1}$ & $ 0 0 1 1 0 0 $ &$10 $ & $ YX$ & $ 0 0 1 1 0 1 $\\
$3 $ & $ Z\mathds{1}$ & $ 0 0 1 0 0 0 $ &$11 $ & $ YY$ & $ 0 0 1 1 1 1 $\\
$4 $ & $ \mathds{1} X$ & $ 0 0 0 0 0 1 $ &$12 $ & $ YZ$ & $ 0 0 1 1 1 0 $\\
$5 $ & $ \mathds{1} Y$ & $ 0 0 0 0 1 1 $ &$13 $ & $ ZX$ & $ 0 0 1 0 0 1 $\\
$6 $ & $ \mathds{1} Z$ & $ 0 0 0 0 1 0 $ &1$4 $ & $ ZY$ & $ 0 0 1 0 1 1 $\\
$7 $ & $ XX$ & $ 0 0 0 1 0 1 $ & $15 $ & $ ZZ$ & $ 0 0 1 0 1 0 $\\
\end{tabular}
\end{ruledtabular}
\caption{\label{tab:located_errors}
Error syndromes $e_i$ for states $\ket{i} = E_i \ket{0}$ indexed by the integer $i$ for the group of located errors $E_i \in \mathds{E}_{\bf P}$.
As evident through the one-to-one correspondence between the located error operators ($E_i$) and the syndromes ($e_i$), the code $\mathcal{S}_1$ is non-degenerate with respect to located errors
Errors involving weight-one ancilla operators ($\mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A} \in \mathds{E}$) are associated with error syndromes whose first two values are either $01,10$ or $11$. These errors are filtered out and do not affect the constructed $\chi_{mn}$.}
\end{table}
In its current form, DCQD assumes the ancilla system is noiseless and therefore any non-trivial syndrome measurements are attributed to the process acting solely on the principal system.
This provides a justification for interpreting the syndrome statistics in terms of the quantum process defined by Eq.~(\ref{eq:PM}) acting on the principal system.
Schemes assuming noiseless {\bf A} require the operator set $\mathds{E}$ have support only on {\bf P} in order to decode the syndrome. \cite{Mohseni_PRL,Omkar1}
Relaxing the assumption of ideal ancilla would introduce ambiguity into syndrome interpretation.
For example, in the Bell state previously used for DCQD \cite{Mohseni_PRL}, a stabilizer measurement cannot discriminate between a process that invokes no error on {\bf A} and a nontrivial error on {\bf P} and one that induces an error on {\bf A} while {\bf P} is unaffected.
\par
In practice, this ambiguity leads to errors in the characterization of the principal system as realistic ancilla also undergo quantum process.
Note that the error ambiguity, seen for example in Bell state DCDQ \cite{Mohseni_PRL}, comes from the invariance of the error syndrome under the interchange ${\bf A } \Leftrightarrow {\bf P}$.
To resolve this ambiguity, and differentiate between the different physical scenarios that lead to the same syndrome, we concatenate the ancilla {\bf A} qubits.
Concatenation invalidates the mapping ${\bf A } \Leftrightarrow {\bf P}$, thus removing the syndrome ambiguity.
Concatenated ancilla qubits involve additional stabilizer generators such that the code detects low weight processes exclusive to system {\bf A}.
These new syndromes can be used to filter the characterization data by rejecting those values that indicate errors on the ancilla.
This offers an improvement to a fundamental limitation of code-based process tomography.
Moreover, the concatenated ancilla further partition the set $\mathds{E}$ into a set of {\em located} errors with support on only the principal system and a set of {\em unlocated} errors whose support is the composite system.
\par
We now outline our main result, the construction of a code characterizing a $n_{\bf P}=2$ qubit principal subsystem with noisy ancilla.
The characterization code must first satisfy the located quantum Hamming bound $\sum_{j=0}^2 \binom{n_{\bf P}}{j} 2^k \leq 2^n$ \cite{Haselgrove}.
The $k=0$ located Hamming bound (we choose $k=0$ since we wish to minimize overhead and are note interested in encoding logical information) is saturated for $n=4$, so we use the $[[4,0,2]]$ code $\mathcal{S}_0 = \<XIXI,IXIX,ZIZI,IZIZ\>$ to characterize {\bf P}.
However, as discussed in the last section, this code leads to the mistaken interpretation that processes acting on {\bf A} (qubits 3,4) characterize {\bf P} (qubits 1,2) under the interchange ${\bf A } \Leftrightarrow {\bf P}$ as is obvious from the symmetry of the generators.
\par
We can remove the syndrome degeneracy by encoding each physical qubit in {\bf A} with a second error detection code.
This form of code concatenation enables the detection of processes that occur on only the ancilla.
Concatenation of the ancilla is also compatible with DCQD, as the first level stabilizers $\mathcal{S}_0$ are still used for direct process characterization.
A schematic of this process is shown in Fig.~\ref{fig:schematic}.
The additional resources required for encoding the ancilla can be managed by adjusting the error detection properties of the second code.
We encode the two ancilla qubits from the original characterization code with a $[[4,2,2]]$ code that is capable of detecting weight-one operators \cite{Gottesman_97,Terhal_RMP}.
This brings the total number of qubits to six and forms a $[[6, 0, 2]]$ code.
\par
The encoding $[[4,2,2]]$ stabilizer group is $\mathcal{S}_E = \left \< XXXX,ZZZZ \right\>$, where we choose $\bar{X}_1 = X X II , \bar{Z}_1=ZI Z I, \bar{X}_2 = I X I X, \bar{Z}_2 = I I ZZ$ as representative logical operators for the two encoded qubits.
Ancilla concatenation means replacing the ancilla qubits in $\mathcal{S}_0$ with the logical qubits from $\mathcal{S}_E$, that is, $X(Z)_{3,4} \mapsto \bar{X}(\bar{Z})_{
|
1,2}$ and $\mathcal{S}_0 \mapsto \<XI\bar{X}\bar{I},IX\bar{I}\bar{X},ZI\bar{Z}\bar{I},IZ\bar{I}\bar{Z}\>$.
Expressed in terms of its generators, the newly formed code is
\begin{eqnarray}
\label{eq:S_1}
\mathcal{S}_1 = &\<& IIXXXX, IIZZZZ, XIXXII,\\ \nonumber
&& ZIZIZI, IXIXIX, IZIIZZ\>.
\end{eqnarray}
It is clear from Eq.~(\ref{eq:S_1}) that either one or both of the first two generators anti-commute with all weight-one errors on the four qubit ancilla subsystem.
The remaining generators associate a unique error syndrome to {\em all} located errors on {\bf P} when {\bf A} is noisless.
Additionally, as detailed below, this code detects all errors occurring on the first two qubits {\bf P} simultaneous to any weight-one errors on {\bf A}.
\par
The group $\mathcal{S}_1$ stabilizes the (unnormalized) one dimensional codespace
\begin{eqnarray}
\label{eq:codeket}
\ket{0} & = & \ket{000000}+\ket{001111}+\ket{010101} + \ket{011010} \\ \nonumber
&+ & \ket{100011} +\ket{101100} +\ket{110110}+\ket{111001}.
\end{eqnarray}
In addition, $\mathcal{S}_1$ partitions the Hilbert space into $64$ one-dimensional orthonormal subspaces $\mathscr{H}_i$, each of which is identified by a unique error syndrome $e_i$.
\par
We now detail the set of errors $\mathds{E}$ for which QEC condition in Eq.~(\ref{eq:QECC}) is satisfied.
We begin by structuring $\mathds{E}$ into two disjoint sets based on location of the induced errors or process.
The first set consists of located errors acting on the principal system {\bf P}, i.e., operators of the form $\sigma^i \sigma^j \mathds{1} \mathds{1} \mathds{1} \mathds{1}$.
We denote this set of 16 errors, which forms the Pauli two-qubit group modulus phases, by $\mathds{E}_{\bf P} \equiv \mathcal{P}_2/ \{\pm i, \pm1\}$.
It shall also be useful to refer to the 12 possible weight-one ancilla errors as $\mathds{E}_{\bf A} \equiv \{X_i,Y_i,Z_i\}$ where $i=3,4,5,6$ are the sites comprising {\bf A}.
The second set of errors, with 192 elements, consists of the tensor product of located and ancilla errors $\mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A} $.
Using the above definitions, the set of detectable processes is the disjoint union $\mathds{E} = \mathds{E}_{\bf P}\otimes \mathds{1}_{\bf A} \cup \mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A} $.
\par
The noisy ancilla filtering properties of this code are evident upon inspecting the syndromes pertaining to the sets $\mathds{E}_{\bf P} \otimes \mathds{1}_{\bf A}$ and $\mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A}$.
The code is {\em non-degenerate} for the set of located processes, i.e., choosing $E_a,E_b \in \mathds{E}_{\bf P} $ the code matrix in Eq.~(\ref{eq:QECC}) becomes $C_{ab}=\delta_{ab}$ for the state in Eq.~(\ref{eq:codeket}).
Elements $E_i \in \mathds{E}_{\bf P} $ map the codestate to distinct orthogonal states $\ket{i} = E_i \ket{0}$, with distinct syndromes $e_{i}$ for $i \in \[0,15\]$.
This group of located processes commutes with the first two generators in Eq.~(\ref{eq:S_1}), so the corresponding syndromes are of the form $e_i=(0,0,e_{i3},e_{i4},e_{i5},e_{i6})$.
Table \ref{tab:located_errors} enumerates the syndromes associated with all located errors.
Syndromes that begin with ``00" indicate {\bf A} is error free and that the corresponding measurement is accurate for characterizing $\chi$ as described in Eqs.~\ref{eq:syn_prob_diag},\ref{eq:syn_prob_coherence},\ref{eq:syn_prob_coherence_2}.
\par
The code is degenerate for processes $E_a, E_b \in \mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A} $.
This result is expected since there are $2^6=64$ syndromes and $208$ operator elements in the set $\mathds{E}$.
The remaining 192 processes $E_j \in \mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A} $ ($j \in \[16,207\]$) map the codeword onto the remaining 48 orthogonal states, $\ket{i}$ with $i \in \[16,63\]$.
Those processes that have an odd weight support on {\bf A} anti-commute with either one or both of the first two generators of $\mathcal{S}_1$.
Consequently, syndromes that begin with $01,10,11$ indicate that noise was detected on the ancilla.
Because these syndromes are degenerate we cannot know exactly which process corrupted the ancilla.
Therefore, this data is {\em filtered} out from characterizing the principal system.
\section{Numerical Characterization of Amplitude Damping Channel}
\label{sec:Monte-Carlo}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{Chi_Simulation.pdf}
\caption{\label{fig:chi}
Simulated AD channel process matrices constructed from ensembles of syndrome measurements according to the DCQD procedure, namely Eqs.~\ref{eq:syn_prob_diag},\ref{eq:syn_prob_coherence},\ref{eq:syn_prob_coherence_2}.
Probabilities are determined from $10^6$ Monte-Carlo events using the numerical parameters $\gamma=.4$, $p=.1$ were used for the amplitude damping and depolarizing channels respectively.
The real and imaginary parts for $\chi$ constructed with a noiseless ancilla is given in panels (a,b) and its difference from the theoretical value appears in panel (c).
A noisy ancilla reduces the accuracy of the DCQD procedure as seen in panels (d-f) for which a standard $[[4,0,2]]$ has been used.
As seen in panels (g-i), weight-one ancilla errors are mitigated by utilizing a concatenated $[[6,0,2]]$ code.
The constructed $\chi$ matrix is characterized by a high degree of fidelity, $F(\mathcal{E}^{AD}_{\bf P} (\rho),\mathcal{E}^{[[6,0,2]]}(\rho))=.9884$. }
\end{center}
\end{figure}
We now test the procedure outlined in the previous section (and Fig.~\ref{fig:DCQD_flowchart}) by numerically constructing the process matrix for the well known amplitude damping (AD) channel.
While we could characterize arbitrary noise on {\bf P}, for clarity we consider the case when only the first qubit experiences AD.
The principal system process matrix is $\mathcal{E}^{AD}_{\bf P} (\rho) = \sum_{a} E_a^{AD} \rho E_a^{AD\dagger}$ where AD channel is written in terms of the Kraus operators $E_0^{AD} = (1 + \sqrt{1-\gamma})\mathds{1}/2 + (1 - \sqrt{1-\gamma}) Z_1/2, E_1^{AD} = \sqrt{\gamma} (X_1+i Y_1)/2$.
Expressing the $\chi$ matrix in the Pauli basis $F_i=\{I, X, Y, Z\}$ the only non zero process matrix elements appear along the diagonals $\chi_{II}= (1 + \sqrt{1-\gamma})^2/4, \chi_{XX} = \chi_{YY} = \gamma /4,\chi_{ZZ}= (1 - \sqrt{1-\gamma})^2/4$ and anti-diagonals $\chi_{IZ} = \chi_{ZI} = \gamma/4, \chi_{YX} = - \chi_{XY} = i\gamma /4$ (see Fig.~\ref{fig:chi}).
To test our code in the presence of a noisy {\bf A} subsystem we take the state $\mathcal{E}^{AD}_{\bf P} (\rho) $ and subject it to an additional depolarizing (DP) channel acting independently on each ancilla qubit.
We construct the channel via a composition of single qubit DP channels so that $\mathcal{E}_{\bf A}^{DP}(\rho) = \mathcal{E}_3^{DP}(\rho) \circ \mathcal{E}_4^{DP}(\rho) \circ \mathcal{E}_5^{DP}(\rho) \circ \mathcal{E}_6^{DP}(\rho)$ where $\mathcal{E}_i^{DP} (\rho) = (1-p) \rho + p (X_i \rho X_i + Y_i \rho Y_i + Z_i \rho Z_i)/3$ and $ f(\rho) \circ g(\rho) = f ( g (\rho))$ denotes the usual functional composition of mappings.
The resulting state is $\mathcal{E}^{DP}_{\bf A}\(\mathcal{E}^{AD}_{\bf P} (\rho) \)$ where for order of the the independent {\bf P}, {\bf A} channels is arbitrary.
To simulate experimental measurement statistics we perform a Monte-Carlo simulation in which we project the state $\mathcal{E}^{DP}_{\bf A}\(\mathcal{E}^{AD}_{\bf P} (\rho) \)$ into the $\pm 1$ eigenstate of each generator in Eq.~\ref{eq:S_1} with probability $\text{Tr} \[ (1\pm g_l) \mathcal{E}^{DP}_{\bf A}\(\mathcal{E}^{AD}_{\bf P} (\rho) \)\]$.
This procedure is repeated for all six syndromes with the $\pm 1$ eigenvalues for each generator defining a single measured syndrome generator.
The probabilities for each clean syndrome, i.e., ``00" syndromes, with respect to all clean results is used to determine the $\chi$ elements by Eqs.~(\ref{eq:syn_prob_diag}), (\ref{eq:syn_prob_coherence}), and (\ref{eq:syn_prob_coherence_2}).
Following this procedure we perform a Monte-Carlo simulation of the following three scenarios:
(i) the AD channel $\mathcal{E}^{AD}_{\bf P} (\rho)$ acting on qubit 1 with a noiseless ancilla {\bf A} ($\mathcal{E}_{\bf A}^{DP}(\rho)= \rho$),
(ii) AD on qubit 1 with a noisy {\bf A} implemented with detection being done by a non-concatenated $[[4,0,2]]$ DCQD code and
(iii) AD on qubit 1 with a noisy {\bf A} using the $[[6,0,2]]$ code given in Eq.~\ref{eq:S_1} to determine $\chi$.
The process matrix constructed in scenario (i), i.e. for a noiseless ancilla system, is shown in Fig.~\ref{fig:chi} panels (a,b) and is compared to the theoretical result in panel (c).
Finite sampling causes a small discrepancy between the theoretical and the simulated result as seen in panel (c).
Next, we simulate case (ii) involving the four qubit non-concatenated DCQD code in which every possible syndrome is used to determine the $\chi$.
The absence of a filtering process means that each error occurring on the ancilla system corrupts the syndrome probabilities which, in turn, determine $\chi_{i,j}$.
The simulated $\chi$ matrix is presented in Fig.~\ref{fig:chi} panels (d,e) and its distance from the clean $\chi$ is given in panel (f).
Finally, for case (iii), we construct $\chi$ using the ancilla concatenated code (Eq.\ref{eq:S_1}) and present the results in panels (g-i).
Inspecting panels (c), (f), and (i) it is obvious that the $\chi$ matrix constructed with the concatenated code is more accurate than the standard non-concatenated code.
To quantify this difference we calculate the fidelity,defined as $F(\rho,\sigma) = \text{Tr}[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}]$ for two density matrices $\rho, \sigma$, between the theoretical and numerically constructed $\chi$ coefficients.
The states we calculate the fidelity of are one qubit states subjected to our constructed amplitude damped channels and the theoretical channel, that is: $\mathcal{E}^{[[6,0,2]]}(\rho)= \sum_{mn} \chi_{mn}^{[[6,0,2]]} F_m \rho F_n^\dagger$, $\mathcal{E}^{[[4,0,2]]}(\rho)= \sum_{mn} \chi_{mn}^{[[4,0,2]]} F_m \rho F_n^\dagger$, and $\mathcal{E}^{AD}_{\bf P} (\rho)$.
Using an initial single qubit state $\rho = \ket{0}\bra{0}$ we find $F(\mathcal{E}^{AD}_{\bf P} (\rho),\mathcal{E}^{[[6,0,2]]}(\rho))=.9884$ and $F(\mathcal{E}^{AD}_{\bf P} (\rho),\mathcal{E}^{[[4,0,2]]}(\rho))=.9165$ which represents a 10\% improvement in the fidelity for the constructed process matrix for the specific case of $p=0.1$.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=8cm]{Failure_Rate_New.pdf}
\caption{\label{fig:failure}
Probabilities of located error syndrome to occur in the presence of a noiseless principal system {\bf P} in the presence of independent depolarizing channels acting on the ancilla {\bf A} with probability $p$.
The blue dashed line ($\mathcal{P}_\mathds{1} = (1-p)^4$) shows the probability for the identity operator to occur while the probability for the identity syndrome to appear in the Monte-Carlo simulation is given by the blue circles (denoted by $p_\mathds{1}$).
Stabilizer operators therefore occur with probability $\Delta p_{\mathds{1}} = p_\mathds{1}- \mathcal{P}_\mathds{1}$.
Orange circles ($p_{00}$) denote the rate at which the remaining located (``00") syndromes occur.
Green circles (dashed lines) denote the simulated (theoretical) failure rate $p_F = p_{00} + \Delta p_{\mathds{1}}$ ($\mathcal{P}_\mathds{1} = p_2/3 + 2p_3/9+ 21p_4/81$, see text for derivation) indicating a located error syndrome from Tab.~\ref{tab:located_errors} is caused by an operator different than $E_i$. }
\end{center}
\end{figure}
\section{Filtering Failure Rate}
\label{sec:analysis}
Figure~\ref{fig:chi} shows how encoded ancilla can improve process characterization as compared to previous QEC-based schemes.
This works by detecting noisy ancilla operations and filtering out those measurements, thus improving the accuracy of the constructed process matrix.
We can explain the improvements in characterization fidelity in terms of a gain in the signal to noise ration for the characterization process.
In particular, we define the signal as the constructed process matrix elements, which are directly related to a measured syndrome $e_i$.
Noise is then those syndromes in which an error $E_j$ occurs on {\bf P} but we measure a syndrome $e_i$ with $j\neq i$.
The rate at which this occurs is represented by $p_F$ in Fig.~\ref{fig:failure}.
To quantify the improvement in the signal to noise ratio we now analyze a simple model, the DP channel $\mathcal{E}_{\bf A}^{DP}(\rho)$, and compare the probability of failure for concatenated and non-concatenated characterization codes.
\par
Using the definition of noise, we say that the filter fails if data collected from a noisy event is used to characterize $\chi_{mn}$.
In the $[[6,0,2]]$ code, failure cannot be due to any weight-one errors on {\bf A} since they, and their product with all located errors ($\mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A}$), are within the detectable errors set $\mathds{E}$.
The filter does however fail in the presence of some weight-two errors which commute with the first two generators in Eq.~\ref{eq:S_1}.
In the DP channel $\mathcal{E}_{\bf A}^{DP}(\rho)$ the probability for a weight $j$ to occur is $p_j=(1-p)^{4-j}\binom{4}{j}p^j$.
Notably, the probability for the weight 0 ``error" $\mathds{1}$ to occur is the probability that the identity occur on each qubit $p_{0} = (1-p)^4$ which appears as the dashed blue line labeled $\mathcal{P}_{\mathds{1}}$ in Fig.~\ref{fig:failure}.
Ancilla errors outside the correctable error set $\mathds{E}_{\bf A}$ occur with probability $p_{\geq 2} = p_2+p_3+p_4$.
However, not all of the errors with weight $\geq 2$ will lead to faulty characterization data since many most of them will still lead to syndromes beginning with one of $01,10,11$ and therefore do not corrupt the constructed $\chi$.
In these cases we discard the data point because it is (correctly) assumed that some error has occurred on {\bf A}.
To confirm our estimates, we numerically calculate the failure rate with $10^6$ Monte-Carlo simulations of the composite depolarizing channel $\mathcal{E}_{\bf A}^{DP}(\rho)$.
With a single exception, the failure probability is by definition the number of syndromes beginning with $00$ divided by the total number of randomly generated errors.
The exception comes from the ambiguity of whether the syndrome $e_0=\{0,0,0,0,0,0\}$ should count towards the error rate, as $e_0$ may be generated by the identity mapping or by any element in the normalizer group $\mathcal{N}(\mathcal{S}_1)$, i.e. the group of errors commuting the all stabilizer elements.
However, we know that the identity operator ($\mathds{1}^{\otimes 4}$) occurs with probability $\mathcal{P}_{\mathds{1}} = (1-p)^4$ as illustrated by the blue dashed curve in Fig.~\ref{fig:failure}.
We determine the rate for erroneous identity-like syndromes to be $\Delta p_{\mathds{1}} = p_\mathds{1}- \mathcal{P}_\mathds{1}$, the difference between $\mathcal{P}_{\mathds{1}}$ and the numerical rate at which we measure the identity syndrome (blue circles in Fig.~\ref{fig:failure}).
In Fig.~\ref{fig:failure} the green circles represent the total failure rate obtained by adding the identity probabilities difference to the probability with which all other located syndromes occur.
Enumerating the number of weight 2,3, and 4 errors which commute with the first two generators of $\mathcal{S}_1$ and the probability with which they occur we find the probability of failure to be $\mathcal{P}_F=2 p_2/3 + 2p_3/9+ 21p_4/81$ where $p_{2,3,4}$ is the for probability for an error of weight 2,3, or 4 to occur.
This function of $\mathcal{P}_F$ is plotted as the dashed green line in Fig.~\ref{fig:failure} and exactly matches our numerical data.
The leading term in $\mathcal{P}_F$ goes as $O(p^2)$ in contrast to to non-concatenated DCQD schemes whose failure rate gores as $O(p)$, the probability for weight-one errors.
explains the sharp contrast in the constructed process matrices in the second and third rows of Fig.~\ref{fig:chi}.
\section{Discussion}
\label{sec:conclusion}
We have introduced a DCQD code that directly characterizes the quantum dynamics of a principal system with assistance from a noisy ancilla system.
Within the stabilizer framework, we show that ancilla noise can be distinguished from processes acting on the principal system by using syndrome value as a filter for non-trivial ancilla processes.
For the example of DCQD with a $[[4,2,2,]]$, we have concatenated the ancilla qubits for purposes of detecting weight-one processes.
and compared the characterization of an amplitude damping process on the principal system using three different approaches: (i) clean ancilla system, (ii) noisy ancilla using a standard DCQD, and (iii) noisy ancilla using our concatenated $[[6,0,2]]$ code.
Our numerical simulations found that the process matrix constructed using the six-qubit code shows a marked improvement in fidelity over the non-concatenated approaches.
\par
Our motivation for encoding the ancilla qubits has been to filter out those measurements that correspond to unwanted data.
From this perspective, ancilla encoding represents a form of filtering the dynamics to isolate non-trivial processes acting only on the principal system.
We have argued that filtering increases the signal-to-noise ratio for process characterization, as measured by the gain in fidelity of the constructed matrix.
Of course, the gain for process characterization depends strongly on the details of the ancilla filter.
For example, the 6-qubit code introduced here detects only weight-one ancilla errors and their product with located principal system errors $\mathds{E}_{\bf P} \otimes \mathds{E}_{\bf A}$.
When higher weight errors are common, the benefit of this ancilla encoding diminishes, and larger distance codes are needed to filter higher weights processes.
For example, a distance 4 code that detects all weight-2 ancilla errors will have a filter failure rate that scale as $O(p^3)$ with $p$ the ancilla error rate.
We could also have used a non-degenerate $[[5,1,3]]$ code to encode the ancilla, where each detectable error would have a unique error syndrome.
In this case, each syndrome would be used without a filtering procedure.
In general, one can improve the signal to noise ratio at the expense of additional ancilla qubits and larger codes.
\par
Additionally, we have taken $k=0$ throughout thought this work, but we could have used a $k\neq0$ code satisfying a generalized Hamming bound \cite{Haselgrove}.
For example, a non-concatenated code performing error correction on two qubits with another encoded is provided in Ref.~\citenum{Omkar1}.
Equations (\ref{eq:syn_prob_diag})-(\ref{eq:syn_prob_coherence_2}) are easily generalized using the higher dimensional projectors $Pi_i$
resulting in a code which detects ancilla errors while encoding some non-trivial quantum information.
\par
Recent progress in realizing stabilizer QEC circuits with 9 and 4 qubits on different lattice configurations suggest that the implementation of these ideas should be experimentally feasible in the near future \cite{Martinis_15,IBM_15}. In particular, it is worth noting that the characterization processes described here and in earlier DCQD work do not require active, feed-forward error correction for purposes of implementation. Consequently, the use of QEC-based DCQD appears to be a natural way point toward the demonstration of error-corrected computation.
\section*{Acknowledgments}
E. D. and T. S. H. acknowledge support from the Intelligence Community Postdoctoral Research Fellowship Program.
This manuscript has been authored by UT-Battelle, LLC, under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy.
The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes.
The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.
|
\section{Conclusions and Future Directions}
\label{sec:conclusion}
It is not uncommon a meta-heuristic algorithm is accompanied by some parameters, the settings of which largely influence its performance on various problems. Tweaking the parameter configuration of a meta-heuristic algorithm to achieve its peak performance on a certain problem can be treated as an optimisation process, as known as PO. Due to the stochastic property of most meta-heuristic algorithms, evaluating the quality of a particular parameter configuration usually requires to run the target algorithms several times. Therefore, it is inarguably that PO is computationally expensive. Building a cheap-to-evaluate surrogate model in lieu of a computationally expensive experiment has been widely accepted as a major approach for expensive optimisation. Instead of developing a new algorithm for PO, this paper aims to study a fundamental issue --- investigating the ability of four prevalent regression algorithms for building a surrogate model of empirical performance. From our extensive experiments, we find that surrogate models built by GP and RF have shown promising generalisation ability for predicting the empirical performance of unseen parameter configurations. In particular, the prediction accuracy depends on the quality of the original performance data. This implies that it needs to be careful to use a surrogate model in the early stage of a PO process. Furthermore, we find that although SVR does not show a promising performance for predicting the approximation error of a parameter configuration, it is able to differentiate the order of two parameter configurations.
Generally speaking, we hope this work will be useful to a wide variety of researchers who seek to model algorithm performance for algorithm analysis, scheduling, algorithm portfolio construction, automated algorithm configuration, and other applications. As for the coming next step, we plan to explore the following three aspects.
\vspace{-0.2em}
\begin{itemize}
\item We would like to apply the regression algorithms investigated in this paper in the context of model-based PO. Although using design and analysis of computer experiments in the context of PO has already been studied in some previous work (e.g. sequential PO~\cite{Bartz-BeielsteinLP05}), it is still worthwhile to see whether the observations in the offline training are directly applicable to online PO.
\item Since collecting a performance data in PO is computationally expensive, it might be interesting to use the offline trained surrogate models to generate pseudo data. In this rigour, semi-supervised learning~\cite{SunGJC13} can be useful to address a small data challenge.
\item Here we set the PO as a per-instance scenario. In the prevalent algorithm configuration literature~\cite{HutterHL11}, it is more interesting to combine the problem feature into the surrogate modelling process so that we can generalise the PO to a range of similar problems.
\item In addition, assessing the performance of evolutionary multi-objective optimisation algorithms, e.g.~\cite{WuKZLWL15,WuLKZZ17,WuLKZ18,LiCFY18,LiDY18}, is even more difficult. Therefore, it is also interesting to investigate appropriate surrogate modelling methods to analyse and understanding the parameter versus algorithm performance in the context of multi-objective optimisation.
\end{itemize}
\vspace{-0.5em}
\section{Experiments and Results}
\label{sec:experiments}
In this section, we will present and compare experimental evaluations of the quality of surrogates constructed by different regression algorithms introduced in~\prettyref{sec:surrogate}. The experimental results are analysed according to the following three research questions (RQs).
\begin{description}
\item[\underline{\textbf{RQ1:}}] \textbf{\textit{Which surrogate model works best for empirical performance modelling on various kinds of benchmark problems?}}
\item[\underline{\textbf{RQ2:}}] \textbf{\textit{Does the empirical performance predicted by a surrogate model follow the order as the ground truth?}}
\item[\underline{\textbf{RQ3:}}] \textbf{\textit{How does the empirical performance landscape fit by a surrogate model compare with the ground truth?}}
\end{description}
\subsection{Comparisons of Different Surrogate Models}
\label{sec:RMSE_comparison}
Bearing the RQ1 in mind, this section empirically compares the generalisation performance of four regression algorithms on unseen parameter configurations. In particular, the root mean square error (RMSE) is used to measure the generalisation performance and it is calculated as:
\begin{equation}
RMSE=\sqrt{\frac{\sum_{i=1}^{\hat{n}}(\hat{\mathcal{L}}(f(\mathbf{x}),\theta_i)-\mathcal{L}(f(\mathbf{x}),\theta_i))^2}{\hat{n}}}
\label{eq:rmse}
\end{equation}
where $\hat{\mathcal{L}}(f(\mathbf{x}),\theta_i)$ is the approximation error of a parameter configuration $\theta_i$ estimated by a surrogate model; while $\mathcal{L}(\mathbf{x},\theta_i)$ is the observed approximation error of $\theta_i$, $i\in\{1,\cdots,\hat{n}\}$ and $\hat{n}$ is the number of data in the testing set.
From the results shown in Tables~\ref{tab:metric_2D} to \ref{tab:metric_30D}, we clearly see that GP and RF are the best regression algorithms to build the surrogate for modelling the empirical performance. RBFN is slightly worse than GP and RF, while SVR is the worst choice except on F14 when $d=2$. Note that our observations of promising performance of GP and RF are also in line with some results reported in the contemporary algorithm configuration literature~\cite{HutterXHL14}. Furthermore, we find that the performance of different regression algorithms are consistent across different dimensions. This makes sense as a surrogate model is built upon the parameter configurations themselves, which are independent from the problem instances. In addition, we find that the RMSE dramatically increases with the dimensionality of the underlying problem. This can be explained as the significant degeneration of the performance of DE with the dimensionality which in term largely increases the approximation errors.
\begin{table*}[htbp]
\centering
\caption{Comparisons of RMSE, PCC and SRCC obtained by four regression algorithms on benchmark problems $(d=2)$}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c|c|c|c|c|c}
\hline
\textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} & \textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} \\
\hline
\multirow{1}[6]{*}{F1} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.3605E-1} & 1.3692E-1 & 1.5620E-1 & 6.4804E-1 & \multirow{1}[6]{*}{F11} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.1228E+1} & 1.2321E+1 & 1.1611E+1 & 3.0824E+1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9025E-1} & 9.8920E-1 & 9.8818E-1 & 7.1022E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8613E-1} & 9.8325E-1 & 9.8563E-1 & 9.1249E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.0219E-1 & 8.4648E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.6439E-1} & 7.2813E-1 & & SRCC & 8.1685E-1 & 8.0941E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.6094E-1} & 8.1033E-1 \\
\hline
\multirow{1}[6]{*}{F2} & RMSE & 5.6003E+0 & 6.8769E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.8678E+0} & 1.0089E+1 & \multirow{1}[6]{*}{F12} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.8104E+7} & 1.9118E+7 & 2.0662E+7 & 1.1799E+8 \\
\cline{2-6}\cline{8-12} & PCC & 9.7771E-1 & 9.6623E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8320E-1} & 9.3107E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8769E-1} & 9.8643E-1 & 9.8516E-1 & 3.0080E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 8.4210E-1 & 7.4855E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.4101E-1} & 8.5432E-1 & & SRCC & 6.0502E-1 & 4.0852E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.6027E-1} & 7.2927E-1 \\
\hline
\multirow{1}[6]{*}{F3} & RMSE & 4.6287E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.5074E+2} & 4.8185E+2 & 6.4441E+2 & \multirow{1}[6]{*}{F13} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4518E+0} & 1.9035E+0 & 2.9745E+0 & 1.1412E+1 \\
\cline{2-6}\cline{8-12} & PCC & 7.4558E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6157E-1} & 7.5379E-1 & 3.9173E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9909E-1} & 9.9843E-1 & 9.9647E-1 & 9.5266E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 8.6529E-1 & 7.2109E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7141E-1} & 9.4045E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.9138E-1} & 7.7192E-1 & 8.8296E-1 & 6.3004E-1 \\
\hline
\multirow{1}[6]{*}{F4} & RMSE & 5.6182E-1 & 9.3965E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.2925E-1} & 9.7513E-1 & \multirow{1}[6]{*}{F14} & RMSE & 1.0581E+0 & 1.3978E+0 & 1.0491E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.0448E+0} \\
\cline{2-6}\cline{8-12} & PCC & 9.8979E-1 & 9.7194E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9103E-1} & 9.7017E-1 & & PCC & 9.3824E-1 & 8.9687E-1 & 9.3839E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.3922E-1} \\
\cline{2-6}\cline{8-12} & SRCC & 9.7495E-1 & 9.6736E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8303E-1} & 9.5503E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.3023E-1} & 9.0572E-1 & 9.2777E-1 & 9.2696E-1 \\
\hline
\multirow{1}[6]{*}{F5} & RMSE & 1.4404E-2 & 1.9011E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.3800E-2} & 1.5263E-2 & \multirow{1}[6]{*}{F15} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.3994E-1} & 6.1380E-1 & 5.6273E-1 & 7.0871E-1 \\
\cline{2-6}\cline{8-12} & PCC & 9.4829E-1 & 9.1330E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5213E-1} & 9.4119E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8694E-1} & 9.8310E-1 & 9.8595E-1 & 9.7769E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.5309E-1 & 9.3783E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5406E-1} & 9.4455E-1 & & SRCC & 9.8333E-1 & 9.8193E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8491E-1} & 9.7681E-1 \\
\hline
\multirow{1}[6]{*}{F6} & RMSE & 5.1460E-1 & 5.8459E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.8990E-1} & 8.5190E-1 & \multirow{1}[6]{*}{F16} & RMSE & 8.2311E-1 & 9.4311E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6776E-1} & 1.0465E+0 \\
\cline{2-6}\cline{8-12} & PCC & 9.8631E-1 & 9.8242E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8764E-1} & 9.6360E-1 & & PCC & 9.8415E-1 & 9.7927E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8628E-1} & 9.7440E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.8310E-1 & 9.8082E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8486E-1} & 9.7742E-1 & & SRCC & 9.8511E-1 & 9.8031E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8572E-1} & 9.7966E-1 \\
\hline
\multirow{1}[6]{*}{F7} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.3570E+1} & 5.7449E+1 & 6.3230E+1 & 2.8793E+2 & \multirow{1}[6]{*}{F17} & RMSE & 9.0189E-2 & 1.2394E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6712E-2} & 9.3559E-2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9132E-1} & 9.9003E-1 & 9.8804E-1 & 7.5629E-1 & & PCC & 9.8626E-1 & 9.7430E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9011E-1} & 9.8523E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.2604E-1 & 9.1303E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5654E-1} & 8.9479E-1 & & SRCC & 9.8428E-1 & 9.7754E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8806E-1} & 9.8360E-1 \\
\hline
\multirow{1}[6]{*}{F8} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.3072E+1} & 6.1123E+1 & 6.5377E+1 & 2.9297E+2 & \multirow{1}[6]{*}{F18} & RMSE & 1.0509E+2 & 1.0808E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5303E+1} & 2.5797E+2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9178E-1} & 9.8911E-1 & 9.8790E-1 & 7.5211E-1 & & PCC & 9.6875E-1 & 9.6675E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7438E-1} & 8.0866E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.6614E-1 & 9.5066E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.6749E-1} & 9.0941E-1 & & SRCC & 9.6516E-1 & 9.5743E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7273E-1} & 9.2020E-1 \\
\hline
\multirow{1}[6]{*}{F9} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.0347E+7} & 1.4066E+7 & 1.1402E+7 & 6.3483E+7 & \multirow{1}[6]{*}{F19} & RMSE & 4.5497E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.4772E+0} & 4.9874E+0 & 1.0352E+1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8560E-1} & 9.7416E-1 & 9.8328E-1 & 2.8465E-1 & & PCC & 9.1451E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.1843E-1} & 9.1196E-1 & 4.0436E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 8.3050E-1 & 8.1905E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.1239E-1} & 6.9996E-1 & & SRCC & 8.9866E-1 & 8.7992E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8220E-1} & 8.4578E-1 \\
\hline
\multirow{1}[6]{*}{F10} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.8167E+1} & 7.1668E+1 & 7.0629E+1 & 2.6264E+2 & \multirow{1}[6]{*}{F20} & RMSE & 4.8658E-2 & 5.8129E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.4185E-2} & 5.2539E-2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8591E-1} & 9.7842E-1 & 9.8005E-1 & 6.9381E-1 & & PCC & 9.8499E-1 & 9.7863E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8761E-1} & 9.8251E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.6279E-1 & 9.5453E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.6651E-1} & 9.0516E-1 & & SRCC & 9.8416E-1 & 9.7760E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8710E-1} & 9.8196E-1 \\
\hline
\end{tabular}
}
\label{tab:metric_2D}
\end{table*}
\begin{table*}[htbp]
\centering
\caption{Comparisons of RMSE, PCC and SRCC obtained by four regression algorithms on benchmark problems $(d=10)$}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c|c|c|c|c|c}
\hline
\textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} & \textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} \\
\hline
\multirow{1}[6]{*}{F1} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.7299E+0} & 1.7636E+0 & 1.9689E+0 & 1.0508E+1 & \multirow{1}[6]{*}{F11} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.5782E+1} & 1.2276E+2 & 8.4930E+1 & 5.8035E+2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9861E-1} & 9.9856E-1 & 9.9828E-1 & 9.5223E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9431E-1} & 9.8604E-1 & 9.9342E-1 & 6.5587E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.8381E-1 & 9.9032E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9443E-1} & 9.2279E-1 & & SRCC & 9.9059E-1 & 9.8960E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9059E-1} & 7.9249E-1 \\
\hline
\multirow{1}[6]{*}{F2} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.0182E+3} & 1.2786E+3 & 1.1599E+3 & 8.9599E+3 & \multirow{1}[6]{*}{F12} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.5225E+7} & 1.4132E+8 & 1.5177E+8 & 1.2503E+9 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9822E-1} & 9.9729E-1 & 9.9770E-1 & 8.5938E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9739E-1} & 9.9281E-1 & 9.9270E-1 & 3.0347E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.8394E-1 & 9.8794E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9256E-1} & 5.9707E-1 & & SRCC & 9.7218E-1 & 9.6638E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8772E-1} & 4.9073E-1 \\
\hline
\multirow{1}[6]{*}{F3} & RMSE & 8.4958E+3 & 1.4600E+4 & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.2164E+3} & 3.9288E+4 & \multirow{1}[6]{*}{F13} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.6751E+1} & 7.0974E+1 & 3.8205E+1 & 4.6031E+2 \\
\cline{2-6}\cline{8-12} & PCC & 9.9385E-1 & 9.8265E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9408E-1} & 8.6001E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9854E-1} & 9.9019E-1 & 9.9725E-1 & 8.1323E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.6171E-1 & 9.6198E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9279E-1} & 5.3054E-1 & & SRCC & 7.5720E-1 & 6.5339E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.0342E-1} & 5.4879E-1 \\
\hline
\multirow{1}[6]{*}{F4} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.0935E-1} & 6.3263E-1 & 8.1344E-1 & 2.2203E+0 & \multirow{1}[6]{*}{F14} & RMSE & 3.4421E-2 & 6.1959E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.8171E-2} & 6.4088E-2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9604E-1} & 9.9575E-1 & 9.9319E-1 & 9.4638E-1 & & PCC & 9.3622E-1 & 8.1933E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5841E-1} & 7.7445E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9633E-1} & 9.9446E-1 & 9.9475E-1 & 9.5691E-1 & & SRCC & 9.4234E-1 & 8.1654E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.6084E-1} & 8.0800E-1 \\
\hline
\multirow{1}[6]{*}{F5} & RMSE & 4.6539E-2 & 4.8853E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.5680E-2} & 1.1754E-1 & \multirow{1}[6]{*}{F15} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1974E+0} & 3.7984E+0 & 3.4427E+0 & 8.1132E+0 \\
\cline{2-6}\cline{8-12} & PCC & 9.7564E-1 & 9.7311E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7708E-1} & 8.3876E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9203E-1} & 9.8875E-1 & 9.9118E-1 & 9.5124E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.5180E-1 & 9.5233E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.5597E-1} & 8.3469E-1 & & SRCC & 9.9141E-1 & 9.9001E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9205E-1} & 9.6200E-1 \\
\hline
\multirow{1}[6]{*}{F6} & RMSE & 4.0360E+0 & 5.4285E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.8827E+0} & 1.4874E+1 & \multirow{1}[6]{*}{F16} & RMSE & 4.4141E+0 & 4.7725E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.2158E+0} & 1.2342E+1 \\
\cline{2-6}\cline{8-12} & PCC & 9.9643E-1 & 9.9355E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9676E-1} & 9.5154E-1 & & PCC & 9.9089E-1 & 9.8940E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9210E-1} & 9.3015E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9510E-1} & 9.9377E-1 & 9.9415E-1 & 9.1485E-1 & & SRCC & 9.9171E-1 & 9.9066E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9244E-1} & 9.3564E-1 \\
\hline
\multirow{1}[6]{*}{F7} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1640E+2} & 3.2116E+2 & 4.0646E+2 & 3.9881E+3 & \multirow{1}[6]{*}{F17} & RMSE & 3.6772E-1 & 3.9961E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1696E-1} & 4.5470E-1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9671E-1} & 9.9663E-1 & 9.9537E-1 & 3.3030E-1 & & PCC & 9.8841E-1 & 9.8631E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9147E-1} & 9.8230E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.8179E-1 & 9.8652E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9245E-1} & 4.6616E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8741E-1} & 9.8447E-1 & 9.8739E-1 & 9.8099E-1 \\
\hline
\multirow{1}[6]{*}{F8} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.2136E+2} & 4.9296E+2 & 5.0804E+2 & 4.5107E+3 & \multirow{1}[6]{*}{F18} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1235E+3} & 3.6255E+3 & 4.0628E+3 & 3.3285E+4 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9603E-1} & 9.9456E-1 & 9.9450E-1 & 4.6619E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9533E-1} & 9.9380E-1 & 9.9284E-1 & 4.0551E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.9165E-1 & 9.9146E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9396E-1} & 5.0729E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9208E-1} & 9.9109E-1 & 9.9188E-1 & 5.7869E-1 \\
\hline
\multirow{1}[6]{*}{F9} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1078E+6} & 3.9717E+6 & 4.8207E+6 & 3.5654E+7 & \multirow{1}[6]{*}{F19} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.3832E+0} & 4.1114E+0 & 4.2199E+0 & 1.0164E+1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9601E-1} & 9.9335E-1 & 9.9111E-1 & 3.6766E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7775E-1} & 9.6694E-1 & 9.6695E-1 & 7.9914E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.8857E-1 & 9.8852E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9138E-1} & 5.1935E-1 & & SRCC & 9.7974E-1 & 9.7945E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9234E-1} & 9.4797E-1 \\
\hline
\multirow{1}[6]{*}{F10} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.7094E+2} & 6.1865E+2 & 5.5723E+2 & 4.6398E+3 & \multirow{1}[6]{*}{F20} & RMSE & 6.3115E-2 & 7.6011E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.9801E-2} & 1.1880E-1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9545E-1} & 9.9220E-1 & 9.9390E-1 & 5.1505E-1 & & PCC & 9.8540E-1 & 9.7913E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9116E-1} & 9.4940E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.9190E-1 & 9.9131E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9376E-1} & 5.3709E-1 & & SRCC & 9.7823E-1 & 9.7534E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8558E-1} & 9.8114E-1 \\
\hline
\end{tabular}
}
\label{tab:metric_10D}
\end{table*}
\begin{table*}[htbp]
\centering
\caption{Comparisons of RMSE, PCC and SRCC obtained by four regression algorithms on benchmark problems $(d=30)$}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c|c|c|c|c|c}
\hline
\textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} & \textbf{Problem} & \textbf{Metric} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} \\
\hline
\multirow{1}[6]{*}{F1} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.1664E+0} & 4.3754E+0 & 5.9464E+0 & 2.6140E+1 & \multirow{1}[6]{*}{F11} & RMSE & 6.1026E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.7617E+2} & 7.6408E+2 & 7.1468E+3 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9970E-1} & 9.9942E-1 & 9.9894E-1 & 9.8077E-1 & & PCC & 9.9656E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9696E-1} & 9.9484E-1 & 4.9608E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9955E-1} & 9.9933E-1 & 9.9832E-1 & 9.6490E-1 & & SRCC & 9.9691E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9696E-1} & 9.9399E-1 & 6.4670E-1 \\
\hline
\multirow{1}[6]{*}{F2} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.3927E+3} & 8.1327E+3 & 1.1700E+4 & 2.9682E+5 & \multirow{1}[6]{*}{F12} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.5205E+8} & 1.0921E+9 & 1.4697E+9 & 1.3268E+10 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9967E-1} & 9.9960E-1 & 9.9918E-1 & 7.8920E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9766E-1} & 9.9618E-1 & 9.9369E-1 & 1.4329E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9949E-1} & 9.9937E-1 & 9.9847E-1 & 8.1683E-1 & & SRCC & 9.8387E-1 & 9.8806E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9281E-1} & 1.7774E-1 \\
\hline
\multirow{1}[6]{*}{F3} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.5734E+4} & 6.8867E+4 & 5.7877E+4 & 1.1203E+6 & \multirow{1}[6]{*}{F13} & RMSE & 1.6142E+2 & 2.6088E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4616E+2} & 1.9077E+3 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9909E-1} & 9.9792E-1 & 9.9855E-1 & 7.5709E-1 & & PCC & 9.9608E-1 & 9.8951E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9675E-1} & 7.3189E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9906E-1} & 9.9806E-1 & 9.9804E-1 & 8.4369E-1 & & SRCC & 8.5153E-1 & 7.8392E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.4461E-1} & 6.0091E-1 \\
\hline
\multirow{1}[6]{*}{F4} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4897E-1} & 3.7718E-1 & 3.7555E-1 & 1.3550E+0 & \multirow{1}[6]{*}{F14} & RMSE & 2.5218E-2 & 8.6535E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.1899E-2} & 7.9183E-2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9874E-1} & 9.9183E-1 & 9.9204E-1 & 8.9427E-1 & & PCC & 9.7414E-1 & 7.8203E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8112E-1} & 7.1034E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9888E-1} & 9.7018E-1 & 9.9810E-1 & 9.3530E-1 & & SRCC & 9.5246E-1 & 7.4453E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.6295E-1} & 4.7142E-1 \\
\hline
\multirow{1}[6]{*}{F5} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.8351E-2} & 1.0707E-1 & 8.1513E-2 & 3.1414E-1 & \multirow{1}[6]{*}{F15} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.1107E+1} & 1.7293E+1 & 1.3455E+1 & 7.1742E+1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9892E-1} & 9.9641E-1 & 9.9793E-1 & 9.6869E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9482E-1} & 9.8756E-1 & 9.9271E-1 & 7.6305E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.9664E-1 & 9.9348E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9705E-1} & 9.7934E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9433E-1} & 9.9250E-1 & 9.9274E-1 & 7.6027E-1 \\
\hline
\multirow{1}[6]{*}{F6} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.4422E+0} & 1.2316E+1 & 8.1210E+0 & 4.0464E+1 & \multirow{1}[6]{*}{F16} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4891E+1} & 1.9211E+1 & 1.9092E+1 & 1.1678E+2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9896E-1} & 9.9624E-1 & 9.9837E-1 & 9.6152E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9520E-1} & 9.9201E-1 & 9.9250E-1 & 6.5834E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9883E-1} & 9.9741E-1 & 9.9851E-1 & 9.6091E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9588E-1} & 9.9333E-1 & 9.9332E-1 & 6.9728E-1 \\
\hline
\multirow{1}[6]{*}{F7} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.8187E+3} & 2.5706E+3 & 2.1993E+3 & 2.2890E+4 & \multirow{1}[6]{*}{F17} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2600E+0} & 2.8985E+0 & 1.5206E+0 & 4.4456E+0 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9671E-1} & 9.9367E-1 & 9.9540E-1 & 4.4227E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8696E-1} & 9.3534E-1 & 9.8134E-1 & 8.2768E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.9188E-1 & 9.9305E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9386E-1} & 5.9837E-1 & & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7828E-1} & 9.6772E-1 & 9.7314E-1 & 8.5502E-1 \\
\hline
\multirow{1}[6]{*}{F8} & RMSE & 2.4025E+3 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.3311E+3} & 3.1974E+3 & 2.6548E+4 & \multirow{1}[6]{*}{F18} & RMSE & 4.8301E+4 & 5.8401E+4 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.6411E+4} & 5.1075E+5 \\
\cline{2-6}\cline{8-12} & PCC & 9.9597E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9620E-1} & 9.9333E-1 & 5.8200E-1 & & PCC & 9.9530E-1 & 9.9312E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9585E-1} & 4.6182E-1 \\
\cline{2-6}\cline{8-12} & SRCC & 9.9596E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9608E-1} & 9.9406E-1 & 6.2086E-1 & & SRCC & 9.9473E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9510E-1} & 9.9470E-1 & 6.4254E-1 \\
\hline
\multirow{1}[6]{*}{F9} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.9825E+7} & 2.6982E+7 & 3.8388E+7 & 3.1390E+8 & \multirow{1}[6]{*}{F19} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.5569E+1} & 9.6097E+1 & 7.5740E+1 & 7.0118E+2 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9790E-1} & 9.9611E-1 & 9.9251E-1 & 1.8441E-1 & & PCC & 9.9469E-1 & 9.9170E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9478E-1} & 6.1791E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9316E-1} & 9.9320E-1 & 9.9271E-1 & 3.9600E-1 & & SRCC & 9.7199E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8092E-1} & 9.9193E-1 & 8.1939E-1 \\
\hline
\multirow{1}[6]{*}{F10} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.3746E+3} & 2.6324E+3 & 3.1720E+3 & 2.5763E+4 & \multirow{1}[6]{*}{F20} & RMSE & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.4363E-2} & 1.8844E-1 & 1.1060E-1 & 3.0212E-1 \\
\cline{2-6}\cline{8-12} & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9576E-1} & 9.9480E-1 & 9.9279E-1 & 5.8482E-1 & & PCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.8325E-1} & 9.3592E-1 & 9.7715E-1 & 8.1226E-1 \\
\cline{2-6}\cline{8-12} & SRCC & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.9487E-1} & 9.9380E-1 & 9.9214E-1 & 6.5200E-1 & & SRCC & 9.7602E-1 & 9.4643E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.7782E-1} & 8.3298E-1 \\
\hline
\end{tabular}
}
\label{tab:metric_30D}
\end{table*}
To have a better understanding of the generalisation performance of different surrogate models (especially the relationship between the predicted performance and its ground truth given a particular parameter configuration), we calculate the Pearson correlation coefficient (PCC) of the results:
\begin{equation}
PCC=\frac{\mathtt{cov}(X,Y)}{\sigma_X\sigma_Y}
\label{eq:pcc}
\end{equation}
where $X$ represents the set of observed approximation errors of all parameter configurations in the testing set while $Y$ is the set of approximation errors estimated by a surrogate model. $\mathtt{cov}(X,Y)$ is the covariance of $X$ and $Y$, $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$. In particular, a higher PCC indicates a better correlation between the predicted performance and the ground truth.
From the results shown in Tables~\ref{tab:metric_2D} to~\ref{tab:metric_30D}, we can see that the observations are in line with the RMSE. The performance of GP and RF are the most competitive regression algorithms in almost all cases, where the correlation between the predicted performance and its ground truth is relatively high. The performance of RBFN is very close to those of GP and RF, while the PCC obtained by SVR is the worst. To have a visual understanding of this point, we also provide the scatter plots of \textit{ground truth vs predicted performance} in Figures~\ref{fig:F1} to \ref{fig:F14}\footnote{More comprehensive figures are moved to the supplementary document, which can be downloaded from \href{https://coda-group.github.io/cec19-supp.pdf}{http://coda-group.github.io/cec19-supp.pdf}.}. According to the observations from these figures and Tables~\ref{tab:metric_2D} to~\ref{tab:metric_30D}, we summarise our findings as follows.
\begin{itemize}
\item As shown in Tables~\ref{tab:metric_2D} to~\ref{tab:metric_30D}, the RMSEs of all four regression algorithms are huge (over $10^7$) on F9 and F12. This is because the performance of DE are miserable on these two test problems with almost all sampled 5,940 parameter configurations. Accordingly, the deviations of the predicted empirical performance are in a relatively large scale. This also explains the increase of RMSEs with the problem dimensionality. However, according to PCCs, we find that the correlation between the predicted empirical performance and the ground truth of GP, RBFN and RF are acceptable.
\item The RMSEs of the first six elementary test problems (i.e. F1 to F6), which are relatively simple, are better than those from CEC 2005 competition. Accordingly, the deviations between the predicted performance and the ground truth are small. This indicates that most parameter configurations are able to lead to an acceptable performance of DE. In other words, DE is not sensitive to its configurations on these problems.
\item As shown in~\prettyref{fig:F8}, we find that SVR largely underestimates the approximation error on F8. Similar observations can be found on F7, F9, F10, F12 and F18 as shown in the supplementary document.
\item As shown in~\prettyref{fig:F14}, we find that scatter plots are crowded in the middle region of the diagonal line. This implies that all parameter configurations fail to lead to a decent result. Similar observations can be found on F13 and F20 when the number of variables becomes large in the supplementary document.
\end{itemize}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/PCC_F1.pdf}
\caption{Scatter plots of the empirical performance predicted by a surrogate model \textit{vs} the observed empirical performance on the testing set (i.e. unseen parameter configurations). In particular, three rows respectively represent results on F1 where $d=2,10,30$.}
\label{fig:F1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/PCC_F8.pdf}
\caption{Scatter plots of the empirical performance predicted by a surrogate model \textit{vs} the observed empirical performance on the testing set (i.e. unseen parameter configurations). In particular, three rows respectively represent results on F8 where $d=2,10,30$.}
\label{fig:F8}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/PCC_F14.pdf}
\caption{Scatter plots of the empirical performance predicted by a surrogate model \textit{vs} the observed empirical performance on the testing set (i.e. unseen parameter configurations). In particular, three rows respectively represent results on F14 where $d=2,10,30$.}
\label{fig:F14}
\end{figure}
Based on the above discussions, we come up with the following response to RQ1:
\vspace{0.5em}
\noindent
\framebox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{
\underline{\textbf{Response to RQ1}}: \textbf{\textit{GP and RF are the best regression algorithms for building the surrogate model of empirical performance. In addition, the quality of the surrogate model depend on the quality of the performance data.}}
}}
\subsection{Comparisons of Performance Ranks Obtained by Different Surrogate Models}
\label{sec:rank}
When using a surrogate in a sequential model-based PO, the prediction accuracy of this model is not utterly important. Instead, reliably differentiating the promising ones with respect to their unpromising counterparts can also provide useful information to guide the optimisation process. In other words, for a set of parameter configurations, we expect that the ranks (or the order) of the empirical performance predicted by a surrogate model can follow those of the ground truth. To this end, we consider using the Spearman's rank correlation coefficient (SRCC) to measure the statistical dependence between the ranks of the predicted performance and the ground truth. Note that the calculation of SRCC is almost the same as that of PCC, except that the raw data is replaced by the corresponding ranks.
\begin{equation}
SRCC=\frac{\mathtt{cov}(r_X,r_Y)}{\sigma_{r_X}\sigma_{r_Y}}
\label{eq:srcc}
\end{equation}
where $r_X$ indicates the ranks of the observed approximation errors of all parameters configurations in the testing set while $r_Y$ is the ranks of those estimated approximation errors. A higher SRCC indicates a better dependency between the predicted performance and the ground truth.
From the results shown in Tables~\ref{tab:metric_2D} to~\ref{tab:metric_30D}, we can still come up with the conclusion that GP and RF are the most reliable regression algorithms for building the surrogate model of the empirical performance. They almost dominate the top two positions in terms of SRCC. It is interesting to note that the SRCCs obtained by SVR are not as poor as its performance on RMSE and PCC. It is even comparable with GP and RF in some cases, e.g. on F20. This suggests that the prediction made by SVR has a decent chance to differentiate the order between two parameter configurations. In this case, SVR might be useful in a model-based PO process where it can be used as a comparison-based surrogate~\cite{LoshchilovSS10}. Furthermore, we also notice that RBFN does not show a good performance on SRCC. It is even sometimes worse than SVR. This indicates that although the prediction made by RBFN is numerically close to the ground truth, it may still mislead a model-based PO as it messes up the order of similar parameter configurations.
Based on the above discussion, we come up with the following response to RQ2:
\vspace{0.5em}
\noindent
\framebox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{
\underline{\textbf{Response to RQ2}}: \textbf{\textit{GP and RF are able to preserve the order of the empirical performance of different parameter configurations. In particular, SVR, which performs poorly on predicting the empirical performance, shows comparable performance for order preservation.}}
}}
\subsection{Comparisons of Landscape Approximation}
\label{sec:landscape}
In previous subsections, we mainly focus on investigating the quality of surrogate models from the approximation accuracy perspective. For the last RQ, we plan to study of the quality of surrogate models from a landscape analysis perspective. Considering the testing data set, we compare the landscapes of the empirical performance predicted by different regression algorithms to the landscape of the ground truth. To this end, we use the kernel density estimation (KDE) method\footnote{https://uk.mathworks.com/help/stats/ksdensity.html} to estimate a probability density function (PDF) of the empirical performance. To have a visual comparison, Figs~\ref{fig:KL_2D} to~\ref{fig:KL_30D} shows the plots of the estimated PDFs of four different regression algorithms and the ground truth. From these figures, we can see that the prediction made by GP, RF and RBFN almost fit the distribution of the ground truth. In contrast, the estimated PDF of SVR deviates from the
|
ground truth in many cases. This becomes more evident when the dimensionality of the underlying problem becomes large.
Since the surrogate model considered in this paper is a mapping between a parameter configuration and its corresponding empirical performance, it is interesting to consider a more complex landscape that is a joint probability distribution of parameter configuration and empirical performance. As it is non-trivial to visualise a multi-dimensional distribution, we try to understand the proximity of the landscape approximated by the surrogate model and that of the ground truth from a statistical distance perspective. To this end, we apply the earth mover's distance (EMD)~\cite{RubnerTG00}, also known as Wasserstein metric, to evaluate the dissimilarity between two multi-dimensional distributions. Generally speaking, given two distributions, the EMD measures the minimum cost of turning one distribution into the other. In our context, similar landscapes are expected to have a relatively small EMD whereas large EMD values will imply that the landscapes are significantly different from each other. Due to the page limit, we do not intend to elaborate the calculation procedure of EMD, interested readers can refer to~\cite{RubnerTG00} for more details. From the comparison results of EMD values shown in~\prettyref{tab:emd}, we find that GP, RF and RBFN have the same level of approximation to the ground truth whereas the divergence values obtained by SVR are relatively large in almost all cases. All these observations are also in line with the RMSEs discussed in~\prettyref{sec:RMSE_comparison}.
Based on the above discussion, we come up with the following response to RQ3:
\vspace{0.5em}
\noindent
\framebox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{
\underline{\textbf{Response to RQ3}}: \textbf{\textit{The landscapes of the empirical performance predicted GP, RF and RBFN well approximate the ground truth; while the landscapes obtained by SVR deviate from the ground truth to a certain extent.}}
}}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/KDE_2D.pdf}
\caption{Estimated probability density distribution of the empirical performance predicted by four different regression algorithms and the ground truth ($d=2$).}
\label{fig:KL_2D}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/KDE_10D.pdf}
\caption{Estimated probability density distribution of the empirical performance predicted by four different regression algorithms and the ground truth ($d=10$).}
\label{fig:KL_10D}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{figs/KDE_30D.pdf}
\caption{Estimated probability density distribution of the empirical performance predicted by four different regression algorithms and the ground truth ($d=30$).}
\label{fig:KL_30D}
\end{figure}
\begin{table*}[htbp]
\centering
\caption{Comparisons of EMD between the surrogate model built by four regression algorithms and the ground truth}
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c|c||c|c|c|c|c|c}
\hline
\textbf{Problem} & \textbf{$d$} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} & \textbf{Problem} & \textbf{$d$} & \textbf{GP} & \textbf{RBFN} & \textbf{RF} & \textbf{SVR} \\
\hline
\multirow{1}[6]{*}{F1} & 2 & 3.9123E-2 & 4.1131E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.8218E-2} & 2.1449E-1 & \multirow{1}[6]{*}{F11} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.6881E+0} & 2.0925E+0 & 2.1808E+0 & 1.0413E+1 \\
\cline{2-6}\cline{8-12} & 10 & 7.4359E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.0765E-1} & 8.3284E-1 & 4.3693E+0 & & 10 & 1.8064E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.6778E+1} & 1.8437E+1 & 2.7649E+2 \\
\cline{2-6}\cline{8-12} & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4450E+0} & 1.8732E+0 & 2.7342E+0 & 1.0298E+1 & & 30 & 1.2526E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{9.1694E+1} & 1.6699E+2 & 5.3056E+3 \\
\hline
\multirow{1}[6]{*}{F2} & 2 & 7.7335E-1 & 8.1710E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6167E-1} & 1.9532E+0 & \multirow{1}[6]{*}{F12} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2890E+6} & 1.3150E+6 & 2.1687E+6 & 2.3287E+7 \\
\cline{2-6}\cline{8-12} & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.7967E+2} & 1.9940E+2 & 2.4552E+2 & 3.6687E+3 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2284E+7} & 1.2908E+7 & 1.7883E+7 & 4.9220E+8 \\
\cline{2-6}\cline{8-12} & 30 & 1.8648E+3 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.5890E+3} & 2.9031E+3 & 2.4368E+5 & & 30 & 1.7468E+8 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.7299E+8} & 2.3724E+8 & 7.7375E+9 \\
\hline
\multirow{1}[6]{*}{F3} & 2 & 2.5151E+1 & 2.6593E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.7417E+1} & 3.0550E+1 & \multirow{1}[6]{*}{F13} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.0126E-1} & 6.3972E-1 & 8.5819E-1 & 3.8741E+0 \\
\cline{2-6}\cline{8-12} & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{8.5900E+2} & 1.1733E+3 & 9.8913E+2 & 1.5110E+4 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.4477E+0} & 1.8491E+1 & 7.6669E+0 & 1.6785E+2 \\
\cline{2-6}\cline{8-12} & 30 & 8.8047E+3 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.7415E+3} & 1.2159E+4 & 7.9610E+5 & & 30 & 4.5265E+1 & 6.9909E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.6507E+1} & 7.9910E+2 \\
\hline
\multirow{1}[6]{*}{F4} & 2 & 2.2263E-1 & 2.5603E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.1234E-1} & 4.1061E-1 & \multirow{1}[6]{*}{F14} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.4411E-1} & 3.5736E-1 & 3.9440E-1 & 4.1817E-1 \\
\cline{2-6}\cline{8-12} & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.4946E-1} & 2.6379E-1 & 3.5331E-1 & 1.1068E+0 & & 10 & 2.3350E-2 & 7.1733E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.0398E-2} & 4.7365E-2 \\
\cline{2-6}\cline{8-12} & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6201E-2} & 1.5856E-1 & 1.4125E-1 & 6.4510E-1 & & 30 & 1.6935E-2 & 7.1612E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.4585E-2} & 6.7035E-2 \\
\hline
\multirow{1}[6]{*}{F5} & 2 & 1.0446E-2 & 1.1772E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.0122E-2} & 1.1300E-2 & \multirow{1}[6]{*}{F15} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.3576E-1} & 2.4196E-1 & 2.6502E-1 & 3.1265E-1 \\
\cline{2-6}\cline{8-12} & 10 & 2.7905E-2 & 2.7986E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.7347E-2} & 9.4473E-2 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.0227E+0} & 1.0634E+0 & 1.2373E+0 & 3.0674E+0 \\
\cline{2-6}\cline{8-12} & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.3163E-2} & 3.6446E-2 & 4.8344E-2 & 2.5692E-1 & & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.2431E+0} & 4.1342E+0 & 5.0520E+0 & 3.4736E+1 \\
\hline
\multirow{1}[6]{*}{F6} & 2 & 2.1599E-1 & 2.3193E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.0715E-1} & 3.0823E-1 & \multirow{1}[6]{*}{F16} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.8404E-1} & 3.1241E-1 & 3.2114E-1 & 3.9340E-1 \\
\cline{2-6}\cline{8-12} & 10 & 1.4138E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.3799E+0} & 1.5950E+0 & 4.9727E+0 & & 10 & 1.3076E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2970E+0} & 1.5427E+0 & 4.6604E+0 \\
\cline{2-6}\cline{8-12} & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.6476E+0} & 2.8931E+0 & 3.7200E+0 & 1.5060E+1 & & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.3627E+0} & 5.3324E+0 & 6.6650E+0 & 6.0207E+1 \\
\hline
\multirow{1}[6]{*}{F7} & 2 & 5.6405E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.5008E+0} & 7.5562E+0 & 6.5047E+1 & \multirow{1}[6]{*}{F17} & 2 & 6.5516E-2 & 7.2802E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.8908E-2} & 6.8641E-2 \\
\cline{2-6}\cline{8-12} & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.8407E+1} & 6.8467E+1 & 8.7470E+1 & 1.8612E+3 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.8298E-1} & 1.9186E-1 & 1.9132E-1 & 2.3036E-1 \\
\cline{2-6}\cline{8-12} & 30 & 5.1245E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.4561E+2} & 5.4400E+2 & 1.7949E+4 & & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.1992E-1} & 5.7923E-1 & 4.9740E-1 & 1.8528E+0 \\
\hline
\multirow{1}[6]{*}{F8} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.4421E+0} & 8.2598E+0 & 1.0246E+1 & 7.5015E+1 & \multirow{1}[6]{*}{F18} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.9171E+0} & 8.8528E+0 & 9.5477E+0 & 5.0562E+1 \\
\cline{2-6}\cline{8-12} & 10 & 7.3016E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.8773E+1} & 9.2672E+1 & 2.8420E+3 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.8751E+2} & 3.9933E+2 & 6.9504E+2 & 1.9780E+4 \\
\cline{2-6}\cline{8-12} & 30 & 5.8388E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.7055E+2} & 7.4258E+2 & 2.1170E+4 & & 30 & 6.8663E+3 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.9223E+3} & 9.8100E+3 & 3.7050E+5 \\
\hline
\multirow{1}[6]{*}{F9} & 2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2270E+6} & 1.4330E+6 & 1.5548E+6 & 2.0123E+7 & \multirow{1}[6]{*}{F19} & 2 & 2.9912E-1 & 2.8090E-1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{2.6204E-1} & 8.0894E-1 \\
\cline{2-6}\cline{8-12} & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.1771E+5} & 4.7615E+5 & 7.5221E+5 & 8.1209E+6 & & 10 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.5440E-1} & 7.6484E-1 & 8.2718E-1 & 2.5802E+0 \\
\cline{2-6}\cline{8-12} & 30 & 3.9609E+6 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.8603E+6} & 5.1305E+6 & 1.6692E+8 & & 30 & 1.3788E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{1.2999E+1} & 1.5938E+1 & 2.7295E+2 \\
\hline
\multirow{1}[6]{*}{F10} & 2 & 9.5085E+0 & \cellcolor[rgb]{ .651, .651, .651}\textbf{6.9783E+0} & 1.2881E+1 & 6.7627E+1 & \multirow{1}[6]{*}{F20} & 2 & 3.5455E-2 & 3.9468E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.3128E-2} & 3.7422E-2 \\
\cline{2-6}\cline{8-12} & 10 & 7.8455E+1 & \cellcolor[rgb]{ .651, .651, .651}\textbf{7.6291E+1} & 8.3358E+1 & 2.9098E+3 & & 10 & 4.0848E-2 & 4.3134E-2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{3.3434E-2} & 5.6918E-2 \\
\cline{2-6}\cline{8-12} & 30 & 5.7160E+2 & \cellcolor[rgb]{ .651, .651, .651}\textbf{4.3714E+2} & 7.0516E+2 & 2.0198E+4 & & 30 & \cellcolor[rgb]{ .651, .651, .651}\textbf{5.6556E-2} & 7.5748E-2 & 5.7657E-2 & 1.8357E-1 \\
\hline
\end{tabular}
}
\label{tab:emd}
\end{table*}
\section{Introduction}
\label{sec:introduction}
Meta-heuristic algorithms are normally accompanied by some parameters which can influence their search behaviour on various optimisation problems. Parameter optimisation (PO) aims to find a best possible parameter configuration $\mathbf{\theta}^{\ast}$ from the parameter space $\Theta$, which consists of all possible configurations, of the target algorithm and helps it achieve its peak performance on a black-box optimisation problem. Formally, given an algorithm, PO can be defined as the following black-box meta-optimisation problem:
\begin{equation}
\begin{array}{l l}
\mathrm{minimize} \quad \mathcal{L}(f(\mathbf{x}),\theta)\\
\mathrm{subject\ to} \quad \theta\in\Theta
\end{array}
\label{eq:loss}
\end{equation}
where $f(\mathbf{x})$ is the optimisation problem under consideration, and $\mathbf{x}\in\mathbb{R}^d$ is a decision variable. $\mathcal{L}(f(\mathbf{x}),\theta)$ is the performance measure associated with a configuration $\theta$ of the target algorithm. In particular, it can either be the runtime cost (e.g. the CPU wall time and/or the number of function evaluations) or the error of the solution found by the target algorithm.
PO is a challenging black-box meta-optimisation problem. First, its landscape is complex and change with the target algorithm when solving different problems. Second, the parameters associated with the target algorithm can have various types (e.g. numerical, integer and categorical) and the number of parameters can be potentially large depending on the algorithm specification. In addition, PO is intrinsically expensive as it requires to explore $\Theta$ by running the target algorithm with different configurations, where evaluating the effectiveness of a configuration will in turn cost a large amount of function evaluations and/or CPU wall time. In the evolutionary computation (EC) community, constructing a cheap-to-evaluate surrogate in lieu of calling the physically expensive objective function has been widely accepted as an effective way for expensive optimisation~\cite{Jin11}. The design and analysis of computer experiments in statistics also uses surrogate models to either fit a global model of the overall landscape or sequentially identify the global optimum of the underlying function~\cite{SantnerWN03}. In the automatic parameter configuration field, sequential model-based Bayesian optimisation methods~\cite{Bartz-BeielsteinLP05,HutterHL11,ThorntonHHL13} have shown strong performance in PO, compared to some traditional methods like grid search and random search~\cite{BergstraB12} and can compete or even surpass the results tuned by experienced human experts. Moreover, regression models have been extensively used in meta-learning to predict the algorithm performance across various datasets~\cite{Reif14}. It is worth to note that all these lines of research need to construct surrogate models of a computationally expensive and complex function in order to inform an active learning criterion that identifies new inputs to evaluate.
The problem of PO has a long history dating back to the 90s~\cite{KohaviJ95}. Recently, it becomes increasingly popular in both meta-heuristics (e.g.~\cite{Bartz-BeielsteinLP05,HutterHL11,BlotHJKT16,LopDubPerStuBir2016irace,LiFKZ14}) and machine learning (e.g.~\cite{SnoekLA12,ThorntonHHL13,SandersG17,CaoKWL12,LiWKC13,CaoKWL14,CaoKWLLK15}) communities, especially with the development of emerging automated machine learning~\cite{automl}. In this paper, instead of developing new algorithms for PO, we focus on studying surrogate models, which sit in the core of the model-based PO framework. We take the differential evolution (DE)~\cite{StornP97,LiKWCR12}, one of the most popular black-box optimiser in the EC community, as the baseline algorithm. To obtain the empirical performance data on a given optimisation problem, we evaluate the performance of DE with respect to 5,940 parameter configurations in an expensive offline phase. The collected performance data are used to train a regression model and to validate its generalisation ability for predicting empirical performance of unseen parameter configurations. Here we consider four off-the-shelf regression algorithms for empirical performance modelling. In particular, we evaluate and compare their abilities in terms of how well they predict the empirical performance with respect to a particular parameter configuration, and also how well they approximate the parameter configuration \textit{versus} the empirical performance landscapes. We envisage that this aspect will shed light on the study of the characteristics of surrogate models in future.
The rest of this paper is organised as follows. \prettyref{sec:method} describes the methodologies that we used to setup the experiments. \prettyref{sec:experiments} presents and analyses the experimental results. Finally, \prettyref{sec:conclusion} concludes this paper and provides some future directions.
\section*{Acknowledgment}
This work was supported by the Royal Society (Grant No. IEC/NSFC/170243).
\bibliographystyle{IEEEtran}
\section{Methodology}
\label{sec:method}
This section mainly describes the benchmark problems chosen in our empirical studies, the baseline algorithm DE and its corresponding parameters, the performance measure used to evaluate the quality of a particular parameter configuration, the method used to collect the algorithm performance data, and the regression algorithms used to build surrogates for modelling the empirical performance.
\subsection{Benchmark Problems}
\label{sec:benchmark}
In this paper, we consider choosing six widely used elementary test problems (i.e. sphere, ellipsoid, rosenbrock, ackley, griewank and rastrigin) and the first fourteen test problems (i.e. excluding those hybrid composite functions) from the CEC 2005 competition~\cite{SuganthanHDLCAT05} to constitute the benchmark problems. To facilitate the notation in~\prettyref{sec:experiments}, the six elementary functions are denoted as F1 to F6 and those from the CEC 2005 competition are denoted as F7 to F20. Note that these test problems have various characteristics. In particular, F1, F2 and F7 to F11 are unimodal functions while the others are multi-modal functions. All test problems have analytically defined continuous objective functions with a known global optimum. The number of variables of each test problem varies from 2 to 30 (in particular $d\in\{2,10,30\}$) and the range of variables is set according to their original paper.
\subsection{DE and its Parameters}
\label{sec:de}
DE~\cite{StornP97} is one of the most popular black-box optimisation algorithm in the EC community including evolutionary multi-objective optimisation~\cite{LiZKLW14,LiKD15,LiKZD15,LiDZK15,LiDZZ17,ChenLY18,LiCMY18}. One of the major reasons that contributes to its success is its simple structure. For a vanilla DE, an offspring solution $\mathbf{x}^c$ is generated by a two-step procedure. First, a trial vector $\overline{\mathbf{x}}$ is generated as:
\begin{equation}
\overline{\mathbf{x}}=\mathbf{x}^1+F\times(\mathbf{x}^2-\mathbf{x}^3)
\label{eq:trial}
\end{equation}
where $F\in(0,3]$, known as the evolution step size, is a parameter of DE. $\mathbf{x}^1$, $\mathbf{x}^2$ and $\mathbf{x}^3$ are randomly chosen from the parent population. Afterwards, $\mathbf{x}^c$ is generated as:
\begin{equation}
x^c_i=
\begin{cases}
\overline{x_i} & \quad \text{if } (rand<CR)\lor(i=j)\\
x_i & \quad \text{otherwise}
\end{cases}
\end{equation}
where $i\in\{1,\cdots, d\}$, $j$ is an integer randomly chosen from 1 to $d$. $\mathbf{x}$ is the parent solution under consideration. $rand$ is a random number chosen from 0 to 1, and $CR\in[0,1]$, known as the crossover rate, is another parameter of DE. In addition, the population size $NP\in\mathbb{N}$ is also a parameter.
Many studies have demonstrated that the performance of DE is highly sensitive to its parameter settings~\cite{DasS11}. During the past decade, many efforts have been devoted to the development of advanced DE variants that are able to adaptively set the parameters on the fly~\cite{BrestGBMZ06,QinHS09,LiFK11} and/or find a good configuration in an offline manner~\cite{BelkhirDSS16}. Since the major purpose of this paper is to investigate the ability of building the surrogate for modelling the empirical performance of an algorithm with respect to its corresponding parameter configurations, we focus on the vanilla DE~\cite{StornP97} which is simple yet without losing the generality of the observations. Obviously, $NP$ is an integer parameter, while $F$ and $CR$ are numerical parameters.
\subsection{Performance Measure}
\label{sec:measure}
As the global optimum of each test problem is known a priori, this paper uses the approximation error to evaluate the empirical performance of a particular parameter configuration. Specifically, it is computed as:
\begin{equation}
\Psi(f(\mathbf{x}),\theta)=f(\mathbf{x})-f(\mathbf{x}^{\ast})
\end{equation}
where $\theta$ is a parameter configuration of DE, $\mathbf{x}$ is the best-so-far solution found by the DE with the parameter configuration $\theta$, and $\mathbf{x}^\ast$ is the global optimum. Since DE is a stochastic algorithm, each parameter configuration needs to be repeated more than one time in practice. Thus, the performance of a parameter configuration $\theta$ is measured as an averaged approximation error:
\begin{equation}
\mathcal{L}(f(\mathbf{x}),\theta)=\frac{1}{n}\sum_{i=1}^n\Psi_i(f(\mathbf{x}),\theta)
\end{equation}
where $\Psi_i(f(\mathbf{x}),\theta)$ is the approximation error of a configuration $\theta$ at the $i$-th run and $n$ is the number of repetitions of experiments with $\theta$ where we set $n=31$ in our experiments.
\subsection{Data Collection}
\label{sec:data}
In principle, algorithm performance data used to construct the surrogate model of an algorithm's empirical performance can be obtained by any means. Since this paper aims to investigate the overall surrogate modelling ability of an algorithm's performance with respect to its parameter space, we are interested in every corner of the space. To this end, the parameter space is sampled in a grid manner, where we chose 9 different $NP$ settings, i.e. $NP=i\times d$, $i\in\{2,\cdots,10\}$, 60 different values for $F\in(0,3]$ with a step size 0.05, and 11 different values for $CR\in[0,1]$ with a step size 0.1. Therefore, there are 5,940 different parameter configurations in total.
\subsection{Regression Algorithms for Surrogate Modelling}
\label{sec:surrogate}
In this paper, four regression algorithms, i.e. Gaussian process (GP), random forest (RF), support vector machine for regression (SVR), radial basis function networks (RBFN), are considered as the candidates for surrogate modelling of DE's empirical performance. Note that these regression algorithms have been widely used in the model-based PO in the algorithm configuration literature~\cite{HutterXHL14,WuKJLZ17,WuLKZZ18}.
To construct a surrogate model on a particular problem instance, each of these four models is trained on the performance data (only 70\% of them are used for training while the remaining 30\% are used for testing) collected by running the DE algorithm with various parameter configurations on each problem instance as introduced in~\prettyref{sec:data}. Note that learning a surrogate model is no free lunch, as each regression algorithm also requires some hyper-parameters to be tuned. To identify the best possible configurations for each regression algorithm, we apply the random search~\cite{BergstraB12} to explore the hyper-parameter space. Specifically, as for GP, we need to choose an appropriate kernel among RBF, rational quadratic and Mat\'ern; as for RF, the number of trees in a forest is chosen from 2 to 100, the minimum number of samples required to split an internal node is chosen from 2 to 11, the number of features to consider when looking for the best split is set in the range $[0.001,1]$, the criterion used to measure the quality of a split is either mean squared error or mean absolute error and the minimum number of samples required to be at a leaf node is chosen from 1 to 11; as for SVR, the kernel is chosen between RBF and Sigmoid, the maximal margin $\epsilon$ is chosen from $[0.01,1]$, the regularisation parameter $C$ is set in between 1 and 10, and $\gamma$ is chosen from $[0.01,1]$ if RBF is used as the kernel. A 5-fold cross-validation (using 80\% of the training data for training and the remaining 20\% data for testing) is used to evaluate the training performance of a particular hyper-parameter configuration of a regression algorithm. To have a fair comparison, all surrogate modelling procedures are implemented by \texttt{scikit-learn}, a machine learning toolbox in Python\footnote{\url{https://scikit-learn.org/stable/}}.
|
\section{Introduction}
Highly nonlinear functions over finite vector spaces have attracted much interest in the last several years, for both their applications to cryptography (see~\cite{cha1} for example) and their connections to a variety of different combinatorial structures. The functions that are furthest from linear are called {\it perfect nonlinear}; unfortunately, none exist for binary vector spaces, which are the most cryptographically useful. However functions do exist in several lesser categories of nonlinearity, such as {\it almost perfect nonlinear}, {\it almost bent}, and {\it crooked}. We focus on the latter, which is the most specialized of the three.
Crooked functions were introduced by Bending and Fon~Der~Flaass~\cite{ben1}, who, building on the graphs of de~Caen, Mathon and Moorhouse~\cite{dec2}, showed that every crooked function defines a distance-regular graph of diameter $3$ with a particular intersection array. Shortly thereafter, van~Dam and Fon~Der~Flaass~\cite{van3} observed that every crooked function defines a binary code of minimum distance $5$, similar to the classical Preparata code. In this paper, we show that the converse of each of these results is also true: crooked functions can be characterized using both Preparata-like codes (Theorem~\ref{thm:crookedcode}) and distance-regular graphs (Theorem~\ref{thm:crookedgraph}). Those codes and graphs offer a more combinatorial way of understanding the nature of nonlinear binary functions.
\section{Almost Perfect Nonlinear Functions}
Before considering crooked functions we need to characterize a more general class, namely {\it almost perfect nonlinear} functions. Throughout this article, let $V := V(m,2)$, a vector space of dimension $m$ over $\ff{2}$, with $m$ odd. Given a function $f:V \rightarrow V$, consider the following system of equations:
\begin{equation}\label{eqn:apn}
\left\{ \begin{array}{rcl}
x + y & = & a \\
f(x) + f(y) & = & b
\end{array} \right\}.
\end{equation}
Note that solutions to \eqref{eqn:apn} come in pairs: if $(x,y)$ is a solution, then so is $(y,x)$. If $f$ is a linear function, then equation \eqref{eqn:apn} has $2^m$ solutions when $b = f(a)$. We say $f$ is {\it almost perfect nonlinear} if, for every $(a,b) \neq (0,0)$, the system has at most two solutions. Equivalently, $f$ is almost perfect nonlinear if and only if for all $a \neq 0$ in $V$, the set
\[
H_a(f) := \{f(x) + f(x+a) \mid x \in V\}
\]
has cardinality $2^{m-1}$.
We may construct a binary code from a function on $V$ in the following manner. Identify $V$ with the finite field $\ff{2^m}$, and let $\al$ be a primitive element of $\ff{2^m}$. Also let $n = 2^m-1$, and assume $f:V \rightarrow V$ is a function such that $f(0) = 0$. We define a parity check matrix $H_f$ by
\[
H_f := \left( \begin{matrix}
1 & \al & \al^2 & \ldots & \al^{n-1} \\
f(1) & f(\al) & f(\al^2) & \ldots & f(\al^{n-1})
\end{matrix} \right),
\]
and define the code $C_f$ to be the kernel of $H_f$ over $\ff{2}$.
The code $C_f$ can be thought of as a generalization of the double error-correcting BCH code, which is the specific case of $f(\al) := \al^3$. It is clear from the parity check matrix that the minimum distance of $C_f$ is at least $3$, and it can be shown that the minimum distance is at most $5$. The following characterization is due to Carlet, Charpin, and Zinoviev~\cite[Theorem~5]{ccz}.
\begin{theorem}\label{thmparam}
The minimum distance of $C_f$ is $5$ if and only if $f$ is almost perfect nonlinear. In this case, the dimension of $C_f$ is
\[
k = 2^m - 2m - 1.
\]
\end{theorem}
In the next section, we give a similar characterization of crooked functions, which are a special class of almost perfect nonlinear functions.
\section{Crooked Functions and Preparata-like Codes}
A function $f: V \rightarrow V$ is {\it crooked} if the following three conditions hold:
\begin{enumerate}
\item $f(0) = 0$; \label{itm:crooked1}
\item $f(x) + f(y) + f(z) \neq f(x+y+z) \quad$ for distinct $x$, $y$, and $z$; \label{itm:crooked2}
\item $f(x) + f(y) + f(z) \neq f(x+a) + f(y+a) + f(z+a) \quad$ for all $a \neq 0$. \label{itm:crooked3}
\end{enumerate}
Condition~\ref{itm:crooked2} is equivalent to almost perfect nonlinearity; thus every crooked function is almost perfect nonlinear. Condition~\ref{itm:crooked3} states that for every $a \neq 0$, no three points in $H_a(f)$ are collinear. It follows that $f$ is crooked if and only if $f(0) = 0$ and $H_a(f)$ is the complement of a hyperplane for all $a \neq 0$. Note that we are using the original definition of crooked functions given in~\cite{ben1}, rather than the generalization appearing in Byrne and McGuire~\cite{bm1} or Kyureghyan~\cite{kyu1}.
The canonical example of a crooked function is the {\it Gold} function. Identify $V$ with $\ff{2^m}$ for odd $m$; then $f(x) := x^{2^k+1}$ is called a Gold function if $\nobreak{\gcd(k,m) = 1}$. More generally, $f(x) := x^{2^k+2^j}$ is crooked provided that $\nobreak{\gcd(k-j,m) = 1}$, and Kyureghyan~\cite{kyu1} has shown that all crooked power functions have this form. For recent progress in constructing nonlinear functions which are not equivalent to the Gold functions, see~\cite{bcfl, bcp1, ekp1}.
Just as almost perfect nonlinear functions give rise to BCH-like codes, crooked functions give to Preparata-like codes. Given $f: V \rightarrow V$ such that $f(0) = 0$, let $P_f$ be the code whose codewords are the characteristic vectors of $(S,T)$, for $S \subset V^*$ and $T \subset V$, such that the following three conditions hold:
\begin{itemize}
\item $|T|$ is even,
\item ${\displaystyle \sum_{r \in S} r = \sum_{r \in T} r}$, and
\item ${\displaystyle f\Big(\sum_{r \in S} r \Big) = \sum_{r \in S}f(r) + \sum_{r \in T}f(r)}$.
\end{itemize}
Identifying $V$ with $\ff{2^m}$, we get the actual Preparata code when $f(x)~:=~x^3$ and the generalized Preparata code when $f(x)~=~x^{2^k+1}$ (see~\cite{bak1}). In general $P_f$ is not linear, and it is easy to verify that $P_f$ always has minimum distance at least $3$. The following result is due to Van~Dam and Fon~Der~Flaass~\cite[Theorem~7]{van3}.
\begin{theorem}
If $f$ is crooked, then $P_f$ has minimum distance $5$ and size $2^{2^{m+1}-2m-2}$.
\label{thmcrkpf}
\end{theorem}
If $P_f$ has minimum distance $5$, then it is {\it nearly perfect}: it satisfies the Johnson bound~\cite[Theorem~17.13]{mac1} with equality. Hence $P_f$ has minimum distance at most $5$ for any $f$. We show the converse of Theorem~\ref{thmcrkpf}.
\begin{theorem}
If $P_f$ has minimum distance $5$, then $f$ is crooked.
\label{thm:crookedcode}
\end{theorem}
\begin{proof}
We assumed in the definition of $P_f$ that $f(0) = 0$, so condition~\ref{itm:crooked1} of crookedness is satisfied. If $P_f$ has minimum distance $5$, then there is no pair $(\phi, T)$ in $P_f$ with $|T| = 4$. That is, for any distinct $w,x,y,z$ such that $w+x+y+z = 0$,
\begin{equation}
f(w) + f(x) + f(y) + f(z) \neq 0.
\label{eqncrkapn}
\end{equation}
Thus condition~\ref{itm:crooked2} of crookedness is also satisfied, and it remains to show condition~\ref{itm:crooked3}. Since condition~\ref{itm:crooked2} is saitsfied, $f$ is almost perfect nonlinear and $C_f$ has dimension $2^m - 2m - 1$ by Theorem~\ref{thmparam}. But $C_f$ is the kernel of $H_f$, so it follows that $H_f$ has a column space of dimension $2m$, namely $V \times V$. This implies that for any $(a,b)$ in $V \times V$, there is a subset $S$ of $V^*$ such that
\begin{equation}
\left(\sum_{r \in S} r, \sum_{r \in S} f(r)\right) = (a,b).
\label{eqncfspan}
\end{equation}
Given any $x \in V$, let $T = \{x,0\}$, so that $|T|$ is even and $\sum_{r \in T} r = x$. Then from equation \eqref{eqncfspan}, there exists some $S \subset V^*$ such that
\[
\Big(\sum_{r \in S} r, \sum_{r \in S} f(r)\Big) = \left(x, 0 \right).
\]
Choosing $S$ in this way, $(S,T)$ is in $P_f$. Now given any $y$, $z$ and $a \neq 0$, consider
\[
(S',T') := (S \oplus \{y\} \oplus \{y+a\}, T \oplus \{z\} \oplus \{z+a\}).
\]
This vector is at distance $4$ from $(S,T)$. Since $P_f$ has distance $5$, $(S',T')$ must not be in $P_f$. But $|T'|$ is even, and
\[
\sum_{r \in S'} r = \sum_{r \in T'}
|
r;
\]
hence for $(S',T') \notin P_f$ it must be the case that
\[
f\Big(\sum_{r \in S'}r\Big) \neq \sum_{r \in S'}f(r) + \sum_{r \in T'} f(r).
\]
This implies
\[
f(x+a) \neq \sum_{r \in S} f(r) + f(y) + f(y+a) + \sum_{r \in T} f(r) + f(z) + f(z+a),
\]
or in other words
\[
f(x+a) \neq f(y) + f(y+a) + f(x) + f(z) + f(z+a).
\]
Thus condition~\ref{itm:crooked3} of crookedness is satisfied for $f$.
\end{proof}
\section{Crooked Graphs}
\begin{comment}
If $f$ is a function from $V$ to $V$, the {\it Fourier transform} of $f$ is the function $\mu_f: V \times V \rightarrow \re$ given by
\[
\mu_f(a,b) = \sum_{x \in V} (-1)^{a^Tx + b^Tf(x)}.
\]
See~\cite{ben1} or~\cite{ccz} for information about functions mapping from $V$ to $\ff{2}$. For $m$ odd, $f$ is {\it almost bent} if
\[
\mu_f(a,b) \in \{0, \pm 2^{(m+1)/2} \}
\]
for all values of $(a,b)$ except $(0,0)$. In~\cite[Theorem~6]{ccz}, it is shown that $C_f$ is uniformly packed if and only if $f$ is almost bent.
The {\it coset graph} $\Ga(C_f)$ is the graph whose vertices are the cosets of $C_f$, with two cosets adjacent if and only if they contain words at Hamming distance $1$. The following theorem relates the function to the graph.
\begin{theorem}
If $f$ is almost bent, then $\Ga(C_f)$ is distance-regular with intersection array
\[
\{ 2^m -1, 2^m -2, 2^{m-1}+1; 1, 2, 2^{m-1}-1 \}.
\]
\label{cordreg}
\end{theorem}
Conversely, if $\Ga(C_f)$ is distance-regular, then $C_f$ is completely-regular. If $C_f$ is completely regular with covering radius $t \leq e+1$, then $C_f$ is uniformly packed and $f$ is almost bent.
\end{comment}
As usual, assume $f(0) = 0$. Define the {\it crooked graph} of $f$, denoted $G_f$, to have vertex set $V \times \ff{2} \times V$ with the following adjacency condition: distinct $(a,i,\al)$ and $(b,j,\be)$ are adjacent if and only if
\[
\al + \be = f(a + b) + (i+j+1)(f(a) + f(b)).
\]
It is not difficult to show that any two vertices in the subset
\[
F_{ai} := \{ (a,i,\al) \mid \al \in V\}
\]
are at distance at least three, and that any two distinct subsets $F_{ai}$ and $F_{bj}$ are joined by a perfect matching. It follows that $G_f$ is a $2^m$-cover of the complete graph $K_{2^{m+1}}$, and each $F_{ai}$ is a fibre (for background on covers of complete graphs, see~\cite{god7}). The following theorem is given by Bending and Fon-Der-Flaass~\cite[Proposition 13]{ben1}.
\begin{theorem}
If $f$ is crooked, then $G_f$ is an antipodal distance-regular graph with intersection array
\[
\{ 2^{m+1} -1, 2^{m+1} -2, 1; 1, 2, 2^{m+1}-1 \}.
\]
\label{thmcrookedskew}
\end{theorem}
For background on distance-regular graphs, see~\cite{bcn1}. Again, we show the converse.
\begin{theorem}
If $G_f$ is distance-regular with intersection array
\[
\{ 2^{m+1} -1, 2^{m+1} -2, 1; 1, 2, 2^{m+1}-1 \},
\]
then $f$ is crooked.
\label{thm:crookedgraph}
\end{theorem}
\begin{proof}
For convenience, consider the graph $G'_f$ which consists of $G_f$ with a loop added to every vertex. This can be done by removing the restriction $(a,i,\al) \neq (b,j,\be)$ from the adjacency condition of $G_f$. If $G_f$ is distance-regular with $a_1 = 0$ and $c_2 = 2$, then $G'_f$ is a graph with the property that any two vertices at distance $1$ or $2$ have exactly two common neighbours.
That is, for any two vertices $(a,i,\al), (b,j,\be)$ such that $(a,i) \neq (b,j)$, there are exactly two vertices $(c,k,\ga)$ such that
\begin{align}
\al + \ga & = f(a + c) + (i+k+1)(f(a) + f(c)), \label{eqn:cond1} \\
\be+\ga & = f(b + c) + (j+k+1)(f(b) + f(c)). \label{eqn:cond2}
\end{align}
We restrict our attention to the cases in which $i = j$, so that $a \neq b$.
Adding \eqref{eqn:cond1} and \eqref{eqn:cond2} together, there are exactly two pairs $(c,k)$ such that
\[
\al + \be = f(a + c) + f(b + c) + (i+k+1)(f(a) + f(b)).
\]
Running over all values of $\al + \be$, we see that for fixed $(a,b,i)$, the multiset
\[
\{ f(a + c) + f(b + c) + (i+k+1)(f(a) + f(b)) \mid c \in V, k \in \ff{2} \}
\]
\begin{equation}
= \{ f(a+c) + f(b+c) \mid c \in V \} \cup \{ f(a+c) + f(b+c) + f(a) + f(b) \mid c \in V \}
\label{eqndd}
\end{equation}
contains each element of $V$ exactly twice.
Now for some fixed $c$, consider $f(a+c) + f(b+c)$. Letting $c' := c + a + b$, we have
\[
f(a+c) + f(b+c) = f(a+c') + f(b+c').
\]
However, the value $f(a+c) + f(b+c)$ only occurs twice in \eqref{eqndd}, so there is no third solution $c'' \neq c,c'$ such that
\[
f(a+c) + f(b+c) = f(a+c'') + f(b+c'').
\]
In other words, letting $x = a+c$, $y = b+c$, and $z = a+c''$, we have
\[
f(x) + f(y) \neq f(z) + f(x+y+z)
\]
for $z \neq x,y$. This is condition~\ref{itm:crooked2} of crookedness for $f$. Also because $\nobreak{f(a+c) + f(b+c)}$ has already occured twice in \eqref{eqndd}, there is no $c''$ such that
\[
f(a+c) + f(b+c) = f(a+c'') + f(b+c'') + f(a) + f(b).
\]
Setting $x = a+c$, $y = a+c''$, $z = a$ and $w = a+b$, we have
\[
f(x) + f(x+w) \neq f(y) + f(y+w) + f(z) + f(z+w)
\]
for any $x,y,z$ and $w$, with $w \neq 0$. This is the condition~\ref{itm:crooked3} of crookedness, so $f$ is crooked.
\end{proof}
\begin{comment}
For convenience, let
\[
D(a,i,b,j) := f(a + b) + (i+j+1)(f(a) + f(b)).
\]
Then for any two $(a,i,\al), (b,j,\be)$ such that $(a,i) \neq (b,j)$, there are exactly two vertices $(c,k,\ga)$ such that
\[
\al + \ga = D(a,i,c,k), \; \be+\ga = D(b,j,c,k).
\]
This implies there are exactly two pairs $(c,k)$ such that
\[
\al + \be = D(a,i,c,k) + D(b,j,c,k).
\]
Running over all values of $\al + \be$, we see that for fixed $(a,i,b,j)$, the multiset
\[
\{ D(a,i,c,k) + D(b,j,c,k) \mid c \in V, k \in \ff{2} \}
\]
contains each element of $V$ exactly twice. If $i = j$, then $a \neq b$, and this multiset becomes
\[
\{ f(a+c) + f(b+c) + (i + k + 1)(f(a) + f(b)) \mid c \in V, k \in \ff{2} \}
\]
\begin{equation}
= \{ f(a+c) + f(b+c) \mid c \in V \} \cup \{ f(a+c) + f(b+c) + f(a) + f(b) \mid c \in V \}.
\label{eqndd}
\end{equation}
Now for some fixed $c$, let
\[
t := f(a+c) + f(b+c).
\]
Then with $c' = c + a + b$, we also have
\[
t = f(a+c') + f(b+c').
\]
Since $t$ only occurs twice in \eqref{eqndd}, there is no third solution $c'' \neq c,c'$ such that
\[
t = f(a+c'') + f(b+c'').
\]
In other words, letting $x = a+c$, $y = b+c$, and $z = a+c''$,
\[
f(x) + f(y) \neq f(z) + f(x+y+z)
\]
for $z \neq x,y$. This is the second condition of crookedness for $f$. Furthermore, since $t$ occurs only twice in \eqref{eqndd}, there is also no $c''$ such that
\[
t = f(a+c'') + f(b+c'') + f(a) + f(b).
\]
Letting $x = a+c$, $y = a+c''$, $z = a$ and $w = a+b$, we have
\[
f(x) + f(x+w) \neq f(y) + f(y+w) + f(z) + f(z+w)
\]
for any $x,y,z$ and $w$, with $w \neq 0$. This is the third condition of crookedness for $f$, so $f$ is crooked.
\end{comment}
|
\section{Introduction}
Pulsars produce highly relativistic winds that consist of electrons and positrons, and possibly a hadronic component \citep[e.g.,][]{Cheng1980, Bednarek2003}. Due to the nature of the wind, the ideal magnetohydrodynamic limit is satisfied, leading to the pulsar's magnetic field being frozen into the out-flowing wind \citep[e.g.,][]{Kirk2009}. When the ram pressure of the wind is equal to the confining pressure of the ambient medium, a termination shock is formed \citep{Rees1974} where the charged particles are re-accelerated \citep[e.g.,][]{Reynolds1984}.
Downstream of the termination shock the electrons (and positrons) interact with the frozen-in magnetic field, leading to synchrotron radiation that is observed from radio to X-ray wavelengths. Additionally, the electrons can also inverse Compton (IC) scatter ambient photons to high-energy (HE) and very-high-energy (VHE) gamma-ray wavelengths. These ambient photons can have a number of origins, including the cosmic microwave background radiation (CMBR), infra-red (IR) radiation from dust, starlight, and even the radiated synchrotron photons. The non-thermal emission leads to a luminous nebula, commonly known as a pulsar wind nebula (PWN).
A growing number of PWNe are observed that are located inside shell-type supernova remnants (SNRs), with this type of system commonly known as composite remnants. Notable examples include the PWN G21.5--0.9 \citep[e.g.,][]{Bocchino2005}, and the Vela PWN \citep[e.g.,][]{Bock1998}. The presence of a shell component plays an important role in determining the morphological evolution of a PWN, which in turn influences the non-thermal emission. The evolution of the nebula can broadly be divided into three phases:
\begin{itemize}
\item
In the initial phase the pulsar injects energy at a constant rate into the nebula, resulting in a PWN that expands supersonically into the slow-moving stellar material that was ejected during the preceding supernova explosion \citep[e.g.,][]{Vanderswaluw2001, Gaensler2006}. Theoretical models predict that the expansion of the outer boundary of the PWN can be described by $R\s{pwn} \propto t^{1.1-1.2}$, while the average magnetic field decreases as $B\propto t^{-1.3}$ \citep[e.g.,][]{Reynolds1984}.
\item
The shock wave of the SNR consists of both a forward and reverse shock. The reverse shock initially expands outwards along with the forward shock, but when the pressure in the remnant become sufficiently small, the reverse shock will start to propagate towards the centre of the shell \citep{McKee1974}. The PWN will enter the next evolutionary phase when the reverse shock reaches $R\s{pwn}$. The reverse shock compresses the PWN \citep[e.g.,][]{Vanderswaluw2001, Bucciantini2003}, leading to a larger magnetic field, and consequently larger synchrotron losses and a brighter radio/X-ray nebula \citep[e.g.,][]{Reynolds1984}.
\item
After the compression of $R\s{pwn}$, the PWN enters a second expansion phase. In contrast to the initial expansion phase, $R\s{pwn}(t)$ does not evolve smoothly, but has an oscillatory nature \citep[e.g.,][]{Vanderswaluw2001, Bucciantini2003}, i.e., the PWN goes through various contractions and expansions. As the ejecta surrounding the PWN has been heated by the reverse shock, the expansion of $R\s{pwn}$ in this phase is subsonic. If the energy output of the pulsar has significantly declined at the onset of the last phase, then the conservation of magnetic flux implies that an expanding nebula will lead to a decrease in the average magnetic field. As a result, the synchrotron component of the non-thermal emission will grow fainter with time.
\end{itemize}
It is well-known that the X-ray synchrotron emission observed from PWNe is produced by a young population of electrons, as these particles have a relatively short lifetime \citep[e.g.,][]{Shklovsky1957}. The evolution of the X-ray emission is therefore correlated with the evolution of the magnetic field, which in turn is dependent on the morphological evolution of the nebula. By contrast, the electrons producing VHE gamma-ray emission have a significantly longer lifetime, implying that the TeV emission observed from PWNe is produced by particles that have accumulated over the lifetime of the pulsar \citep[e.g.,][]{Dejager2009}. This is strikingly illustrated by the energy-dependent morphology of the $\sim 21\,\text{kyr}$ old nebula HESS J1825-137, where VHE gamma-ray observations reveal a PWN that is significantly larger than the associated X-ray nebula \citep{H_Aharonian2006}. For a PWN with an average magnetic field of $B=5\,\mu\text{G}$, the lifetime of an electron emitting $1\,\text{keV}$ X-rays is $\sim 3\,\text{kyr}$, whereas the corresponding lifetime of an electron producing $1\,\text{TeV}$ gamma-rays is $\sim 19\,\text{kyr}$ \citep[e.g.,][]{Dejager2009}.
Based on the information presented in the previous paragraphs, \citet{Dejager2008a} proposed that the average magnetic field in an aged PWN could evolve below the $B\sim 3\,\mu\text{G}$ value of the interstellar medium (ISM), resulting in these sources being undetectable at synchrotron frequencies. However, due to the longer lifetimes of the VHE gamma-ray producing electrons, these ancient PWNe may still be visible at TeV energies. As PWNe count among the more common TeV sources, the ancient PWN scenario could offer an explanation for a number of unidentified TeV sources that lack a synchrotron counterpart \citep{H_Aharonian2008} .
Two unidentified sources that were proposed by \cite{Tibolla2011} as ancient PWNe candidates are HESS J1427--608 \citep{H_Aharonian2008} and HESS J1507--622 \citep{H_Aharonian2011}, and in this paper we extend the initial time-dependent modeling of \cite{Tibolla2011} for these sources. Before introducing the model, the two-component nature of the PWN electron spectrum is briefly reviewed in Section \ref{sec:two_com}, as this has some implications for the model. In Section \ref{sec:model}, the spatially independent model used to calculate the time evolution of the electron spectrum is presented. In order to test the model, it is first applied to the young PWN G21.5--0.9, with the modeling results discussed at the beginning of Section \ref{sec:results}. This section also focuses on modeling HESS J1427--608 and HESS J1507--622 within a PWN framework. The aim is to not only investigate the ancient PWN hypothesis, but also to determine whether a clear argument can be made for identifying HESS J1427--608 and HESS J1507--622 as PWNe. The final section of the paper deals with the discussion and main conclusions drawn from the modeling.
\section{The two-component electron spectrum}\label{sec:two_com}
Observations of PWNe indicate that the energy spectrum of the electrons (and positrons) responsible for the non-thermal emission can be separated into two distinct components \citep[e.g.,][]{Weiler1978, Gaensler2006, Dejager2009}: (1) a low-energy component producing the radio synchrotron and GeV IC emission, and (2), a high-energy component producing the X-ray synchrotron and TeV IC emission. It further follows from observations that these two components can both be described by a power-law $N_e\propto E^{-\alpha}$, with $\alpha_R \sim 1.0-1.3$ for the low-energy component \citep[e.g.,][]{Weiler1978}, and $\alpha_X \sim 2$ for the high-energy component. The stated value of $\alpha_X$ is specifically in the vicinity of the shock, as synchrotron losses and diffusion will lead to an evolution of $\alpha_X$ as the particles propagate away from the shock \citep[][]{Bocchino2005, Mangano2005, Schock2010}. A similar evolution is not expected for the low-energy component, as diffusion and synchrotron losses are markedly more effective at higher energies. For a discussion on the spatial evolution of the particle spectra in PWNe, \citet{Vorster2013} can be consulted. Although it has never been measured directly, particle evolution models predict that the transition between the two components should occur at an energy of $E\lesssim 0.3\,\text{TeV}$ \citep{Zhang2008, Fang2010, Tanaka2011}.
Motivated by the above-mentioned considerations, particle evolution models often use a broken power-law to describe the spectrum of electrons injected into the PWN at the termination shock \citep[e.g.,][]{Venter2006, Zhang2008, Tanaka2011}. While these models typically assume that the two components connect smoothly, i.e., have the same intensity at the transition, \citet{Dejager2008c} concluded from their modeling of Vela X that the two components do not connect smoothly, but that the low-energy component should cut off steeply in order to connect to the high-energy component. In this paper it will be demonstrated that this discontinuity is not limited to the particle spectrum of the aged Vela PWN, but also seems to be present in the spectrum of the young nebula G21.5--0.9.
The question naturally arises as to how these two electron populations are formed. As demonstrated by \citet{Axford1977}, \citet{Krymskii1977}, \citet{Bell1978}, and \citet{Blandford1978}, diffusive shock acceleration leads to a power law energy spectrum $N_e \propto E^{-\alpha}$, with $\alpha\geq 2$. It is therefore possible to associate the origin of the high-energy component with this process. To explain the low-energy component is more difficult, as $\alpha_R=1$, and one would not naturally associate the origin of this component with diffusive shock acceleration. \citet{Summerlin2012} recently showed that it is nevertheless possible for relativistic magnetohydrodynamic shocks to produce this hard spectrum if particles are subjected to shock drift acceleration.
An alternative explanation for the origin of the low-energy component has also been proposed by \citet{Spitkovsky2008}. Results from particle-in-cell simulations show that the acceleration of particles at the termination shock leads to a relativistic Maxwellian spectrum with a non-thermal power-law tail. This result is also indirectly supported by the modeling of \citet{Fang2010}, and \citet{Grondin2011}, where these authors were able to reproduce the non-thermal emission from four PWNe using the spectrum predicted by \citet{Spitkovsky2008}. The advantage of this spectrum is that it provides a natural explanation for the discrepancy in intensity between the two components. A Maxwellian spectrum will therefore also be used for the modeling of G21.5--0.9.
Extending the simulations of \citet{Spitkovsky2008}, \citet{Sironi2011} found that magnetic reconnection occurring in the striped pulsar wind \citep[e.g.,][]{Coroniti1990} can accelerate particles at the termination shock, leading to a deviation from a Maxwellian spectrum. From their simulations it follows that this modified low-energy component can be described by a power-law with $\alpha_R \sim 1.5$. These results may be supported by the observations of \citet{Dodson2003}. Focusing on the very inner regions of the Vela PWN, these authors found radio lobes in the equatorial plane of the nebula. As magnetic reconnection will also occur around the equatorial plane, it is possible that these lobes are formed by the accelerated particles that have been injected into the nebula.
In summary, it is clear that two distinct electron populations should be present in PWNe. However, the exact nature of these components is still unknown.
\section{The Model}\label{sec:model}
The temporal evolution of the electron spectrum in a PWN can be calculated using the equation \citep[e.g.,][]{Tanaka2010}
\begin{equation}\label{eq:dn_dt}
\frac{\partial N_e(E_e,t)}{\partial t} = Q(E_e,t) + \frac{\partial}{\partial E}\left[\dot{E}(E_e,t)N_e(E_e,t)\right],
\end{equation}
where $E_e$ represents the electron energy and $N_e(E_e,t)$ the number of electrons per energy interval. The number of electrons injected into the PWN at the termination shock, per time and energy interval, is given by $Q(E_e,t)$, while the second term on the right-hand side of Equation (\ref{eq:dn_dt}) describes continuous energy losses (or gains) suffered by the particles, with $\dot{E}(E_e,t)$ the total energy loss rate.
Emulating \citet{Venter2006}, a broken power-law spectrum is used to model the emission from the sources studied in this paper,
\begin{equation}\label{eq:power-law}
Q(E_e,t) = \begin{cases}
Q_{\rm{R}}\left(E_{\rm{b}}/E_e\right), & \text{if } E_{\min} \le E_e \le E_{\rm{b}}\\
Q_{\rm{X}}\left(E_{\rm{b}}/E_e\right)^2, & \text{if } E_{\rm{b}} < E_e \le E_{\max}
\end{cases},
\end{equation}
where $Q_{\rm{R}}$ and $Q_{\rm{X}}$ are normalisation constants, $E_{\min}$ and $E_{\max}$ the minimum and maximum electron energy, respectively, and $E_{\rm{b}}$ the energy where the spectrum transitions between the two components. Note that the indices chosen for the components of the broken power-law follow from the discussion presented in Section \ref{sec:two_com}. Keeping in mind that \citet{Dejager2008c} derived a discontinuous spectrum for Vela X, it is not an a priori requirement that the two components should have the same intensity at $E\s{b}$.
The normalisation constants are determined by the prescription that the total energy in a given component should be some fraction $\eta_i$ ($i=\mbox{ R,X}$) of the pulsar's spin-down luminosity $L(t)$ \citep[e.g.,][]{Venter2006}
\begin{equation}\label{eq:Q_calc}
\int Q_i\left(E_{\rm{b}}/E_e\right)^{p_i} E_e dE_e = \eta_i L(t).
\end{equation}
Assuming that the pulsar is a pure dipole radiator with a braking index of $3$, the time-dependence of the luminosity is given by
\begin{equation}\label{eq:L_t}
L(t) = \frac{L_0}{\left(1+t/\tau\right)^2}.
\end{equation}
In the expression above $L_0$ represents the initial luminosity and $\tau$ the characteristic spin-down time scale of the pulsar.
The total energy loss rate in the model, $\dot{E}$ in Equation (\ref{eq:dn_dt}), includes both synchrotron radiation and IC scattering, as well as adiabatic cooling/heating. The energy loss rate as a result of synchrotron radiation and IC scattering is given by \citep[e.g.,][]{Longair2011}
\begin{equation}\label{eq:E_dot_syn}
\dot{E}\s{n-t}(E_e,t) = \frac{4}{3}\frac{\sigma\s{T}}{\left(m_e c\right)^2 c}E_e^2 U_B\left(1+\frac{U\s{IC}}{U_B}\right),
\end{equation}
where $\sigma\s{T}$ is the Thomson cross-section, $U_B=B^2/8\pi$ the energy density of the magnetic field, and $U\s{IC}$ the energy density of the target photon field. Although Equation (\ref{eq:E_dot_syn}) describes IC scattering in the Thomson regime, \citet{Moderski2005} have shown that this expression is still valid if $U_{\rm{IC}}/U_B \lesssim 3$, and Klein-Nishina effects can be neglected. For the CMBR with an energy density of $U\s{IC}\sim 0.3\,\text{eV}\,\text{cm}^{-3}$, this condition is satisfied for an average magnetic field of $B > 2\,\mu\text{G}$.
Anticipating the modeling results, it was however found that this condition is violated for both HESS J1427--608 and HESS J1507--622, and Klein-Nishina effects are therefore taken into account by multiplying Equation (\ref{eq:E_dot_syn}) with a correction factor $F\s{KN}$. \citet{Moderski2005} have further shown that when the target photon field is described by a black-body spectrum, the modification factor can be approximated by
\begin{equation}\label{eq:KN_correction}
F\s{KN} \sim \frac{1}{\left(1+4\gamma\epsilon\right)},
\end{equation}
where $\gamma=E_e/m_ec^2$ is the Lorentz factor of the particle, and $\epsilon = 2.8k\s{B} T/m_ec^2$. In the last expression $k\s{B}$ represents Boltzmann's constant and $T$ the temperature of the black-body spectrum. The approximation given in Equation (\ref{eq:KN_correction}) is appropriate as the photon fields used for the modeling of the unidentified TeV sources are described by a black-body spectrum.
For adiabatic cooling (heating), the energy loss rate is given by
\begin{equation}\label{eq:E_dot_ad}
\dot{E}\s{ad}(E_e,t) = \frac{1}{3}\left(\nabla \cdot \mathbf{v}\right)E_e,
\end{equation}
where $\mathbf{v}$ is the convection velocity downstream of the termination shock. If the system has a spherical symmetry, then Equation (\ref{eq:E_dot_ad}) can be simplified to
\begin{equation}\label{eq:ad_r}
\dot{E}\s{ad}(E_e,t) = \frac{1}{3r^2}\frac{\partial}{\partial r}\left[r^2 v(r)\right]E_e.
\end{equation}
From the above expression it follows that particles suffer the largest amount of adiabatic losses in the inner part of the system. Furthermore, if $v\propto 1/r^2$, then the particles will suffer no adiabatic losses. Therefore, to correctly include adiabatic losses requires $v(r)$ to be known. \citet{Tanaka2010} included adiabatic losses in their spatially independent PWN using the approximation
\begin{equation}\label{eq:E_dot_ad2}
\dot{E}\s{ad}(E_e,t) = \frac{v\s{pwn}(t)}{R\s{pwn}(t)}E_e,
\end{equation}
where $v\s{pwn}(t)$ and $R\s{pwn}(t)$ are respectively the expansion velocity and radius of the PWN. This approximation is also used for the present model.
As discussed in the Introduction, the PWN goes through three evolutionary phases, with the expansion/compression of $R\s{pwn}(t)$ approximated using the power-laws
\begin{equation}\label{eq:R_t}
R_{\rm{pwn}}(t) = \begin{cases}
R_0(t/t_0)^{r_1} & \text{if } t<t_{\rm{rs}}\\
R_0(t_{\rm{rs}}/t_0)^{r_1}(t/t_{\rm{rs}})^{r_2} & \text{if } t_{\rm{rs}}\le t<t_{\rm{se}}\\
R_0(t_{\rm{rs}}/t_0)^{r_1}(t\s{se}/t_{\rm{rs}})^{r_2}(t/t_{\rm{se}})^{r_3} & \text{if } t\ge t_{\rm{se}}
\end{cases}.
\end{equation}
Here $t\s{rs}$ represents the time needed for the reverse shock of the SNR to reach the PWN, and $t\s{se}$ the time when the PWN enters the second expansion phase. For the initial condition, $R_0=0.01\,\text{pc}$ when $t_0=10\,\text{yr}$ \citep{Gelfand2009}. The values $r_1$, $r_2$, and $r_3$ are not linearly independent, as the size of the PWN predicted by the model must be equal to the observed size. Note that the distance to the source $d$ influences the values of $r_1$, $r_2$ and $r_3$, as a larger value of $d$ implies a larger source, and hence a faster expansion. As a point of reference, \citet{Reynolds1984} calculated that $R_{\rm{pwn}} \propto t^{1.2}$ when $t < \tau$.
Apart from $R\s{pwn}$, the adiabatic loss rate described by Equation (\ref{eq:E_dot_ad2}) is also a function of $v\s{pwn}(t)=dR_{\rm{pwn}}(t)/dt$. \citet{Gelfand2009} calculated that the expansion velocity increases from $v\s{pwn}(t)\sim 1300\,\text{km}\,\text{s}^{-1}$ at $t=0.01\,\text{kyr}$ to $v\s{pwn}(t)\sim 2300\,\text{km}\,\text{s}^{-1}$ at $t=5\,\text{kyr}$. However, these values were calculated for a specific scenario, and are only provided as a point of reference.
The reverse shock time scale $t_{\rm{rs}}$ is given by \cite[e.g.][]{Reynolds1984, Ferreira2008}
\begin{equation}\label{eq:t_rev}
t_{\rm{rs}} = 4 \left(\frac{M_{\rm{ej}}}{3M_{\odot}}\right)^{3/4} \left(\frac{E_{\rm{ej}}}{10^{51}\,\text{erg}}\right)^{-45/100} \biggl(\frac{n_{\rm{ism}}}{ 1\,\text{cm}^{-3}}\biggr)^{-1/3}\,\text{kyr},
\end{equation}
where $M_{\rm{ej}}$ and $E_{\rm{ej}}$ are respectively the mass and kinetic energy of the supernova ejecta, and $n_{\rm{ism}}$ the particle number density of ISM. Using the fiducial value of $E_{\rm{ej}}=10^{51}\,\text{erg}$, along with the values of $M_{\rm{ej}}=5 M_{\odot}$ and $n_{\rm{ism}}= 1\,\text{cm}^{-3}$, leads to the estimate $t\s{rs} \approx 6\,\text{kyr}$. Using a smaller value for the ISM density, $n_{\rm{ism}}= 0.1\,\text{cm}^{-3}$, increases the reverse shock time scale to $t\s{rs} \approx 11\,\text{kyr}$.
The evolution of the average magnetic field in the nebula $B\s{pwn}(t)$ is calculated using the conservation of magnetic flux \citep[e.g.,][]{Tanaka2010}
\begin{equation}\label{eq:B_evolve}
\int_0^{t} \eta_B L(t)dt = V\s{pwn}(t)\frac{B^2_{\rm{pwn}}(t)}{8\pi},
\end{equation}
where $\eta_B$ is the fraction of the pulsar's spin-down luminosity converted into magnetic energy, and $V\s{pwn}(t)$ the volume of the PWN. Using $R_{\rm{pwn}} \propto t^{1.2}$, along with the fact that $L$ is effectively time-independent when $t < \tau$, leads to the time evolution of the magnetic field $B \propto t^{-1.3}$, identical to the time-dependence derived by \cite{Reynolds1984}.
An important parameter in the study of PWNe is the ratio of electromagnetic to particle energy in the nebula $\sigma$. In terms of the present model, this ratio is defined as
\begin{equation}
\sigma = \frac{\eta_B}{\eta\s{R}+\eta\s{X}},
\end{equation}
and is subjected to the constraint $\sigma \lesssim 1$ \citep[e.g.,][]{Dejager2009}. An additional constraint follows from $\eta_B+\eta\s{R}+\eta\s{X} \lesssim 1$. This sum is not set strictly equal to unity to allow for the fact that a fraction $\eta\s{rad}$ of the pulsar's spin-down luminosity is radiated away in the form of pulsed emission, i.e., $\eta_B+\eta\s{R}+\eta\s{X}+\eta\s{rad} = 1$. The value of $\eta\s{rad}$ is difficult to determine, but the results from the first \emph{Fermi}-LAT pulsar catalogue \citep{Abdo2010} suggest that $\eta\s{rad}\sim 1\%-10\%$ is reasonable. For the modeling $\eta\s{rad} \lesssim 1\%$ is used, similar to the value derived for the Crab pulsar \citep{Abdo2010}. Note that this small value used for $\eta\s{rad}$ in the model effectively implies that $\sigma \simeq \eta_B$.
Apart from energy losses, the model also takes into account that particles can escape from the PWN as a result of diffusion. The escape time scale $\tau\s{esc}$ is given by \citep{Parker1965}
\begin{equation}\label{eq:tau_esc}
\tau\s{esc}(t) = \frac{R^2_{\rm{pwn}}(t)}{6\kappa(t)},
\end{equation}
where $\kappa(t)$ is the diffusion coefficient. Diffusion in a PWN results from particles interacting with irregularities in the magnetic field, and it may be argued that $\kappa(t) \propto 1/B\s{pwn}(t)$ \citep[e.g.,][]{Lerche1981}. Furthermore, the diffusion coefficient is chosen to scale linearly with energy, i.e., $\kappa\equiv \kappa (E_e/1\mbox{ TeV})$. This functional form of $\kappa$ is similar to the form derived for Bohm diffusion
\begin{equation}\label{eq:kappa_bohm}
\kappa\s{Bohm} = \frac{cE_e}{3qB},
\end{equation}
where $q$ is the electric charge of the particle.
The temporal evolution of the electron spectrum is not obtained by solving Equation (\ref{eq:dn_dt}) directly, but rather in the following fashion: the amount of particles with energy $E_e$ injected into the PWN over the time interval $dt$ is given by $Q(E_e,t)dt$, where $Q(E_e,t)$ is specified using Equation (\ref{eq:power-law}). The injected particles are then added to the current value of $N_e(E,t)$ to obtain the total number of particles in the nebula $N_e(E_e,t+dt)=N_e(E_e,t)+Q(E_e,t)dt$. In the time interval $dt$ the particles also suffer energy losses as a result of adiabatic cooling and non-thermal radiation. The new energy of the particles is given by $E_e'=E_e-dE_e$, where $dE_e$ is the sum of the loss rates given in Equations (\ref{eq:E_dot_syn}) and (\ref{eq:E_dot_ad2}). The particles $N_e(E_e,t+dt)$ will therefore evolve to a new position in energy space $N'_e(E_e',t+dt)$. During the interval $dt$ a fraction of the particles will also have escaped from the nebula. Assuming that the particles are distributed uniformly throughout the nebula, this fraction is given by $\xi\s{esc}=dt/\tau\s{esc}$, with the escape time scale calculated using Equation (\ref{eq:tau_esc}). A value $\xi\s{esc} \ge 1$ indicates that all particles have effectively escaped from the system. Note that if $\xi\s{esc} > 1$, then the value $\xi\s{esc} = 1$ is used in the model. The number of particles remaining in the nebula after a time $t+dt$ is thus given by $N'_e(E_e',t+dt)(1-\xi\s{esc})$. This approach is essentially similar to the those used by, e.g., \citet{Gelfand2009}, \citet{Schock2010}, and \citet{Vanetten2011}, and is adequate provided that $dt$ is chosen sufficiently small.
In order to determine if the model predicts the correct evolution of the electron spectrum, it was tested using four well-known criteria: (1) adiabatic losses only lead to a reduction in the intensity of the spectrum, but does not lead to any spectral changes \citep[e.g.,][]{Vorster2013}; (2) if the system is in a steady-state, synchrotron losses lead to a spectrum that is steeper by one power of $E_e$ when compared to the source spectrum \citep[e.g.,][]{Pacholczyk1970}; (3) adiabatic losses primarily affect the low- energy electrons, while synchrotron losses are more important for the high-energy electrons \citep[e.g.,][]{Vorster2013}; and (4) in the absence of synchrotron losses, particles escaping from a system will result in a softer spectrum, with this softening directly related to the energy dependence of the diffusion coefficient \citep[e.g.,][]{Lerche1981}. In the present model $\kappa \propto E_e$, and the spectrum should again be softer by one power of $E_e$ (compared to the source spectrum). It was found that the model predicts all of the above-mentioned behaviour.
In reality, the evolution of a PWN could be considerably more complex than described by the model. One might, for example, think of a nebula that expands in a very inhomogeneous ISM. Simulations by, e.g., \citet{Blondin2001} and \citet{Vorster2013b}, show that in such a scenario the reverse shock of the SNR will be asymmetric, leading to a cigar or bullet-shaped PWN. One might also argue, with merit, that a more realistic model should include a spatial dependence, like the model presented by \citet{Vorster2013}. However, in the case where only spatially integrated observations are available, it is not clear how useful a spatially-dependent model would be. In this regard, the present model should be viewed as a first-order approximation. Furthermore, any time dependence in the conversion efficiencies $\eta\s{R}$, $\eta\s{X}$, and $\eta\s{B}$, as well as the values of the energy spectrum $E\s{min}$, $E\s{max}$, and $E\s{b}$, is not taken into account. It is unknown, to the authors at least, if any theoretical calculations exist that predict the time-dependence of the above-mentioned parameters.
The non-thermal emission is calculated using the appropriate expressions given in \cite{Blumenthal1970}. This implies that Klein-Nishina effects are taken into account when calculating the IC spectrum.
\section{Results}\label{sec:results}
\subsection{A Test-case PWN}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.85\textwidth]{Test_PWN.eps}
\caption{Model PWN showing the influence of the various parameters on the evolution of the non-thermal radiation spectra. The discontinuous spectrum is used as a reference scenario (see text for parameter values used), while Scenarios A--D are identical to the basis scenario, with the exception of one varied parameter. The parameters that were varied are listed in Table \ref{tab:param_vary}.}
\label{fig:test}
\end{center}
\end{figure}
It is instructive to first apply the model to a general PWN in order to illustrate the effect that the various free parameters have on the evolution of the non-thermal radiation spectra. For the test case, a PWN placed at a distance of $1\,\text{kpc}$ is allowed to expand for $t\s{age}=1\,\text{kyr}$, with the nebula having a present-day size of $R\s{pwn}=1\,\text{pc}$, and an expansion velocity of $v\s{age}=1000\,\text{km}\,\text{s}^{-1}$. The conversion efficiency of spin-down luminosity to magnetic energy is chosen to be $\eta_B=0.03$, leading to a present-day magnetic field of $B\s{age}=50\,\mu\text{G}$. For this specific PWN, both adiabatic and escape losses are neglected.
The first important distinction that can be made is the effect of a continuous/discontinuous particle spectrum on the broadband emission. Note that both these source spectra are described by Equation (\ref{eq:power-law}), but with the following difference: for the continuous spectrum the two components of the broken power-law have the same intensity at $E\s{b}$. In the case of the discontinuous spectrum, the high-energy component has a lower intensity, compared to the low-energy component, at $E\s{b}$. The continuous spectrum requires only a single conversion efficiency, and consequently a single conversion efficiency $\eta\equiv\eta_R+\eta_X$. For both the continuous and discontinuous spectra the values $\eta_R=0.92$ and $\eta_X=0.05$ are chosen, implying that the same fraction of the pulsar's spin-down luminosity is converted to particle energy for the two cases.
The radiation spectra obtained with the two types of source spectra are shown in Figure \ref{fig:test}, from which it can be seen that the continuity/discontinuity in the particle spectra is preserved in the non-thermal emission. Another salient feature visible in Figure \ref{fig:test} is that the different source spectra predict fluxes for the non-thermal emission that differ by more than an order of magnitude. This discrepancy is easy to explain if it is kept in mind that in both scenarios the same amount of spin-down energy is converted to particle energy. To transform a discontinuous source spectrum into continuous spectrum requires that a fraction of the energy stored in the low-energy component must be transferred to the high-energy component, thereby reducing the intensity of the former component, while increasing the intensity of the latter component. As the low-energy component is described by $N_e\propto E_e^{-1}$, and the high-energy component by $N_e\propto E_e^{-2}$, a small change in the intensity (or equivalently energy) of the former component will lead to a larger change in the intensity of the latter component. This can also be seen from Figure \ref{fig:test} where the discrepancy between the X-ray spectra is larger than the discrepancy between the radio spectra. With these differences taken into account, it is clear that the two source spectra will lead to the derivation of different parameters for the same PWN. This will be discussed in a more qualitative fashion when the modelling results of G21.5--0.9 are presented in Section \ref{sec:results_G21.5}.
Having demonstrated the effect of the source spectrum on the non-thermal emission, it is also necessary to demonstrate the effect that different values for the various parameters have on the evolution of the non-thermal spectra. For this purpose a number of alternative scenarios are chosen, with these alternative scenarios having only one varied parameter compared to the discontinuous source spectrum scenario. The parameters varied are listed in Table \ref{tab:param_vary}, with the quantity in brackets indicating the value of the reference scenario.
\begin{table}[t]
\textbf{\caption[]{\label{tab:param_vary}
{\textnormal{Parameter values used for the scenarios depicted in Figure \ref{fig:test}. The value in brackets indicates the value used for the discontinuous source spectrum scenario.}}}}
\begin{center}
\begin{tabular}{ll}
\hline\midrule
Scenario & Difference to reference scenario \\
\midrule
Scen A & PWN has an age of $t\s{age}=10\,\text{kyr}$ ($t\s{age}=1\,\text{kyr}$) \\
Scen B & spin-down time scale of the pulsar is $\tau=0.3\,\text{kyr}$ ($\tau=1\,\text{kyr}$)\\
Scen C & adiabatic losses are included (adiabatic losses are neglected)\\
Scen D & escape losses are included (escape losses are neglected) \\
\midrule
\end{tabular}
\end{center}
\end{table}
The first quantity varied is the age of the system, corresponding to Scen A in Figure \ref{fig:test}. As the PWN ages, the intensity of the X-ray and TeV spectra decrease as a result of synchrotron and IC losses, while the spectral index becomes steeper by one power of energy. By contrast, the low-energy component remains largely unaffected by non-thermal losses. As mentioned at the end of Section \ref{sec:model}, this is the theoretically expected effect of non-thermal losses, thus indicating that the model works correctly.
In the next scenario, Scen B, the spin-down time scale is reduced to $\tau=0.3\,\text{kyr}$. Figure \ref{fig:test} shows that the influence of this parameter is to reduce the non-thermal flux at all wavelengths. At times $t < \tau$, the spin-down luminosity of the pulsar is approximately constant, but decreases rapidly after $t > \tau$. As the number of particles injected into the nebula is determined by the time integral over $L$, which in turn is dependent on the value of $\tau$, a smaller number of particles have been injected into the nebula in Scen B (compared to the reference scenario), leading to the reduced flux.
The next parameter investigated is the effect of adiabatic losses on the radiation spectra, Scen C. It can be seen from Figure \ref{fig:test} that the low-energy component of the particle spectrum is primarily affected by these losses, leading to a decrease in the intensity of the radio/GeV spectra. For the high-energy component of the source spectrum, synchrotron losses are more important, and the effect of adiabatic losses becomes negligible. One noteworthy point is that adiabatic losses do not affect the spectral index.
Lastly, Scen D shows the effect of escape losses on the radiation spectra. As $\kappa$ scales with energy, the high-energy component of the particle spectrum is primarily affected, leading to a decrease in the intensity of the X-ray/TeV spectra. One important feature of escape losses in a spatially independent model is that this process leads to a spectral evolution that is very similar to that of synchrotron losses, and one might argue that it would not be possible to distinguish between the two loss processes. Therefore, to find a model prediction that is compatible with the data, the escape losses are initially fixed using the Bohm diffusion coefficient (\ref{eq:kappa_bohm}) while all other parameters are varied. Only after a reasonable agreement has been found is the diffusion coefficient varied to improve the model prediction. The same is also true for the parameters $E\s{min}$, $E\s{max}$, and $E\s{b}$.
\begin{table}[!ht]
\textbf{\caption[]{\label{tab:parameters}
{\textnormal{Values derived with the model for the various free parameters. Values marked with an * represent parameters that were kept fixed, or parameters that follow from the derived model parameters.}}}}
\begin{center}
\begin{tabular}{lcccc}
\hline\midrule
Parameter & Symbol & G21.5--0.9 & J1427--608 & J1507--622 \\
\midrule
Initial spin-down luminosity ($10^{38}\,\text{erg}\,\text{s}^{-1}$) & $L_0$ & $0.54*$ & $5.5$ & $1.2$ \\
Spin-down time scale (kyr) & $\tau$ & $3^*$ & $3$ & $0.5$ \\
Age of nebula (kyr) & $t\s{age}$ & $0.87^*$ & $10$ & $24$ \\
Present-day magnetic field ($\mu\text{G}$) & $B\s{age}$ & $230$ & $0.4$ & $1.7$ \\
Radio conversion efficiency & $\eta\s{R}$ & $0.68$ & $0.81$ & $0.8$ \\
X-ray conversion efficiency & $\eta\s{X}$ & $0.13$ & $0.18$ & $0.17$ \\
Ratio: particle conversion efficiencies & $\eta\s{R}/\eta\s{X}$ & $5.4^*$ & $4.5^*$ & $4.7^*$ \\
Ratio: magnetic to particle energy ($10^{-3}$) & $\sigma$ & $180$ & $0.01$ & $30$ \\
Minimum electron energy ($10^{-3}\,\text{TeV}$) & $E\s{min}$ & $0.3$ & $100$ & $1$ \\
Maximum electron energy ($10^2\,\text{TeV}$) & $E\s{max}$ & $2.7$ & $3$ & $2$ \\
Break energy (TeV) & $E\s{b}$ & $0.1$ & $0.18$ & $0.5$ \\
Diffusion coefficient ($10^{25} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$) & $\kappa$ & $2.2$ & $7$ & $15$\\
Ratio: diffusion coefficients & $\kappa/\kappa\s{Bohm}$ & $390^*$ & $2.3^*$ & $20^*$\\
Distance to source (kpc) & $d$ & $4.8^*$ & $11^*$ & $6^*$ \\
\midrule
\end{tabular}
\end{center}
\end{table}
\subsection{G21.5--0.9}\label{sec:results_G21.5}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.85\textwidth]{G21.5.eps}
\caption{Model prediction for the young nebula G21.5--0.9. Radio data are taken from \citet{Goss1970}, \citet{Becker1975}, \citet{Morsi1987}, \citet{Salter1989b, Salter1989a}, \citet{Bock2001}, \citet{Bandiera2001}, \citet{Bietenholz2008}, and \citet{Bietenholz2011}, infra-red data from \citet{Gallant1999}, X-ray data from \citet{Slane2000}, \citet{DeRosa2009}, and the INTEGRAL Science Data Centre (http://www.isdc.unige.ch/heavens webapp/integral), and TeV data from \citet{Dejager2008b}. The IC prediction is calculated using both the CMBR and an SSC component, although the former component leads to a negligible flux.}
\label{fig:G21.5}
\end{center}
\end{figure}
With a spin-down luminosity of $L=3.3 \times 10^{37}\,\text{erg}\,\text{s}^{-1}$ \citep{Camilo2006}, the pulsar in the SNR G21.5--0.9 is one of the most energetic pulsars in the Galaxy. The PWN is located at a distance of $4.8\,\text{kpc}$ \citep{Tian2008}, with an estimated age of $870\,\text{yr}$ \citep{Bietenholz2008}. Radio \citep[e.g.,][]{Goss1970, Becker1975, Morsi1987, Salter1989a, Salter1989b, Bock2001, Bandiera2001, Bietenholz2008, Bietenholz2011}, IR \citep[e.g.,][]{Gallant1999, Zajczyk2012}, and X-ray \citep{Slane2000, DeRosa2009, Tsujimoto2011} observations of the PWN show a bright nebula with a radius of $\sim 40''$. The nebula is embedded in diffuse X-ray emission, believed to be the result of dust-scattered X-rays from the PWN \citep{Bocchino2005}. At GeV energies, \emph{Fermi}-LAT only detected upper limits for the nebula \citep{Ackermann2011}, while a detection at TeV energies has been reported by the H.E.S.S. Collaboration \citep{Djannati2007, Dejager2008b}.
For the modelling of G21.5--0.9, a number of constraints on the parameters follow from observations. As the age and present-day luminosity are known, the initial luminosity is fixed for a choice of $\tau$ using Equation (\ref{eq:L_t}). For the spin-down time scale, the value of $\tau=3\,\text{kyr}$ estimated by \citet{Dejager2009b} is used. The derived distance to the source implies a PWN radius of $R\s{pwn}=0.93\,\text{pc}$. The PWN is too young to have interacted with the reverse shock, and must therefore still be in the first expansion phase. With the age and size of the PWN taken into account, the expansion rate in Equation (\ref{eq:R_t}) has the value $r_1=1.02$, leading to a present-day expansion velocity of $v\s{age}=1060\,\text{km}\,\text{s}^{-1}$.
Figure \ref{fig:G21.5} shows the
|
model prediction for the non-thermal radiation spectra, with the derived parameters listed in Table \ref{tab:parameters}. In order to find an agreement between the TeV data and the model prediction, it is necessary to include synchrotron self-Compton scattering in the model. \citet{Atoyan1996} calculated that the density of the synchrotron photon field is
\begin{equation}\label{eq:ssc}
n_{\rm{ssc}} = \frac{Q_{\rm{syn}}}{4\pi R_{\rm{syn}}c}\frac{\bar{U}}{h\nu},
\end{equation}
where $Q\s{syn}$ is the synchrotron emissivity, $R_{\rm{syn}}$ the radius within which most of the synchrotron emission is produced, and $\bar{U} \simeq 2.24$. Additionally, photons from the CMBR were also taken into account, but it was found that this radiation field is significantly less important than the SSC component.
From the model a present-day average magnetic field of $B\s{age}=230\,\mu\text{G}$ is derived. This is comparable to the value of $B\s{age}=300\,\mu\text{G}$ inferred for the $\sim 1\,\text{kyr}$ old Crab Nebula \citep[e.g.,][]{Trimble1982}. To obtain the model prediction presented in Figure \ref{fig:G21.5}, the diffusion coefficient should not be larger than $\kappa= 2.2 \times 10^{25} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$. For Vela X with an average magnetic field of $B\s{age}=5\,\mu\text{G}$ \citep[e.g.,][]{Dejager2008c}, a value of $\kappa=10^{27} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$ has been estimated by \cite{Hinton2011} to explain the absence of particles with an energy $E_e > 100$ GeV from the relic PWN observed by \emph{Fermi}-LAT. Treating the magnetic field and diffusion coefficient of Vela X as fiducial values, the scaling $\kappa \propto 1/B$ implies that the magnetic field $B\s{age}=230\,\mu\text{G}$ should lead to $\kappa= 2.2 \times 10^{25} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$, in agreement with the value derived from the model. In terms of the Bohm diffusion coefficient (\ref{eq:kappa_bohm}), the derived diffusion coefficient has the value of $\kappa=390\kappa\s{Bohm}$.
Among the parameters derived for the nebula G21.5--0.9 is the ratio of conversion efficiencies $\eta\s{R}/\eta\s{X}=5.4$, significantly smaller than the ratio $\eta\s{R}/\eta\s{X}=116-150$ derived by \citet{Dejager2008c} for Vela X. The ratio of magnetic to particle energy is found to be $\sigma=0.18$, larger than the value of $\sigma \sim 0.003$ calculated by \citet{Kennel1984b} for the Crab Nebula. The well-known steady-state magnetohydrodynamic model of \cite{Kennel1984a} predicts a radial velocity that is almost independent of $r$ when $\sigma=0.25$. Based on this result, Equation (\ref{eq:E_dot_ad2}) should be a reasonable approximation for the adiabatic losses when $\sigma=0.18$, as the velocity in the largest part of the PWN will not differ significantly from the expansion velocity.
To illustrate the effect of a continuous/discontinuous source spectrum on the derived parameters, one need only consider the modeling results of \citet{Tanaka2010}. Using a continuous source spectrum and a model similar to the present one, these authors derived a smaller present-day magnetic field of $B\s{age}\le 64\,\mu\text{G}$ for G21.5--0.9. However, \citet{Tanaka2010} were unable to predict the $1-10$ keV X-ray observations, while the present model with the different normalisation constants is a very good description of the broadband spectra. Furthermore, it was found that a magnetic field much smaller than $B\s{age} = 230\,\mu\text{G}$ cannot be used in the present model, as this leads to the requirement $\eta\s{R} > 1$.
Following the discussion presented in Section \ref{sec:two_com}, the non-thermal emission from G21.5--0.9 is also modelled using a Maxwellian spectrum with a power-law tail
\begin{equation}\label{eq:thermal}
Q(E_e,t) = \begin{cases}
Q_{\rm{T}}\left(E_e/E_{\rm{ts}}\right)\exp\left[-E_e/E_{\rm{ts}}\right], & \text{for all } E_e\\
Q_{\rm{N}}\exp\left[(-E-E_{\max})/\Delta E_{\max}\right]\left(E_e/E_{\rm{ts}}\right)^{-\alpha_N}, & \text{if } E_{\rm{b}}<E_e\le E_{\max}
\end{cases},
\end{equation}
where $Q\s{T}$ and $Q\s{N}$ respectively represent the normalisation constants for the thermal and non-thermal components, and $E_{\rm{ts}}=0.26(\gamma/10^6)\,\text{TeV}$. Here $\gamma$ represents the Lorentz factor of the electrons upstream of the termination shock, while $E_{\rm{b}}=7E_{\rm{ts}}$. Note that Equation (\ref{eq:thermal}) is a slightly modified \citet{Spitkovsky2008} spectrum introduced by \citet{Fang2010}. The aim is not to model G21.5--0.9 time-dependently using Equation (\ref{eq:thermal}), but merely to determine whether a Maxwellian spectrum can be used to explain the radio/GeV data.
Figure \ref{fig:G21.5} shows that Equation (\ref{eq:thermal}) can also be used to model the broadband data, except at radio frequencies where a much harder spectrum is predicted. From this modeling values for $E\s{min}$ and $E\s{max}$ are derived that are similar to the values listed in Table \ref{tab:parameters}. Other parameters that are derived include a Lorentz factor of $\gamma=5\times 10^4$, implying a break energy of $E\s{b}=0.09\,\text{TeV}$, along with $\alpha_N=2.7$ and $\Delta E_{\max}=160\,\text{TeV}$ for the power-law tail. These last two values are comparable to the values of $\alpha_N=2.5$ and $\Delta E_{\max}=100\,\text{TeV}$ predicted by \cite{Spitkovsky2008}.
\subsection{HESS J1427--608}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.85\textwidth]{1427.eps}
\caption{Model prediction for the unidentified source HESS J1427--608. The radio data is taken from \cite{Murphy2007}, the X-ray data from \cite{Fujinaga2012}, the \emph{Fermi} data from \cite{Nolan2012}, and the TeV data from \cite{H_Aharonian2008}. For this source the GeV/TeV spectrum is produced by the IC scattering of both the CMBR and an IR component.}
\label{fig:1427}
\end{center}
\end{figure}
One of the sources discovered in a H.E.S.S. Galactic Plane Survey is HESS J1427--608, with an intrinsic source extension of $2'.4-4'.8$ \citep{H_Aharonian2008}. Observations by \emph{Fermi}-LAT detected an associated GeV source, 2FGL J1427.6-6048 \citep{Nolan2012}, while \cite{Fujinaga2012} recently reported an X-ray detection with \emph{Suzaku} in the $2-10$ keV band. The X-ray emission is spatially coincident with the TeV emission, and has a radius of $2'$. Based on the well-known fact that the VHE nebula is typically larger than the X-ray nebula \citep[e.g.,][]{Kargaltsev2010}, it was suggested by \cite{Fujinaga2012} that the \emph{Suzaku} observation represents the X-ray counterpart to HESS J1427--608. Using the X-ray observations, \cite{Fujinaga2012} estimated that the source is located at a distance of $d\sim 11\,\text{kpc}$, and has an age of $t\s{pwn} \sim 6.4\,\text{kyr}$. A possible radio counterpart, MGPS J142755-605038 with a radius of $0.56'-0.77'$, has been observed in a Molonglo Sky Survey \citep{Murphy2007}. As the radio nebula of a PWN is typically larger than the X-ray nebula \citep[e.g.,][]{Gaensler2006}, it is difficult to simultaneously associate both the radio and \emph{Suzaku} sources with the VHE emission, assuming that HESS J1427--608 is indeed a PWN, and that the \emph{Suzaku} detection represents the X-ray nebula. This incompatibility is also strongly underlined by the model.
For the modeling a spherical source with a radius of $2'.4$ is used, along with the distance and age estimates derived by \cite{Fujinaga2012}. The observed size and estimated distance lead to a radius of $R\s{pwn}=7.7\,\text{pc}$. For a $\sim 6\,\text{kyr}$ source it is entirely possible that the PWN has not yet interacted with the reverse shock, and it is thus assumed that HESS J1427--608 is still in the first expansion phase. The rate of expansion is $r_1=1.03$, leading to present-day expansion velocity of $v\s{age}=1200\,\text{km}\,\text{s}^{-1}$. To model the GeV/TeV data it is necessary to not only include IC scattering of the CMBR, but also scattering of an IR photon field. The energy spectrum of the IR photons is taken as a black-body spectrum with a temperature of $T=50\,\text{K}$ and an energy density of $U\s{IC}=2\,\text{eV}\,\text{cm}^{-3}$. This is comparable to the values of $T=46\,\text{K}$ and $U\s{IC}=5\,\text{eV}\,\text{cm}^{-3}$ used by \cite{Zhang2008} to model the TeV data of MSH 15-52 and HESS J1825-137.
Apart from the radio measurement, Figure \ref{fig:1427} shows that the model prediction is in good agreement with the data if adiabatic losses are neglected. The scenario presented in Figure \ref{fig:1427} requires a large initial luminosity ($L_0=1.2\times 10^{39}\,\text{erg}\,\text{s}^{-1}$) and spin-down time scale ($\tau=3\,\text{kyr}$), leading to a present-day luminosity of $L=1.2 \times 10^{38}\,\text{erg}\,\text{s}^{-1}$. From the model prediction a present-day magnetic field of $B\s{age}=4.2\,\mu\text{G}$ is derived. The ratio of particle conversion efficiencies is $\eta\s{R}/\eta\s{X}=13.8$, comparable to the value of $\eta\s{R}/\eta\s{X}=5.4$ derived for G21.5--0.9. Moreover, the model predicts that the ratio of magnetic to particle energy is $\sigma = 4\times 10^{-4}$. The small $\sigma$ value derived implies that the energy content in this source is predominantly stored in the particles, in contrast to G21.5--0.9 ($\sigma=0.18$) where the electromagnetic energy is an important fraction of the total energy. For the escape losses, the model predicts a present-day diffusion coefficient of $\kappa=10^{26} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$, or equivalently, $\kappa=30\kappa\s{Bohm}$.
\citet{Vorster2013b} calculated that when $\sigma<0.01$, the radial convection velocity in the PWN decreases as $v\propto 1/r^2$. It follows from Equation (\ref{eq:ad_r}) that the particles will not be subjected to adiabatic losses for such a profile, thereby motivating the neglect of this energy loss process from the modeling. An agreement between the model and data could also be found with adiabatic losses included. In this scenario a marginally larger present-day magnetic field of $B\s{age}=3.9\,\mu\text{G}$ is derived. The largest difference, compared to the scenario presented in Figure \ref{fig:1427}, is that a very large initial luminosity of $L_0=6.5\times 10^{39}$ is required to make up for the adiabatic losses suffered by the low-energy particles.
The present-day value of $L=1.2 \times 10^{38}\,\text{erg}\,\text{s}^{-1}$ predicted by the model (with adiabatic losses neglected) is remarkably similar to a pulsar that has recently been detected by \cite{Arzoumanian2011} near the Galactic plane. The authors measured a value of $L=1.2\times 10^{38}\,\text{erg}\,\text{s}^{-1}$ for PSR J2022+3842, and estimated the age of the source to be $t\s{age}=8.9\,\text{kyr}$. Although there is some uncertainty regarding the distance to PSR J2022+3842, \cite{Arzoumanian2011} placed the pulsar at $d=10\,\text{kpc}$. Furthermore, the authors also detected a very faint elliptical PWN in X-rays with the total dimensions of $29 \times 35\,\text{pc}$. The only difference between this PWN and HESS J1427--608 is that a bright radio nebula, G76.9+1.0 \citep{Landecker1993}, is associated with the X-ray PWN. Although the nature of G76.9+1.0 is not entirely clear, \cite{Landecker1993} argued that the filled centre of the radio source is more indicative of a PWN than an SNR. However, the authors derived a spectral index of $\alpha_R=0.62$, much steeper than the values $\alpha_R=0-0.3$ typically associated with PWNe \citep[e.g.,][]{Weiler1980}.
Even though the values derived from the model prediction shown in Figure \ref{fig:1427} are compatible with PWN parameters, the over-prediction of the radio data makes it difficult to unambiguously accept this scenario. The model predicts a bright radio source that has thus far not been observed in the region of the sky spatially coincident with the position of HESS J1527-608. A model prediction compatible with the radio data can be obtained from the scenario presented above, provided that the minimum electron energy is $E\s{min} > 0.1\,\text{TeV}$. This seems an unnatural high value, and this solution is therefore disfavoured.
An alternative solution would be to decrease the value of the magnetic field. Figure \ref{fig:1427} also shows a scenario where an agreement between the model prediction and radio data has been obtained using a present-day magnetic field of $B\s{age}=0.42\,\mu\text{G}$, with the parameters derived from this scenario listed in Table \ref{tab:parameters}. In order for the magnetic field to reach such a low value, the PWN must be older than the value of $t\s{age}=6.4\,\text{kyr}$ estimated by \cite{Fujinaga2012}, and the larger value of $t\s{age}=10\,\text{kyr}$ is chosen as the age of the PWN.
For this alternative scenario, a smaller (compared to the $B\s{age}=4.2\,\mu\text{G}$ scenario) expansion rate of $r_1=0.96$ is derived, leading to a present-day expansion velocity of $v\s{age}=720\,\text{km}\,\text{s}^{-1}$. This scenario requires a smaller initial luminosity ($L_0=5.5\times 10^{38}\,\text{erg}\,\text{s}^{-1}$), leading to a present-day luminosity of $L=2.9 \times 10^{37}\,\text{erg}\,\text{s}^{-1}$. The ratio of the conversion efficiencies is $\eta\s{R}/\eta\s{X}=4.5$, while the ratio of particle to magnetic energy is $\sigma =10^{-5}$. Lastly, a diffusion coefficient of $\kappa= 5.6 \times 10^{25} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$ is predicted by the model, or $\kappa=2.3\kappa\s{Bohm}$. Note that the prediction of the radio data requires a large minimum energy ($E\s{min}=0.1\,\text{TeV}$) that can be reduced to $E\s{min}=10^{-2}\,\text{TeV}$ if the magnetic field is decreased to $B\s{age}=0.1\,\mu\text{G}$. While these parameters may lead to an acceptable agreement between the model and radio data, Figure \ref{fig:1427} shows that this scenario significantly under-predicts the \emph{Suzaku} spectrum. This discrepancy can be explained if the X-ray observations are not related to the TeV nebula.
To understand why the model has difficulty in predicting both the radio and X-ray synchrotron data, one need only consider the two quantities that eventually determine the synchrotron flux: the number of particles that produce the non-thermal emission, and the magnetic field strength. As the magnetic field is the same for both the radio and X-ray producing particles, the only way to predict both the radio and X-ray data would be to increase the number of high-energy particles. However, it follows from Figure \ref{fig:test} that this would also increase the TeV flux, leading to an over-prediction of the H.E.S.S. data.
\subsection{HESS J1507--622}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.85\textwidth]{1507.eps}
\caption{Model prediction for the unidentified source HESS J1507--622. The radio upper limit is taken from \cite{Green1999}, and the GeV data from \cite{Nolan2012}. The X-ray upper limit and the TeV data are taken from the \cite{H_Aharonian2011}. The IC spectrum is produced by only taking into account the scattering of CMBR photons.}
\label{fig:1507}
\end{center}
\end{figure}
Also discovered in a H.E.S.S. Galactic Plane Survey is the bright ($\sim 8\%$ of the Crab flux) VHE source HESS J1507--622, with a radius of $\sim 9'$ \citep{H_Aharonian2011}. A possible synchrotron counterpart may be provided by the extended, diffuse X-ray emission (with a radius of $10''-13''$) observed with \emph{Chandra} \citep{H_Aharonian2011}. However, the identification of this X-ray source with the VHE emission region remains inconclusive \citep{H_Aharonian2011}. An additional synchrotron upper limit is provided by the source MGPS J150850-621025 discovered in a Molonglo Galactic Plane Survey \citep{Green1999}. Lastly, a GeV counterpart, 2FGL J1507.0-6223, has recently been discovered by the \emph{Fermi}-LAT Collaboration and is reported in \cite{Nolan2012}.
HESS J1507--622 is unique in the sense that it lies $\sim 3^{\circ}.5$ from the Galactic plane, whereas all other unidentified source lie within $\pm 1^{\circ}$ from the Galactic equator. Most Galactic VHE sources are connected to young stellar populations (located in the disk), and one would therefore not expect a bright VHE source at the observed position. Furthermore, the absence of a bright X-ray counterpart is surprising as the comparably low hydrogen column density at $\sim 3.5^{\circ}$ leads to a considerably lower absorption of X-rays, as well as reduced background emission \citep{H_Aharonian2011}.
To explain the uniqueness of the source, the \cite{H_Aharonian2011} considered a number of possible scenarios. On the one hand, the absence of counterparts, especially in X-rays, suggests a hadronic scenario. Given the low density of target material off the Galactic plane \citep[see e.g.,][]{Lockman1984}, this scenario was disfavoured by the \cite{H_Aharonian2011} unless the source could be placed at a very small distance of $d < 1\,\text{kpc}$. Although unlikely, the hadronic scenario can not be fully excluded \citep{H_Aharonian2011}. An alternative scenario is that HESS J1507--622 is an ancient PWN. As a result of its small angular extension, the leptonic scenario would place the source at a distance of $d > 6\,\text{kpc}$ \citep{H_Aharonian2011}.
For the modeling the hints of diffuse X-ray emission detected by \emph{Chandra} are taken as an upper limit. The source is placed at a distance of $d=6\,\text{kpc}$, leading to a radius of $R\s{pwn}=15.7\,\text{pc}$. For the first modelling attempts, it was assumed that HESS J1507--622 is still in the initial expansion phase. In order to model the GeV data, this scenario requires a break energy of $E\s{b}=5\,\text{TeV}$. This is an order of magnitude larger than the values derived for G21.5--0.9 ($E\s{b}=0.1\,\text{TeV}$) and HESS J1427--608 ($E\s{b}=0.18\,\text{TeV}$). Values similar to those presented in Table \ref{tab:parameters} have also been derived for a number of known PWNe, including the $\sim 21\,\text{kyr}$ old nebula HESS J1825-137. \cite{Zhang2008} found that $E\s{b}\le 0.15\,\text{TeV}$, while \cite{Tanaka2011} derived values that where $E\s{b}\le 0.3\,\text{TeV}$. The exception to the results of \cite{Tanaka2011} is Kes 75, where the authors derived a possible value of $E\s{b}=2.6\,\text{TeV}$. Using a Maxwellian source spectrum, \cite{Fang2010} derived the even smaller values of $E\s{b}=0.02-0.09\,\text{TeV}$.
Although it is not excluded that such a large break energy is the result of shock acceleration, an alternative scenario is favoured in the present paper where HESS J1507--622 has been compressed by the reverse shock. As $\dot{E_e}/E_e$ is constant in Equation (\ref{eq:E_dot_ad2}), the effect of adiabatic losses is to shift the electron spectrum to lower energies without affecting the spectral shape, as illustrated in Figure \ref{fig:test}. During the compression phase, the exact opposite will occur. The particles will be subjected to adiabatic heating, causing the electron spectrum to shift to higher energies, thereby leading to an increase in the value of $E\s{b}$.
At an offset of $3^{\circ}.5$ one would expect the ISM to have a lower density compared to the Galactic plane, and from the reverse shock time scale given by Equation (\ref{eq:t_rev}), it follows that smaller ISM densities lead to larger time scales. Inserting the values of $E_{\rm{ej}}=10^{51}$ erg, $M_{\rm{ej}}=9 M_{\odot}$, and $n_{\rm{ism}}= 0.1 \mbox{ cm}^{-3}$ into Equation (\ref{eq:t_rev}) leads to an estimate of $t_{\rm{rs}} = 19.6\,\text{kyr}$. For the compression scenario the value of $t_{\rm{rs}} = 20\,\text{kyr}$ is used, while the compression phase is chosen to last for $4\,\text{kyr}$. It is assumed that the nebula has not yet entered the second expansion phase, implying that the current age of the PWN is $t\s{age}=24\,\text{kyr}$. In the initial phase the PWN expands to a radius of $R\s{pwn}=20\,\text{pc}$ with a constant velocity of $v\s{pwn}=980\,\text{km}\,\text{s}^{-1}$. In the next phase the interaction with the reverse shock compresses the PWN, thereby causing $v\s{pwn}$ to reverse direction. During this compression phase $R\s{pwn}$ initially moves inward with a velocity of $v\s{pwn}=1300\,\text{km}\,\text{s}^{-1}$, reducing to $v\s{pwn}=850\,\text{km}\,\text{s}^{-1}$ after $t=24\,\text{kyr}$. Given the offset from the Galactic plane, it seems reasonable to assume that the ISM is homogeneous. This will lead to a symmetric reverse shock and a preservation of the spherical nature of the PWN. Furthermore, one would not expect any photon field other than the CMBR to be present at the position of HESS J1507--622, and only this component is taken into account.
The model prediction resulting from the compression scenario is shown in Figure \ref{fig:1507}, with the derived parameters listed in Table \ref{tab:parameters}. With the compression taken into account, the break energy is reduced to $E\s{b}=0.5\,\text{TeV}$. Note that this is the break energy of the source spectrum, while the particle spectrum in the PWN has a break at $5\,\text{TeV}$. The effect of the compression is also reflected in the derived value of $E\s{min}$. Comparing Figures \ref{fig:1427} and \ref{fig:1507} shows that the radio data for the two unidentified TeV sources are very similar, yet the values of $E\s{min}$ listed in Table \ref{tab:parameters} differ significantly between the two sources. Although the model predicts a value of $E\s{min}=10^{-3}\,\text{TeV}$ at the termination shock of HESS J1507--622, the compression also increases the minimum electron energy in the nebula, thereby making it possible for the radio synchrotron spectrum to be compatible with the upper limit.
Other parameters derived include a relatively large initial luminosity ($L_0=1.2\times 10^{38}\,\text{erg}\,\text{s}^{-1}$) and a short spin-down time scale ($\tau=0.5\,\text{kyr}$). As the expansion of the PWN is driven by the continual injection of the pulsar's spin-down energy into the nebula, the small $\tau$ value (compared to the other two scenarios investigated) implies that the energy input declines rapidly with time, thereby motivating the constant expansion velocity in the initial phase. A present-day magnetic field of $B\s{age}=1.7\,\mu\text{G}$ is derived from the model, larger than the value of $B\s{age}=0.5\,\mu\text{G}$ estimated by \cite{Tibolla2011} and the \cite{H_Aharonian2011}. Additionally, the ratios $\eta\s{R}/\eta\s{X}=4.7$ and $\sigma=0.03$ are derived. A present-day diffusion coefficient of $\kappa=1.5 \times 10^{26} E\s{TeV}\,\text{cm}^2\,\text{s}^{-1}$ is derived, comparable to the value derived for HESS J1427--608. In terms of the Bohm coefficient (\ref{eq:kappa_bohm}), the derived diffusion coefficient has the value of $\kappa=20\kappa\s{Bohm}$. For the expansion phase, adiabatic losses are calculated using Equation (\ref{eq:E_dot_ad2}), while the compression phase requires adiabatic heating that is ten times larger than that predicted by Equation (\ref{eq:E_dot_ad2}).
\section{Discussion and Conclusions}
In this paper a time-dependent PWN particle evolution model is presented and applied to the young PWN G21.5--0.9, as well as to the unidentified TeV sources HESS J1427--608 and HESS J1507--622. For the three sources sets of parameters are derived that are reasonable within a PWN framework, thereby strengthening the argument that HESS J1427--608 and HESS J1507--622 can be identified as PWNe. The robustness of the derived parameter sets was tested by considering a large number of alternative scenarios. It was found that markedly changing the values of the parameters, compared to those given in Table \ref{tab:parameters}, leads to model predictions that are not compatible with the observations.
As discussed in \citet{Possenti2002}, observations indicate that the X-ray luminosity of the PWN decreases with age, while \citet{Mattana2009} found that there is no correlation between the pulsar's spin-down luminosity and the observed TeV luminosity of the PWN. The corollary is that the PWN will remain bright at TeV energies, even if the pulsar's spin-down luminosity has reached an undetectable level \citep[e.g.,][]{Dejager2009b,H_Aharonian2011}. This evolutionary trend is also predicted by the model, as the synchrotron luminosity of the evolved PWNe fades away below the sensitivity of the current generation of X-ray satellites, while nevertheless remaining bright at TeV energies.
Apart from detecting pulsars that can be associated with the unidentified sources, additional multi-wavelength observations may further strengthen the PWN identification. A characteristic of PWNe is that electrons and positrons are responsible for the observed non-thermal emission. Although it will not directly confirm an unidentified TeV source as a PWN, the detection of a $511\,\text{keV}$ annihilation line by future sub-MeV experiments, e.g., the proposed GRIPS satellite \citep{Greiner2012}, will at least indicate that the particles responsible for the TeV emission in the unidentified sources are leptonic \citep{Tibolla2011b}.
As such, only a few alternative explanations for the unidentified TeV sources have thus far been proposed, including the suggestion by \citet{Yamazaki2006} that these sources can be associated with old SNRs. Arguing that SNRs can only confine multi-TeV particles for a very short period ($t \lesssim 1\,\text{kyr}$), the aforementioned proposal has however been questioned by \citet{Gabici2007}. As a second alternative, \citet{Gabici2007} have suggested that the unidentified sources can still be identified with SNRs if multi-TeV particles that have escaped from the remnant interact with nearby dense clouds. This scenario seems unlikely for the unidentified sources discovered so far, given the absence of dense molecular clouds spatially coincident with most of these sources. Given its location above the Galactic plane, this is particularly true for HESS J1507--622.
Motivated by observations, a broken power-law is used as the source spectrum for the electrons injected into the PWN at the termination shock. In contrast to previous PWN models of a similar nature \citep[see e.g.,][]{Zhang2008,Tanaka2010}, the source spectrum in the present model has a discontinuity in intensity at the transition between the low and high-energy components. The choice of a discontinuous source spectrum leads to a better model prediction of the data at all wavelengths, in contrast to a continuous one. A similar conclusion has also been drawn by \cite{Dejager2008c} from their modeling of Vela X. As a discontinuous spectrum is also required for the young ($t\s{age}\sim 1\,\text{kyr}$) nebula G21.5--0.9, the discrepancy between the two components cannot be an artifact of PWN evolution. A characteristic of the discontinuous spectrum is that a particle conversion efficiency must be specified for both the low ($\eta\s{R}$) and high-energy ($\eta\s{X}$) components, with a ratio of $\eta\s{R}/\eta\s{X}\sim 4.5-5.4$ derived for the three sources.
The data for G21.5--0.9 were also modeled using a Maxwellian source spectrum with a non-thermal tail. The aim was not to model the evolution of the PWN time-dependently, but rather to illustrate that a Maxwellian source spectrum can be used to predict the data. Although the Maxwellian spectrum supplies a natural explanation for the ratio $\eta\s{R} /\eta\s{X} > 1$, a synchrotron radio spectrum is predicted that is harder than the one observed. However, as discussed in Section \ref{sec:two_com}, the results of \citet{Sironi2011} indicate that magnetic reconnection at the termination shock can accelerate particles, leading to a modification of the Maxwellian that would produce a softer radio synchrotron spectrum.
For HESS J1427--608 two possible scenarios were investigated. In the first, a present-day magnetic field of $B\s{age}=4.2\,\mu\text{G}$ is derived, along with a present-day luminosity of $L=1.2\times 10^{38}\,\text{erg}\,\text{s}^{-1}$. This scenario predicts PWN values that are very similar to the recently discovered pulsar PSR J2022+3842 and its associated PWN \citep{Arzoumanian2011}. However, the $B\s{age}=4.2\,\mu\text{G}$ scenario predicts a bright radio nebula that has thus far not been observed. An alternative scenario is considered where the magnetic field in the nebula has evolved to a very low value of $B\s{age}=0.4\,\mu\text{G}$, leading to a synchrotron spectrum that is compatible with radio upper limits. However, this scenario significantly under-predicts the \emph{Suzaku} observations presented by \citet{Fujinaga2012}. The $B\s{age}=0.4\,\mu\text{G}$ scenario represents an ancient PWN as the very small present-day magnetic field leads to a low level of synchrotron emission, while still remaining bright at GeV/TeV energies. The fact that the model cannot simultaneously predict both the radio
|
mathcal{H}^1(\Gamma_a)<\infty$, which implies that $\dim_H(\Gamma_a)=1$ for all $a\in(0,1/2]$.
\end{proof}
\end{theorem}
For $a\in(1/2,1)$, on the other hand, $F_a(x)$ is not nondecreasing, so we cannot find an upper bound for $L$ using the Triangle Inequality. In fact, no such upper bound exists when $a\in(1/2,1)$; to prove this, we must show that the graph's Hausdorff dimension is greater than 1, but in order to do this, we first must calculate the box-counting dimension of the graph.
\begin{theorem}
If $\Gamma_a$ is the graph of $F_a$, then its box-counting dimension $\dim_B(\Gamma_a)=\log_3(12a-3)$ for all $a\in(1/2,1)$.
\begin{proof}
Suppose $\Gamma_a$ is the graph of $F_a$, and suppose $a\in(1/2,1)$.
To determine the box-counting dimension of $\Gamma_a$, we will use a variant of the more familiar method of box-counting. Instead of counting squares of side length $\delta$, we will determine the smallest area $A(f_i)$ needed to cover the graph of iteration $f_i$ using rectangles that share a horizontal side length of $\delta$. Then the ``number'' of squares with side length $\delta$ needed to cover the graph of $f_i$ can be expressed as $N_\delta(f_i)=A(f_i)/\delta^2$. This variant works by the same principle as box-counting dimension with squares does, but it gives more precise answers for each $\delta$ we use.
Letting $\delta=1/3^i$, we see that for all $\Gamma_a$, $A(f_0)=1$. Using the horizontal and vertical ratios of contraction for (5), (6), and (7), we see that
\begin{flalign*}
A(f_{i+1})&=\frac{1}{3}aA(f_i)+\frac{1}{3}(2a-1)A(f_i)+\frac{1}{3}aA(f_i)\\
&=A(f_i)\left(\frac{4a-1}{3}\right).
\end{flalign*}
So by induction,
\begin{equation*}
A(f_i)=\left(\frac{4a-1}{3}\right)^i
\end{equation*}
and thus,
\begin{flalign*}
N_\delta(f_i)&=\frac{([4a-1]/3)^i}{(1/3)^{2i}}\\
&=(12a-3)^i.
\end{flalign*}
So for $a\in(1/2,1)$, the box-counting dimension of $\Gamma_a$ is
\begin{flalign*}
\dim_B(\Gamma_a)&=\lim_{i\to\infty}\frac{\log([12a-3]^i)}{-\log(1/3^i)}\\
&=\log_3 (12a-3).
\end{flalign*}
\end{proof}
\end{theorem}
Next, we will show that the Hausdorff dimension of $\Gamma_a$ is equal to its box-counting dimension by the Mass Distribution Principle (see, e.g. \cite{Falconer03}):
\begin{theorem}
If $\Gamma_a$ is the graph of $F_a$, then its Hausdorff dimension $\dim_H(\Gamma_a)=\log_3(12a-3)$ for all $a\in(1/2,1)$.
\begin{proof}
We consider a small variation on Katsuura's construction of $\Gamma_a$ using contraction mappings. Let $E_0=[0,1]\times[0,1]$ and define further levels of the construction by $E_{i+1}=w_1(E_i)\cup w_2(E_i)\cup w_3(E_i)$ where $i>0$ and $w_1$, $w_2$, and $w_3$ are mappings (5)--(7). Clearly $E_{i+1}\subset E_i$ for all $i\geq0$, and $\displaystyle\bigcap_{i=0}^\infty E_i=\Gamma_a$. This last relationship between the $E_i$s can be understood as follows: While in Okamoto's construction, linear segments are constructed ``upwards'' to a graph with infinite length, in this construction, rectangular regions are constructed ``downwards'' to the same graph which has zero area. Moreover, we see that each $E_i$ can be covered by $3^i$ rectangles of length $(1/3)^i$.
Using methods related to the box-counting process in Theorem 4, it can be shown that if $a\in(1/2,1)$, then the area of $E_{i+
|
1}$ can be expressed as
\begin{flalign*}
A(E_{i+1})&=a\left(\frac{1}{3}\right)A(E_i)+(2a-1)\left(\frac{1}{3}\right)A(E_i)+a\left(\frac{1}{3}\right)A(E_i)\\
&=\frac{4a-1}{3}A(E_i),
\end{flalign*}
and since $A(E_0)=1$, we have $\displaystyle A(E_i)=\left(\frac{4a-1}{3}\right)^i$ for all $i\geq0$.
Now, let $\mu$ be the natural mass distribution on $\Gamma_a$; we start with unit mass on $E_0$ and repeatedly ``spread'' this mass over the total area of each $E_i$. Also, let $U$ be any set whose diameter $|U|<1$. Then there exists some $i\geq0$ such that
\begin{equation*}
\left(\frac{1}{3}\right)^{i+1}\leq|U|<\left(\frac{1}{3}\right)^i,
\end{equation*}
an inequality that applies to any $U$ satisfying $0<|U|<1$. Given these conditions on the diameter of $U$, it is clear that for every $U$, there is some $i$ such that $U$ is contained in an open square of side length $(1/3)^i$ and $U$ contains points in at most two level-$i$ ``sub-rectangles.''
Hence, the area of $U$ is bounded above by the area of the open square containing it; that is, $A(U)\leq(1/9)^i$. In terms of measure, we know that the entire area of $U$ can be contained in $E_i$, so
\begin{flalign*}
\mu(U)&\leq\frac{A(U\cap E_i)}{A(E_i)}\\
&\leq\frac{(1/9)^i}{([4a-1]/3)^i}\\
&\leq\left(\frac{1}{12a-3}\right)^i.
\end{flalign*}
And since $\displaystyle\left(\frac{1}{3}\right)^{i+1}\leq|U|$ implies that $\displaystyle\left(\frac{1}{3}\right)^i\leq3|U|$, we have
\begin{equation*}
\mu(U)\leq\left(\frac{1}{12a-3}\right)^i=\left(\frac{1}{3^i}\right)^{\log_3 (12a-3)}\leq(3|U|)^{\log_3 (12a-3)}=(12a-3)|U|^{\log_3 (12a-3)},
\end{equation*}
and therefore, by the Mass Distribution Principle, $\log_3 (12a-3)\leq\dim_H(\Gamma_a)\leq\dim_B(\Gamma_a)$, and given the upper bound obtained in Theorem 4, we have $\dim_H(\Gamma_a)=\log_3 (12a-3)$.
\end{proof}
\end{theorem}
This means that if $a>1/2$, then $\dim_H(\Gamma_a)>1$. And if the Hausdorff dimension of $\Gamma_a$ exceeds 1, then its one-dimensional Hausdorff measure---that is, its length---must be infinite. So for $a>1/2$, the graph's arc length $L$ indeed has no upper bound.
Given the result of Theorem 5, it is clear that $\dim_H(\Gamma_a)$ ranges continuously between 1 and 2 for all $a\in(1/2,1)$. This means that for $a\in(1/2,1)$, $\Gamma_a$ meets the criterion of a fractal, being a self-similar set whose Hausdorff dimension exceeds its topological dimension.
\section{Conclusions}
In his paper, Okamoto \cite{Okamoto05} posed another question related to $F_a$'s differentiability: If $a\in(0,1/3)$, there are infinitely many $x$ for which $F_a'(x)$ is finite but nonzero, but how do we classify them? In the corollary to Theorem 1, we have provided a partial answer to this question. Since $F_a'(x)=0$ if $a\in(0,1/3)$ and $x$ satisfies $\gamma=1/3$, we know that the points in question at least must be non-normal numbers in $[0,1]$. It follows that the set of such points must occupy a subset of $[0,1]$ with Lebesgue measure 0.
\section{Acknowledgments}
The author would like to thank Professor Daniel Jackson for direction and assistance with this project.
\newpage
|
\section{Introduction}
\emph{Dynamical systems} are the classical constructive formalism for
behaviour arising from the deterministic evolution of system state
over time \cite{Birkhoff1927}, dating back to the works of Newton and
Laplace. Clearly \emph{temporal logics}, with operators such as
`next', `always', `eventually' and `for-at-least', constitute a
companion descriptive formalism. However, the relation is not
one-to-one: One one hand, there is a unifying theory underlying the
various perspectives on dynamical systems as monoid actions, which
uniformly covers discrete and continuous, as well as hybrid systems
\cite{Jacobs2000}. But on the other hand, the diversity of temporal
logics in literature is immense, cf.~\cite{Venema2001}, and the choice
for a particular system is often justified by ad-hoc pragmatic
arguments. The present article explores a systematic and fairly
generic approach to the construction of temporal logics for dynamical
systems, via the rather recent mathematical field of \emph{universal
coalgebra} which appears to be intimately connected to both
dynamical systems \cite{Rutten2000} and modal logics \cite{Kurz2008}.
A different approach also based on coalgebras and the Stone duality
has been suggested \cite{Bonsangue2005} for constructing modal logics
of \emph{transition systems}, a close relative of dynamical systems in
computer science.
The method outlined in the remainder of this article, while
theoretically simple, touches on many different fields of mathematics:
order theory, category theory, algebra, coalgebra, classical modal
logics à la Kripke, and coalgebraic logics à la Moss \cite{Moss1999}.
Thus a significant proportion of the available space is dedicated to
reviewing the relevant definitions and propositions from the
respective standard literature. This review makes up the sections
\ref{ingredients1} and \ref{ingredients2}. The expert reader is
encouraged to skip ahead: Section~\ref{constructions} ties up all the
loose ends and gives a novel contribution. There a selection of
obvious coalgebraic perspectives on dynamical systems is explored, and
the respective logics entailed by applying Moss's construction are
characterized.
\section{Review: Classical Ingredients}
\label{ingredients1}
This section reviews some basic definitions and propositions.
\subsection{Order Relations}
We assume that the reader is familiar with basic order-theoretic
properties of binary relations, namely with \emph{reflexive},
\emph{transitive}, \emph{symmetric} relations, and with
\emph{preorders}, \emph{partial orders} and \emph{equivalences}. We
give two additional related definitions that are not quite as
universal:
\begin{definition}
Let $X$ be a set. A binary relation $R \subseteq X^2$ is called
\begin{itemize*}
\item \emph{non-branching} if and only if $x \mathrel R y$ and $x
\mathrel R z$ imply $y \mathrel R z$ or $z \mathrel R y$, and
\item \emph{linear} if and only if $x \mathrel R y$ or $y \mathrel R
x$,
\end{itemize*}
respectively, for all $x, y, z \in X$.
\end{definition}
\subsection{Monoids}
We assume that the reader is familiar with the notions of a
\emph{monoid} $\mathbb M = (M, 0, +)$, and of monoid \emph{generators}
and \emph{cyclic} monoids. Every monoid induces an ordering relation.
\begin{definition}[Monoid Order]
Let $\mathbb M = (M, 0, {+})$ be a monoid. For any elements $a, b
\in M$, we write $a \leq_{\mathbb M} b$ if and only if there is some
$c \in M$ such that $a + c = b$. We say that $a \leq_{\mathbb M} b$
\emph{via} $c$. It follows directly from the monoid axioms that
$\leq_{\mathbb M}$ is reflexive and transitive, hence a preorder.
By extension, $\mathbb M$ itself is called
\emph{symmetric}/\emph{non-branching}/\emph{linear} if and only if
$\leq_{\mathbb M}$ is symmetric/non-branching/linear, respectively.
\end{definition}
Note that being symmetric in this sense is different from being
Abelian. In fact, symmetry characterizes a subclass of monoids, the
groups.
\begin{lemma}[Groups]
A monoid $\mathbb M$ is a group if and only if it is symmetric.
Every symmetric monoid is trivially linear, with the degenerate
order $(\leq_{\mathbb M}) = M^2$, the full relation.
\end{lemma}
\subsection{Dynamical Systems}
\begin{definition}[Dynamical System]
Let $\mathbb T = (T, 0, {+})$ be a monoid called \emph{time}. A
\emph{dynamical system} is an enriched structure $\mathbb S =
(\mathbb T, S, \Phi)$ with
\begin{itemize*}
\item a set $S$ called \emph{state space}, and
\item a map $\Phi : S \times T \to S$ called \emph{dynamics},
\end{itemize*}
such that
\begin{align}
\Phi(s, 0) &= s & \Phi\bigl(\Phi(s, t), u\bigr) &= \Phi(s, t + u)
\end{align}
In other words, $\Phi$ is a \emph{right monoid action} of $\mathbb
T$ on $S$. $\mathbb S$ is called
\begin{itemize*}
\item \emph{linear-time} if and only if $\mathbb T$ is linear,
otherwise \emph{nonlinear-time}, and
\item \emph{invertible} if and only if $\mathbb T$ is symmetric.
\end{itemize*}
\end{definition}
\begin{corollary}
There are no invertible nonlinear-time dynamical systems.
\end{corollary}
Dynamical systems are a fundamental model class of many natural and
social sciences. In comparison with their younger counterpart in
computer science, automata and transition systems, dynamical systems
are typically
\begin{itemize}
\item behaviourally weaker -- deterministic, non-pointed (without
distinguished initial states) and total (without spontaneous
termination), but
\item structurally stronger -- with additional features of time
(density, completeness) and state space (topology, metric,
differential geometry, measures).
\end{itemize}
Automata-like construction can be emulated by dynamical systems; see
examples below.
\begin{definition}[Step, Trajectory, Orbit]
From the dynamics map we may derive three forms of auxiliary functions:
\begin{align*}
\Phi^t &: S \to S
&
\Phi_s &: T \to S
&
\Phi^\circ &: S \to \mathcal PS
\\
\Phi^t(s) &= \Phi(s, t) &
\Phi_s(t) &= \Phi(s, t) &
\Phi^\circ(s) &= \mathrm{Img}(\Phi_s) = \{ \Phi(s, t) \mid t \in T \}
\end{align*}
\begin{itemize*}
\item $\Phi^t$ is called the \emph{step} of \emph{duration} $t$, or
just the $t$-step.
\item $\Phi_s$ is called the \emph{trajectory} of \emph{initial}
state $s$.
\item $\Phi^\circ(s)$ is called the \emph{orbit} of state $s$.
\end{itemize*}
\end{definition}
\begin{lemma}[Homomorphic Steps]
The dynamical systems with time $\mathbb T$ are precisely those
systems $(\mathbb T, S, \Phi)$ such that the step construction is a
monoid homomorphism from $\mathbb T$ into the monoid of functions of
type $S \to S$ with right composition.
\begin{align}
\Phi^0 &= \mathrm{id}_S & \Phi^{t+u} &= \Phi^u \circ \Phi^t
\end{align}
where $\mathrm{id}_X(x) = x$ and $(f \circ g)(x) = f(g(x))$ for all $x$.
\end{lemma}
\begin{corollary}[Generating Steps]
If $G \subseteq T$ is a generator of $\mathbb T$, then $\Phi$ is
determined uniquely by the collection of steps $\{ \Phi^t \mid t \in
G \}$.
\end{corollary}
\begin{example}[Instances of Time]
\label{ex:time}%
\leavevmode
\begin{itemize}
\item The time monoid $(\mathbb{N}, 0, +)$ yields standard
non-invertible, discrete-time dynamical systems. The step
$\Phi^1$ is generating. Trajectories are (one-sided) infinite
sequences.
\item The time monoid $(\mathbb{Z}, 0, +)$ yields standard
invertible, discrete-time dynamical systems. The step $\Phi^1$ is
generating and must be invertible. Trajectories are two-sided
infinite sequences.
\item The time monoid $(\mathbb{R}_+, 0, +)$ yields standard
non-invertible, continuous-time dynamical systems. No simple step
generator exists. Trajectories are one-sided parametric curves.
\item The time monoid $(\mathbb{R}, 0, +)$ yields standard
invertible, continuous-time dynamical systems. No simple step
generator exists; classical definitions are given as solutions to
ordinary differential equations. Trajectories are two-sided
parametric curves.
\item The ``time'' monoid $(\Sigma^*, \varepsilon, \cdot)$ over some
finite alphabet $\Sigma$ yields total semiautomata, or
deterministic finitely-labelled transition systems. The steps $\{
\Phi^a \mid a \in \Sigma \}$ (columns of the transition table) are
generating. Trajectories are big-step transition functions of
total automata, mapping input words to final states.
\end{itemize}
\end{example}
\subsection{Propositional Modal Logics}
We assume that the reader is familiar with the syntax and semantics of
classical propositional logics and their presentation in terms of the
connectives $\neg$ and $\to$.
\begin{definition}[Syntax of Propositional Modal Logics]
The modal extension of classical propositional logics adds two unary
connectives $\Box$ and $\lozenge$, taking $\Box$ as primitive and defining
\begin{equation*}
\lozenge A = \neg \Box \neg A
\end{equation*}
\end{definition}
\begin{definition}[Semantics of Propositional Modal Logics]
A \emph{normal} modal extension of classical propositional logics
adds at least the deduction rule of \emph{necessitation} or
\emph{generalization}, and the axiom of \emph{distribution}:
\begin{align*}
A \vdash \Box A &&
\Box(A \to B) \to (\Box A \to \Box B)
\end{align*}
\end{definition}
\begin{example}
Important normal modal logics are obtained by adding certain axioms:
\begin{itemize}
\item $\Box A \to A$ added to the minimal system results in the logic $T$.
\item $\Box A \to \Box \Box A$ added to $T$ results in the logic
$S4$.
\item $\Box(\Box A \to B) \lor \Box(\Box B \to A)$ added to $S4$
results in the logic $S4.3$.
\item $\lozenge A \to \Box \lozenge A$ added to $S4$ or $S4.3$
results in the logic $S5$.
\end{itemize}
\end{example}
\subsection{Kripke Semantics}
\begin{definition}[Kripke Frame]
A Kripke frame is a structure $(W, R)$ with a set $W$ of
\emph{worlds} and a relation $R$ on $W$ called \emph{accessibility}.
\end{definition}
\begin{definition}[Kripke Model]
Let $(W, R)$ be a Kripke frame. A Kripke model (of propositional
modal logic) is an extended structure $(W, R, \Vdash)$, where
$\Vdash$ is a relation between $W$ and the language $\mathit{Prop}$
of logical formulas, such that
\begin{equation*}
\begin{aligned}
w &\Vdash \neg A && \iff && w \not\Vdash A
\\
w &\Vdash A \to B && \iff && w \not\Vdash A \text{~or~} w \Vdash B
\\
w &\Vdash \Box A && \iff && v \Vdash A \text{~whenever~} w \mathrel{R} v
\end{aligned}
\end{equation*}
\nopagebreak
We say that $w$ \emph{satisfies} $A$ in $(W, R, \Vdash)$ if and only
if $w \Vdash A$.
\end{definition}
\begin{lemma}
The satisfaction relation $\Vdash$ of a Kripke frame is determined
uniquely by the satisfaction of atomic propositions.
\end{lemma}
\begin{definition}[Validity]
A formula $A$ is called \emph{valid} in
\begin{itemize}
\item a world $w$ if and only if $w$ satisfies $A$,
\item a Kripke model $(W, R, {\Vdash})$ if and only if it is valid
in all worlds $w \in W$,
\item a Kripke frame $(W, R)$ if and only if it is valid in all
Kripke models $(W, R, \Vdash)$,
\item a class $C$ of Kripke frames if and only if it is valid in all
members of $C$.
\end{itemize}
\end{definition}
\begin{definition}[Soundness/Completeness]
A propositional modal logic $L$ is called, with respect to a class
$C$ of Kripke frames,
\begin{itemize*}
\item \emph{sound} if and only if truth in $L$ implies validity in
$C$, and
\item \emph{complete} if and only if validity in $C$ implies truth
in $L$.
\end{itemize*}
\end{definition}
\begin{theorem}[Soundness/Completeness]
\label{theorem:log-frame}
The modal logics $S4$/$S4.3$/$S5$ are sound and complete for the
class of Kripke frames $(W, R)$ where $R$ is an
arbitrary/non-branching/symmetric preorder, respectively.
\end{theorem}
\begin{definition}[Finite Model Property]
A propositional modal logic $L$ is said to have the \emph{finite
model property}, if and only if it is complete for a class of
finite Kripke frames.
\end{definition}
\begin{theorem}
\label{theorem:log-fmp}
The modal logics $S4$/$S4.3$/$S5$ have the finite model property,
for subclasses of the respective classes given in
Theorem~\ref{theorem:log-frame}.
\end{theorem}
\section{Review: Additional Ingredients}
\label{ingredients2}
This section reviews some definitions and propositions that are also
basic, but from less well-known fields. See
\cite{Rutten2000,Moss1999} for greater detail.
\subsection{Category Theory}
\begin{definition}[Set Endofunctor]
A \emph{functor} $F$ on the category of sets, or \emph{set
endofunctor}, is a map that assigns
\begin{itemize*}
\item to every set $X$ a set $FX$, and
\item to every function $h : X \to Y$ a function $Fh : FX \to FY$,
\end{itemize*}
such that
\begin{align*}
F(\mathrm{id}_X) &= \mathrm{id}_{FX} & F(g \circ h) &= Fg \circ Fh
\end{align*}
where $\mathrm{id}_X(x) = x$ and $(g \circ f)(x) = g\bigl(f(x)\bigr)$.
\end{definition}
\noindent All functors considered in the following are tacitly set
endofunctors.
\begin{definition}[Monotonic Functor]
A functor $F$ is called \emph{monotonic} if and only if $X \subseteq
Y$ implies $FX \subseteq FY$.
\end{definition}
Coalgebraic logics deal with a class of functors called
\emph{standard}, which are essentially monotonic, plus an additional
condition, namely preservation of weak pullbacks, that is rather
technical but fortunately inessential for the present discussion.
\begin{definition}[Finitary Functor]
A functor is called \emph{finitary} if and only if
\begin{equation*}
FX \subseteq \textstyle\bigcup \{ FY \mid Y \subseteq X; Y \text{~finite} \}
\end{equation*}
otherwise \emph{infinitary}. For monotonic finitary functors, the above
is necessarily an equality. A standard, infinitary functor $F$
has a \emph{finitary restriction} $F\fin$ defined by
\begin{align*}
F\fin X &= \textstyle\bigcup \{ FY \mid Y \subseteq X; Y \text{~finite} \}
&
F\fin (h : X \to Y) &= F h \rvert_{F\fin X}
\end{align*}
\end{definition}
\begin{definition}[Functor Product]
The pointwise Cartesian product of functors is again a functor.
\begin{align*}
(F \times G)X &= FX \times GX & \bigl((F \times G)h\bigr)(x, y) &= \bigl((Fh)(x), (Gh)(y)\bigr)
\end{align*}
\end{definition}
\begin{example}
The following are standard functors:
\begin{itemize}
\item The \emph{identical} functor $\mathcal I$
\begin{align*}
\mathcal IX &= X & \mathcal Ih &= h
\end{align*}
$\mathcal I$ is finitary; hence $\mathcal I\fin = \mathcal I$.
\item The \emph{constant} functor $\_ @ C$ for some set $C$
\begin{align*}
X @ C &= C & h @ C &= \mathrm{id}_C
\end{align*}
$\_ @ C$ is finitary.
\item the \emph{powerset} functor $\mathcal P$
\begin{align*}
\mathcal PX &= \{ W \mid W \subseteq X \} & (\mathcal Ph)(W) &=
\{ h(x) \mid x \in W \}
\end{align*}
$\mathcal P$ is not finitary; its finitary restriction is the
\emph{finite powerset} functor $\mathcal P\fin$.
\item the \emph{Hom} functor $\_^C$ for some set $C$
\begin{align*}
X^C &= \{ f \mid f : C \to X \} & (h^C)(g) &= h \circ g
\end{align*}
$\_^C$ is finitary if and only if $C$ is finite; its finitary
restriction is the \emph{image-finite} functor $\_^C\fin$.
\end{itemize}
\end{example}
Clearly, a relation $R \in \mathcal P(X \times Y)$ is precisely the
set of pairs $(x, y)$ for which there is some $r \in R$ such that
$\pi_1(r) = x$ and $\pi_2(r) = y$, where $\pi_1, \pi_2$ are the
natural projections from the binary Cartesian product. This seemingly
redundant presentation suggests an interaction of relations and
functors.
\begin{definition}[Relational Lifting]
Let $F$ be a functor. Every relation $R \in \mathcal P(X \times Y)$
has a \emph{lifting} $F[R] \in \mathcal P(FX \times FY)$ defined as
the set of pairs $(\hat x, \hat y)$ for which there is some $\hat r
\in FR$ such that $(F \pi_1)(\hat r) = \hat x$ and $(F \pi_2)(\hat
r) = \hat y$.
\end{definition}
\begin{example}
The liftings for the functors discussed above are as follows:
\begin{itemize}
\item The identical functor lift a relation to itself: $x
\mathrel{\mathcal I[R]} y$ if and only if $x \mathrel R y$.
\item The constant functor lifts to the identity relation: $c
\mathrel{[R] @ C} c'$ if and only if $c = c' \in
C$.
\item $Y \mathrel{\mathcal P[R]} Z$ if and only if for all $y \in Y$
there is a $z \in Z$, and vice versa, such that $y \mathrel{R} z$.
\item $f \mathrel{[R]^C} g$ if and only if $f(c) \mathrel R g(c)$
for all $c \in C$.
\end{itemize}
\end{example}
\subsection{Universal Coalgebra}
\begin{definition}[Coalgebra]
Let $F$ be a functor. An $F$-\emph{coalgebra} is a structure $(X,
f)$ with an object $X$ and an arrow $f : X \to FX$.
\end{definition}
\begin{definition}[Homomorphism]
Let $F$ be a functor. Let $(X, f)$ and $(Y, g)$ be $F$-coalgebras.
An $F$-\emph{coalgebra homomorphism} from $(X, f)$ to $(Y, g)$ is an
arrow $h : X \to Y$ such that $Fh \circ f = g \circ h$. We write $h
: (X, f) \to (Y, g)$ or simply $h : f \to g$.
\end{definition}
\begin{definition}[Final Coalgebra]
Let $F$ be a functor. An $F$-coalgebra $(Z, z)$ is called
\emph{final} if and only if there is a unique homomorphism $f! : f
\to z$ from any other $F$-coalgebra.
\end{definition}
\begin{theorem}
Every finitary functor has a final coalgebra.
\end{theorem}
\begin{definition}[Bisimulation]
Let $F$ be a functor. Let $(X, f)$ and $(Y, g)$ be $F$-coalgebras.
A \emph{bisimulation} between $(X, f)$ and $(Y, g)$ is a relation $R
\subseteq X \times Y$ that can be extended to an $F$-coalgebra $(R,
r)$ such that the projections are coalgebra homomorphisms $\pi_1 : r
\to f$ and $\pi_2 : r \to g$. We say that states $x \in X$ and $y
\in Y$ are \emph{bisimilar} if and only if there is a bisimulation
relating them.
\end{definition}
The final coalgebra can be seen as a system of representatives of
equivalence classes modulo bisimilarity.
\begin{theorem}
Let $F$ be a standard functor. If a final $F$-coalgebra $(Z, z)$
exists then, for given $F$-coalgebras $(X, f)$ and $(Y, g)$, two
states $x \in X; y \in Y$ are bisimilar if and only if $f!(x) =
g!(y)$.
\end{theorem}
\begin{definition}[Parallel Coalgebra Composition]
\label{def:coalg-par}%
Coalgebras with the same carrier can be combined in parallel: Let
$(X, f)$ be an $F$-coalgebra and $(X, g)$ be a $G$-coalgebra. Then
$(X, \langle f, g \rangle)$ is an $(F \times G)$-coalgebra, where
\begin{equation*}
\langle f, g \rangle(x) = \bigl(f(x), g(x)\bigr)
\end{equation*}
\end{definition}
\subsection{Moss's Coalgebraic Logic}
The idea of Moss's coalgebraic logic \cite{Moss1999} is to replace
Kripe frames by $F$-coalgebras for some functor $F$, and to derive a
universal and natural modality from $F$ itself.
\begin{definition}[Moss's Coalgebraic Logic, Abstract]
Fix a standard functor $F$. Extend the syntax of propositional
logic by a pseudo-unary connective $\nabla$ that, unlike the
classical modalities like $\Box$, applies not to a single formula $A
\in \mathit{Prop}$ but to an expression of type either $\widehat A
\in F(\mathit{Prop})$ or $\widehat A \in F\fin(\mathit{Prop})$. For
infinitary $F$ where the choice makes a difference, the cases are
called \emph{infinitary
|
} and \emph{finitary} $F$-coalgebraic logics,
respectively. A Moss model is a structure $(X, f, \Vdash)$ where
$(X, f)$ is an $F$-coalgebra and $\Vdash$ is a relation between
coalgebra states and formulas, such that
\begin{align*}
x \Vdash \neg A \iff x \not\Vdash A
&&
x \Vdash A \to B \iff x \not\Vdash A \text{~or~} x \Vdash B
\end{align*}
as for Kripke models, but
\begin{equation*}
x \Vdash \nabla \widehat A \iff f(x) \mathrel{F[\Vdash]} \widehat A
\end{equation*}
\end{definition}
Moss's coalgebraic logic as presented here specifies satisfaction only
up to atomic propositions, in analogy to Kripke frames. In Moss's
original presentation, the specification is unique, in analogy to
Kripke models.
\begin{definition}[Moss's Coalgebraic Logic, Concrete]
Let $(X, f)$ be an $F$-coalgebra. Let $s : X \to \mathcal
P(\mathit{Prop}_0)$ be the map that assigns to each state $x \in X$
the desired set of valid atomic propositions. Then $(X, s)$ is a
$\mathrm{Const}\bigl(\mathcal P(\mathit{Prop}_0)\bigr)$-coalgebra.
For the parallel composite coalgebra $(X, g = \langle f, s
\rangle)$, a unique Moss model is specified by the additional clause
\begin{align*}
x \Vdash A \iff A \in s(x) \qquad (A \in \mathit{Prop}_0)
\end{align*}
\end{definition}
The following two propositions state that traditional Kripke frames
are essentially equivalent to the special case $F = \mathcal P$.
\begin{lemma}
\label{lemma:coalg-kripke-correspondence}%
$\mathcal{P}$-coalgebras $(X, f)$ are in one-to-one correspondence
to relations $R$ on $X$ by putting $x \mathrel{R} y$ if and only if
$y \in f(x)$.
\end{lemma}
\begin{theorem}
The Kripke modalities $\Box, \lozenge$ and the Moss modality
$\nabla$ for finitary $\mathcal P$-coalgebraic logics are
equivalent. For infinitary $\mathcal P$-coalgebraic logics, they
are also equivalent in the presence of infinitary conjunction and
disjunction; otherwise $\nabla$ is generally more expressive.
\begin{align*}
&w \Vdash_{\mathrm K} \Box A \iff w \Vdash_{\mathrm M} \nabla \{ A
\} \lor \nabla \emptyset & &w \Vdash_{\mathrm K} \lozenge A \iff w
\Vdash_{\mathrm M} \nabla \{ A, \top \}
\\
&w \Vdash_{\mathrm M} \nabla \widehat A \iff w \Vdash_{\mathrm K}
\Box \left({\textstyle\bigvee \widehat A}\right) \land {\textstyle
\bigwedge \lozenge \widehat A} & &\text{where}\quad \lozenge
\widehat A = \{ \lozenge B \mid B \in \widehat A \}
\end{align*}
where $\Vdash_{\mathrm K}$/\/$\Vdash_{\mathrm M}$ denote satisfaction
{\`a} la Kripke/Moss, respectively.
\end{theorem}
In general, the infinitary version of the operator $\nabla$ is better
matched with a logic where conjunction and disjunction are also
infinitary. While an uncommon topic classically, infinitary logics
are an important topic in modal logic because of their connection to
bisimulation. The following theorem generalizes a theorem of
Kripke-style logic, where bisimilarity is defined ad-hoc but
equivalently to the coalgebraic notion specialized as in
Lemma~\ref{lemma:coalg-kripke-correspondence}.
\begin{theorem}
In fully ($\land, \lor, \nabla$) infinitary $F$-coalgebraic logic,
two states $s, t \in S$ satisfy the same set of formulas if and only
if they are bisimilar.
\end{theorem}
\section{Constructions}
\label{constructions}
This section gives novel theoretical results by invetigating the
ramifications of the following recipe:
\begin{enumerate}
\item identify some generic $F$-coalgebraic view on dynamical systems,
\item use Moss's construction to obtain logics with $\nabla_F$
modality, depending on the functor $F$,
\item relate $\nabla_F$ to established temporal logic operators.
\end{enumerate}
Note that all of the following constructions have the state space $S$
of a fixed dynamical system as the carrier of some coalgebra for
various functors. Hence the associated logical languages can coexist
naturally in a single system, by the parallel composition given in
Definition~\ref{def:coalg-par}.
\subsection{Step Logics}
\begin{definition}[Step Coalgebra]
Let $\mathbb S = (\mathbb T, S, \Phi)$ be a dynamical system. For
any element $t \in T$, the $\mathcal I$-coalgebra $(S, \Phi^t)$ is
called the $t$-\emph{step coalgebra} of $\mathbb S$.
\end{definition}
\begin{definition}[Multi-Step Coalgebra]
Let $\mathbb S = (\mathbb T, S, \Phi)$ be a dynamical system. For
any subset $U \subseteq T$, the $\_^U$-coalgebra $(S, s \mapsto
\Phi_s \circ \mathrm{in})$, given the inclusion map $\mathrm{in} : U
\to T$, is called the $U$-\emph{multi-step coalgebra} of $\mathbb
S$.
\end{definition}
\begin{lemma}
The $\nabla$ modality of step coalgebras amounts to
\begin{itemize}
\item for the $t$-step:
\begin{equation*}
s \Vdash \nabla A \iff \Phi(s, t) \Vdash A
\end{equation*}
\item for the $U$-multi-step:
\begin{equation*}
s \Vdash \nabla \widehat A \iff \Phi(s, t) \Vdash \widehat A(t) \text{~for all~} t \in U
\end{equation*}
\end{itemize}
The functors for $t$-steps and finite $U$-multi-steps are finitary;
hence no additional distinction between finitary and infinitary
logics arises.
\end{lemma}
\begin{definition}[Step Modality]
\begin{align*}
\bigcirc A &= \nabla A &
\bigcirc_t A &= \nabla u \mapsto
\begin{cases}
A & (t = u)
\\
\top & (t \neq u)
\end{cases}
\end{align*}
\end{definition}
\begin{example}
\label{ex:steps}
(Multi-)Step coalgebras are of particular interest for finite
generators, since they specify the dynamics uniquely and concisely.
The following are generating, cf.\ Example~\ref{ex:time}:
\begin{itemize}
\item For time $(\mathbb N, 0, +)$, the $1$-step coalgebra maps
every state to its successor. The resulting temporal logic has
$\bigcirc$ as the \emph{next} operator of traditional unidirectional
discrete-time temporal logic.
\item For time $(\mathbb Z, 0, +)$, the $(\pm 1)$-step coalgebra
maps every state to its successor/predecessor, respectively. The
resulting temporal logic has $\bigcirc_{\pm1}$ as the
\emph{next}/\emph{previously} operators of traditional
bidirectional discrete-time temporal logic, respectively.
\item For ``time'' $(\Sigma^*, \varepsilon, \cdot)$, the
$\Sigma$-multi-step coalgebra maps every automaton state to its
response function (row of the transition table). The resulting
logic has $(\bigcirc_a)_{a\in\Sigma}$ as the generating cases of
Pratt's \emph{necessity} operators $[a]$ in dynamic
logic \cite{Pratt1976}, where they are extended to the free Kleene
algebra over $\Sigma$.
\end{itemize}
Interesting infinite, non-generating examples include:
\begin{itemize}
\item For time $(\mathbb R, 0, +)$ and $\delta > 0$, let $U$ denote
the open interval $(-\delta, \delta)$. The $U$-multi-step
coalgebra maps every state to its temporal $\delta$-neighbourhood.
\end{itemize}
\end{example}
\begin{lemma}
The modality $\nabla$ and the family of modalities $(\bigcirc_t)_{t
\in U}$ for generating $U$ are straightforwardly equivalent if $U$
is finite, and equivalent in the presence of infinitary conjunction
otherwise.
\begin{equation*}
x \Vdash \nabla \widehat A \iff x \Vdash \bigwedge_{t \in U} \bigcirc_t \widehat A(t)
\end{equation*}
\end{lemma}
The following construction is the multi-step limit case $U = T$.
\subsection{Trajectory Logics}
\begin{definition}[Trajectory Coalgebra]
Let $\mathbb S = (\mathbb T, S, \Phi)$ be a dynamical system. The
$\_^T$-coalgebra $(S, s \mapsto \Phi_s)$ is called the
\emph{trajectory coalgebra} of $\mathbb S$.
\end{definition}
\begin{lemma}
The $\nabla$ modality of trajectory coalgebras amounts to
\begin{equation*}
s \Vdash \nabla \widehat A \iff \Phi(s, t) \Vdash \widehat A(t) \text{~for all~} t \in T
\end{equation*}
\end{lemma}
The $\nabla$ trajectory modality is a surprisingly powerful logical
operator, with the severe disadvantage that there is no canonical
syntactic representation. The following examples are but a small
subset of useful special cases.
\begin{example}
Arguments of the $\nabla$ trajectory modality are functions of type
$T \to \mathit{Prop}$. Various intensional notations for such
functions, or time-dependent formulas, give rise to well-known
temporal operators. Note that all following examples work for
finitary $\nabla$.
\begin{itemize}
\item Consider discrete time $(\mathbb N, 0, +)$ or $(\mathbb Z, 0,
+)$. Define a \emph{zip} operator as
\begin{equation*}
A \leftrightharpoons B = \nabla t \mapsto
\begin{cases}
A & t \text{~even}
\\
B & t \text{~odd}
\end{cases}
\end{equation*}
Then a dynamic system is bipartite, with characteristic
formula $A$, if and only if $(A \leftrightharpoons
\neg A) \lor (\neg A \leftrightharpoons A)$ is valid in the Moss
model associated with its trajectories.
\item Consider automaton time $(\Sigma^*, \varepsilon, \cdot)$.
Define a \emph{consumption} operator as
\begin{equation*}
\mathit{eat}(L, A, B) = \nabla t \mapsto
\begin{cases}
A & t \in L
\\
B & t \not\in L
\end{cases}
\end{equation*}
for languages $L \subseteq \Sigma^*$ and formulas $A, B$. Now let
$A$ be a formula characterizing accepting states. Then an
automaton, as a dynamical system, accepts
\begin{itemize}
\item at least the language $L \subseteq \Sigma^*$ if and only if
$\mathit{eat}(L, A, \top)$
\item exactly the language $L \subseteq \Sigma^*$ if and only if
$\mathit{eat}(L, A, \neg A)$
\end{itemize}
is valid for its initial state(s) in the Moss model associated
with its trajectories.
\item Consider time with a linear antisymmetric order $<$. Define a
\emph{change} operator as
\begin{equation*}
\mathit{chg}(t, A, B, C) = \nabla u\mapsto
\begin{cases}
A & u < t
\\
B & u = t
\\
C & u > t
\end{cases}
\end{equation*}
for time duration $t$ and formulas $A, B, C$. Then
minimum/maximum-duration operators can be defined directly, in two
variants differing in the inclusion of boundary cases:
\begin{align*}
\mathrm{min}\, t.\; A &= \mathit{chg}(t, A, \top, \top) &
\mathrm{max}\, t.\; A &= \mathit{chg}(t, \top, \top, \neg A)
\\
\mathrm{min}'\, t.\; A &= \mathit{chg}(t, A, A, \top)
&
\mathrm{max}'\, t.\; A &= \mathit{chg}(t, \top, \neg A, \neg A)
\end{align*}
Imprecise operators such as \emph{until} can be expressed as
infinitary disjunctions:
\begin{equation*}
A \mathbin{\mathbf{U}} B = \bigvee_{t \in T} \mathit{chg}(t, A, B, \top)
\end{equation*}
\end{itemize}
\end{example}
\subsection{Orbit Logics}
The following construction shifts the coalgebraic focus from
trajectories to orbits which are images of trajectories, hence
abstracting from durations. The result is a family of qualitive
temporal logics that can be expressed naturally in the classical modal
operators, uniformly for all kinds of time structure.
\begin{definition}[Orbit Coalgebra]
Let $\mathbb S = (\mathbb T, S, \Phi)$ be a dynamical system. The
$\mathcal P$-coalgebra $(S, \Phi^\circ)$ is called the \emph{orbit
coalgebra} of $\mathbb S$. We say that in $\mathbb S$, $y$ is
\emph{reachable} from $x$, written $x \leadsto_{\mathbb S} y$, if
and only if $y \in \Phi^\circ(x)$.
\end{definition}
\begin{lemma}
\label{lemma:dyn2rel}%
For dynamical systems $\mathbb S$, the reachability relation
$\leadsto_{\mathbb S}$ is
\begin{enumerate}
\item always a preorder,
\item additionally non-branching, but not generally linear, if
$\mathbb S$ is linear-time,
\item additionally symmetric if $\mathbb S$ is invertible.
\end{enumerate}
\end{lemma}
\begin{proof}
We have $x \leadsto_{\mathbb S} y$ if and only if there is some $t$
such that $\Phi(x, t) = y$. We say $x \leadsto_{\mathbb S} y$ via
$t$.
\begin{enumerate}
\item Reflexivity and transitivity follow directly from the monoid
axioms: $x \leadsto_{\mathbb S} x$ via $0$, and if $x
\leadsto_{\mathbb S} y$ via $t$ and $y \leadsto_{\mathbb S} z$ via
$u$, then $x \leadsto_{\mathbb S} z$ via $t + u$.
\item Assume that $x \leadsto_{\mathbb S} y$ via $t$ and $x
\leadsto_{\mathbb S} z$ via $u$. By linearity of $\mathbb T$
assume, without loss of generality, that $t \leq_{\mathbb T} u$
via $v$. Then $y \leadsto_{\mathbb S} z$ via $v$.
\item For symmetric $\mathbb T$, if $x \leadsto_{\mathbb S} y$ via
$t$, then $y \leadsto_{\mathbb S} x$ via $-t$. \qedhere
\end{enumerate}
\end{proof}
\noindent The weakening in case 2 of the preceding proposition is necessary.
\begin{example}[Nonlinear Linear-Time Dynamical System]
Set $T = \{0\}$, giving rise to the singleton monoid which is
trivially linear. This fixes $\Phi$ completely as $\Phi(s, t) =
\Phi(s, 0) = s$, giving rise to a ``still-life'' structure of time.
Then neither $x \leadsto_{\mathbb S} y$ nor $y \leadsto_{\mathbb S}
x$ for $x \neq y$.
\end{example}
\begin{definition}[Orbital Frame]
A Kripke frame is called \emph{orbital} if and only if it
corresponds, in the sense of
Lemma~\ref{lemma:coalg-kripke-correspondence}, to the orbital
coalgebra of some dynamical system. An orbital frame is called
linear-time/invertible if and only if it corresponds to the orbital
coalgebra of some linear-time/invertible dynamical system,
respectively.
\end{definition}
\noindent Using the preceding definition, Lemma~\ref{lemma:dyn2rel}
extends to Kripke frames.
\begin{lemma}
\label{lemma:dyn2rel2}%
For any orbital Kripke frame $\mathbb F = (W, R)$, the relation $R$ is
\begin{enumerate}
\item always a preorder,
\item additionally non-branching if $\mathbb F$ is linear-time,
\item additionally symmetric if $\mathbb F$ is invertible.
\end{enumerate}
\end{lemma}
\noindent This statement has a partial, finitary converse.
\begin{lemma}
\label{lemma:rel2dyn}%
A finite Kripke frame $(W, R)$ is
\begin{enumerate}
\item always orbital if $R$ is a preorder,
\item additionally linear-time if $R$ is non-branching,
\item additionally invertible if $R$ is symmetric.
\end{enumerate}
\end{lemma}
\begin{proof}
Construct a dynamical system $\mathbb S = (\mathbb T, S, \Phi)$ with
$(\leadsto_{\mathbb S}) = R$. In any case, clearly $S = W$.
Proceed in reverse order and increasing flexibility of cases. For
the latter two, consider the partition of $W$ into \emph{strongly
connected components} (sccs) of the preorder $R$: maximal subsets
$C \subseteq X$ such that $x \mathrel R y$ for all $x, y \in C$. We
write $x \sim y$ if and only if $x, y$ are in the same scc, that is
$x \mathrel R y$ and $y \mathrel R x$.
\begin{enumerate}
\setcounter{enumi}{2}
\item Set $\mathbb T = (\mathbb Z, 0, +)$. By symmetry of $R$ there
are no related pairs across sccs. For each component $C$ choose
an arbitrary cyclic permutation. Set $\Phi^1$ to their
composition.
Then
\begin{itemize}
\item $x \leadsto_{\mathbb S} y$ via some $i < k$, where $k$ is the
size of the scc containing both, if $x \mathrel R y$, and
\item otherwise $x \not\leadsto_{\mathbb S} y$.
\end{itemize}
\setcounter{enumi}{1}%
\item Set $\mathbb T = (\mathbb N, 0, +)$. We say that $y$ is a
\emph{successor} of $x$, writing $x \ll y$, if and only if $x
\mathrel R y$ but not $y \mathrel R x$. Clearly, $x \mathrel R y$
if and only if either $x \sim y$ or $x \ll y$. We say that $x$ is
\emph{transient} if it has successors. Since $W$ is finite and
$R$ is non-branching, every transient $x$ has a unique least
successor $x'$, and all elements reachable from $x$ are
successors. Set $\Phi^1(x) = x'$. For non-transient $x$, all
elements reachable from $x$ are in the same scc. Proceed as
above. Then
\begin{itemize}
\item $x \leadsto_{\mathbb S} y$ via some $i < k$, where $k$ is the
number of successors of $x$, if $x \ll y$,
\item $x \leadsto_{\mathbb S} y$ via some $i < k$, where $k$ is the
size of the scc containing both, if $x \sim y$, and
\item otherwise $x \not\leadsto_{\mathbb S} y$.
\end{itemize}
\setcounter{enumi}{0}%
\item There are in general no least successors, and there may
non-successors reachable from transient elements. A more basic
construction is needed: Set $\mathbb T = (\mathbb N^*,
\varepsilon, \cdot)$, the free monoid over $\mathbb N$. For each
$x \in W$ choose some infinite sequence $y = (y_0, y_1, \dots) \in
W^\omega$ such that $x \mathrel R z$ if and only if $z = y_i$ for
some $i$. This is always possible since the set $\{ z \mid x
\mathrel R z\}$ is finite and nonempty. For the generating steps
$\{ \Phi^n \mid n \in \mathbb N\}$, set $\Phi^n(x) = y_n$. Then
\begin{itemize}
\item $x \leadsto_{\mathbb S} y$ via $1$, if $x \mathrel R y$, and
\item otherwise $x \not\leadsto_{\mathbb S} y$. \qedhere
\end{itemize}
\end{enumerate}
\end{proof}
\begin{theorem}
The modal logics $S4$/$S4.3$/$S5$ are sound and complete for
arbitrary/linear-time/invertible orbital frames, respectively.
\end{theorem}
\begin{proof}
$S4$/$S4.3$/$S5$ are sound for the class of Kripke frames $(W, R$)
where $R$ is an arbitrary/non-branching/symmetric preorder,
respectively. By Lemma~\ref{lemma:dyn2rel2}, they are also sound for
the subclasses of arbitrary/linear-time/invertible orbital frames,
respectively.
$S4$/$4.3$/$S5$ are complete for the class of Kripke frames $(W, R$)
where $R$ is an arbitrary/non-branching/symmetric preorder,
respectively, and have the finite model property. By
Lemma~\ref{lemma:rel2dyn}, they are also complete for the
subclasses of arbitrary/linear-time/invertible orbital frames,
respectively.
\end{proof}
\begin{example}
The operators $\Box$ and $\lozenge$ are well-suited to express
``long-term'' behavioral properties of dynamical systems. For
instance, let $A$ be the characteristic formula of a subset $U
\subseteq S$ of the state space. Then $U$ is a stationary solution
of a dynamical system if and only if $A \to \Box A$ is valid in the
Moss model associated with its orbits.
\end{example}
\section{Conclusion}
Many operators discussed in the temporal logic literature can be
subsumed under a common framework by viewing them as instances of
Moss's modality $\nabla$, for some coalgebraic presentation of the
underlying dynamical system models. As a rule of thumb,
\begin{itemize}
\item step coalgebras go with discrete time,
\item trajectory coalgebras go with quantitative operators for either
discrete or dense time, and
\item orbit coalgebras go with arbitrary time and qualitative
operators, in particular the classical modal operators and the
framework of normal modal logics.
\end{itemize}
The examples given in this article are of course only a small
selection to prove the viability of the approach. There is
considerable potential for generalization. The trajectory modality is
an extremely expressive tool, and it is likely that many other
temporal operators can be shown to coincide with particular
intensional notations for it. Besides, coalgebraic perspectives on
dynamical systems other than the three detailed above could be
considered. An interesting open problem and direction for future
research is the integration of measure-theoretic temporal operators,
for instance in duration calculus \cite{Chaochen1991}, into the
framework.
|
\section{Introduction}
Paper \cite{1} considered the possibility of a
rapid variation of the
neutrino magnetic
form factor at small momentum transfer.
Such a rapid variation would allow comparatively
large magnetic
(transition) moments at negligible
momentum transfer
(absorption/emission of almost real photon
by neutrino and neutrino
propagation in external magnetic field)
despite severe bounds
from $\nu e$-scattering experiments and
the lack of pair production at
$e^+ e^- $ colliders.
A neutrino magnetic form factor
appears at the 1-loop level and varies
with momentum transfer with a typical
scale set by the virtual particles masses.
In scattering experiments momentum
transfer varies from several MeV
(for $\bar{\nu}_e$ from reactors) to
100 MeV ($\nu_{\tau}$ scattering),
and a low-energy enhancement can only
be achieved if very
light particles run in the loop
and couple to the photon. The only existing
charged particle with a mass in the
right range is the electron, and it indeed
couples with $\nu_e$. However, the
real momentum scale is set
by the $W$-boson running in the graph,
and its huge mass determines the
value of $q^2$ at which the neutrino magnetic
form factor starts to diminish.
The way out of this was suggested in ref
\cite{1} and consists in
using a virtual neutral light spinor particle
with nonzero magnetic moment and
a light scalar instead of
charged particles in the loop.
Such particles could
have escaped detection, and the origin
of the magnetic moment of the
new fermion (a non-renormalizable interaction)
could be found at a much
higher scale, involving particles
of mass $M \gg M_W$.
The result of the calculation of the
neutrino magnetic form factor was
given in \cite{1} in the following form:
\begin{equation}
\mu_{\nu}= \frac{f^2 \mu_N}{16\pi^2}
I(m_N, m_{\varphi}, m_{\nu},
q^2) \;\; ,
\label{1}
\end{equation}
where $\varphi$ is the light scalar
particle while $f$ is Yukawa
coupling constant. The interaction $f
\bar{\nu} N \varphi$ induces
$\mu_{\nu}$ at one loop
(in this paper we
consider both $\nu$ and $N$ as Dirac fermions)
(this coupling breaks the elecroweak SU(2)
symmetry, but this
can indeed occur after
the usual standard model symmetry breaking).
In general $I \sim 1$, but in the special
case where $\nu$ and $N$
are nearly degenerate while the scalar
mass is negligeable,
($m_{\nu} \approx m_N
=m$, $m_{\varphi} \ll m_{\nu} -m_N$)
a logarithmic enhancement takes place
at small
$q^2$ \cite{1}:
\begin{equation}
I = 4\ln \frac{m}{2(m_N -m_{\nu})} -\frac{7}{2}
\;\; , \;\; q^2 \mathrel{\mathpalette\fun <} m^2
\label{2}
\end{equation}
\begin{equation}
~~~~I = -1/2 ~~~~~~~~~~~~~~~~~~~~~~ ,
\;\; q^2 \gg m^2 \;\; .
\label{3}
\end{equation}
When deriving the expressions
(\ref{2}) and (\ref{3}) we dealt both with
ultraviolet-convergent and
-divergent integrals
over the virtual particles 4-momentum $k_{\mu}$.
The ultraviolet
divergent integral has the following form:
\begin{equation}
J_{\mu} = \int \frac{d^4 k}{(k^2 +a^2)^3} \hat{k}
\sigma_{\mu\nu}\hat{k} q_{\nu} \;\; ,
\;\; \sigma_{\mu\nu}
=\frac{1}{2}(\gamma_{\mu}\gamma_{\nu} -
\gamma_{\nu}\gamma_{\mu}) \;\;
.
\label{4}
\end{equation}
We calculated this integral in 4 space-time
dimensions with a sharp
symmetrical cut-off
$\Lambda$ and got zero when averaging over the
directions of 4-momentum
$k_{\mu}$:
\begin{equation}
\gamma_{\alpha} \frac{1}{2}(\gamma_{\mu}\gamma_{\nu}
-\gamma_{\nu}\gamma_{\mu})\gamma_{\alpha} =
2g_{\mu\nu} -2g_{\nu\mu}
=0
\label{5}
\end{equation}
We also remarked that using instead
dimensional regularization adds
a non-trivial constant due to a $d-4$ factor
from the $\gamma$-matrices
algebra
($\gamma_{\alpha}\frac{1}{2}(\gamma_{\mu}\gamma_{\nu}-
\gamma_{\nu}\gamma_{\mu})\gamma_{\alpha} = (d-4)
\frac{1}{2}(\gamma_{\mu}\gamma_{\nu} -
\gamma_{\nu}\gamma_{\mu})$)
multiplies the pole in $1/(d-4)$ from
the integration over $|k|$.
So, in this way of
performing calculations the function $I$
shifts by a constant.
This is of course neither unexpected nor
discouraging, since we are
dealing with the effective interaction
of a particle
$N$ with bare magnetic moment, which
is nonrenormalizable.
In \cite{1} it was naturally assumed
that the $N$ magnetic moment
was due in turn to a
loop with virtual heavy charged
particles of mass $M$, $M\mathrel{\mathpalette\fun >} 100$
GeV. At scale $M$ we have a perfectly
renormalizable theory while
for $q\ll M$ we
had to deal with a nonrenormalizable
effective theory.
In such an effective theory the
logarithmic term in (\ref{2}) which
originates from small momentum of
virtual particles is calculated
unambiguously while the precise value
of the constant may
depend on how the theory looks at the scale $M$.
Quite intriguing is the fact that the
dimensional regularization
of the effective one loop calculation
discussed above produces exaclty an
additional factor $\delta I =+1/2$,
which would result in a precocious
decrease of the magnetic moment,
long before the $M$ scale is reached:
$I \sim m^2/q^2$ for $q^2 > m^2$ !
We thus have three possibilities:
a) The constant term in $I$ depends on
the form of the theory at scale
$M$ at which magnetic moment of
particle $N$ is determined;
b) The constant term is universal
and equals 0 (as dimensional
regularization suggests),
leading to precocious power suppression
of the magnetic form factor
c) The constant term is universal
and equals $-1/2$ as suggested by
our naive sharp symmetric cut-off,
that is the superficially ultraviolet
divergent contribution from the loop vanishes identically.
To determine which of these possibilities is realized,
we study in the present note the
simplest renormalizable model for inducing the
magnetic moment of $N$:
we assume a simple Yukawa coupling of $N$
with charged spinor and scalar
fields of equal masses $M$.
In the next Section we will calculate the
magnetic moment of $N$ in this
model. This will also bring us to discuss
the charge form factor of $N$.
The 2-loops diagrams which produce the
ordinary neutrino magnetic
formfactor will be calculated in the third Section.
Finally a comparison with the result obtained in
Sect. 2 will allow us to present expressions for the
neutrino magnetic
formfactor in a form, given by eq. (\ref{1}).
We will see that option
(c) realizes; which confirms the exact
expression given in \cite{1}.
At the end of this introduction let us make
the following technical
remark: as we are interested here only in
the neutrino magnetic form factor
behavior in the domain $m^2 \ll q^2 \ll M^2$ we will
neglect the masses of $\nu$ and of $N$ in what follows.
\section{Electromagnetic form factors of the particle $N$.}
The coupling of $N\bar{N}$ pair with a photon
is generated by two one-loop
diagrams (see Fig. 1). The corresponding
amplitude is (in order to
simplify our formulas we take the masses
$M$ of the heavy virtual spinor and scalar
particles to be equal):
\begin{eqnarray}
M_{\mu} &=&\int \frac{d^4 p(i)^6\sqrt{4\pi\alpha}}{(2\pi)^4 (p^2
-M^2)} \bar{N}\frac{1}{\hat{p}+\hat{k}_2 -M} \gamma_{\mu}
\frac{1}{\hat{p} +\hat{k}_1 -M} N+ \nonumber \\
& + &\int \frac{d^4 p (i)^6
\sqrt{4\pi\alpha}}{(2\pi)^4[(p-k_1)^2 -M^2][(p-k_2)^2 -M^2]}
\times
\nonumber \\
&\times & \bar{N}\frac{1}{\hat{p} -M} N(2p -k_1 -k_2)_{\mu}
|
\;\; ,
\label{6}
\end{eqnarray}
where for simplicity we also put the
Yukawa coupling equal to unity.
In what
follows we will neglect mass of the
external particle $N$. Performing
integration we obtain the charge and
magnetic formfactors of $N$.
Here we meet with the first surprise
-- due to the loop correction $N$
seems to get finite non-zero charge.
Such a behavior is obviously
forbidden by
charge conservation. However,
multiplying $M_{\mu}$ by $(k_1
-k_2)_{\mu}$ to check electromagnetic
current conservation we
encounter differences of linear
divergent integrals. So in spite of
finiteness of the result for the
amplitude $M_{\mu}$, to preserve gauge
invariance we need a proper
regularization scheme. Using dimensional
regularization we automatically obtain
zero charge for $N$. After simple
transformations using Feynman
parametrization for the propagators we get:
\begin{eqnarray}
M_{\mu} &=& -i \int \frac{2ydydxd^d p\sqrt{4\pi\alpha}}{(2\pi)^4 [p^2
+2x(1-x)y^2(k_1 k_2)+M^2]^3} \times \nonumber \\
&\times& \bar{N}[\frac{4-d}{d}\gamma_{\mu}p^2 -M^2 \gamma_{\mu}
-2x(1-x)y^2 \times \nonumber \\
& \times & (k_1 k_2)\gamma_{\mu} +4x(1-x)y^2(k_1 k_2)\gamma_{\mu}
+(k_1 +k_2)_{\mu} M]N \;\; .
\label{7}
\end{eqnarray}
The sum of the first three terms in brackets is zero
the fourth term
generates a charge form factor while
the last term is the magnetic
form factor we want to study
(for massless fermions $(k_1 +k_2)_{\mu}\bar{N} N =
\bar{N}\sigma_{\mu\nu}q_{\nu} N$).
For small momentum transfer
($q^2 \equiv (k_1 -k_2)^2 \ll M^2$) we get:
\begin{equation}
M_{\mu} =\frac{i\sqrt{4\pi\alpha}}{192\pi^2} \frac{q^2}{M^2}
\bar{N} \gamma_{\mu} N -\frac{i\sqrt{4\pi\alpha}}{32\pi^2 M}
\bar{N} N (k_1 +k_2)_{\mu} \;\; ,
\label{8}
\end{equation}
here the first term describes the charge radius
of particle $N$, while
the second describes its magnetic moment.
To end this Section let us note that for
large momentum transfer $q^2
\gg M^2$ the magnetic form factor falls
down as expected like$\sim M^2/q^2
\ln^2(q^2/M^2)$, while the charge form factor
tends to a (non-zero)
constant.
(It is interesting to consider the charge form
factor in the light
of dispersion relations. As the imaginary part of
the form factor falls down,
an unsubtracted dispersion relation can be written.
It
produces a form factor which falls down for
$q^2 \gg M^2$, but contains
nonzero charge for $N$. Subtracting this
"charge" we get constant
behavior for $q^2 \gg M^2$.)
\section{Neutrino magnetic form factor for $q^2 \gg m^2$.}
The two Feynman diagrams shown in Fig. 2
are in turn responsible for the
ordinary neutrino
magnetic form factor. Since we neglect
both $\nu$ and $N$ masses, helicity
flip (necessary for a magnetic form factor)
can only occur in the inner
fermion line (with mass $M$). As the first
part of calculation we perform
the integral over momentum $p$. Taking into
account only the
terms proportional to
$M$ (they are the only ones contributing to
the magnetic moment)
we obtain:
\begin{equation}
A_{\mu}=i \int \frac{2ydydx}{32\pi^2}\frac{M[\hat{k}_2\gamma_{\mu}
+\gamma_{\mu}\hat{k}_1 -(k_1 +k_2)_{\mu}] \sqrt{4\pi\alpha}}
{[M^2 -y(1-y)(k-xk_2 -(1-x)k_1)^2 +2yx(1-x)k_1 k_2]} \;\; ,
\label{9}
\end{equation}
where $0 <y,x <1$. When expression
(\ref{9}) is inserted into the
external loop we get as expected
from renormalizability an
extra suppression of the integrand over $k_{\mu}$ in
the domain $k^2 > M^2$. Taking into
account the propagators of fields
$\varphi$ and $N$ we observe that
the integral over $k_{\mu}$ is now u.v.
convergent by power counting.
In this way, our microscopic
model of the $N$ particle magnetic moment
regularizes the expression for
the light neutrino magnetic form factor.
This leads as announced to the convergence of the
analog of integral (\ref{4}) in the
2-loop approach;
so even making {\it the full integration}
in space-time dimensions
$d \neq 4$ we get zero for the
integral proportional to $\hat{k}\sigma_{\mu\nu}\hat{k}$.
In this way
the result obtained in \cite{1} for the ordinary neutrino
magnetic moment is
justified; our option c) realizes.
Armed with the qualitative arguments given
above let us proceed with
the calculation of the second loop.
Making use of Feynman parametrization
for propagators of particles $N$ and
$\varphi$ (we use parameters $u$
and $v$, $0 < u$, $v < 1$) after
simple transformations we get:
\begin{equation}
T_{\mu} = f^2 \int \frac{2ydydx \sqrt{4\pi\alpha}}{32\pi^2}
\frac{d^4 k2v dvdu}{(2\pi)^4[k^2 -2k_1 k_2 uv^2(1-u)]^3}
\times
\label{10}
\end{equation}
$$
\times \frac{M\bar{\nu}_2 [-(k_1 +k_2)_{\mu}2k_1 k_2 u(1-u)v^2
+\hat{k}((k_1 +k_2)_{\mu} -\hat{k}_2 \gamma_{\mu}
-\gamma_{\mu}\hat{k}_1)\hat{k}]\nu_1} {[M^2 -y(1-y)(k-xk_2
-(1-x)k_1 + uvk_2 +(1-u)v k_1)^2 +2xy(1-x)k_1 k_2]} .
$$
The second term in brackets in the
nominator produces corrections to
the
neutrino magnetic moment of order $\frac{1}{M}\times
\frac{q^2}{M^2}$ and is negligible for $q^2 \ll M^2$.
To calculate
the
contribution of the first term, let us begin
with comparatively small
momentum transfer, $q^2 \ll M^2$. For such momentum
transfer the
second bracket in the denominator equals $M^2$
(let us mention that
integral over $k_{\mu}$ is u.v. convergent for the part of
$T_{\mu}$ now considered) and we readily get:
\begin{equation}
T_{\mu} =\frac{i\sqrt{4\pi\alpha}f^2}{32\pi^2 M}\bar{\nu}_2 \nu_1
(k_1 +k_2)_{\mu} \frac{1}{32\pi^2} \;\; , \;\; \mbox{\rm or} \;\;
I=-\frac{1}{2} \;\; \mbox{\rm for} \;\; q^2 \ll M^2 \;\; .
\label{11}
\end{equation}
For large momentum transfer, $q^2 \gg M^2$
the second bracket in
denominator in (\ref{10}) is proportional to
$q^2$ which means that the
neutrino magnetic moment falls down,
$I\sim M^2/q^2$ according to
general expectations for a renormalizable field theory.
\section{Conclusions.}
In this paper we demonstrated that an
effective theory with a
neutral spinor
particle $N$ with nonzero magnetic moment allows to
calculate the
induced
neutrino magnetic form factor in the domain $q^2 \ll M^2$,
where $M$
is scale at which $N$ magnetic moment is generated if loop
calculations are made with naive ultraviolet cut-off.
This is
fully consistent with a dimensional regularization treatment {\it
of the full renormalisable underlying theory},
but {\it not} of the
effective model.
All authors were supported by INTAS grant 94-2352;
RBN, VAN, MIV acknowledge support of INTAS
grants INTAS-93-3316-ext,
INTAS-RFBR 95-0567 and grant RFBR-96-02-18010 as well.
|
\subsection{Related Work}
Several existing works analyze ground-based images with different meteorological observations. Most of them correlate the cloud coverage obtained from sky images with meteorologists' observations. \citet{Silva2016} validated cloud coverage measurements obtained from ground-based automatic imager and human observations for two meteorological stations in Brazil. \citet{Huo2012} also performed such field experiments for three sites in China. The computation of such cloud coverage percentage is important in solar energy generation. It can hugely impact the amount of solar radiation falling at a particular place.
The correct estimation of solar irradiance, is particularly important in tropical countries like Singapore, where the amount of received solar irradiance is high. \citet{rizwan2012generalized} demonstrated that tropical countries are conducive for installing large central power stations powered by solar energy, because of the large amount of incident sunlight throughout the year. Several attempts have been made to estimate the solar radiation from general meteorological measurements via temperature, humidity and precipitation~\citep{HSmodel,DCmodel,BCmodel,Huntmodel}. These existing models aim to provide global solar radiation using different sensors. \citet{alsadi2017estimation} demonstrated such estimation models from the perspective of a photovoltaic (PV) solar field, showing that succeeding rows of PV panels receive less solar radiation than that of first row. They also provided an analytical solution by including the design parameters in the estimation model.
In addition to solar irradiance estimation, there have been several efforts in forecasting the solar irradiance, with a lead time of few minutes. \citet{baharin2016short} proposed a machine-learning forecast model for PV power output, using Malaysia as the case study. Similarly \citet{chu2015short} used a reforecasting method to improve the PV power output forecasts with a lead time of $5$, $10$, and $15$ minutes.
Satellite images have also been used in the realm of solar analytics. \citet{mueller2004rethinking} proposed a clear sky model that is based on radiative transfer models obtained from Meteosat's atmospheric parameters. However, satellite data have lower temporal and spatial resolutions.
Recently, with the development of low-cost photogrammetric techniques, sky camera are being deployed for such purposes. These sky cameras have both high temporal and high spatial resolutions, and are able to provide a more localized information about the atmospheric events. \citet{alonso2015} used sky cameras to quantify the total solar radiation. \citet{yang2015expanding} studied these solar irradiance variability using entropy and covariance. \citet{dev2018solar} used triple exponential smoothing for analyzing the seasonality of the solar irradiance. However, these approaches do not model the sharp short-term variations of solar radiation.
\subsection{Outline}
In this paper, we use images obtained from WSIs to accurately model the fluctuations of the solar radiation. There are several advantages of using a WSI for this instead of a pyranometer. Common weather stations generally use a solar sensor that measures the total solar irradiance. It is a point measurement providing scalar information for a particular location and does not provide information on clouds and their evolution over time. On the other hand, the wide-angle view of a ground-based sky camera provides us extensive information about the sky. It allows for the tracking of clouds over successive image frames, and also to predict their future location. In this paper, we attempt to model solar irradiance from sky images. This can also help in solar energy forecasting, which is useful in photovoltaic systems~\citep{solar_PV}.
The main contributions of this paper are as follows:
\begin{itemize}
\item We develop a framework to accurately estimate and track the rapid fluctuations of solar irradiance;
\item We propose a method for estimating solar irradiance using ground-based sky camera images;
\item We conduct extensive benchmarking of our proposed method with other solar irradiance estimation models.
\end{itemize}
The rest of the paper is organized as follows. Section~\ref{sec:data} describes our experimental setup that captures the sky/cloud images and collects other meteorological sensor data. Our framework for estimating solar irradiance is presented in Section~\ref{sec:solarmodel}. Section~\ref{sec:comparemodels} discusses the evaluation of our approach, and its benchmarking with other existing solar estimation models. We discuss the possible applications of our approach in Section~\ref{sec:discuss}. We also point out a few limitations of our approach, and ways to address them. Section~\ref{sec:conclusion} concludes the paper.
\section{Data Collection}
\label{sec:data}
Our experimental setup consists of weather stations and ground-based WSIs. These devices are co-located on the rooftop of our university building in Singapore ($1.34^{\circ}$N, $103.68^{\circ}$E). They continuously capture various meteorological data, and we archive them for subsequent analysis.
\subsection{Whole Sky Imager (WSI)}
Commercial WSIs are available in the market. However, those imagers have high cost, low image resolution, and little flexibility in operation. In our research group, we have designed and built custom low-cost high-resolution sky imagers, which we call WAHRSIS, i.e.\ Wide Angle High Resolution Sky Imaging System~\citep{WAHRSIS}. A WAHRSIS imager essentially consists of a high-resolution digital single-lens reflex (DSLR) camera with a fish-eye lens and an on-board micro-computer.
The entire device is contained inside a weather-proof box with a transparent dome for the camera. Over the years, we have built several versions of WAHRSIS~\citep{WAHRSIS,IGARSS2015a}. They are now deployed at several locations around our university campus, capturing images of the sky at regular intervals.
Our WAHRSIS camera is calibrated with respect to white balancing, geometric distortions and vignetting. The imaging system in WAHRSIS is modified so that it captures the near-infrared region of the spectrum. Hence, the red channel of the captured image is more prone to saturation, which renders the captured image reddish in nature. Therefore, we employ custom white balancing in the camera, such that it compensates the alteration owing to the near-infrared capture. Figure~\ref{fig:white-balance} depicts the captured images obtained from automatic and custom white balancing.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{f01.png}\\
\makebox[0.35\textwidth][c]{\small (a) With auto white-balancing}
\makebox[0.35\textwidth][c]{\small (b) With custom white-balancing}
\caption{We use custom white-balancing for correcting the white balance.
\label{fig:white-balance}}
\end{center}
\end{figure}
We use the popular toolbox by \citet{scaramuzza2006toolbox} for the geometric calibration of WAHRSIS. This process involves the computation of the intrinsic parameters of the camera. We use a black-and-white regular checkerboard pattern, and position it at various points around the camera. Figure~\ref{fig:calibration_images} illustrates a few sample positions of the checkerboard in the calibration process. Using the corner points and the known dimensions of the pattern, we can estimate the intrinsic parameters of the camera.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{f02.png}
\caption{We position the checkerboard at various locations for geometric camera calibration.
\label{fig:calibration_images}}
\end{center}
\end{figure}
Finally, we apply vignetting correction to the images captured by our sky camera. Owing to the fish-eye nature of the lens, the area around the centre of the lens is brighter than at the sides. We use an integrating sphere to correct this variation of illumination. Figure~\ref{fig:sphere} depicts an image captured inside an integrating sphere that provides a uniform illumination distribution in all directions. We use luminance characteristics from this reference image to correct the images captured by our sky camera.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{f03.png}
\caption{Reference image captured inside the uniformly-illuminated integrating sphere.
\label{fig:sphere}}
\end{center}
\end{figure}
\subsection{Weather Station}
In addition to the sky imagers, we have also installed co-located weather stations. We use \emph{Davis Instruments 7440 Weather Vantage Pro} for our recordings. It measures rainfall, total solar radiation, temperature and pressure at intervals of $1$ minute. The resolution of the tipping-bucket rain gauge is $0.2$ mm/tip.
It also includes a solar pyranometer measuring the total solar irradiance flux density in W/$\mbox{m}^2$. This consists of both direct and diffuse solar irradiance components. The solar sensor integrates the solar irradiance across all angles, and provides the total solar irradiance. On a clear day with no occluding clouds, the solar sensor can be approximated by a typical cosine response, shown in Figure~\ref{fig:sensor-reading} for varying degrees of solar incident angle. The solar sensor reading is highest around noon when the incident angle of sun rays is at the minimum, whilst the reading is low during morning and evening hours.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.5\textwidth]{f04.pdf}
\caption{Response of the solar sensor on a clear day with varying solar incident angle.
\label{fig:sensor-reading}}
\end{center}
\end{figure}
The solar radiation under a clear sky can be modeled using the solar zenith angle and the earth's eccentricity. Several clear sky models have been developed for various regions. The best clear-sky model for Singapore is provided by \citet{dazhi2012estimation}. We performed a comparison of various clear sky models in Singapore~\citep{dev2017study}, and found that the \citet{dazhi2012estimation} model provides a good estimate of the clear sky irradiance. It models clear-sky Global Horizontal Irradiance (GHI) $G_c$ as follows:
\begin{align}
G_c = 0.8277E_{0}I_{sc}(\cos\alpha)^{1.3644}e^{-0.0013\times(90-\alpha)},
\end{align}
where $E_{0}$ is the eccentricity correction factor for earth, $I_{sc}$ is the solar irradiance constant ($1366.1$Watt/$\mbox{m}^2$), and $\alpha$ is the solar zenith angle (measured in degrees). The factor $E_{0}$ is calculated as:
\begin{equation*}
\begin{aligned}
\label{eq:E0value}
E_0 = 1.00011 + 0.034221\cos(\Gamma) + 0.001280\sin(\Gamma) + 0.000719\cos(2\Gamma) + 0.000077\sin(2\Gamma),
\end{aligned}
\end{equation*}
where $\Gamma = 2\pi(d_n-1)/365$ is the day angle (measured in radians) and $d_n$ is the day number of the year.
As an illustration, we show the clear-sky radiation for the 1st of September 2016 in Figure~\ref{fig:Cos_response}, compared to the actual solar irradiance measured by our weather station. We also show the deviation of the measured solar radiation from the clear-sky model. We observe that there are rapid fluctuations in the measured readings. In our previous work~\citep{IGARSS16_solar}, we observed that these rapid fluctuations are caused by the incoming clouds that obstruct the sun from direct view. Such information about the cloud profile and its formation cannot be obtained from a point-source solar recording. Therefore, we aim to model these rapid fluctuations in the measured solar radiation from wide-angle images captured by our sky cameras.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=1\textwidth]{f05.png}\\
\makebox[0.45\textwidth][c]{(a) \small Measured solar irradiance along with clear-sky model.}
\makebox[0.45\textwidth][c]{(b) \small Percentage deviation of solar irradiance from clear sky data.}
\caption{Solar irradiance measurements on the 1st of September 2016. Note the rapid fluctuations of high magnitude in the measurements.
\label{fig:Cos_response}}
\end{center}
\end{figure}
\section{Modeling Solar Irradiance}
\label{sec:solarmodel}
This section presents our model for computing solar irradiance from images captured by a whole sky imager. We sample pixels using a cosine weighted hemispheric sampling to simulate the behavior of a pyranometer based on the fisheye camera lens. We then compute the relative luminance using the image capturing parameters after gamma correction. We finally derive an empirical fitting function to scale the computed luminance estimates to match measured irradiance values.
\subsection{Cosine Weighted Hemispheric Sampling}
The behavior of our fisheye lens with focal length $f$ is modeled by the equisolid equation $r=2f \sin(\theta/2)$, relating the distance ($r$) of any pixel from the center of the image to its incident light ray elevation angle ($\theta$). This allows to project a captured image onto the unit hemisphere, as shown in Figure~\ref{fig:sampling}.
The solar irradiance is composed of a direct component relating the sun light reaching the earth without interference, as well as diffuse and reflected components. Given the high resolution of our images, we consider randomly sampled pixel locations on the hemisphere as input to the luminance computation. We follow a cosine weighted hemispheric distribution function, the center of which is at the location of the sun. This is because clouds in the circumsolar region have the highest impact on the total solar irradiance received on the earth's surface~\citep{IGARSS16_solar}. We provide more emphasis to the clouds around the sun, as compared to those near the horizon. In our previous work~\citep{dev2016estimation}, we used a cloud mask around the sun to estimate the solar irradiance. However, this requires the additional step of optimizing the size of the cropped image for best results. Therefore, we adopt the strategy of cosine weighted hemispheric sampling.
The first step is to compute the sampled locations from the top of the unit hemispheric dome. Each of the locations are computed as follows, using two random floating points $R_1$ and $R_2$ as input, where $(0 \leq R_1, R_2 \leq 1)$:
\[\phi = 2\pi R_1,\ \theta = \arccos(\sqrt{R_2})\]
\begin{equation}
\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}=\begin{bmatrix}
\sin(\theta)\cdot \cos(\phi) \\
\sin(\theta)\cdot \sin(\phi) \\
\cos(\theta)
\end{bmatrix}
\end{equation}
This is represented in Figure~\ref{fig:sampling}.
The second step is to detect the location of the sun using a thresholding method. This is needed to align the center of the previously computed distribution (i.e.\ top of the hemispheric dome) to the actual sun location in the unit sphere. We choose a threshold of $240$ in the red channel $R$ of the $RGB$ captured image, and compute the centroid of the largest area above the threshold~\citep{IGARSS16_calib}. We then compute the rotation matrix transforming the z-axis unit vector to the unit vector pointing towards the sky. We apply this rotation to all the sampled points, resulting in Figure~\ref{fig:sampling}. This means that the number of sampled points in a region of the hemisphere is proportional to the cosine of the angle between the sun direction and the direction to that region. We experimentally concluded that this achieves a good balance between all irradiance components. We consider the pixel values of a total of $5000$ points sampled using this method as input for the irradiance estimation.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{f06a.png}\\
\parbox{0.3\textwidth}{(a) Projection of the original image on a hemisphere.}
\parbox{0.3\textwidth}{(b) Cosine hemispheric sampling with origin at the top.}
\parbox{0.3\textwidth}{(c) Applying a rotation matrix to center at the sun location.}\\
\vspace{0.3cm}
\includegraphics[width=0.9\textwidth]{f06b.png}\\
\parbox{0.3\textwidth}{(d) Original image with detected sun location in red.}
\parbox{0.3\textwidth}{(e) Projection of the sampled points onto the image.}
\parbox{0.3\textwidth}{(f) Projection of the rotated sampled points onto the image.}
\caption{Cosine weighted hemispheric sampling process.
\label{fig:sampling}}
\end{center}
\end{figure}
\subsection{Relative Luminance Calculation}
For each of the $i$ sampled pixels in the $RGB$ image, we compute its luminance value using the following formula. The formula is proposed in SMPTE Recommended Practice 177~\citep{smpte1993rp} to compute the luminance of an image from the \emph{R}, \emph{G} and \emph{B} values of the \emph{RGB} image.
\[Y_i = 0.2126\cdot R_i + 0.7152\cdot G_i + 0.0722\cdot B_i\]
The JPEG compression format encodes images after applying a gamma correction. This non-linearity mimics the behavior of the human eye. This needs to be reversed in order to compute the irradiance. We use a gamma correction factor of $2.2$, which is most commonly used in imaging systems~\citep{Poynton03}. We thus apply the following formula, assuming pixel values normalized between $0$ and $255$:
\[Y_i' = 255{(Y_i/255)}^{2.2}\]
We then average the pixel values across all the $i$ sampled points in the image, and denote it by $\mathcal{N} = (1/n)\sum_{i=1}^{n} Y_i'$, the average luminance value of the sampled points from the image.
However, each image of the sky camera is captured with varying camera parameters such as ISO, F-number and shutter speed. These camera parameters can be read from the image metadata, and are useful to estimate the scene luminance. The amount of brightness of the sampled points $\mathcal{N}$, is proportional to the number of photons hitting the camera sensor. This relationship between scene luminance and pixel brightness is linear~\citep{hiscocks2011measuring}, and can be modeled using the camera parameters:
\[\mathcal{N} = K_c \left( \frac{e_t\cdot S}{f_s^2}\right) \mathcal{L}_s\]
where $\mathcal{N}$ is the pixel value, $K_c$ is a calibration constant, $e_t$ the exposure time in seconds, $f_s$ the aperture number, $S$ the ISO sensitivity and $\mathcal{L}_s$ the luminance of the scene.
We can thus compute the relative luminance $\mathcal{L}_r$ as follows:
\[\mathcal{L}_r = \mathcal{N} \left( \frac{f_s^2}{e_t\cdot S}\right)\]
\subsection{Modeling Irradiance from Luminance Values}
Using our hemispheric sampling and relative luminance computation, we therefore have one relative
|
luminance value $\mathcal{L}_r$ per image. We use this relative luminance value to estimate the solar radiation. The usual sunrise time in Singapore is between $6$:$40$am and $7$:$05$am, and sunset time is approximately between $6$:$50$pm and $7$:$10$pm local time.
This information is obtained from \citep{nea-weather}. Therefore, we consider images captured in the time interval of $7$:$00$am till $7$:$00$pm.
We use our ground-based whole sky images captured during the period from January $2016$ till August $2016$ to model the solar radiation. The solar irradiance is computed as the flux of radiant energy per unit area normal to the direction of flow. The first step in estimating irradiance from the luminance is thus to cosine weight it according to its direction of flow. We weight our measurements according to the solar zenith angle $\alpha$. This is based on empirical evidences of our experiments on solar irradiance estimation. The modeled luminance $\mathcal{L}$ is expressed as:
\begin{equation*}
\mathcal{L} = \mathcal{L}_r(\cos\alpha)
\end{equation*}
Let us assume that the actual solar radiation recorded by the weather station is $\mathcal{S}$. We check the nearest weather station measurement for all the images captured by WAHRSIS between April $2016$ till December $2016$. Figure~\ref{fig:solar_model} shows the scatter plot between the image luminance and solar radiation. The majority of the data follows a linear relationship between the two. However, it deviates from linearity for higher values of luminance. This is mainly because of the fact that the mapping between scene luminance and obtained pixel value in the camera sensor becomes non-linear for large luminances. A more detailed discussion on this is provided in Section~\ref{sec:discuss}.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{f07.png}\\
\parbox{0.3\textwidth}{\centering (a) Linear model.}
\parbox{0.3\textwidth}{\centering (b) Quadratic model.}
\parbox{0.3\textwidth}{\centering (c) Cubic model.}
\caption{Empirical fit between solar irradiance and image luminance computed with our proposed framework. We observe that it deviates from linearity at higher luminance values. Higher order polynomials are ill-conditioned.
\label{fig:solar_model}}
\end{center}
\end{figure}
We attempt to fit a linear model and other higher-order polynomial regressors to model the relationship between image luminance from sky camera images and the measured solar radiation. Figure~\ref{fig:solar_model} shows the best fit curve for several orders of polynomial function. In order to provide an objective evaluation of the different models, we also compute the RMSE value between the actual and regressed values. Table~\ref{tab:model-fitting} summarises the performance of the different order polynomials. We observe that lower order polynomials of degree $1$ and $2$ perform slightly inferior to those of higher order polynomials.
However, higher-order polynomial models are ill-conditioned. Therefore, we choose the cubic model as our proposed model to model the measured solar radiation $\mathcal{S}$ from the image luminance $\mathcal{L}$. This is based on the assumption that the mapping from scene luminance to pixel values in the captured image is linear for lower luminance values, and it behaves in a non-linear fashion for higher luminance values. We use this selected model in all our subsequent discussions and evaluations.
\begin{table}[htb]
\normalsize
\centering
\caption{Performance evaluation of various polynomial order regressors. We measure the RMSE value for each of the models.}
\label{tab:model-fitting}
\begin{tabular}{rc}
\hline
\textbf{Proposed models} & \textbf{RMSE} (W/$\mbox{m}^2$) \\
\hline
Linear (degree $1$) & 178.27 \\
Quadratic (degree $2$) & 178.26 \\
Cubic (degree $3$) & 176.57 \\
Quartic (degree $4$) & 176.52 \\
Quintic (degree $5$) & 176.49 \\
\hline
\end{tabular}
\end{table}
We model solar radiation as: $\mathcal{S} = a_3\times\mathcal{L}^3 + a_2\times\mathcal{L}^2 + a_1\times\mathcal{L}+ a_0$, with $a_3=-4.25e-12$, $a_2=3.96e-07$, $a_1=0.00397$ and $a_0=7.954$ for our data.
This model is derived specifically for the equatorial region like Singapore, and the regression constants are based on our WAHRSIS sky imaging system. They would have to be fine-tuned for other regions and different imaging systems using our methodology. To facilitate this, we make the source code of all the simulations in this paper available online at \url{https://github.com/Soumyabrata/estimate-solar-irradiance}.
\section{Experimental Validation}
\label{sec:comparemodels}
In this section, we evaluate the accuracy of our proposed approach. It is derived based on WAHRSIS images captured from January to August $2016$. We also use these images to evaluate the accuracy of our proposed model. Furthermore, we benchmark our algorithm with other existing solar radiation estimation models.
\subsection{Evaluation}
One of the main advantages of our approach is that all rapid fluctuations of solar radiation can be accurately tracked from the image luminance. We illustrate this by providing the measured solar readings of 01-Sep-2016 in Figure~\ref{fig:tracksun}. The clear-sky model follows a cosine response and is shown in black, while the measured solar recordings are shown in red. We normalize our computed luminance in such a manner that it matches the measured solar readings. We multiply each data point with a conversion factor, such that the distance between corresponding inter-samples of luminance and weather station recordings is minimized (cf.\ Appendix A for details). We use this normalization factor in order to map the computed image luminance to have a similar scale to the cosine clear sky model. We observe that our computed luminance from the whole-sky image and the measured solar radiation closely follow each other. We emphasize here that it is important to accurately track the rapid solar fluctuations. Unlike other solar estimation models based on meteorological sensor data, our proposed model can successfully estimate the \emph{peaks} and \emph{troughs} of solar readings.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{f08.pdf}
\caption{Measured weather station data (in red) vs.\ clear sky radiation (in black) as on 01-Sep-2016. The sampling interval between two measurements is $2$ minutes.
\label{fig:tracksun}}
\end{figure*}
Using our proposed methodology, we compute the luminance of all the captured images. Subsequently, using our proposed cubic model, we estimate the corresponding solar radiation values. The estimated solar irradiance values are compared with the actual irradiance values obtained from the solar sensors in the co-located weather station, which serve as the ground-truth measurements. Figure~\ref{fig:HOD} shows the histogram of differences between the estimated and actual solar radiation. We observe that the estimates do not deviate much from the actual solar radiation. Nearly half ($47.9\%$) of data points are concentrated in the range [$-100$,$+100$] W/$\mbox{m}^2$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{f09.pdf}
\caption{Histogram of differences between estimated and actual solar irradiance.
\label{fig:HOD}}
\end{center}
\end{figure}
\subsection{Benchmarking}
We benchmark our proposed approach with other existing solar estimation models. To the best of our knowledge, there are no proposed models to estimate short-term fluctuations of solar irradiance from ground-based images. However, most remote sensing analysts have been using other meteorological sensor data, e.g.\ temperature, humidity, rainfall and dew point temperature to estimate daily solar irradiance. One of the pioneer works was done by \citet{HSmodel}, who proposed a model based on daily temperature variations. \citet{DCmodel} improved the model by including clear sky transitivity as one of the factors. On the other hand, \citet{BCmodel} also proposed a new model of solar radiation estimation, by including the atmospheric transmission coefficient. Subsequently, \citet{Huntmodel} showed that the solar estimation model can be further improved by incorporating precipitation data in the model. We benchmark our proposed approach with these different existing models. We illustrate the various benchmarking models in Figure~\ref{fig:othermodels}. Unfortunately, most of these other approaches fail to capture the short-term variations of the solar radiation.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.95\textwidth]{f10.pdf}
\caption{Comparison amongst different benchmarking solar estimation models, along with clear sky model and measured solar irradiance on 01-Sep-2016. Most of the existing algorithms fail to capture the rapid fluctuations of the measured solar irradiance.
\label{fig:othermodels}}
\end{figure*}
We calculate the Root Mean Square Error (RMSE) of the estimated solar radiation and Spearman's rank correlation coefficient as evaluation metrics. The RMSE of an estimation algorithm represents the standard deviation of the actual and estimated solar radiation values. Spearman correlation is a non-parametric measure to characterize the relationship between measured and estimated solar radiation, which does not assume that the underlying dataset are derived from a normal distribution. We report both metrics in Table~\ref{tab:corr_results}. Our proposed approach achieves the best results amongst all methods. Note that the training and testing set of images are identical, and all images are considered for benchmarking purposes.
\begin{table}[htb]
\normalsize
\centering
\caption{Benchmarking of our proposed approach with other solar radiation estimation models.}
\label{tab:corr_results}
\begin{tabular}{rcc}
\hline
\textbf{Methods} & \textbf{RMSE} (W/$\mbox{m}^2$) & \textbf{Correlation}\\
\hline
Proposed approach & \textbf{178.27} & \textbf{0.86}\\
\citet{HSmodel} & 982.35 & 0.67\\
\citet{DCmodel} & 324.48 & 0.67\\
\citet{BCmodel} & 318.07 & 0.68\\
\citet{Huntmodel} & 922.66 & 0.65\\
\hline
\end{tabular}
\end{table}
Furthermore, we check if our proposed model generalizes well with random samples of our captured sky camera images. We choose a random selection of images as the training set, and fit our linear regressor on these selected training images. The RMSE values are then calculated on these training images. We perform this analysis for varying percentages of training images. Each experiment is performed $100$ times to remove any selection bias.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{f11.png}\\
\parbox{0.45\textwidth}{\centering (a) Performance on training set.}
\parbox{0.45\textwidth}{\centering (b) Performance on test set.}
\caption{Effect of the percentage of training images on RMSE values. The lower and upper end of each box represents the $25^{th}$ and $75^{th}$ percentiles of the data, and the red line represents the median. Each experiment is conducted $100$ times with a random choice of training and test sets.
\label{fig:train_test}}
\end{center}
\end{figure}
Figure~\ref{fig:train_test}(a) shows the results on training images. We observe that the variation of the RMSE values gradually decreases as we increase the number of training images. Moreover, we check the variation of RMSE values when the test images are not identical as training images. Once we choose a random selection of images as training set, the remaining images are considered as the test set. We show the RMSE on such images in Fig~\ref{fig:train_test}(b). As expected, the variation of RMSE values increases with higher percentage of training images. The linear regressor model overfits the data, and provides higher variation in the error when tested on a fewer test images. However, the average RMSE does not vary much in all cases. Therefore we conclude that our proposed model is free from selection bias, and generalizes well with random selection of training and testing images.
We show the scatter plot between the measured solar radiation and estimated solar radiation for the different benchmarking algorithms in Figure~\ref{fig:scatter-other}. We observe that there is no strong correlation for most of these existing algorithms. This is because meteorological sensor data alone, with no cloud information cannot determine the sharp fluctuations of the solar radiation.
This is an important limitation of these models, which we have attempted to address in this paper. Our model based on sky images has additional information about cloud movement and its evolution, which is the fundamental reason behind rapid solar radiation fluctuations. In our proposed model, most of these short-term variations are captured (cf.\ Figure~\ref{fig:tracksun}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{f12.png}\\
\parbox{0.22\textwidth}{\centering (a)\small \citet{HSmodel}.}
\parbox{0.22\textwidth}{\centering (b)\small \citet{DCmodel}.}
\parbox{0.22\textwidth}{\centering (c)\small \citet{BCmodel}.}
\parbox{0.22\textwidth}{\centering (d)\small \citet{Huntmodel}.}
\caption{Scatter plot between measured solar irradiance and estimated solar irradiance for the benchmarking algorithms.
\label{fig:scatter-other}}
\end{center}
\end{figure}
\section{Discussion}
\label{sec:discuss}
\subsection{Short-term Forecasts}
Our proposed approach can estimate the solar radiation accurately with the smallest RMSE compared to other models. The main advantage of our approach is that it can be used on predicted images as well, opening the potential for short term solar irradiance forecasting, which is needed in the solar energy field. As an initial case study, we have exploited optical flow techniques to estimate the direction and flow of cloud motion vectors between two successive image frames. We use the $(B-R)/(B+R)$ ratio channel of the sky/cloud image, where $B$ and $R$ are the blue and red channels respectively. We implement an optical flow technique~\citep{flow-pgm} that uses a simpler conjugate gradient solver to obtain the flow field. Figure~\ref{fig:vectorflow} illustrates the estimated flow field.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.9\textwidth]{f13.png}\\
\parbox{0.4\textwidth}{\centering \small (a) Horizontal translation.}
\parbox{0.4\textwidth}{\centering \small (b) Vertical translation.}
\caption{Horizontal and vertical translation of pixels between two successive frames, computed using optical flow.
\label{fig:vectorflow}}
\end{center}
\end{figure}
Using the images captured at $t$ and $t-2$ minutes, we estimate the horizontal and vertical translation. Under the assumption that the flow of cloud motion vectors for the successive $t+2$ minutes is similar to that of previous frames, we estimate the future $t+2$ minutes frame, and subsequently the $t+4$ minutes frame. Figure~\ref{fig:comb-example} illustrates this. We obtain a forecast accuracy of $70\%$ for a prediction lead time upto $6$ minutes.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{f14.png}
\caption{Prediction of sky/cloud image using optical flow technique.
\label{fig:comb-example}}
\end{center}
\end{figure}
In future work, we plan to use our proposed methodology of estimating solar irradiance on this predicted sky/cloud image. This will enable us to provide more stable and reliable forecasts of solar irradiance.
\subsection{Scope for Improvement}
There is still scope for improvement in our approach. First, we use \emph{JPEG} images instead of uncompressed \emph{RAW} images for the computation of scene luminance. The \emph{JPEG} compression algorithm introduces non-linearities in the pixel values, which affects the process of estimating solar irradiance. We can generate more consistent results by using only \emph{RAW} format images. Nevertheless, we still use \emph{JPEG} images, as they have a significantly smaller size, which is more practical from an operational point of view. In contrast, uncompressed \emph{RAW} images are much larger in size, which makes it impossible to capture and store \emph{RAW} images at short intervals due to the processing requirements. In our future work, we intend to explore the use of \emph{RAW} format images for the computation of solar irradiance values from sky cameras.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.45\textwidth]{f15.pdf}
\caption{Distribution of F-numbers of the WAHRSIS images used to derive the proposed model.
\label{fig:fn_dist}}
\end{center}
\end{figure}
Secondly, our captured images have a wide range of camera settings with varying shutter speed, ISO and aperture values. This is disadvantageous because the relationship between pixel value and camera aperture value becomes non-linear for larger F-numbers. The relationship deviates from linearity for F-numbers above $4.0$~\citep{hiscocks2011measuring}. Figure~\ref{fig:fn_dist} depicts the wide range of F-numbers in the captured images used in deriving our proposed model. We observe that a significant percentage of images have large F-numbers,
where the non-linearity sets in. This can be solved by using the aperture priority mode of the sky camera, wherein the F-number is fixed, and the exposure time varies dynamically to match the lighting conditions of the scene.
\section{Conclusions \& Future work}
\label{sec:CLS}
\label{sec:conclusion}
We presented a method for estimating the rapid fluctuations of the solar irradiance using the luminance of images taken by a whole sky imager. We are able to estimate the rapid short-term variations, which significantly improves on the state-of-the-art. This approach is of interest for solar energy generation, because these variations cause a sudden decrease in the electricity output from solar panels. Short-term predictions of such ramp-downs are needed to maintain the stability of the power grid.
Combining our solar irradiance estimation approach with cloud movement tracking in the input images could ultimately lead to better irradiance predictions. Such information on rapid fluctuations of solar irradiance can assist in establishing a high-reliability solar energy generation system. We also plan to explore methodologies from time-series modelling~\citep{dev2018solar} to predict solar irradiance.
\section{Code availability}
The source code of all simulations in this paper is available at \url{https://github.com/Soumyabrata/estimate-solar-irradiance}.
|
\section{Introduction}
In their Comment \cite{Ruhman19} Ruhman and Lee criticize our derivation of
the cutoff frequency in the gap equation, $\omega _{c}$, by arguing that at
energies beyond the Fermi energy $\epsilon _{F}$ an uncontrollable set of
additional contributions in perturbation theory would have to be taken into
account. We object to this statement as unfounded, as explained in detail
below. As for the second part of our paper \cite{Wolfle18}, devoted to the
explanation of the isotope effect, we agree with\ Ruhman and Lee \cite%
{Ruhman19}\ \ to have made an error in evaluating the deformation potential
coupling of carriers to the soft TO phonon mode. This coupling is indeed
negligible and the results for the transition temperatures $T_{c}$ obtained
by using that interaction are incorrect. In the following we present results
of a reevaluation of $T_{c}$ obtained by approximating the pair interaction
by the screened Coulomb interaction. We give the specifics below.
In their paper on superconductivity in lightly doped SrTiO$_{3}$ \cite%
{Ruhman16} Ruhman and Lee, (1) apply the usual method of separating pair
interaction into a longitudinal optical (LO) phonon mediated attractive part
and a repulsive Coulomb interaction part, which they treat by subtracting a $%
\mu ^{\ast }$ parameter of conventional magnitude to the pair coupling. This
conventional procedure requires the Fermi energy $\epsilon_{F}\gg\omega_{LO}$%
, the optical phonon frequency and thus is no longer valid once the Fermi
energy falls below the longitudinal phonon energy, as is the case for
densities $n< 3*10^{20}$cm$^{-3}$. They further assume, (2) a frequency
cutoff in the gap equation of order Fermi energy $\epsilon _{F}$, without
justification. The resulting transition temperatures calculated by them are
orders of magnitude too low at low density \cite{Behnia13,Behnia14,Schooley64}.
They conjecture that the doping
reduces the dielectric constant by a factor of $\approx 20$ even at the low
doping end. There is no experimental indication for such a drastic change of
the dielectric properties induced by light doping.
We believe to have resolved both difficulties, (1) and (2), in our work \cite%
{Wolfle18}, in which (1) we use the fully screened Coulomb interaction, thus
avoiding (a) the splitting off of the phonon mediated interaction and (b)
treating the remaining static Coulomb interaction only approximately. We (2)
show that the ionic screening is so strong as to keep the dimensionless
screened Coulomb interaction in the weak coupling regime not only for
energies less than the Fermi energy, but up to the LO phonon energy scale. A
detailed evaluation of the dimensionless coupling function is presented
below, demonstrating that the coupling is indeed much less than unity for
typical momenta and frequencies up to the phonon scale. The frequency cutoff
$\omega _{c}$\ in the gap equation is provided by the energy beyond which
the electron self energy exceeds the energy itself. We present below a
calculation of the transition temperature again using a hard cutoff
approximation, for calculational convenience. The resulting transition
temperature as a function of doping is in good agreement with
experiment \cite{Behnia13,Behnia14,Schooley64} .
As for the isotope effect \cite{vdMarel16} we do not yet have a convincing derivation of the
coupling of charge carriers to the soft TO phonon of the required strength,
ready to be reported here. We note, however, that additional support for the
existence of the strong isotope effect is coming from the observed gigantic
increase of $T_{c}$ by applying pressure and tensile strain, moving the
system closer to the QCP \cite{Sochnikov18}.
\section{Superconductivity mediated by the screened Coulomb interaction}
The superconducting state of an interacting Fermi system is characterized by
a gap parameter $\Delta (\mathbf{k,}i\omega _{n})$ obtainable from the gap
equation \cite{Eliashberg60}
\begin{equation}
\Delta (\mathbf{k,}i\omega _{n})=-T\sum_{\omega _{m},p}\frac{V_{pair}(%
\mathbf{k,p;}i\omega _{n},i\omega _{m})\Delta (\mathbf{p},i\omega _{m})}{%
(\omega _{m}+i\Sigma (\mathbf{p,}i\omega _{m}))^{2}+\xi _{\mathbf{p}%
}^{2}+\Delta ^{2}(\mathbf{p},i\omega _{m})} \label{Eq:gap_eq}
\end{equation}%
assuming spin singlet pairing (for a review see \cite{Allen83}) . The pair
interaction is given as the sum of the screened Coulomb interaction and a
exchange contribution
\begin{equation}
V_{pair}(k\mathbf{,}p)=V_{C}(k-p)+V_{X}(k,p)
\end{equation}%
where $k=(\mathbf{k,}i\omega _{n}),$ etc., are momentum and Matsubara
frequency variables.
In doped SrTiO$_{3}$ the Coulomb interaction is screened by ionic and
electronic charges
\begin{equation}
V_{C}(q)=\frac{4\pi e^{2}}{q^{2}\varepsilon _{ion}(q)+4\pi e^{2}\chi _{el}(q)%
}
\end{equation}%
where $\varepsilon _{ion}(q)$ is the dielectric function of the undoped
system and $\chi _{el}(q)$ accounts for the screening effected by the
itinerant electronic charges. The ionic screening is dominated by the soft
transverse optical phonon mode ($\omega _{TO}$) and its longitudinal partner
($\omega _{LO}$, for a discussion see e.g. \cite{Vanderbilt94})
\begin{equation}
\varepsilon _{ion}(q)\approx \varepsilon _{\infty }\frac{\omega
_{n}^{2}+\omega _{LO}^{2}(\mathbf{q)}}{\omega _{n}^{2}+\omega _{TO}^{2}(%
\mathbf{q)}}
\end{equation}%
where $\varepsilon _{\infty }$\ is the optical dielectric constant. The
electronic response $\chi _{el}(q)$ is given by the bubble diagram and may
be approximated by
\begin{equation}
\chi _{el}(q)\approx \frac{n}{\frac{2}{3}\epsilon _{F}+\frac{\omega _{n}^{2}%
}{2\epsilon _{q}}+\frac{1}{2}\epsilon _{q}}
\end{equation}%
where $\epsilon _{q}=q^{2}/2m_{1}$ and $m_{1}$ is the carrier mass. At the
lowest densities $n$ for which superconductivity has been observed \cite%
{Behnia14} $n=5\times 10^{17}$cm$^{-3}$ , the Fermi energy is $\epsilon
_{F}\approx 14$K (we use energy units of Kelvin, and also take $\hbar =1$).
This should be compared with a typical phonon frequency $\omega _{D}=380$K .
As shown below, the exchange terms $V_{X}(k,p)$ of the interaction are
systematically small and may be neglected, except for a contribution
involving exchange of the soft phonon. The latter will be important near the
ferroelectric transition, but we will omit this contribution here.
Let us first discuss the behavior of the interaction function $V_{C}(\mathbf{%
q},i\omega _{l})$ for large momenta and frequencies. In the limit of large $%
q $ we have $V_{C}\propto q^{-2}$, which together with the fall off of the
Green's function provides convergence of the momentum integration. In the
limit of large frequency one finds $V_{C}=4\pi e^{2}/(q^{2}\varepsilon
_{\infty })$ , such that the frequency summation is not convergent unless
the high frequency behavior of the self energy $\Sigma (\mathbf{p,}i\omega
_{m})$ is taken into account. We estimated in our paper that the imaginary
part of $\Sigma (\mathbf{p,}\omega +i0)$ ($\Sigma (\mathbf{p,}i\omega _{m})$
analytically continued to the real frequency axis) grows with $\omega $ ,
exceeding $\omega $ beyond a frequency $\omega _{c}$. For $\omega >\omega
_{c}$ , $\Sigma (\mathbf{p,}i\omega _{m})>\omega _{m}$ and the Green's
function falls of faster than $\omega _{m}^{-2}$ thus ensuring convergence
of the $\omega _{m}$ summation in the gap equation. We therefore used $%
\omega _{c}$ as a density dependent frequency cutoff $\omega _{c}(n)$. We
found $\omega _{c}$ to vary with density in the range of $600$K to $2500$K,
which is much higher than the Fermi energy at low doping (see Fig.\ \ref%
{fig:coupl_vs_logn}) . A full-scale evaluation of the selfenergy is beyond
the scope of the present work.
In order to justify the neglect of higher order correction terms to the gap
equation at such elevated frequencies we now show that the system indeed
remains in the weak coupling regime for frequencies up to $\omega _{c}(n)$.
We define a dimensionless coupling function $\lambda (q,\omega )=N(\epsilon
_{F})V_{C}(q\mathbf{,}i\omega )$ as a measure of the strength of the
interaction ($N(\epsilon _{F})=m_{1}k_{F}/\pi ^{2}$ is the density of states
at the Fermi energy). We adopt this definition of the coupling as the
relevant one, rather than the alternative definition $\lambda _{1}(q,\omega
)=N(\epsilon _{q})V_{C}(q\mathbf{,}i\omega )$, where $N(\epsilon
_{q})=m_{1}q/\pi ^{2}$, because $\lambda _{1}(q,\omega )$ is usually
multiplied by a factor $k_{F}/q$, and $\lambda _{1}(q,\omega
)(k_{F}/q)=\lambda (q,\omega )$. The factor $k_{F}/q$ arises in higher order
expressions such as vertex corrections, crossed interaction terms, and more,
from the angular integral of the vector $\mathbf{q}$ in the arguments of
single particle Green's functions, e.g. $G(\mathbf{k+q},i(\omega _{n}+\nu
_{m}))$. At higher electron densities the electron spectrum deviates from
the simple parabolic form, leading to a slower growth of the density of
states than assumed above, which also supports the choice of $\lambda
(q,\omega )$ as a preferable measure of the interaction strength. In Fig.\ %
\ref{fig:coupl_vs_logn} we first show the coupling function at the Fermi
energy $\lambda (k_{F},\epsilon _{F})$ versus electron density $n$ ($n$ in
units of cm$^{-3}$). At low density $\lambda (k_{F},\epsilon _{F})$ is seen
to be very much less than unity, getting larger at higher density, but
remaining less than unity in the whole density regime.\ One should note that
the cutoff frequency $\omega _{c}(n)$ approaches the Fermi energy for
densities $n\gtrsim 10^{20}$cm$^{-3}$ anyway, and then the issue of \ the
cutoff exceeding the Fermi energy is absent. The smallness of\ $\lambda
(k_{F},\epsilon _{F})$\ at low density suggests that $\lambda $ will remain
small at much higher frequencies and momenta.\ The coupling function
increases with frequency, but as shown in Fig.\ \ref{fig:coupl_vs_om} it
never exceeds unity, for frequencies $\omega <6\omega _{D}=2300$K. In Fig.\ %
\ref{fig:coupl_vs_om} the coupling is shown for the typical momentum at
frequency $\omega $, defined as $q_{typ}=k_{F}+\sqrt{2m_{1}\omega }$ . Also
shown is the dependence on density. For completeness we also show the
coupling function $\lambda _{1}(q_{typ},\omega )$\ in the inset of Fig.\ \ref%
{fig:coupl_vs_om}, which is still less than unity in the relevant
frequency range $\omega <\omega _{c}$.\ We may conclude from Figs.\ \ref%
{fig:coupl_vs_logn},\ \ref{fig:coupl_vs_om} that in the frequency range
up to $6\omega _{D}$ higher order contributions such as vertex corrections
and exchange interaction terms, which are of second or higher order in the
coupling $\
|
lambda $ may be safely neglected. This conclusion holds provided
there is no additional instability in the system such as a charge or spin
density wave instability, in the neighborhood of which even a small
interaction may be critically enhanced. There is no experimental indication
of any other instability in addition to the ferroelectric instability. The
latter is expected to give rise to an enhanced pairing interaction mediated
by the soft transverse optical phonon close to the ferroelectric quantum
critical point.
\begin{figure}[tbp]
\includegraphics[width=1.2\columnwidth]{coupl_vs_logn.pdf}
\caption{Coupling function $\protect\lambda(k_{F},\protect\epsilon_{F})$
versus the logarithm of density}
\label{fig:coupl_vs_logn}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=1.2\columnwidth]{coupl_vs_om.pdf}
\caption{Coupling function $\protect\lambda(q_{typ},\protect\omega)$ versus
frequency at various densities $n$. The inset shows coupling function $%
\protect\lambda_{1}(q_{typ},\protect\omega)$ versus frequency in the
relevant region $\protect\omega < \protect\omega_{c}$.}
\label{fig:coupl_vs_om}
\end{figure}
In our paper we presented the results obtained for the combined pair
interaction $V_{C}+V_{X}$ with $V_{X}$ given by the TO-phonon mediated
exchange interaction, which was unfortunately not correctly derived. Now we
assume that $V_{X}$ is negligible if the system is not yet close to the
ferroelectric transition, i.e. in the absence of isotope doping or applied
pressure, but will be important close to the transition. We reevaluate the
transition temperature $T_{c}$ from Eq.\ \ref{Eq:gap_eq} above in the limit $%
\Delta \rightarrow 0$. For calculational convenience we again use a hard
frequency cutoff $\omega _{c}(n).$\ In our paper we distinguished three
different density regimes, for which we found different density power laws,
which we combined into an interpolation formula
\begin{equation}
\omega _{c}(n)=\frac{\omega _{D}}{(c_{1}(\frac{k_{F}}{q_{R}})^{-1/2}+c_{2}(%
\frac{k_{F}}{q_{R}})^{2})^{-1}+c_{3}(\frac{k_{F}}{q_{R}})^{-1}}
\end{equation}%
with parameters $c_{1}$, $c_{2}$, $c_{3}$ and where the Fermi wave number $%
k_{F}(n)=(3\pi ^{2}n)^{1/3}$and $q_{R}^{2}/2m=\omega _{D}$ have been used.
The prefactors of these power laws were estimated by order of magnitude in
Appendix A of \cite{Wolfle18}. In the numerical solution of the linearized
gap equation we assumed reasonable values for the prefactors such that the
resulting transition temperatures $T_{c}$ were in agreement with the data.
In Fig.\ \ref{fig:omc_vs_logn} the cutoff frequency is plotted as a function
of $n$, using the parameters $c_{j}$ specified below. For comparison the
Fermi energy is shown as well. At low density $\omega_{c}$ is seen to be
orders of magnitude larger than the Fermi energy. At high density the cutoff
frequency approaches the Fermi energy from above.
\begin{figure}[tbp]
\includegraphics[width=1.2\columnwidth]{omc_vs_logn.pdf}
\caption{Cutoff frequency $\protect\omega_{c}$ (red line) and Fermi energy
(blue line) in units of $\protect\omega_{D}$ versus the logarithm of density}
\label{fig:omc_vs_logn}
\end{figure}
In Matsubara space $V_{C}(\mathbf{q},i\omega _{l})$ is a positive definite
function, which implies that the eigenfunctions $\Delta (\mathbf{k,}i\omega
_{n})$ of the linearized gap equation must necessarily have zero's in
frequency space\cite{Bogoliubov58,Morel-Anderson62}. We find an even
frequency eigenfunction $\Delta (\mathbf{k,}i\omega _{n})$ featuring two
zero's at $\pm \omega _{n_{0}}$.
In Fig.\ \ref{fig:Tc_vs_logn} the result for $T_{c}$ versus $log_{10}n$ is
shown for the choice of parameters $(c_{1},c_{2},c_{3})=(0.49,1.55,0.118)$
as compared to the values $(1.1,0.6,0.036)$ used in our earlier evaluation
\cite{Wolfle18} for the combined pair interaction in which the attractive TO
phonon mediated interaction was taken sufficiently strong so that the
eigenfunction did not have zeros. It should be noted that the two components
of the pair interaction in that calculation \cite{Wolfle18} partially
compensate each other, leading to a distinctly different frequency
dependence, which explains the change in the $c_{j},$ $j=1,2,3$ . A more
quantitative evaluation of the parameters $c_{j}$, for the two cases of pair
interaction, $V_{C}$, and $V_{C}+V_{X}$ would lead to somewhat different
values.\ Also shown in Fig.\ \ref{fig:Tc_vs_logn} are experimental data, which
are reasonably well accounted for by our theory.
\begin{figure}[tbp]
\includegraphics[width=1.2\columnwidth]{Tc_vs_logn.pdf}
\caption{Transition temperature Tc in Kelvin versus logarithm of electron
density in cm$^{-3}$. Theory: solid line; Experiment: crosses (Schooley et
al., 1964); filled circles: Nb-doped; + symbols: O-reduced (Lin et al.,
2014) }
\label{fig:Tc_vs_logn}
\end{figure}
\section{The weakly attractive Fermi gas}
In order to clarify the question of the energy cutoff in the gap equation
further it is useful to consider a system of weakly interacting fermions
allowing for controlled approximations. We consider a Fermi gas with
quasiparticle energy $\xi _{\mathbf{k}}=(k^{2}-k_{F}^{2})/2m$ and weak
attractive interaction of the form
\begin{equation}
V_{pair}(\mathbf{k},\mathbf{k}^{\prime })=\left\{
\begin{array}{c}
-V_{0},\text{ \ \ \ \ }|\xi _{\mathbf{k}}|,|\xi _{\mathbf{k}^{\prime
}}|<p_{c}^{2}/2m \\
0,\text{ \ \ \ \ \ \ else}%
\end{array}%
\right.
\end{equation}%
We now assume that $p_{c}\gg k_{F}$ and that the dimensionless coupling $%
\lambda (\epsilon )=V_{0}N(\epsilon )\ll 1$ , for $\epsilon <\epsilon
_{c}=p_{c}^{2}/2m$, where $N(\epsilon )=mk/\pi ^{2}$ and $\epsilon =k^{2}/2m$%
. In this case higher order terms in perturbation theory contributing to
both, the irreducible particle-particle vertex (the full pair interaction)
and the self energy are negligible. The linearized gap equation takes the
form
\begin{equation}
\Delta =-\sum_{\mathbf{p}}V_{pair}(\mathbf{k},\mathbf{p})\frac{\tanh \frac{%
\xi _{\mathbf{p}}}{2T_{c}}}{2\xi _{\mathbf{p}}}\Delta
\end{equation}%
The transition temperature follows as \cite{Gorkov61}
\begin{equation}
T_{c}\approx \epsilon _{F}\exp [-\frac{1}{N_{0}|\Gamma _{0}|}]
\label{Tc_weak}
\end{equation}%
where $\Gamma _{0}$ may be interpreted as scattering amplitude. For the
above model one finds
\begin{equation}
\frac{1}{N_{0}|\Gamma _{0}|} =(\frac{1}{N_{0}V_{0}}-\frac{p_{c}}{k_{F}}%
)(1+O(\lambda(\epsilon_{c}))
\end{equation}%
where $N_{0}=mk_{F}/2\pi ^{2}$ is the density of states (of one spin
component) at the Fermi level. In the usual limit of attraction only within
a narrow energy shell about the Fermi level , $p_{c}\ll k_{F}$ , the weak
coupling result is recovered, $T_{c}\approx \epsilon _{c}\exp [-\frac{1}{%
N_{0}V_{0}}]$. In the opposite case, $p_{c}\gg k_{F}$, which is the
situation realized in weakly doped SrTiO$_{3}$, the transition temperature
may be enhanced by orders of magnitude due to the strong renormalization of
the scattering amplitude $\Gamma _{0}$ by virtual processes involving states
in the high energy region $\epsilon _{F}<\xi _{\mathbf{p}}<\epsilon _{c}$.
\section{Conclusion}
The observed superconductivity of lightly doped STO challenges the
"conventional wisdom" accumulated over 60 years of theoretical effort to
understand many different manifestations of this phenomenon. Fortunately,
the situation here is much simpler than that posed by the enigmatic strong
coupling superconductors, in particular the cuprates, in that the charge
carriers in STO are in the weak coupling limit. The extremely strong ionic
screening reduces the strength of the (dimensionless) dynamic Coulomb
interaction to about $0.01$ to $0.2$, depending on density and energy, up to
the LO phonon energy. As a consequence, the contribution of high energy ($%
\epsilon >\epsilon _{F}$) virtual processes to pairing is not cut off by
higher order contributions such as vertex corrections, but is found to
enhance the critical temperature at the lowest densities by orders of
magnitude.This physics has been known by the pioneers of superconductivity,
e.g. Gorkov and Melik-Barkhudarov, who stated that for a Fermi gas with weak
attraction the bare interaction has to be replaced by the scattering
amplitude, which may be much larger than the bare interaction owing to
scattering into high energy intermediate states. The question is then what
determines the frequency cutoff in the gap equation. Here we propose that
the growth of the electron self\ energy $\Sigma (\omega )$ with energy
provides a cutoff at $\omega _{c}=\Sigma (\omega _{c})$. The strong fall off
of the Green's \ function with energy beyond that point ensures convergence
of the frequency summation in the gap equation. The ultimate energy limit $%
\omega _{x}$ beyond which the weak coupling treatment would no longer be
valid is given by the energy at which the dimensionless coupling $%
\lambda(\omega _{x})\approx 1$. As shown above (see Figs.\ \ref%
{fig:coupl_vs_logn},\ \ref{fig:coupl_vs_om_v2}), $\omega _{c}\ll \omega _{x}$
, implying that $\omega _{x}$ is not relevant.\ The effect of the cutoff $%
\omega_{x}$ has been discussed in the context of the plasmon exchange
mechanism for conventional metals \cite{Grabowski84}. A different lesson to
be learned from the pioneers, here Bogoliubov and Anderson, is that in the
case of the pair interaction given by the screened Coulomb interaction,
which is a positive definite function (in Matsubara frequency space) the gap
function must change sign as a function of frequency (leading for
even-frequency pairing to a kind of "d-wave" pairing in frequency space).
The observed "dome" in $T_{c}$ versus $\log n$ can be fully accounted for by
the limiting effect of strong electronic screening on the high density side
and the cutoff energy $\omega _{c}\propto k_{F}^{1/2}$ derived from
electron-phonon scattering at low density. As for the observed strong
isotope effect, which may be expected to arise from a pair interaction
contribution mediated by the soft TO phonon mode, we have to withdraw our
earlier proposal of a deformation potential electron-phonon coupling. We do,
however, take the present observations of the isotope effect at face value,
noting that in the meantime new data on the effect of strain on $T_{c}$ also
suggest a strong coupling of carriers to the soft mode\cite{Sochnikov18}.
Work on deriving a sufficiently strong alternative e-ph coupling is in
progress.
\section{Acknowledgements}
PW acknowledges support by a Distinguished Senior Fellowship of Karlsruhe
Institute of Technology. AVB is supported by ERC Synergy HERO (810451) and
KAW 2018-0104.
|
\section{Introduction}
In network inference applications, it is important to detect community structure, i.e., cluster vertices into potential blocks. However, it can be prohibitively expensive to observe the entire graph in many cases, especially for large graphs. Thus it becomes essential to identify vertices that have the most impact on block structure and only check whether there are edges between them to save significant resources but still recover the block structure.
Many classical methods only consider the adjacency or Laplacian matrices for community detection~\cite{Fortunato2016}. By contrast, vertex covariates can also be taken into consideration for the inference. These covariate-aware methods rely on either variational methods~\cite{Choi2012,Roy2019,Sweet2015} or spectral approaches~\cite{Binkiewicz2017,Huang2018,Mele2019,Mu2022}. However, none of them focus on the problem of clustering vertices for partially observed graphs. To address this issue, existing methods propose different types of random and adaptive sampling strategies to minimize the information loss from the data reduction~\cite{Yun2014,Purohit2017}.
We propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel (SBM) when we only have limited resources to check whether there are edges between certain selected vertices. The innovation of our approach is the application of Chernoff information. To our knowledge, this is the first time that it has been applied to network sampling problems. Motivated by the Chernoff analysis, we not only propose a dynamic network sampling scheme to optimize block recovery, but also provide the framework and justification for using Chernoff information in subsequent inference for graphs.
The structure of this article is summarized as follows. Section~\ref{sec:2} reviews relevant models for random graphs and the basic idea of spectral methods. Section~\ref{sec:3} introduces the notion of Chernoff analysis for analytically measuring the performance of block recovery. Section~\ref{sec:4} includes our dynamic network sampling scheme and theoretical results. Section~\ref{sec:5} provides simulations and real data experiments to measure the algorithms' performance in terms of actual block recovery results. Section~\ref{sec:6} discusses the findings and presents some open questions for further investigation. Appendix provides technical details for our theoretical results.
\section{Models and Spectral Methods}
\label{sec:2}
In this work, we are interested in the inference task of block recovery (community detection). To model the block structure in edge-independent random graphs, we focus on the SBM and the generalized random dot product graph (GRDPG).
\begin{definition}[Generalized Random Dot Product Graph~\cite{Rubin-Delanchy2017}]
\label{def:GRDPG}
Let $ \mathbf{I}_{d_+ d_-} = \mathbf{I}_{d_+} \bigoplus \left(-\mathbf{I}_{d_-} \right) $ with $ d_+ \geq 1 $ and $ d_- \geq 0 $. Let $ F $ be a $ d $-dimensional inner product distirbution with $ d = d_+ + d_- $ on $ \mathcal{X} \subset \mathbb{R}^d $ satisfying $ \mathbf{x}^\top \mathbf{I}_{d_+ d_-} \mathbf{y} \in [0, 1] $ for all $ \mathbf{x}, \mathbf{y} \in \mathcal{X} $. Let $ \mathbf{A} $ be an adjacency matrix and $ \mathbf{X} = [\mathbf{X}_1, \cdots, \mathbf{X}_n]^\top \in \mathbb{R}^{n \times d} $ where $ \mathbf{X}_i \sim F $, i.i.d. for all $ i \in \{ 1, \cdots, n \} $. Then we say $ (\mathbf{A}, \mathbf{X}) \sim \text{GRDPG}(n, F, d_+, d_-) $ if for any $ i, j \in \{ 1, \cdots, n \} $
\begin{equation}
\mathbf{A}_{ij} \sim \text{Bernoulli}(\mathbf{P}_{ij}) \qquad \text{where} \qquad \mathbf{P}_{ij} = \mathbf{X}_{i}^\top \mathbf{I}_{d_+ d_-} \mathbf{X}_j.
\end{equation}
\end{definition}
\begin{definition}[$ K $-block Stochastic Blockmodel Graph~\cite{Holland1983}]
\label{def:SBM}
The $ K $-block stochastic blockmodel (SBM) graph is an edge-independent random graph with each vertex belonging to one of $ K $ blocks. It can be parametrized by a block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $ summing to unity. Let $ \mathbf{A} $ be an adjacency matrix and $ \bm{\tau} $ be a vector of block assignments with $ \tau_i = k $ if vertex $ i $ is in block $ k $ (occuring with probability $ \pi_k $). We say $ (\mathbf{A}, \bm{\tau}) \sim \text{SBM}(n, \mathbf{B}, \bm{\pi}) $ if for any $ i, j \in \{ 1, \cdots, n \} $
\begin{equation}
\mathbf{A}_{ij} \sim \text{Bernoulli}(\mathbf{P}_{ij}) \qquad \text{where} \qquad \mathbf{P}_{ij} = \mathbf{B}_{\tau_i \tau_j}.
\end{equation}
\end{definition}
\begin{remark}
\label{remark:GRDPG-SBM}
The SBM is a special case of the GRDPG model. Let $ (\mathbf{A}, \bm{\tau}) \sim \text{SBM}(n, \mathbf{B}, \bm{\pi}) $ as in Definition~\ref{def:SBM} where $ \mathbf{B} \in (0, 1)^{K \times K} $ with $ d_+ $ strictly positive eigenvalues and $ d_- $ strictly negative eigenvalues. To represent this SBM in the GRDPG model, we can choose $ \bm{\nu}_1, \cdots, \bm{\nu}_K \in \mathbb{R}^d $ where $ d = d_+ + d_- $ such that $ \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_\ell = \mathbf{B}_{k \ell} $ for all $ k, \ell \in \{ 1, \cdots, K \} $. For example, we can take $ \bm{\nu} = \mathbf{U}_B |\mathbf{S}_B|^{1/2} $ where $ \mathbf{B} = \mathbf{U}_B \mathbf{S}_B \mathbf{U}_B^\top $ is the spectral decomposition of $ \mathbf{B} $ after re-ordering. Then we have the latent position of vertex $ i $ as $ \mathbf{X}_i = \bm{\nu}_k $ if $ \tau_i = k $.
\end{remark}
The parameters of the models can be estimated via spectral methods~\cite{Von2007}, which have been widely used in random graph models for community detection~\cite{Lyzinski2014,Lyzinski2016,McSherry2001,Rohe2011}. Two particular spectral embedding methods, adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE), are popular since they enjoy nice propertices including consistency~\cite{Sussman2012} and asymptotic normality~\cite{Athreya2016,Tang2018}.
\begin{definition}[Adjacency Spectral Embedding]
Let $ \mathbf{A} \in \{0, 1 \}^{n \times n} $ be an adjacency matrix with eigendecomposition $ \mathbf{A} = \sum_{i=1}^{n} \lambda_i \mathbf{u}_i \mathbf{u}_i^\top $ where $ |\lambda_1| \geq \cdots \geq |\lambda_n| $ are the magnitude-ordered eigenvalues and $ \mathbf{u}_1, \cdots, \mathbf{u}_n $ are the corresponding orthonormal eigenvectors. Given the embedding dimension $ d < n $, the adjacency spectral embedding (ASE) of $ \mathbf{A} $ into $ \mathbb{R}^d $ is the $ n \times d $ matrix $ \mathbf{\widehat{X}} = \mathbf{U}_A |\mathbf{S}_A|^{1/2} $ where $ \mathbf{S}_A = \text{diag}(\lambda_1, \cdots, \lambda_d) $ and $ \mathbf{U}_A = [\mathbf{u}_1 | \cdots | \mathbf{u}_d] $.
\end{definition}
\begin{remark}
\label{remark:dhat}
There are different methods for choosing the embedding dimension~\cite{Hastie2009,Jolliffe2016}; we adopt the simple and efficient profile likelihood method~\cite{Zhu2006} to automatically identify ``elbow", which is the cut-off between the signal dimensions and the noise dimensions in scree plot.
\end{remark}
\section{Chernoff Analysis}
\label{sec:3}
To analytically measure the performance of algorithms for block recovery, we consider the notion of Chernoff information among other possible metrics. Chernoff information enjoys the advantages of being independent of the clustering procedure, i.e., it can be derived no matter which clustering methods are used, and it is intrinsically relating to the Bayes risk~\cite{Tang2018,Athreya2017,Karrer2011}.
\begin{definition}[Chernoff Information~\cite{Chernoff1952,Chernoff1956}]
Let $ F_1 $ and $ F_2 $ be two continuous multivariate distributions on $ \mathbb{R}^d $ with density functions $ f_1 $ and $ f_2 $. The Chernoff information is defined as
\begin{equation}
\begin{split}
C(F_1, F_2) & = - \log \left[\inf_{t \in (0,1)} \int_{\mathbb{R}^d} f_1^t(\mathbf{x}) f_2^{1-t}(\mathbf{x}) d\mathbf{x} \right] \\
& = \sup_{t \in (0, 1)} \left[- \log \int_{\mathbb{R}^d} f_1^t(\mathbf{x}) f_2^{1-t}(\mathbf{x}) d\mathbf{x} \right].
\end{split}
\end{equation}
\end{definition}
\begin{remark}
Consider the special case where we take $ F_1 = \mathcal{N}(\bm{\mu}_1, \bm{\Sigma}_1) $ and $ F_2 = \mathcal{N}(\bm{\mu}_2, \bm{\Sigma}_2) $; then the corresponding Chernoff information is
\begin{equation}
C(F_1, F_2) = \sup_{t \in (0, 1)} \left[ \frac{1}{2} t (1-t) (\bm{\mu}_1 - \bm{\mu}_2)^\top \bm{\Sigma}_t^{-1} (\bm{\mu}_1 - \bm{\mu}_2) + \frac{1}{2} \log \frac{\lvert \bm{\Sigma}_t \rvert}{\lvert \bm{\Sigma}_1 \rvert^t \lvert \bm{\Sigma}_2 \rvert^{1-t}} \right],
\end{equation}
where $ \bm{\Sigma}_t = t \bm{\Sigma}_1 + (1-t) \bm{\Sigma}_2 $.
\end{remark}
The comparsion of block recovery via Chernoff information is based on the statistical information between the limiting distributions of the blocks and smaller statistical information implies less information to discriminate between different blocks of the SBM. To that end, we also review the limiting results of ASE for SBM, essential for investigating Chernoff information.
\begin{theorem}[CLT of ASE for SBM~\cite{Rubin-Delanchy2017}]
\label{thm:CLT-ASE-SBM}
Let $ (\mathbf{A}^{(n)}, \mathbf{X}^{(n)}) \sim \text{GRDPG}(n, F, d_+, d_-) $ be a sequence of adjacency matrices and associated latent positions of a $ d $-dimensional GRDPG as in Definition~\ref{def:GRDPG} from an inner product distribution $ F $ where $ F $ is a mixture of $ K $ point masses in $ \mathbb{R}^d $, i.e.,
\begin{equation}
F = \sum_{k=1}^{K} \pi_k \delta_{\bm{\nu}_k} \qquad \text{with} \qquad \forall k, \; \pi_k > 0 \quad \text{and} \quad \sum_{k=1}^{K} \pi_k = 1,
\end{equation}
where $ \delta_{\bm{\nu}_k} $ is the Dirac delta measure at $ \nu_k $. Let $ \Phi(\mathbf{z}, \bm{\Sigma}) $ denote the cumulative distribution function (CDF) of a multivariate Gaussian distribution with mean $ \bm{0} $ and covariance matrix $ \bm{\Sigma} $, evaluated at $ \mathbf{z} \in \mathbb{R}^d $. Let $ \mathbf{\widehat{X}}^{(n)} $ be the ASE of $ \mathbf{A}^{(n)} $ with $ \mathbf{\widehat{X}}^{(n)}_i $ as the $ i $-th row (same for $ \mathbf{X}^{(n)}_i $). Then there exists a sequence of matrices $ \mathbf{M}_n \in \mathbb{R}^{d \times d} $ satisfying $ \mathbf{M}_n \mathbf{I}_{d_+ d_-} \mathbf{M}_n^\top = \mathbf{I}_{d_+ d_-} $ such that for all $ \mathbf{z} \in \mathbb{R}^d $ and fixed index i,
\begin{equation}
\mathbb{P} \left\{ \sqrt{n} \left(\mathbf{M}_n \mathbf{\widehat{X}}^{(n)}_i - \mathbf{X}^{(n)}_i \right) \leq \mathbf{z} \; \big| \; \mathbf{X}^{(n)}_i = \bm{\nu}_k \right\} \to \Phi(\mathbf{z}, \bm{\Sigma}_k),
\end{equation}
where for $ \bm{\nu} \sim F $
\begin{equation}
\label{eq:Sigmax}
\bm{\Sigma}_k = \bm{\Sigma}(\bm{\nu}_k) = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \mathbb{E} \left[ \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \bm{\nu} \bm{\nu}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-},
\end{equation}
with
\begin{equation}
\label{eq:Delta}
\bm{\Delta} = \mathbb{E} \left[ \bm{\nu} \bm{\nu}^\top \right].
\end{equation}
\end{theorem}
For a $ K $-block SBM, let $ \mathbf{B} \in (0, 1)^{K \times K} $ be the block connectivity probability matrix and $ \bm{\pi} \in (0, 1)^K $ be the vector of block assignment probabilities. Given an $ n $ vertex instantiation of the SBM parameterized by $ \mathbf{B} $ and $ \bm{\pi} $, for sufficiently large $ n $, the large sample optimal error rate for estimating the block assignments using ASE can be measured via Chernoff information as~\cite{Tang2018,Athreya2017}
\begin{equation}
\label{eq:rho}
\rho = \min_{k \neq l} \sup_{t \in (0, 1)} \left[ \frac{1}{2} n t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) + \frac{1}{2} \log \frac{\lvert \bm{\Sigma}_{k \ell}(t) \rvert}{\lvert \bm{\Sigma}_k \rvert^t \lvert \bm{\Sigma}_\ell \rvert^{1-t}} \right],
\end{equation}
where $ \bm{\Sigma}_{k\ell}(t) = t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell $, $ \bm{\Sigma}_k = \bm{\Sigma}(\bm{\nu}_k) $ and $ \bm{\Sigma}_\ell = \bm{\Sigma}(\bm{\nu}_\ell) $ are defined as in Eq.~\eqref{eq:Sigmax}. Also note that as $ n \to \infty $, the logarithm term in Eq.~\eqref{eq:rho} will be dominated by the other term. Then we have the approximate Chernoff information as
\begin{equation}
\label{eq:rhoapprox}
\rho \approx \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}),
\end{equation}
where
\begin{equation}
\label{eq:C_kl}
C_{k ,\ell}(\mathbf{B}, \bm{\pi}) =\sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \right].
\end{equation}
We also introduce the following two notions, which will be used when we describe our dynamic network sampling scheme.
\begin{definition}[Chernoff-active Blocks]
For $K$-block SBM parametrized by the block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and the vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. The Chernoff-active blocks $ (k^*, \ell^*) $ are defined as
\begin{equation}
(k^*, \ell^*) = \arg \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}),
\end{equation}
where $ C_{k ,\ell}(\mathbf{B}, \bm{\pi}) $ is defined as in Eq.~\eqref{eq:rhoapprox}.
\end{definition}
\begin{definition}[Chernoff Superiority]
For $K$-block SBMs, given two block connectivity probability matrices $ \mathbf{B}, \mathbf{B}^\prime \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. Let $ \rho_B $ and $ \rho_{B^\prime} $ denote the Chernoff information obtained as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively. We say that $ \mathbf{B} $ is Chernoff superior to $ \mathbf{B}^\prime $, denoted as $ \mathbf{B} \succ \mathbf{B}^\prime $, if $ \rho_B > \rho_{B^\prime} $.
\end{definition}
\begin{remark}
If $ \mathbf{B} $ is Chernoff superior to $ \mathbf{B}^\prime $, then we can have a better block recovery from $ \mathbf{B} $ than $ \mathbf{B}^\prime $. In addition, Chernoff superiority is transitive, which is straightforward from the definition.
\end{remark}
\section{Dynamic Network Sampling}
\label{sec:4}
Consider the $ K $-block SBM parametrized by the block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and the vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $ with $ K > 2 $. Given initial sampling parameter $ p_0 \in (0, 1) $, initial sampling is uniformly at random, i.e.,
\begin{equation}
\label{eq:B0}
\mathbf{B}_0 = p_0 \mathbf{B}.
\end{equation}
\begin{theorem}
\label{thm:Chernoff-Superiority}
For $K$-block SBMs, given two block connectivity probability matrices $ \mathbf{B}, p\mathbf{B} \in (0, 1)^{K \times K} $ with $ p \in (0, 1) $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $, we have $ \mathbf{B} \succ p \mathbf{B} $.
\end{theorem}
The proof of Theorem~\ref{thm:Chernoff-Superiority} can be found in Appendix. As an illustration, consider a 4-block SBM parametrized by block connectivity probability matrix $ \mathbf{B} $ as
\begin{equation}
\label{eq:exampleB}
\mathbf{B} =
\begin{bmatrix}
0.04 & 0.08 & 0.10 & 0.18 \\
0.08 & 0.16 & 0.20 & 0.36 \\
0.10 & 0.20 & 0.25 & 0.45 \\
0.18 & 0.36 & 0.45 & 0.81
\end{bmatrix}.
\end{equation}
Figure~\ref{fig:rho0} shows Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and $ p \mathbf{B} $ for $ p \in (0, 1) $. In addition, Figure~\ref{fig:rho0a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho0b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. As suggested by Theorem~\ref{thm:Chernoff-Superiority}, for any $ p \in (0, 1) $ we have $\rho_{B} > \rho_{pB} $ and thus $ \mathbf{B} \succ p \mathbf{B} $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho0a}]{
[width=0.45\textwidth]{figures/rho0a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho0b}]{
[width=0.45\textwidth]{figures/rho0b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and $ p \mathbf{B} $ for $ p \in (0, 1) $.}
\label{fig:rho0}
\end{figure}
Now given dynamic network sampling parameter $ p_1 \in (0, 1-p_0) $, the baseline sampling scheme can proceed uniformly at random again, i.e.,
\begin{equation}
\label{eq:B1}
\mathbf{B}_1 = \mathbf{B}_0 + p_1 \mathbf{B} = (p_0 + p_1) \mathbf{B}.
\end{equation}
\begin{corollary}
\label{cor:Chernoff-Superiority}
For $K$-block SBMs, given block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. We have $ \mathbf{B} \succ \mathbf{B}_1 \succ \mathbf{B}_0 $ where $ \mathbf{B}_0 $ is defined as in Eq.~\eqref{eq:B0} with $ p_0 \in (0, 1) $ and $ \mathbf{B}_1 $ is defined as in Eq.~\eqref{eq:B1} with $ p_1 \in (0, 1-p_0) $.
\end{corollary}
The proof of Corollary~\ref{cor:Chernoff-Superiority} can be found in Appendix. This corollay implies that we can have a better block recovery from $ \mathbf{B}_1 $ than $ \mathbf{B}_0 $.
\begin{assumption}
\label{cond:1}
The Chernoff-active blocks after initial sampling is unique, i.e., there exists an unique pair $ \left(k_0^*, \ell_0^* \right) \in \{(k, \ell) \; | \; 1 \leq k < \ell \leq K \} $ such that
\begin{equation}
\left(k_0^*, \ell_0^* \right) = \arg \min_{k \neq l} C_{k ,\ell}(\mathbf{B}_0, \bm{\pi}),
\end{equation}
where $ \mathbf{B}_0 $ is defined as in Eq.~\eqref{eq:B0} and $ \bm{\pi} $ is the vector of block assignment probabilities.
\end{assumption}
To improve this baseline sampling scheme, we concentrate on the Chernoff-active blocks $ \left(k_0^*, \ell_0^* \right) $ after initial sampling assuming Assumption~\ref{cond:1} holds. Instead of sampling from the entire block connectivity probability matrix $ \mathbf{B} $ like the baseline sampling scheme as in Eq.~\eqref{eq:B1}, we only sample the entries associated with the Chernoff-active blocks. As a competitor to $ \mathbf{B}_1 $, our Chernoff-optimal dynamic network sampling scheme is then given by
\begin{equation}
\label{eq:B1tilde}
\widetilde{\mathbf{B}}_1 = \mathbf{B}_0 + \frac{p_1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*},
\end{equation}
where $ \circ $ denotes Hadamard product, $ \pi_{k_0^*} $ and $ \pi_{\ell_0^*} $ denote the block assignment probabilities for block $ k_0^* $ and $ \ell_0^* $ respectively, and $ \mathbf{1}_* $ is the $ K \times K $ binary matrix with 0's everywhere except for 1's associated with the Chernoff-active blocks $ \left(k_0^*, \ell_0^* \right) $, i.e., for any $ i, j \in \{1, \cdots, K \} $
\begin{equation}
\mathbf{1}_{k_0^*, \ell_0^*}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in \left\{ \left(k_0^*, k_0^* \right), \; \left(k_0^*, \ell_0^* \right), \; \left(\ell_0^*, k_0^* \right), \; \left(\ell_0^*, \ell_0^* \right) \right\} \\
0 & \text{otherwise}
\end{cases}
.
\end{equation}
Note that the multiplier $ \frac{1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2} $ on $ p_1 \mathbf{B} \circ \mathbf{1}_* $ assures that we sample the same number of potential edges with $ \widetilde{\mathbf{B}}_1 $ as we do with $ \mathbf{B}_1 $ in the baseline sampling scheme. In addition, to avoid over-sampling with respect to $ \mathbf{B} $, i.e., to ensure $ \widetilde{\mathbf{B}}_1[i, j] \leq \mathbf{B}[i, j] $ for any $ i, j \in \{1, \cdots, K \} $, we require
\begin{equation}
\label{eq:p1max}
p_1 \leq p_1^{\text{max}} = \left( 1 - p_0 \right) \left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2.
\end{equation}
\begin{assumption}
\label{cond:Chernoff-Superiority2}
For $K$-block SBMs, given a block connectivity probability matrix $ \mathbf{B} \in (0, 1)^{K \times K} $ and a vector of block assignment probabilities $ \bm{\pi} \in (0, 1)^K $. Let $ p_1^* \in (0, p_1^{\text{max}}] $ be the smallest positive $ p_1 \leq p_1^{\text{max}} $ such that
\begin{equation}
\arg \min_{k \neq l} C_{k ,\ell}(\widetilde{\mathbf{B}}_1, \bm{\pi})
\end{equation}
is not unique where $ p_1^{\text{max}} $ is defined as in Eq.~\eqref{eq:p1max} and $ \widetilde{\mathbf{B}}_1 $ is defined as in Eq.~\eqref{eq:B1tilde}. If the arg min is always unique, let $ p_1^* = p_1^{\text{max}} $.
\end{assumption}
For any $ p_1 \in (0, p_1^*)$, we can have a better block recovery from $ \widetilde{\mathbf{B}}_1 $ than $ \mathbf{B}_1 $, i.e., our Chernoff-optimal dynamic network sampling sheme is better than the baseline sampling scheme in terms of block recovery.
As an illustaration, consider the 4-block SBM with initial sampling parameter $ p_0 = 0.01 $ and block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. Figure~\ref{fig:rho1} shows the Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1 $ as in Eq.~\eqref{eq:B1tilde} with dynamic network sampling parameter $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}. In addition, Figure~\ref{fig:rho1a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho1b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. Note that for any $ p_1 \in (0, p_1^*) $ we have $\rho_{B} > \rho_{\widetilde{B}_1} > \rho_{B_1} > \rho_{B_0} $ and thus $ \mathbf{B} \succ \widetilde{\mathbf{B}}_1 \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho1a}]{
[width=0.45\textwidth]{figures/rho1a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho1b}]{
[width=0.45\textwidth]{figures/rho1b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1 $ as in Eq.~\eqref{eq:B1tilde} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}.}
\label{fig:rho1}
\end{figure}
As described earlier, it may be the case that $ p_1^* < p_1^{\text{max}} $ at which point Chernoff-active blocks change to $ (k_1^*, \ell_1^*) $. This potential non-uniquess of the Chernoff argmin is a consequence of our dynamic network sampling scheme. In the case of $ p_1 > p_1^* $, our Chernoff-optimal dynamic network sampling scheme is adopted as
\begin{equation}
\label{eq:B1tildestar}
\widetilde{\mathbf{B}}_1^* = \mathbf{B}_0 + \left(p_1 - p_1^* \right) \mathbf{B} + \frac{p_1^*}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*},
\end{equation}
Similarly, the multiplier $ \frac{1}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2} $ on $ p_1^* \mathbf{B} \circ \mathbf{1}_{k_0^*, \ell_0^*} $ assures that we sample the same number of potential edges with $ \widetilde{\mathbf{B}}_1^* $ as we do with $ \mathbf{B}_1 $ in the baseline sampling scheme. In addition, to avoid over-sampling with respect to $ \mathbf{B} $, i.e., $ \widetilde{\mathbf{B}}_1^*[i, j] \leq \mathbf{B}[i, j] $ for any $ i, j \in \{1, \cdots, K \} $, we require
\begin{equation}
\label{eq:p11max}
p_1 \leq p_{11}^{\text{max}} = 1 - p_0 - \frac{p_1^*}{\left(\pi_{k_0^*} + \pi_{\ell_0^*}\right)^2 } + p_1^*.
\end{equation}
For any $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $, we can have a better block recovery from $ \widetilde{\mathbf{B}}_1^* $ than $ \mathbf{B}_1 $, i.e., our Chernoff-optimal dynamic network sampling sheme is again better than the baseline sampling scheme in terms of block recovery.
As an illustration, consider a 4-block SBM with initial sampling parameter $ p_0 = 0.01 $ and block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. Figure~\ref{fig:rho2} shows the Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} with dynamic network sampling parameter $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2} and $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. In addition, Figure~\ref{fig:rho2a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ and Figure~\ref{fig:rho2b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $. Note that for any $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ we have $\rho_{B} > \rho_{\widetilde{B}_1^*} > \rho_{B_1} > \rho_{B_0} $ and thus $ \mathbf{B} \succ \widetilde{\mathbf{B}}_1^* \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:rho2a}]{
[width=0.45\textwidth]{figures/rho2a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:rho2b}]{
[width=0.45\textwidth]{figures/rho2b.png}
}
\caption{Chernoff information $ \rho $ as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}, $ \mathbf{B}_0 $ as in Eq.~\eqref{eq:B0}, $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1}, and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2} and $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}.}
\label{fig:rho2}
\end{figure}
We summarize the uniform dynamic sampling scheme (baseline) as Algorithm~\ref{algo:1} and our Chernoff-optimal dynamic network sampling scheme as Algorithm~\ref{algo:2}. Recall given potential edge set $ E $ and initial sampling parameter $ p_0 \in (0, 1) $, we have the initial edge set $ E_0 \subset E $ with $ \lvert E_0 \rvert = p_0 \lvert E \rvert $. The goal is to dynamically sample new edges from the potential edge set so that we can have a better block recovery given limited resources.
\begin{algorithm}
\label{algo:1}
\SetAlgoNoLine
\KwIn{Number of vertices $ n $; potential edge set $ E = \{(i, j) \; | \; i, j \in \{1, \cdots, n \} \} $; initial edge set $ E_0 \subset E $; dynamic network sampling parameter $ p_1 \in \left(0, 1- \frac{\lvert E_0 \rvert}{\lvert E \rvert} \right) $}
Construct dynamic edge set as
\begin{equation*}
E_1 = \left\{(i ,j) \; | \; (i ,j) \in E \setminus E_0 \right\} \qquad \text{with} \qquad \lvert E_1 \rvert = p_1 \lvert E \rvert.
\end{equation*} \\
Construct dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \bigcup E_1 \;\; \text{or} \;\; (j, i) \in E_0 \bigcup E_1 \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Estimate dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using Gaussian mixture modeling (GMM) to estimate the block assignments as $ \bm{\widehat{\tau}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via Bayesian Information Criterion (BIC).
\KwOut{Block assignments $ \bm{\widehat{\tau}} $.}
\caption{Uniform dynamic network sampling scheme (baseline)}
\end{algorithm}
\begin{algorithm}
\label{algo:2}
\SetAlgoNoLine
\KwIn{Number of vertices $ n $; potential edge set $ E = \{(i, j) \; | \; i, j \in \{1, \cdots, n \} \} $; initial edge set $ E_0 \subset E $; dynamic network sampling parameter $ p_1 \in \left(0, 1- \frac{\lvert E_0 \rvert}{\lvert E \rvert} \right) $}
Construct dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \;\; \text{or} \;\; (j, i) \in E_0 \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Estimate dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using GMM to estimate the initial block assignments as $ \bm{\widehat{\xi}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via BIC. \\
Estimate the dynamic block assignment probability vector as $ \bm{\widehat{\pi}} \in (0, 1)^K $ where for $ k \in \{1, \cdots, K \} $
\begin{equation*}
\widehat{\pi}_k = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1} \{\bm{\widehat{\xi}}_i = k \}.
\end{equation*} \\
Estimate the dynamic block connectivity probability matrix as
\begin{equation*}
\mathbf{\widehat{B}} = \bm{\widehat{\mu}} \mathbf{I}_{d_+ d_-} \bm{\widehat{\mu}}^\top \in [0,1]^{\widehat{K} \times \widehat{K}},
\end{equation*}
where $ \bm{\widehat{\mu}} \in \mathbb{R}^{\widehat{K} \times \widehat{d}} $ is the estimated means of all clusters. \\
Find the Chernoff-active blocks as
\begin{equation*}
\left(k^*, \ell^* \right) = \arg \min_{k \neq l} C_{k ,\ell} \left(\mathbf{\widehat{B}}, \bm{\widehat{\pi}} \right).
\end{equation*} \\
Construct dynamic edge set as
\begin{equation*}
\begin{split}
E_1 \subseteq E_* \qquad & \text{with} \qquad \lvert E_1 \rvert = \min \left\{p_1 \lvert E \rvert \left(\widehat{\pi}_{k^*} + \widehat{\pi}_{\ell^*}\right)^2, \lvert E_* \rvert \right\}, \\
E_{11} \subset E \setminus \left(E_0 \bigcup E_1 \right) \qquad & \text{with} \qquad \lvert E_{11} \rvert = p_1 \lvert E \rvert - \lvert E_1 \rvert,
\end{split}
\end{equation*}
where
\begin{equation*}
E_* = \left\{(i ,j) \; | \; (i ,j) \in E \setminus E_0 \; \text{and} \; \widehat{\xi}_i, \widehat{\xi}_j \in \{k^*, \ell^* \} \right\}.
\end{equation*} \\
Update dynamic adjacency matrix as $ \mathbf{A} \in \{0, 1\}^{n \times n} $ where for any $ i, j \in \{1, \cdots, n \} $
\begin{equation*}
\mathbf{A}[i, j] =
\begin{cases}
1 & \text{if} \;\; (i, j) \in E_0 \bigcup E_1 \bigcup E_{11} \;\; \text{or} \;\; (j, i) \in E_0 \bigcup E_1 \bigcup E_{11} \\
0 & \text{otherwise}
\end{cases}
.
\end{equation*} \\
Update dynamic latent positions as $ \mathbf{\widehat{X}} \in \mathbb{R}^{n \times \widehat{d}} $ using ASE of updated $ \mathbf{A} $ where $ \widehat{d} $ is chosen as in Remark~\ref{remark:dhat}. \\
Cluster $ \mathbf{\widehat{X}} $ using GMM to estimate the block assignments as $ \bm{\widehat{\tau}} \in \{1, \cdots, \widehat{K} \}^{n} $ where $ \widehat{K} $ is chosen via BIC.
\KwOut{Block assignments $ \bm{\widehat{\tau}} $.}
\caption{Chernoff-optimal dynamic network sampling scheme}
\end{algorithm}
\section{Experiments}
\label{sec:5}
\subsection{Simulations}
In addition to Chernoff analysis, we also evalute our Chernoff-optimal dynamic network sampling sheme via simulations. In particular, consider the 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} and dynamic network sampling parameter $ p_1 \in (0, p_{11}^{\text{max}}] $ where $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. We fix initial sampling parameter $ p_0 = 0.01 $. For each $ p_1 \in (0, p_1^*) $ where $ p_1^* $ is defined as in Assumption~\ref{cond:Chernoff-Superiority2}, we simulate 50 adjacency matrices with $ n = 12000 $ vertices from $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1} and $ \widetilde{\mathbf{B}}_1 $ as
|
in Eq.~\eqref{eq:B1tilde} respectively. For each $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $, we simulate 50 adjacency matrices with $ n = 12000 $ vertices from $ \mathbf{B}_1 $ as in Eq.~\eqref{eq:B1} and $ \widetilde{\mathbf{B}}_1^* $ as in Eq.~\eqref{eq:B1tildestar} respectively. In addition, Figure~\ref{fig:sim0a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $, i.e., 3000 vertices in each block, and Figure~\ref{fig:sim0b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $, i.e., 1500 vertices in two of the blocks and 4500 vertices in the other two blocks. We then apply ASE $ \circ $ GMM (Step 3 and 4 in Algorithm~\ref{algo:1}) to recover block assignments and adopt adjusted Rand index (ARI) to measure the performance. Figure~\ref{fig:sim0} shows ARI (\texttt{mean$ \pm $stderr}) associated with $ \mathbf{B}_1 $ for $ p_1 \in (0, p_{11}^{\text{max}}] $, $ \widetilde{\mathbf{B}}_1 $ for $ p_1 \in (0, p_1^*) $, and $ \widetilde{\mathbf{B}}_1^* $ for $ p_1 \in [p_1^*, p_{11}^{\text{max}}] $ where the dashed lines denote $ p_1^* $. Note that we can have a better block recovery from $ \widetilde{\mathbf{B}}_1 $ and $ \widetilde{\mathbf{B}}_1^* $ than $ \mathbf{B}_1 $, which argee with our results from Chernoff analysis.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:sim0a}]{
[width=0.45\textwidth]{figures/sim0a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:sim0b}]{
[width=0.45\textwidth]{figures/sim0b.png}
}
\caption{Simulations for 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} with initial sampling parameter $ p_0 = 0.01 $ and dynamic network sampling parameter $ p_1 \in (0, p_{11}^{\text{max}}] $ where $ p_{11}^{\text{max}} $ is defined as in Eq.~\eqref{eq:p11max}. The dashed lines denote $ p_1^* $ which is defined as in Assumption~\ref{cond:Chernoff-Superiority2}.}
\label{fig:sim0}
\end{figure}
Now we compare the performance of Algorithms~\ref{algo:1} and~\ref{algo:2} by actual block recovery results. In particular, we start with the 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB}. We consider dynamic network sampling parameter $ p_1 \in (0, 1-p_0) $ where $ p_0 $ is the initial sampling parameter. For each $ p_1 $, we simulate 50 adjacency matrices with $ n = 4000 $ vertices and retrieve associated potential edge sets. We fix initial sampling parameter $ p_0 = 0.15 $ and randomly sample initial edge sets. We then apply both algorithms to estimate the block assignments and adopt ARI to measure the performance. Figure~\ref{fig:sim1} shows ARI (\texttt{mean$ \pm $stderr}) of two algorithms for $ p_1 \in (0, 0.85) $ where Figure~\ref{fig:sim1a} assumes $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $, i.e., 1000 vertices in each block, and Figure~\ref{fig:sim1b} assumes $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $, i.e., 500 vertices in two of the blocks and 1500 vertices in the other two blocks. Note that both algorithms tend to have a better performance as $ p_1 $ increases, i.e., as we sample more edges, and Algorithm~\ref{algo:2} can always recover more accurate block structure than Algorithm~\ref{algo:1}.
\begin{figure}[h!]
\subfigure[balanced: $ \bm{\pi} = (\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}) $ \label{fig:sim1a}]{
[width=0.45\textwidth]{figures/sim1a.png}
}
\hfil
\subfigure[unbalanced: $ \bm{\pi} = (\frac{1}{8}, \frac{1}{8}, \frac{3}{8}, \frac{3}{8}) $ \label{fig:sim1b}]{
[width=0.45\textwidth]{figures/sim1b.png}
}
\caption{Simulations for 4-block SBM parameterized by block connectivity probability matrix $ \mathbf{B} $ as in Eq.~\eqref{eq:exampleB} with initial sampling parameter $ p_0 = 0.15 $ and dynamic network sampling parameter $ p_1 \in (0, 0.85) $.}
\label{fig:sim1}
\end{figure}
\subsection{Real Data}
We also evaluate the performance of Algorithms~\ref{algo:1} and~\ref{algo:2} for real application. We conduct real data experiments on a diffusion MRI connectome dataset~\cite{Priebe2019}. There are 114 graphs (connectomes) estimated by the NDMG pipeline~\cite{Kiar2018} in this dataset. Each vertex in these graphs (the number of vertices $ n $ varies from 23728 to 42022) has a \{Left,~Right\} hemisphere label and a \{Gray,~White\} tissue label. We consider the potential 4 blocks as \{LG,~LW,~RG,~RW\} where L and R denote the Left and Right hemisphere label, G and W denote the Gray and White tissue label. Here we consider initial sampling parameter $ p_0 = 0.25 $ and dynamic network sampling parameter $ p_1 = 0.25 $. Let $ \Delta = \text{ARI(Algo2)} - \text{ARI(Algo1)} $ where ARI(Algo1) and ARI(Algo2) denotes the ARI when we apply Algorithms~\ref{algo:1} and~\ref{algo:2} respectively. The following hypothesis testing yields \texttt{p-value=0.0184}.
\begin{equation}
H_0: \; \text{median}(\Delta) \leq 0 \qquad \text{v.s.} \qquad H_A: \; \text{median}(\Delta) > 0.
\end{equation}
Furthermore, we test our algorithms on a Microsoft bing entity dataset~\cite{Agterberg2020}. There are 2 graphs in this dataset where each has 13535 vertices. We treat block assignments estimated from the complete graph as ground truth. We consider initial sampling parameter $ p_0 \in \left\{0.2, \; 0.3 \right\} $ and dynamic network sampling parameter $ p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $. For each $ p_1 $, we sample 100 times and compare the overall performance of Algorithm~\ref{algo:1} and~\ref{algo:2}. Figure~\ref{fig:real2} shows the results where ARI is reported as \texttt{mean($\pm$stderr)}.
\begin{figure}[h!]
\subfigure[$ p_0 = 0.2, \; p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $ \label{fig:real2a}]{
[width=0.45\textwidth]{figures/real2a.png}
}
\hfil
\subfigure[$ p_0 = 0.3, \; p_1 \in \left\{0, \; 0.05, \; 0.1, \; 0.15, \; 0.2 \right\} $ \label{fig:real2b}]{
[width=0.45\textwidth]{figures/real2b.png}
}
\caption{Algorithms' comparative performance on Microsoft bing entity data via ARI with different initial sampling parameter $ p_0 $ and dynamic network sampling parameter $ p_1 $.}
\label{fig:real2}
\end{figure}
We also conduct real data experiments with 2 social network datasets.
\begin{itemize}
\item LastFM asia social network data set~\cite{Leskovec2014,Rozemberczki2020}: Vertices (the number of vertices $ n = 7624 $) represent LastFM users from asian countries and edges (the number of edges $ e = 27806 $) represent mutual follower relationships. We treat 18 different location of users, which are derived from the country field for each user, as the potential block.
\item Facebook large page-page network data set~\cite{Leskovec2014,Rozemberczki2019}: Vertices (the number of vertices $ n = 22470 $) represent official Facebook pages and edges (the number of edges $ e = 171002 $) represent mutual likes. We treat 4 page types \{Politician,~Governmental~Organization,~Television~Show,~Company\}, which are defined by Facebook, as the potential block.
\end{itemize}
We consider initial sampling parameter $ p_0 \in \left\{0.15, \; 0.35 \right\} $ and dynamic network sampling parameter $ p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $. For each $ p_1 $, we sample 100 times and compare the overall performance of Algorithm~\ref{algo:1} and~\ref{algo:2}. Figure~\ref{fig:real3} shows the results where ARI is reported as \texttt{mean($\pm$stderr)}.
\begin{figure}[h!]
\subfigure[LastFM: $ p_0 = 0.15, \; p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $ \label{fig:real3a}]{
[width=0.45\textwidth]{figures/real3a.png}
}
\hfil
\subfigure[Facebook: $ p_0 = 0.35, \; p_1 \in \left\{0.05, \; 0.1, \; 0.15, \; 0.2, \; 0.25 \right\} $ \label{fig:real3b}]{
[width=0.45\textwidth]{figures/real3b.png}
}
\caption{Algorithms' comparative performance on social network data via ARI with different initial sampling parameter $ p_0 $ and dynamic network sampling parameter $ p_1 $.}
\label{fig:real3}
\end{figure}
\section{Discussion}
\label{sec:6}
We propose a dynamic network sampling scheme to optimize block recovery for SBM. Theoretically, we provide justification of our proposed Chernoff-optimal dynamic sampling scheme via the Chernoff information. Practically, we evaluate the performance, in terms of block recovery (community detection), of our method on several real datasets including diffusion MRI connectome dataset, Microsoft bing entity graph transitions dataset and social network datasets. Both theoretically and practically results suggest that our method can identify vertices that have the most impact on block structure and only check whether there are edges between them to save significant resources but still recover the block structure.
As the Chernoff-optimal dynamic sampling scheme depends on the initial clustering results to identify Chernoff-active blocks and construct dynamic edge set. Thus the performance could be impacted if the initial clustering is not very ideal. One of the future direction is to design certain strategy to reduce this dependency such that the proposed scheme is more robust.
\section*{Appendix}
\begin{proof}[Proof of Theorem~\ref{thm:Chernoff-Superiority}.]
Let $ \mathbf{B} = \mathbf{U} \mathbf{S} \mathbf{U}^\top $ be the spectral decomposition of $ \mathbf{B} $ and $ \mathbf{B}^\prime = p \mathbf{B} $ with $ p \in (0, 1) $. Then we have
\begin{equation}
\label{eq:Bprime}
\mathbf{B}^\prime = \mathbf{U}^\prime \mathbf{S} \left(\mathbf{U}^\prime\right)^\top \qquad \text{where} \qquad \mathbf{U}^\prime = \sqrt{p} \mathbf{U}.
\end{equation}
By Remark~\ref{remark:GRDPG-SBM}, to represent these two SBMs parametrized by two block connectivity matrices $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively (with the same block assignment probability vector $ \bm{\pi} $) in the GRDPG models, we can take
\begin{equation}
\label{eq:nunuprime}
\begin{split}
\bm{\nu} & = \begin{bmatrix}
\bm{\nu}_1 & \cdots & \bm{\nu}_K
\end{bmatrix}^\top = \mathbf{U} |\mathbf{S}|^{1/2} \in \mathbb{R}^{K \times d}, \\
\bm{\nu}^\prime & = \begin{bmatrix}
\bm{\nu}_1^\prime & \cdots & \bm{\nu}_K^\prime
\end{bmatrix}^\top = \mathbf{U}^\prime |\mathbf{S}|^{1/2} = \sqrt{p} \mathbf{U} |\mathbf{S}|^{1/2} = \sqrt{p} \bm{\nu} \in \mathbb{R}^{K \times d}.
\end{split}
\end{equation}
Then for any $ k \in \{1, \cdots, K \} $, we have $ \bm{\nu}_k^\prime = \sqrt{p} \bm{\nu}_k \in \mathbb{R}^{d} $. By Theorem~\ref{thm:CLT-ASE-SBM}, we have
\begin{equation}
\begin{split}
\bm{\Delta} & = \sum_{k=1}^{K} \pi_k \bm{\nu}_k \bm{\nu}_k^\top \in \mathbb{R}^{d \times d}, \\
\bm{\Delta}^\prime & = \sum_{k=1}^{K} \pi_k \bm{\nu}_k^\prime \left(\bm{\nu}_k^\prime\right)^\top = p \sum_{k=1}^{K} \pi_k \bm{\nu}_k \bm{\nu}_k^\top = p \bm{\Delta} \in \mathbb{R}^{d \times d}.
\end{split}
\end{equation}
Note that $ \mathbf{B} $ and $ \mathbf{B}^\prime $ have the same eigenvalues, thus we have $ \mathbf{I}_{d_+ d_-} = \mathbf{I}_{d_+ d_-}^\prime $. See also Lemma 2~\cite{Gallagher2019}. Then for $ k \in \{1, \cdots, K \} $, we have
\begin{equation}
\begin{split}
\bm{\Sigma}_k & = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \mathbb{E} \left[ \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu} \right) \bm{\nu} \bm{\nu}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[\sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \in \mathbb{R}^{d \times d}, \\[1em]
\bm{\Sigma}_k^{\prime} & = \frac{1}{p^2} \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[p^2 \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-p \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[p \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \left(1-\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& \qquad + \mathbf{I}_{d_+ d_-} \bm{\Delta}^{-1} \left[(1-p) \sum_{\ell=1}^{K} \pi_{\ell} \left(\bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_{\ell} \right) \bm{\nu}_{\ell} \bm{\nu}_{\ell}^\top \right] \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \\
& = p \bm{\Sigma}_k + \mathbf{V}^\top \mathbf{D}_k(p) \mathbf{V} \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\mathbf{V} & = \bm{\nu} \bm{\Delta}^{-1} \mathbf{I}_{d_+ d_-} \in \mathbb{R}^{K \times d}, \\
\mathbf{D}_k(p) & = (1-p) \text{diag} \left(\pi_1 \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_1, \cdots, \pi_K \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_K \right) \in (0, 1)^{K \times K}.
\end{split}
\end{equation}
Recall that by Remark~\ref{remark:GRDPG-SBM}, we have $ \bm{\nu}_k^\top \mathbf{I}_{d_+ d_-} \bm{\nu}_\ell = \mathbf{B}_{k \ell} \in (0, 1) $ for all $ k, \ell \in \{ 1, \cdots, K \} $. Then we have $ \mathbf{D}_k(p) $ is positive-definite for any $ k \in \{1, \cdots, K \} $ and $ p \in (0, 1) $. For $ k, \ell \in \{1, \cdots, K \} $ and $ t \in (0, 1) $, let $ \bm{\Sigma}_{k\ell}(t) $ and $ \bm{\Sigma}_{k\ell}^{\prime}(t) $ denote the matrics as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively, i.e.,
\begin{equation}
\begin{split}
\bm{\Sigma}_{k\ell}(t) & = t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell \in \mathbb{R}^{d \times d}, \\[1em]
\bm{\Sigma}_{k\ell}^{\prime}(t) & = t \bm{\Sigma}_k^{\prime} + (1-t) \bm{\Sigma}_\ell^{\prime} \\
& = t \left[p \bm{\Sigma}_k + \mathbf{V}^\top \mathbf{D}_k(p) \mathbf{V} \right] + (1-t) \left[p \bm{\Sigma}_\ell + \mathbf{V}^\top \mathbf{D}_\ell(p) \mathbf{V} \right] \\
& = p \left[t \bm{\Sigma}_k + (1-t) \bm{\Sigma}_\ell \right] + \mathbf{V}^\top \left[t \mathbf{D}_k(p) + (1-t) \mathbf{D}_\ell(p) \right] \mathbf{V} \\
& = p \bm{\Sigma}_{k\ell}(t) + \mathbf{V}^\top \mathbf{D}_{k \ell}(p, t) \mathbf{V} \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\mathbf{D}_{k \ell}(p, t) = t \mathbf{D}_k(p) + (1-t) \mathbf{D}_\ell(p) \in \mathbb{R}_+^{K \times K}.
\end{equation}
Recall that $ \mathbf{D}_k(p) $ and $ \mathbf{D}_\ell(p) $ are both positive-definite for any $ k, \ell \in \{1, \cdots, K \} $ and $ p \in (0, 1) $, thus $ \mathbf{D}_{k \ell}(p, t) $ is also positive-definite for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $. Now by the Sherman-Morrison-Woodbury formula~\cite{Horn2012}, we have
\begin{equation}
\begin{split}
\left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} & = \left[p \bm{\Sigma}_{k\ell}(t) + \mathbf{V}^\top \mathbf{D}_{k \ell}(p, t) \mathbf{V} \right]^{-1} \\
& = \frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \left[\mathbf{D}_{k \ell}^{-1}(p, t) + \frac{1}{p} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \right]^{-1} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \\
& = \frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \mathbf{M}_{k \ell}^{-1}(p, t)\mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \in \mathbb{R}^{d \times d},
\end{split}
\end{equation}
where
\begin{equation}
\mathbf{M}_{k \ell}(p, t) = \mathbf{D}_{k \ell}^{-1}(p, t) + \frac{1}{p} \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \in \mathbb{R}^{K \times K}.
\end{equation}
Recall that for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $, $ \mathbf{D}_{k \ell}(p, t) $ and $ \bm{\Sigma}_{k\ell}(t) $ are both positive-definite, thus $ \mathbf{M}_{k \ell}(p, t) $ is also positive-definite. Then for any $ k, \ell \in \{1, \cdots, K \} $ and $ p,t \in (0, 1) $, we have
\begin{equation}
\label{eq:nuSigma}
\begin{split}
(\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime) & = p (\bm{\nu}_k - \bm{\nu}_\ell)^\top \\
& \quad \left[\frac{1}{p} \bm{\Sigma}_{k\ell}^{-1}(t) - \frac{1}{p^2} \bm{\Sigma}_{k\ell}^{-1}(t) \mathbf{V}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) \right] \\
& \quad (\bm{\nu}_k - \bm{\nu}_\ell) \\
& = (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \\
& \quad - \frac{1}{p} \mathbf{x}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{x} \\
& = (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) - h_{k \ell}(p, t),
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
\mathbf{x} & = \mathbf{V} \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \in \mathbb{R}^K, \\
h_{k \ell}(p, t) & = \frac{1}{p} \mathbf{x}^\top \mathbf{M}_{k \ell}^{-1}(p, t) \mathbf{x}.
\end{split}
\end{equation}
Recall that for any $ k, \ell \in \{1, \cdots, K \} $ and $ p, t \in (0, 1) $, $ \mathbf{M}_{k \ell}(p, t) $ is positive-definite, thus we have $ h_{k \ell}(p, t) > 0 $. Together with Eq.~\eqref{eq:nuSigma}, we have
\begin{equation}
t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) > t (1-t) (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime).
\end{equation}
Thus for any $ k, \ell \in \{1, \cdots, K \} $, we have
\begin{equation}
\begin{split}
C_{k ,\ell}(\mathbf{B}, \bm{\pi}) & =\sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k - \bm{\nu}_\ell)^\top \bm{\Sigma}_{k\ell}^{-1}(t) (\bm{\nu}_k - \bm{\nu}_\ell) \right], \\
& > \sup_{t \in (0, 1)} \left[ t (1-t) (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime)^\top \left[\bm{\Sigma}_{k\ell}^{\prime}(t) \right]^{-1} (\bm{\nu}_k^\prime - \bm{\nu}_\ell^\prime) \right] \\
& = C_{k ,\ell}(\mathbf{B}^\prime, \bm{\pi}).
\end{split}
\end{equation}
Let $ \rho_B $ and $ \rho_{B^\prime} $ denote the Chernoff information obtained as in Eq.~\eqref{eq:rhoapprox} corresponding to $ \mathbf{B} $ and $ \mathbf{B}^\prime $ respectively (with the same block assignment probability vector $ \bm{\pi} $). Then we have
\begin{equation}
\rho_{B} \approx \min_{k \neq l} C_{k ,\ell}(\mathbf{B}, \bm{\pi}) > \min_{k \neq l} C_{k ,\ell}(\mathbf{B}^\prime, \bm{\pi}) \approx \rho_{B^\prime}.
\end{equation}
Thus we have $ \mathbf{B} \succ \mathbf{B}^\prime = p \mathbf{B} $ for $ p \in (0, 1) $.
\end{proof}
\begin{proof}[Proof of Corollary~\ref{cor:Chernoff-Superiority}.]
By Eq.~\eqref{eq:B0} and Eq.~\eqref{eq:B1}, we have
\begin{equation}
\begin{split}
\mathbf{B}_0 & = \frac{p_0}{p_0+p_1} \mathbf{B}_1, \\
\mathbf{B}_1 & = (p_0 + p_1) \mathbf{B}.
\end{split}
\end{equation}
Recall that $ p_0 \in (0, 1) $ and $ p_1 \in (0, 1-p_0) $. Then by Theorem~\ref{thm:Chernoff-Superiority}, we have $ \mathbf{B} \succ \mathbf{B}_1 \succ \mathbf{B}_0 $.
\end{proof}
\begin{backmatter}
\section*{Funding
Cong Mu's work is partially supported by the Johns Hopkins Mathematical Institute for Data Science (MINDS) Data Science Fellowship.
\section*{Abbreviations
\textbf{SBM}: Stochastic Blockmodel \\
\textbf{GRDPG}: Generalized Random Dot Product Graph \\
\textbf{ASE}: Adjacency Spectral Embedding \\
\textbf{LSE}: Laplacian Spectral Embedding \\
\textbf{GMM}: Gaussian Mixture Modeling \\
\textbf{BIC}: Bayesian Information Criterion \\
\textbf{ARI}: Adjusted Rand Index \\
\textbf{stderr}: Standard Error \\
\textbf{NDMG}: NeuroData's Magnetic Resonance Imaging to Graphs
\section*{Availability of data and materials
Social network datasets are available at \href{http://snap.stanford.edu/data/}{http://snap.stanford.edu/data/}.
\section*{Authors' information
\bibliographystyle{bmc-mathphys}
|
\section{Introduction}
\label{sec:intro}
Suppose the normal linear regression model is used to relate $y$ to the
potential predictors $x_1, \dots, x_p$,
\begin{equation} \label{full-model}
\bm{y} \sim N_n(\alpha\bm{1}_n+\bm{X}_F\bm{\beta}_F,\sigma^2 \bm{I}_n)
\end{equation}
where $\alpha$ is an unknown intercept parameter,
$\bm{1}_n$ is an $n \times 1$ vector each component of which is one,
$\bm{X}_F=(\bm{x}_1,\dots, \bm{x}_p)$ is an $n \times p$ design matrix,
$\bm{\beta}_F$ is a $p \times 1$ vector of unknown
regression coefficients, $\bm{I}_n$ is an $n\times n$ identity matrix
and $\sigma^2$ is an unknown positive scalar.
(The subscript $F$ denotes the full model).
We assume that the columns of $\bm{X}_F$ have been standardized so that for
$1 \leq i \leq p$, $\bm{x}'_i\bm{1}_n = 0$ and
$\bm{x}'_i\bm{x}_i/n =1$.
We shall be particularly interested in the variable selection
problem where we would like to select an unknown
subset of the important predictors.
It will be convenient throughout to index each of these $2^p$ possible
subset choices by the vector
\begin{equation*}
\bm{\gamma} =(\gamma_1, \dots,\gamma_p)'
\end{equation*}
where $\gamma_i=0$ or $1$.
We use $q_\gamma=\bm{\gamma}'\bm{1}_p$ to denote the size of
the $\bm{\gamma}$th subset.
The problem then becomes that of selecting a submodel
of \eqref{full-model} which has a density of the form
\begin{equation} \label{submodel-gamma}
p(\bm{y}|\alpha, \bm{\beta}_\gamma, \sigma^2,\bm{\gamma})
=\phi_n(\bm{y}; \alpha \bm{1}_n+ \bm{X}_\gamma\bm{\beta}_\gamma,\sigma^2 \bm{I}_n)
\end{equation}
where $\phi_n(\bm{y}; \bm{\mu}, \bm{\Sigma})$
denotes the $n$-variate normal density with mean vector $\bm{\mu}$ and
covariance matrix $\bm{\Sigma}$.
In \eqref{submodel-gamma}, $\bm{X}_\gamma$ is the $n \times q_\gamma$
matrix whose columns correspond to the $\bm{\gamma}$th subset
of $x_1,\dots,x_p$, $\bm{\beta}_\gamma$
is a $q_\gamma \times 1$ vector of unknown regression coefficients.
We assume throughout that $\bm{X}_\gamma$ is of full rank denoted
\begin{equation*}
r_\gamma = \min\{q_\gamma, n-1 \}.
\end{equation*}
Lastly, let $\mathcal{M}_\gamma$ denote the submodel given by \eqref{submodel-gamma}.
A Bayesian approach to this problem entails the specification of
prior distributions on the models $\pi_\gamma =
\mbox{Pr}(\mathcal{M}_\gamma)$, and on the parameters
$p(\alpha,\bm{\beta}_\gamma, \sigma^2)$ of each model.
For each such specification,
of key interest is the posterior probability of $\mathcal{M}_\gamma$
given $\bm{y}$
\begin{equation} \label{posterior-2}
\mbox{Pr}(\mathcal{M}_\gamma|\bm{y})
=\frac{\pi_\gamma m_\gamma(\bm{y})}{\sum_\gamma \pi_\gamma m_\gamma(\bm{y})}
=\frac{\pi_\gamma \mbox{BF}[\mathcal{M}_\gamma; \mathcal{M}_N]}
{\sum_\gamma \pi_\gamma \mbox{BF}[\mathcal{M}_\gamma; \mathcal{M}_N]},
\end{equation}
where $m_\gamma(\bm{y})$ is the marginal density of $\bm{y}$ under $\mathcal{M}_\gamma$.
In \eqref{posterior-2}, $ \mbox{BF}[\mathcal{M}_\gamma; \mathcal{M}_N]$
is so called ``null-based Bayes factor'' for comparing
each of $ \mathcal{M}_\gamma$ to the null model $\mathcal{M}_N$
which is defined as
\begin{equation*}
\mbox{BF}[\mathcal{M}_\gamma; \mathcal{M}_N]= \frac{m_\gamma(\bm{y})}{m_N(\bm{y})},
\end{equation*}
where the null model $\mathcal{M}_N$ is given by
$ \bm{y} \sim N_n(\alpha\bm{1}_n,\sigma^2 \bm{I}_n)$ and $ m_N(\bm{y})$ is the marginal density of $\bm{y}$ under the null model.
For model selection, a popular strategy is to select the model for which
$\mbox{Pr}(\mathcal{M}_\gamma|\bm{y})$ or
$ \pi_\gamma\mbox{BF}[\mathcal{M}_\gamma; \mathcal{M}_N] $ is largest.
Our main focus in this paper is to propose and study specifications for the parameter prior
for each submodel $\mathcal{M}_\gamma$, which we will consider to be of the form
\begin{equation}\label{priorform}
p(\alpha,\bm{\beta}_\gamma, \sigma^2)= p(\alpha)p(\sigma^2)
p(\bm{\beta}_\gamma|\sigma^2)=
p(\alpha)p(\sigma^2)\int p(\bm{\beta}_\gamma|\sigma^2,g)p(g)dg,
\end{equation}
where $g$ is a hyperparameter.
In Section \ref{sec:priors},
we explicitly describe our choices of prior forms for \eqref{priorform}.
Our key innovation there will be to use a generalization of
\begin{equation}\label{g-prior}
p(\bm{\beta}_\gamma|\sigma^2,g)
= \phi_{q_\gamma}(\bm{\beta};\bm{0}, g \sigma^2(\bm{X}'_\gamma \bm{X}_\gamma)^{-1})),
\end{equation}
\citeapos{Zellner-1986} $g$-prior,
a normal conjugate form which leads to tractable marginalization, for example see
\cite{George-Foster-2000,Fernandez-Ley-Steel-2001, Liang-etal-2008}.
Under \eqref{g-prior} and a flat prior on $\alpha$,
the marginal density of $\bm{y}$ given $g$ and $\sigma^2$ under
$\mathcal{M}_\gamma$ is
given by
\begin{equation} \label{marginal-known-1}
m_\gamma(\bm{y}|g,\sigma^2)
\propto \exp\left( \frac{g}{g+1}
\left\{\max_{\alpha,\bm{\beta}_\gamma} \log p(\bm{y}| \alpha, \bm{\beta}_\gamma,\sigma^2)
-q_\gamma H(g) \right\}\right)
\end{equation}
where $H(g)= (2g)^{-1}(g+1)\log(g+1)$,
a special case of the key relation in \cite{George-Foster-2000}.
As they point out, for particular values of $g$, when $\sigma^2$ is known,
the Bayesian strategy of choosing $\mathcal{M}_\gamma$
to maximize \eqref{marginal-known-1} corresponds to common fixed penalty selection criteria.
For example, setting $H(g)=2$, $\log n$ or $2\log p$ (independently of $\bm{y}$)
would correspond to AIC \citep{Akaike-1974}, BIC \citep{Schwarz-1978},
or RIC \citep{Foster-George-1994}, respectively.
For a discussion of recommendations in the literature
for choosing a fixed $g$ depending on $p$ and/or $n$,
see Section 2.4 of \cite{Liang-etal-2008}.
Although the correspondences to fixed penalty criteria are interesting,
as a practical matter,
it is necessary to deal with the uncertainty about $g$ and $\sigma^2$
to obtain useful criteria.
For this purpose, \cite{George-Foster-2000} proposed selecting the model
maximizing $m_\gamma(\bm{y}|g,\sigma^2)$ based on an empirical Bayes estimate
of $g$ and the standard unbiased estimate of $\sigma^2$.
More recently, \cite{Cui-George-2008} proposed margining out $g$
with respect to a prior, and \cite{Liang-etal-2008} proposed margining out $g$
and $\sigma^2$ with respect to priors.
All of these strategies lead to criteria that can be seen
as adapting to the fixed penalty criterion which would be most suitable
for the data at hand.
In this paper, we shall similarly follow a fully Bayes approach,
but with a generalization of the $g$-prior \eqref{g-prior}
and an extension of the considered class of priors on $g$.
After describing our prior forms in Section \ref{sec:priors}
and then calculating the marginals and Bayes factors in Section \ref{sec:marginal+BF},
we ultimately obtain our proposed $g$-prior Bayes Factor ($g$BF),
which is of the form (omitting the $\gamma$ subscripts for clarity)
\begin{equation}\label{gBF}
g\mbox{BF}[\mathcal{M}_\gamma] =
\begin{cases}
\displaystyle C_{n,q}\,\left\{\frac{\bar{d}}{d_q}\right\}^{-q}
\frac{\left\{1- R^2+ d_q^2\,\|\hat{\bm{\beta}}_{LS}\|^2\right\}
^{-\frac{1}{4}-\frac{q}{2}}}
{\left(1- R^2 \right)^{(n-q)/2-3/4}} & \mbox{if }q < n-1,\\
\left\{\bar{d} \times \| \hat{\bm{\beta}}^{MP}_{LS}\|\right\}^{-n+1}
& \mbox{ if }q \geq n-1,
\end{cases}
\end{equation}
where $C_{n,q} \equiv \frac{B(q/2+1/4,(n-q)/2-3/4)}{B(1/4,(n-q)/2-3/4)}$
using the Beta function $B(\cdot,\cdot)$,
$ R^2$ is the familiar $R$-squared statistic under $\mathcal{M}_\gamma$,
$\bar{d}$ and $d_r$ are respectively the geometric mean and minimum
of the singular values of $\bm{X}_\gamma$, $ \| \cdot \|$ is the $L_2$ norm,
and finally, for the standardized response
$(\bm{y}-\bar{y}\bm{1}_n)/\| \bm{y}-\bar{y}\bm{1}_n \|$, $\hat{\bm{\beta}}_{LS}$
is the usual least squares estimator, and $ \hat{\bm{\beta}}^{MP}_{LS}$
is the least squares estimator using the Moore-Penrose inverse matrix.
Two immediately apparent features of \eqref{gBF} should be noted.
First, in contrast to other fully Bayes factors for our selection problem,
$g$BF is a closed form expression which allows for interpretation
and straightforward calculation under any model.
As will be seen in later sections, this transparency reveals that $g$BF
not only rewards explained variation overall,
but also rewards variation explained by the larger principal components
of the design matrix.
Second, $g$BF can be applied to all models
even when the number of predictors $p$ exceeds the number of observations $n$.
This includes $p > n$ which is of increasing interest.
This is not the case for \eqref{g-prior} which requires $p \le n-1$
so that $\bm{X}'_\gamma \bm{X}_\gamma$ will be invertible for all $q_\gamma$,
(recall that $\bm{X}_\gamma$ has dimension at most $n-1$
because its columns have been centered).
Note also that when $p > n-1$, penalized sum-of-squares criteria
such as AIC, BIC and RIC will be unavailable for all submodels.
The organization of this paper is as follows.
In Section \ref{sec:priors},
we will give priors including a special variant of $g$-prior.
In Section \ref{sec:marginal+BF}, we derive the Bayes factor above.
In Section \ref{sec:hyper}, we discuss the choice of hyper-parameters
which appears in the variant of $g$-priors.
In Section \ref{sec:after}, the estimation after selection is discussed.
In Section \ref{sec:consistency}, we show that $g\mbox{BF}$
has consistency for model selection as $n \to \infty$.
In Section \ref{sec:sim}, we give some numerical results.
\section{A Fully Bayes Prior Formulation}
\label{sec:priors}
We now proceed to describe the prior components that form
$p(\alpha,\bm{\beta}_\gamma, \sigma^2)$ in \eqref{priorform}.
Throughout this section and the next,
we will omit the subscript $\gamma$ for notational simplicity when there is no ambiguity.
\subsection{A generalized $g$-prior for $\beta$}
To motivate our proposed generalization of Zellner's $g$-prior,
we begin with a reconsideration the original $g$-prior \eqref{g-prior}
for the case $p \leq n-1$.
The covariance matrix of the $g$-prior, $g\sigma^2(\bm{X}'\bm{X})^{-1}$,
is proportional to the covariance matrix of the least squares estimator
$\hat{\bm{\beta}}_{LS}$.
As a consequence of this choice, the marginal density with respect to the $g$-prior
appealingly becomes a function only of the residual sum-of-squares, RSS.
However, from the ``conditioning'' viewpoint of \cite{Casella-1980, Casella-1985}
which advocates more shrinkage on higher variance estimates,
the original $g$-prior may not be reasonable.
To see why, let us rotate the problem by the $q \times q$
orthogonal matrix
$\bm{W}=(\bm{w}_1,\dots,\bm{w}_q)$ which diagonalizes $\bm{X}' \bm{X}$ as
\begin{equation} \label{evd-f}
\bm{W}'(\bm{X}'\bm{X})\bm{W}=\bm{D}^2
\end{equation}
where $\bm{D}=\mbox{diag}(d_1, \dots, d_q)$
with
\begin{equation}
d_1 \geq \dots \geq d_q >0.
\end{equation}
Thus
\begin{equation*}
\bm{W}'\hat{\bm{\beta}}_{LS} \sim N_q(\bm{W}'\bm{\beta}, \sigma^2\bm{D}^{-2}).
\end{equation*}
For this rotation, we consider priors on $\bm{\beta}$ for which
\begin{equation*}
\bm{W}'\bm{\beta} \sim N_q(\bm{0},\sigma^2 \bm{\Psi}_q)
\end{equation*}
where $\bm{\Psi}_q=\mbox{diag}(\psi_1,\dots,\psi_q)$.
From a Bayesian perspective, it is more sensible to put stronger prior information
on the components $\bm{w}_i'\bm{\beta}$ of $\bm{W}'\bm{\beta}$
which are estimated poorly, (that is, the components with larger sample variance).
Hence, we would like to consider $\bm{\Psi}_q$ for which
\begin{equation} \label{des-order}
\psi_1 \geq \dots \geq \psi_q>0
\end{equation}
are in descending order. In fact, a slightly weaker ordering of the form
\begin{equation} \label{weak-des-order}
d_1^2\psi_1 \geq \dots \geq d_q^2\psi_q >0
\end{equation}
would still be reasonable because
the resulting Bayes estimator of $\bm{w}'_i\bm{\beta}$ would be of the form
\begin{equation*}
(1+\{d_i^2\psi_i\}^{-1})^{-1}\bm{w}'_i\hat{\bm{\beta}}_{LS},
\end{equation*}
so that under \eqref{weak-des-order},
the components of $\bm{W}'\hat{\bm{\beta}}_{LS}$ with larger variance would be shrunk more.
We note that the original $g$-prior \eqref{g-prior},
for which $\psi_i =g d_i^{-2}$, satisfies only the extreme boundary of
\eqref{weak-des-order}, namely
\begin{equation*}
d_1^2\psi_1 = \dots = d_q^2\psi_q=g.
\end{equation*}
This violates \eqref{des-order} whenever $d_i > d_{i+1}$ in which case
$\psi_i< \psi_{i+1}$.
An appealing general form for $\bm{\Psi}_q$ is
$ \bm{\Psi}_q(g,\bm{\nu})=\mbox{diag}(\psi_1(g,\bm{\nu}), \dots , \psi_q(g,\bm{\nu}))$ where
\begin{equation} \label{general-psi}
\psi_i(g,\bm{\nu})=(1/d_i^2)\left\{ \nu_i(1+g)-1 \right\},
\end{equation}
$ \bm{\nu}=(\nu_1,\dots,\nu_q)'$ and $\nu_i \geq 1$ for any $i$, guaranteeing
$\psi_i(g,\bm{\nu}) >0$.
Note that $ \bm{\Psi}_q(g,\bm{\nu})$, like the original $g$-prior,
is controlled by a single hyperparameter $g>0$.
When $\nu_1 = \dots =\nu_q=1$, $ \sigma^2\bm{\Psi}_q(g,\bm{\nu}) $ becomes
$g\sigma^2\bm{D}^{-2}$, yielding
the covariance structure of the original $g$-prior.
Although \eqref{weak-des-order} will be satisfied whenever
$ \nu_1 \geq \dots \geq \nu_q \geq 1$,
in subsequent sections we shall ultimately be interested
in the particular design dependent choice
\begin{equation} \label{our-nu}
\nu_1 =d_1^2/d_q^2, \ \nu_2 =d_2^2/d_q^2, \dots ,\nu_q=1
\end{equation}
which satisfies \eqref{des-order} as well as \eqref{weak-des-order}.
In summary, when $q \leq n-1$, we propose a generalized $g$-prior for $\bm{\beta}$ of the form
\begin{equation}\label{gen-g-psmall}
p(\bm{\beta} \,|\, \sigma^2,g)
=\phi_q(\bm{W}'\bm{\beta};\bm{0},\sigma^2 \bm{\Psi}_q(g,\bm{\nu}))
\end{equation}
where $ \nu_1 \geq \dots \geq \nu_q \geq 1$.
When $ q > n-1$ and the rank of $\bm{X}$ is $n-1$,
there exists a $q \times (n-1)$ matrix
$\bm{W}=(\bm{w}_1,\dots,\bm{w}_{n-1})$ which diagonalizes $\bm{X}'\bm{X}$ as
\begin{equation} \label{evd-m}
\bm{W}'(\bm{X}'\bm{X})\bm{W}=\bm{D}^2
\end{equation}
where $\bm{W}'\bm{W}=\bm{I}_{n-1}$ and $\bm{D}=\mbox{diag}(d_1, d_2, \dots, d_{n-1})$
with $ d_1 \geq d_2 \geq \dots \geq d_{n-1} >0$.
For this case, we propose a generalized $g$-prior of the form
\begin{equation}
p(\bm{\beta}|\sigma^2,g)
=\phi_{n-1}(\bm{W}'\bm{\beta};\bm{0},\sigma^2 \bm{\Psi}_{n-1}(g,\bm{\nu}))\;
p_\#(\bm{W}'_\#\bm{\beta})
\end{equation}
where
$\bm{\Psi}_{n-1}(g,\bm{\nu})=\mbox{diag}(\psi_1,\dots,\psi_{n-1})$
is again given by \eqref{general-psi}
and $\nu_1 \geq \dots \geq \nu_{n-1}\geq 1$. Here, $\bm{W}_\# $ is an arbitrary
matrix which makes the $q\times q$ matrix
$(\bm{W}, \bm{W}_\#)$ orthogonal, and $p_\#(\cdot)$ is an arbitrary probability
density on $\bm{W}'_\#\bm{\beta}$, respectively.
As will be seen, the choices of $\bm{W}_\#$ and $p_\#$
have no effect on the selection criteria we obtain, thus we leave them as arbitrary.
As in \eqref{our-nu},
we shall be ultimately interested in the particular design dependent choice
\begin{equation} \label{our-nu-1}
\nu_1 =d_1^2/d_{n-1}^2, \ \nu_2 =d_2^2/d_{n-1}^2, \dots ,
\nu_{n-1}=1.
\end{equation}
Combining the above two cases by letting
\begin{equation}
r=\min\{q, n-1\},
\end{equation}
our suggested generalized $g$-prior is of the form
\begin{equation} \label{prior-beta-1}
\begin{split}
&p(\bm{\beta}|g,\sigma^2) \\
&=\phi_r(\bm{W}'\bm{\beta};\bm{0},\sigma^2 \bm{\Psi}_r(g,\bm{\nu})) \times
\begin{cases}
1 & \mbox{ if }q \leq n-1 \\
p_\#(\bm{W}'_\# \bm{\beta}), & \mbox{ if }q > n-1,
\end{cases}
\end{split}
\end{equation}
where the $q\times r$ matrix $\bm{W}$
satisfies both
$ \bm{W}'\bm{X}'\bm{X}\bm{W} =\mbox{diag}(d_1^2,\dots,d_r^2)$
and $\bm{W}'\bm{W}=\bm{I}_{r}$,
and $ \bm{\Psi}_r(g,\bm{\nu})=\mbox{diag}(\psi_1(g,\bm{\nu}),\dots,\psi_r(g,\bm{\nu}))$
with \eqref{general-psi}.
\medskip
\begin{remark}
In \eqref{evd-f} and \eqref{evd-m}, let
\begin
|
{equation} \label{u}
\bm{U}=(\bm{u}_1,\dots,\bm{u}_r)=(\bm{Xw}_1/d_1,\dots,\bm{Xw}_r/d_r)=\bm{XWD}^{-1}.
\end{equation}
Then $\bm{U}'\bm{U}=\bm{I}_r$ and
\begin{equation} \label{svd-1}
\bm{X}=\bm{UDW}'=\sum_{i=1}^r d_i \bm{u}_i \bm{w}'_i.
\end{equation}
This is the non-null part of the well-known singular value decomposition (SVD).
The diagonal elements of $\bm{D}$ = $\mbox{diag}(d_1,\dots,d_r)$
are the singular values of $\bm{X}$, and the columns of
$\bm{U}=(\bm{u}_1,\dots,\bm{u}_r)$ are the normalized principal components
of the column space of $\bm{X}$.
Note that the components of the rotated vector $\bm{W}'\bm{\beta}$
are the coefficients for the principal component regression of $\bm{y}$ on $\bm{UD}$.
From the definition of $\bm{W}$ and $\bm{U}$ by \eqref{evd-f}, \eqref{evd-m} and
\eqref{u},
the signs of $\bm{u}_i\bm{w}'_i$ are determinate
although the signs of $\bm{w}_i$ and $\bm{u}_i$
for $1 \leq i \leq r$ are indeterminate.
These indeterminacies can safely be ignored in our development.
\end{remark}
\subsection{A prior for $g$}
Turning to the prior for the hyperparameter $g$, we propose
\begin{equation}
p(g) =
\frac{g^b(1+g)^{-a-b-2}}{B(a+1,b+1)}\,
I_{(0, \infty)}(g)
\label{prior-g}
\end{equation}
with $a> -1$, $ b>-1$, a Pearson Type VI or {\itshape
beta-prime} distribution under which $1/(1+g)$ has a Beta
distribution $Be(a+1,b+1)$.
Choices for the hyperparameters $a$ and $b$ are discussed later.
Although \cite{Zellner-Siow-1980} did not explicitly use a $g$-prior formulation with a
prior on $g$,
their recommendation of a multivariate Cauchy form for $ p(\bm{\beta}|\sigma^2)$
implicitly corresponds to using a $g$-prior with an inverse Gamma prior
\begin{equation*}
(n/2)^{1/2}\{\Gamma(1/2)\}^{-1} g^{-3/2} e^{-n/(2g)}
\end{equation*}
on $g$. Both \cite{Cui-George-2008} and \cite{Liang-etal-2008} proposed
using $g$-priors with priors of the form
\begin{equation} \label{prior-liang}
p(g)=(a+1)^{-1}(1+g)^{-a-2},
\end{equation}
the subclass of \eqref{prior-g} with $b=0$.
Cases for which $b= O(n)$ will be of interest to us in what follows.
\subsection{Priors for $\alpha$ and $\sigma^2$}
For the parameter $\alpha$, we ultimately use the location invariant flat prior
\begin{equation}
p(\alpha)=I_{(-\infty,\infty)}(\alpha) \label{prior-alpha},
\end{equation}
but do so indirectly by first applying the proper uniform prior
\begin{equation}
p(\alpha;h_\alpha)=\frac{1}{2h_\alpha} I_{(-h_\alpha,h_\alpha)}(\alpha)
\label{proper-prior-alpha}
\end{equation}
and then taking the limit as $h_\alpha \to \infty$.
Similarly for $\sigma^2$, we ultimately use the scale invariant prior
\begin{equation}
p(\sigma^2)=(\sigma^2)^{-1}I_{(0,\infty)}(\sigma^2) \label{prior-sigma},
\end{equation}
but do so by first applying the proper truncated version
\begin{equation}
p(\sigma^2;h_\sigma)
=\frac{(\sigma^2)^{-1}}{\int_{h^{-1}_\sigma}^{h_\sigma}
(\sigma^2)^{-1} d \sigma^2}I_{(h^{-1}_\sigma, \ h_\sigma)}(\sigma^2)
=\frac{(\sigma^2)^{-1}}{2\log h_\sigma}
I_{(h^{-1}_\sigma, \ h_\sigma)}(\sigma^2)
\label{proper-prior-sigma^2}
\end{equation}
and then taking the limit as $h_\sigma \to \infty$.
Because $\alpha$ and $\sigma^2$ appear in every model, this approach avoids the arbitrary norming constant
difficulties associated with the use of improper priors in Bayesian model selection.
We note in passing that for the estimation of a
multivariate normal mean, priors equivalent to
\eqref{gen-g-psmall}, \eqref{prior-g}, \eqref{prior-alpha} and \eqref{prior-sigma}
have been considered by
\cite{Strawderman-1971} and extended by
\cite{Maru-Straw-2005}.
\section{Marginal Densities and Bayes Factors}
\label{sec:marginal+BF}
For the model $\mathcal{M}_\gamma $, the marginal density of $\bm{y}$
under the proper priors $p(\bm{\beta}|\sigma^2,g)$,
$p(g)$, $ p(\alpha;h_\alpha)$ and $p(\sigma^2;h_\sigma)$ given by
\eqref{prior-beta-1}, \eqref{prior-g}, \eqref{proper-prior-alpha} and \eqref{proper-prior-sigma^2}, is
obtained as
\begin{equation}\label{full-marginal}
\begin{split}
m_\gamma(\bm{y})= &
\int_{-h_\alpha}^{h_\alpha} \int_{R^{q}}
\int_{h_\sigma^{-1}}^{h_\sigma}
\int_0^{\infty}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)
p(\alpha;h_\alpha)p(\bm{\beta}|\sigma^2,g) \\
& \quad \qquad \times p(\sigma^2;h_\sigma)p(g)
d \alpha \, d \bm{\beta} \, d(\sigma^2) \,dg.
\end{split}
\end{equation}
Going further, it will be useful to consider the limit of a renormalized version
of $m_\gamma(\bm{y})$,
\begin{equation}\label{full-marginal-M}
\begin{split}
& M_\gamma(\bm{y}) \\
&= \lim_{h_\alpha \to \infty}\lim_{h_{\sigma} \to \infty}
\left\{4 h_\alpha \log h_{\sigma} \right\}m_\gamma(\bm{y}) \\
&=\int_{-\infty}^{\infty} \int_{R^{q}}
\int_{0}^{\infty}\int_0^{\infty}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)
p(\bm{\beta}|g, \sigma^2)\frac{1}{\sigma^2}p(g)
d \alpha \, d \bm{\beta} \, d(\sigma^2) \,dg,
\end{split}
\end{equation}
which is the marginal density of $\bm{y}$ with respect to the improper priors
\eqref{prior-alpha} on $\alpha$ and \eqref{prior-sigma} on $\sigma^2$.
Note that the second equality in \eqref{full-marginal-M}
follows from the monotone convergence theorem.
Thus $m_\gamma(\bm{y})/m_N(\bm{y})$, the Bayes factor
for $\mathcal{M}_\gamma $ with respect to the null model $\mathcal{M}_N $, approaches
$M_\gamma(\bm{y})/M_N(\bm{y})$ as $h_\alpha \to \infty$ and
$h_\sigma \to\infty$.
Hence the use of improper priors $p(\alpha)=1$ and
$p(\sigma^2)=1/\sigma^2$ is formally justified
because $M_\gamma(\bm{y})/M_N(\bm{y})$ is well-defined.
Defining $\bm{v}=\bm{y}-\bar{y}\bm{1}_n$, where $\bar{y}$ is the mean of $\bm{y}$,
so that
\begin{equation*}
\| \bm{y} -\alpha \bm{1}_n- \bm{X\beta} \|^2
= n(-\alpha+\bar{y})^2
+ \|\bm{v}-\bm{X} \bm{\beta} \|^2,
\end{equation*}
we obtain
\begin{equation}\label{marginal-alpha}
\int_{-\infty}^{\infty}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)
d \alpha = \frac{n^{1/2}}{(2\pi\sigma^2)^{(n-1)/2}}
\exp\left(
-\frac{\| \bm{v}-\bm{X}\bm{\beta}\|^2}{2\sigma^2}\right) .
\end{equation}
We make the following orthogonal
transformation when integration with respect to $\bm{\beta}$ is considered:
\begin{equation}\label{btrans}
\bm{\beta} \to
\begin{cases}
\bm{W}'\bm{\beta} \equiv \bm{\beta}_* & \mbox{ if }q \leq n-1 \\
\begin{pmatrix}
\bm{W}' \bm{\beta} \\
\bm{W}'_\# \bm{\beta}
\end{pmatrix}
\equiv
\begin{pmatrix}
\bm{\beta}_* \\
\bm{\beta}_\#
\end{pmatrix}
& \mbox{if } q > n-1,
\end{cases}
\end{equation}
so that
\begin{equation*}
\begin{split}
& \int_{-\infty}^{\infty} \int_{R^q}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)p(\bm{\beta}|\sigma^2,g)
d \alpha \, d\bm{\beta} \\
&=
\frac{n^{1/2}}{(2\pi\sigma^2)^{(n-1)/2}}
\frac{|\bm{\Psi}|^{-1/2}}
{(2\pi\sigma^2 )^{r/2}}
\int_{R^{r}}
\exp\left(-\frac{\| \bm{v}-\bm{UD\beta}_* \|^2}{2\sigma^2}
-\frac{\bm{\beta}'_*\bm{\Psi}^{-1}\bm{\beta}_*}{2\sigma^2}\right) d\bm{\beta}_* \\
& \qquad \times \begin{cases}
1 & \mbox{if }q \leq n-1 \\
\int_{R^{q-n+1}}p_\#(\bm{\beta}_\#)d \bm{\beta}_\# \ (=1) & \mbox{if }q > n-1.
\end{cases}
\end{split}
\end{equation*}
Completing the square
$ \|\bm{v} - \bm{UD\beta}_*\|^2+\bm{\beta}'_* \bm{\Psi}^{-1}\bm{\beta}_*$
with respect to $\bm{\beta}_*$, we have
\begin{equation}\label{complete-square}
\begin{split}
& \|\bm{v} - \bm{UD\beta}_*\|^2+\bm{\beta}'_* \bm{\Psi}^{-1} \bm{\beta}_*
\\
& =\{\bm{\beta}_*-(\bm{D}^2+\bm{\Psi}^{-1})^{-1}\bm{D}'\bm{U}'\bm{v}\}'(\bm{D}^2+\bm{\Psi}^{-1})
\{\bm{\beta}_*-(\bm{D}^2+\bm{\Psi}^{-1}
)^{-1}\bm{D}'\bm{U}'\bm{v}\} \\
& \qquad -\bm{v}'\bm{UD}(\bm{D}^2+\bm{\Psi}^{-1})^{-1}\bm{D}'\bm{U}'\bm{v}+\bm{v}'\bm{v},
\end{split}
\end{equation}
where the residual term is rewritten as
\begin{equation*}
\begin{split}
& -\bm{v}'\bm{UD}(\bm{D}^2+\bm{\Psi}^{-1})^{-1}\bm{D}'\bm{U}'\bm{v}+\bm{v}'\bm{v} \\
&\quad =-\bm{v}'\left(\sum_{i=1}^r \bm{u}_i \bm{u}'_i \frac{d_i^2}{d_i^2+\psi_i^{-1}}\right)\bm{v}
+\bm{v}'\bm{v} \\
& \quad =\frac{g}{g+1}\left\{ \bm{v}'\bm{v}- \sum_{i=1}^r (\bm{u}'_i\bm{v})^2\right\}
+\frac{1}{1+g} \left\{ \bm{v}'\bm{v}- \sum_{i=1}^r \left(1-\frac{1}{\nu_i}\right)
(\bm{u}'_i \bm{v})^2\right\} .
\end{split}
\end{equation*}
Hence by
\begin{equation*}
|\bm{\Psi}|=\prod_{i=1}^r \frac{\nu_i+\nu_i g-1}{d_i^2}, \;\;\;
|\bm{D}^2+\bm{\Psi}^{-1}|=\prod_{i=1}^r
\frac{d_i^2\nu_i(1+g)}{\nu_i+\nu_i g-1},
\end{equation*}
we have
\begin{equation}\label{marginal-alpha-beta}
\begin{split}
&\int_{-\infty}^{\infty} \int_{R^q}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)p(\bm{\beta}|g, \sigma^2)
d \alpha \, d\bm{\beta} \notag \\
&=
\frac{n^{1/2}}{(2\pi\sigma^2)^{(n-1)/2}}
\frac{(1+g)^{-r/2}}{\prod_{i=1}^r \nu_i^{1/2}}
\exp\left(
-\frac{\|\bm{v}\|^2\{g(1-R^2)+1-Q^2\}}{2\sigma^2(g+1)} \right)
\end{split}
\end{equation}
where
\begin{equation}\label{R2Q2}
\begin{split}
R^2 &= \sum_{i=1}^r \frac{(\bm{u}'_i\bm{v})^2}{\bm{v}'\bm{v}}
= \sum_{i=1}^r \left\{\mathrm{cor}(\bm{u}_i,\bm{y})\right\}^2, \\
Q^2 &= \sum_{i=1}^r \left(1-\nu_i^{-1}\right)
\frac{(\bm{u}'_i \bm{v})^2}{\bm{v}'\bm{v}}
= \sum_{i=1}^r \left(1-\nu_i^{-1}\right)
\left\{\mathrm{cor}(\bm{u}_i,\bm{y})\right\}^2.
\end{split}
\end{equation}
Note that $R^2$ and $Q^2$ are the usual and a modified version
of the $R$-squared statistics and $\mathrm{cor}(\bm{u}_i,\bm{y})$
is the correlation of the response $\bm{y}$ and the $i$-th
principal component of $\bm{X}$.
Next we consider the integration with respect to $\sigma^2$.
By \eqref{marginal-alpha-beta}, we have
\begin{equation}\label{marginal-alpha-beta-sigma}
\begin{split}
&\int_{-\infty}^{\infty} \int_{R^q} \int_{0}^{\infty}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2)
p(\bm{\beta}|g, \sigma^2)\frac{1}{\sigma^2}
d \alpha \, d\bm{\beta} \, d\sigma^2 \\
& \ = \int_{0}^{\infty}\frac{n^{1/2}}{(2\pi\sigma^2)^{(n-1)/2}}
\frac{(1+g)^{-r/2}}{\prod_{i=1}^r \nu_i^{1/2}}
\exp\left(-\frac{\|\bm{v}\|^2\{g(1-R^2)+1-Q^2\}}{2\sigma^2(g+1)} \right)
\frac{1}{\sigma^2}d\sigma^2
\\
& \ = \frac{K(n,\bm{y})}{\prod_{i=1}^r \nu_i^{1/2}}
(1+g)^{-r/2+(n-1)/2}\left\{ g(1-R^2)+1-Q^2 \right\}^{-(n-1)/2}
\end{split}
\end{equation}
where
\begin{equation*}
K(n,\bm{y})=\frac{n^{1/2}\Gamma(\{n-1\}/2)}
{\pi^{(n-1)/2}\|\bm{y}-\bar{y}\bm{1}_n\|^{n-1}}.
\end{equation*}
When $ q \geq n-1$, $R^2=1$ and $r=n-1$ so that
\begin{equation}\label{marginal-many}
\begin{split}
& \int_{-\infty}^{\infty} \int_{R^q} \int_{0}^{\infty}
p(\bm{y}|\alpha,\bm{\beta}, \sigma^2) p(\bm{\beta}|g, \sigma^2)\frac{1}{\sigma^2}
d \alpha \, d\bm{\beta} \, d\sigma^2 \\
& \quad = \frac{K(n,\bm{y})}{\prod_{i=1}^{n-1} \nu_i^{1/2}}
\left\{ 1-Q^2 \right\}^{-(n-1)/2}
\end{split}
\end{equation}
which does not depend on $g$.
Hence, in this case, $M_\gamma(\bm{y})$ does not depend on the prior density of $g$.
Notice also that if $\nu_1=\dots=\nu_{n-1}=1$ when $q \geq n-1$,
$Q^2=0$ so that $M_\gamma(\bm{y})=K(n,\bm{y})$
will not distinguish between models.
Clearly, the choice of $\bm{\nu}$ matters.
When $q < n-1$, we consider the prior \eqref{prior-g} of $g$
with $-1<a<-1/2$ and $ b =(n-5)/2-q/2-a $,
where $b$ is guaranteed to be strictly greater
than $-1$ for $q < n-1$.
Then we have
\begin{align}\label{marginal-few}
& M_\gamma(\bm{y}) \notag \\
& = \frac{K(n,\bm{y})}{\prod_{i=1}^q \nu_i^{1/2}B(a+1,b+1)} \notag \\
& \qquad \times
\int_0^{\infty} \frac{g^b}{(1+g)^{a+b+2}}
\frac{\left\{g(1-R^2)+1-Q^2 \right\}^{-(n-1)/2}}{(1+g)^{q/2-(n-1)/2}}
\, dg \notag \\
& =
\frac{K(n,\bm{y})(1-Q^2)^{-(n-1)/2}}
{\prod_{i=1}^q \nu_i^{1/2}B(a+1,b+1)}
\int_0^{\infty} g^{b}
\left(\frac{1-R^2}{1-Q^2}g+1 \right)^{-(n-1)/2} dg \\
&=\frac{K(n,\bm{y})(1-Q^2)^{-(n-1)/2+b+1}}
{\prod_{i=1}^q \nu_i^{1/2} \{1-R^2\}^{b+1}}
\frac{B(q/2+a+1,b+1)}{B(a+1,b+1)} \notag \\
&=\frac{K(n,\bm{y})(1-Q^2)^{-q/2-a-1}}
{\prod_{i=1}^q \nu_i^{1/2} \{1-R^2\}^{(n-q-3)/2-a}}
\frac{B(q/2+a+1,(n-q-3)/2-a)}{B(a+1,(n-q-3)/2-a)}. \notag
\end{align}
In the same way, $M_N(\bm{y})$ for the null model,
is obtained as
\begin{equation} \label{marginal-null}
M_{N}(\bm{y}) =K(n,\bm{y}).
\end{equation}
Using \eqref{marginal-many}, \eqref{marginal-few} and
\eqref{marginal-null}, we have a following theorem about the Bayes factor
ratio of the marginal densities under
each of $\mathcal{M}_\gamma$ and $\mathcal{M}_N$.
\begin{thm}\label{main-thm-1}
Under the proper prior distributions
of $\bm{\beta}|(\sigma^2, \, g)$, $\alpha$ and $\sigma^2$
given by \eqref{prior-beta-1}, \eqref{proper-prior-alpha} and
\eqref{proper-prior-sigma^2}, and under the proper prior distribution of $g$ given by
\eqref{prior-g} with $-1<a<-1/2$ and $b=(n-5)/2-q/2-a$ when $q < n-1$, (when $q \geq n-1$, the prior distribution of $g$ is arbitrary), the limit of the Bayes factor for comparing
each of $ \mathcal{M}_\gamma$ to the null model is
\begin{equation*}
\lim_{h_\alpha \to \infty} \lim_{h_\sigma \to \infty}
\frac{m_\gamma(\bm{y})}{m_N(\bm{y})}=
\mathrm{BF}[\mathcal{M}_\gamma; \mathcal{M}_N|a,\bm{\nu}]
\end{equation*}
where
\begin{equation}\label{BC-2}
\begin{split}
& \mathrm{BF}[\mathcal{M}_\gamma; \math
|
U(0)= E(0)$ and $W(0)= I(0)$. It follows that if both the eigenvalues of the Floquet's matrix $F$ associated to Eqs.~\ref{UtraSEIRperiodic}-\ref{WtraSEIRperiodic} are interior to the unit circle in the complex plane, then
\begin{equation}
(U(t),W(t)) \rightarrow (0,0) \Rightarrow (E(t),I(t)) \rightarrow (0,0)
\end{equation}
in turn implying that $ S(t) \rightarrow 1$. As regards the instability of DFE, by linearizing the SEIR model around the DFE one gets a linear time-varying system equivalent to Eqs.~\ref{UtraSEIRperiodic}-\ref{WtraSEIRperiodic}. Thus if the eigenvalues of the matrix $F$ are outside the unit circle of the complex plane, then the DFE is unstable.
It is of interest to note that although $\mathcal{R}_0^{SEIR}(t) \approx \mathcal{R}_0^{SIR}(t)$, for the SEIR endemic model the stability of the DFE is deeply different from the SIR case. Indeed, in the SIR model the behavior of the DFE is uniquely determined by the average of the transmission rate $\beta(t)$, while in the SEIR case the stability of the DFE depends on the "whole shape" of $\beta(t)$.
The computation of Floquet's matrices is usually nontrivial. However, matrix $F$ can be easily computed in the noteworthy case where $\beta(t)$ is switching between two constant values, which mimicks well the realistic school-holiday scenario where students have a large number of contacts during the school period, which remarkably reduces during holidays \cite{Earn_chaos_2000,keeling}. Setting
\begin{equation}
\beta(t) =
\begin{cases}
\beta_1, & \quad \text{if } t \in [0,T],\\
\beta_2, & \quad \text{if } t \in (T,1],
\end{cases}
\end{equation}
the linear time-varying system of period one year is equivalent to the following two systems
\begin{equation}
X_1^{\prime}=M_1 X_1 ; 0\le t \le T,
\end{equation}
\begin{equation}
X_2^{\prime}=M_2 X_2 ; T < t \le 1,
\end{equation}
where for $i=1,2$
\begin{equation}
M_{i} = \begin{bmatrix}
-\alpha & \beta_i \\
\alpha & -\gamma
\end{bmatrix}.
\end{equation}
As a consequence $F$ is the product of two matrix exponentials:
\begin{equation}
F= Exp(M_2(1-T))Exp(M_1 T).
\end{equation}
The generalization to more complex patterns of alternance between school and holidays can also easily handled.
In the general case, if one eliminates the first state variable in the above equivalent linear systems, one obtains the following \textit{family} of Newton's equation\cite{donofrioMBS02}
\begin{equation}\label{Newton}
w^{\prime\prime} + (\gamma+\alpha)w^{\prime} + (\gamma -\alpha \beta(t))w=0.
\end{equation}
Further, by setting $ y(t) = w(t) Exp((\gamma+\alpha)t/2) $ \cite{minorsky}, one yields the Hill family of equations
\begin{equation}\label{Newton2}
y^{\prime\prime} + \left(-\frac{(\gamma+\alpha)^2}{4}+\gamma -\alpha \beta(t)\right)y=0.
\end{equation}
The families of Eqs.~\ref{Newton}-\ref{Newton2} are ubiquitous and widely studied in various fields of mathematical physics \cite{Arscott64,mcl,minorsky,farkas,cesari,arnold,landau}. For example, in mechanics they model the periodic perturbations in the elastic force of damped or undamped oscillators respectively, which are at the base of several theories, such as that of parametric resonance \cite{cesari,arnold,landau}; while in applied electromagnetism Eq.~\ref{Newton2} models the frequency modulation\cite{mcl}. Analytic solutions for $t \in [0,1]$ are known for a number of particular forms of $\beta(t)$. For example, if $\beta(t)$ is linearly increasing, the solutions of Eq.~\ref{Newton2} in $ t \in [n T, n T+1)$ are Airy's functions, whereas if the contact rate is sinusoidally varying around an average value, the solutions of Eq.~\ref{Newton2} are Mathieu's functions.
\subsubsection{Nonlinear contact rates as a phenomenological route to behavior change}
A limitation of mass-action-based SIR and SEIR models with and without vital dynamics, and of other epidemic models, is that they assume that the behavioral component is either constant (resulting in a constant $\beta$) or influenced by exogenous phenomena such as the school/work calendar (resulting in a periodic time-varying $\beta(t)$ with one-year period). Capasso and Serio were first in pointing out this shortcoming \cite{Capasso_Serio}, and in stressing that agents might respond to the threat posed by the outbreak by changing their social behavior (i.e. reducing their average number of contacts $C$), and/or by adopting protections against transmission. They modelled these effects by a \textit{prevalence-dependent} transmission rate of the form $\beta(I)$, $\beta'(I)<0$, showing that behavior change may deeply modify the quantitative dynamics of an outbreak \cite{Capasso_Serio}. In a more general study on the SIR endemic model \cite{noibeta}, it was stressed that prevalence-dependent behavior is influenced by a range of information variables, some of which can be made available to agents with time-lags. The inclusion of delayed prevalence-dependent behavior in the SIR endemic model can destabilize the endemic equilibrium by triggering sustained oscillations, and, in turn, these endogenous oscillations may interact with the one-year periodicity of the contact rate, resulting in chaotic fluctuations \cite{noibeta}.
Finally, another seminal paper is Hadeler and Chavez \cite{hadeler1995core}, which models the spread of a vaccine/education preventable sexually transmitted disease in a population split in two groups: an inactive group with $\beta =0$ and a \textit{core} active group where $\beta >0$. The model included a unidirectional prevalence-depending flux of subjects from the inactive to the \textit{core} group, with tranfer rate $r(I)$ decreasing with $I$.
\subsection{Mass vaccination and herd immunity for vaccine preventable infections: the SIR model with vaccination at birth}
\label{SIR_vaccination}
In modern communities new vaccines, targeting different infections and different populations, appear at a continuously increasing rate, making the mosaic of characteristics of the different available vaccines quite articulated. Nonetheless, much of the main dynamic effects of mass vaccination in controlling communicable diseases can be understood by the following simple variation of the endemic SIR model of Eqs.~\ref{SIR_vitaldyn_eqS}-\ref{SIR_vitaldyn_eqI}
\begin{align}
S^{\prime}&= \mu( 1- x ) - \mu S -\beta S I, \label{SIR_vitaldyn_vacc_eqS}\\
I^{\prime}&= \beta I S -(\gamma +\mu)I, \label{SIR_vitaldyn_vacc_eqI}\\
V^{\prime}&= \mu x - \mu V. \label{SIR_vitaldyn_vacc_eqV}
\end{align}
\noindent
Model of Eqs.~\ref{SIR_vitaldyn_vacc_eqS}-\ref{SIR_vitaldyn_vacc_eqI}-\ref{SIR_vitaldyn_vacc_eqV} (of course $R= 1- S-I-V$) considers vaccination of a time-invariant, randomly chosen, proportion $x$ of newborn susceptible individuals per unit of time by a \textit{perfect vaccine}, i.e. one with $100\%$ take and lifelong duration (see section \ref{BRN}). Under a perfect vaccine the fraction successfully vaccinated coincides with the \textit{vaccine coverage}, i.e. the fraction vaccinated. Vaccinated individuals are fully protected against infection (and ensuing disease) and therefore behave, in practice, as removed individuals. This does not need to be the case if, for example, the vaccine has waning immunity, or leaves some residual susceptibility - as is the case e.g. for the varicella vaccine - in which case further compartments are required.
After discarding the $V$ equation, the analysis of Eqs.~\ref{SIR_vitaldyn_vacc_eqS}-\ref{SIR_vitaldyn_vacc_eqI} goes along exactly the same lines as Eqs.~\ref{SIR_vitaldyn_eqS}-\ref{SIR_vitaldyn_eqI}. Since we are interested in infections that are endemic prior to vaccination, we assume $ \mathcal{R}_0 >1 $, where $ \mathcal{R}_0$ still represents the BRN. The key threshold quantity is now represented by the VRN
\begin{equation}\label{VRN}
\mathcal{R}_V = \frac{\beta}{\gamma +\mu} (1-x).
\end{equation}
It can then be easily shown that Eqs.~\ref{SIR_vitaldyn_vacc_eqS}-\ref{SIR_vitaldyn_vacc_eqI} have the $ DFE = (1-x,0) $, which is globally attractive if $\mathcal{R}_V \le 1 $, and unstable for $ \mathcal{R}_V>1 $. For $\mathcal{R}_V>1$ the model also has the globally attractive (by the same Poincar{\'e}-Bendixson argument) endemic state
\begin{equation}\label{EE_SIR_vacc}
EE_V= (S_e, I_e) = \left( \frac{1}{\mathcal{R}_0}, \frac{\mu}{\mu+\gamma}\left(1-x-\frac{1}{\mathcal{R}_0}\right) \right).
\end{equation}
This endemic state is also characterised by damped oscillations, with pseudo-period
\begin{equation}
T_V = \frac{2 \pi}{\sqrt{\mu(\mu+\gamma)(\mathcal{R}_0 (1-x) -1)}} . \end{equation}\label{T_SIR_vacc}
Though the mathematical analysis is the same as for the basic endemic SIR model, there is a number of substantive implications of vaccination. Indeed, the threshold for the VRN implies the already met coverage threshold: $ x_{c}= 1-1/\mathcal{R}_0$ (i.e. formula \ref{thepc}), so that if $x\ge x_{c}$ then the infection is eliminated (technically, the introduction of mass vaccination above threshold causes the EE to disappear, leaving only the DFE, which becomes GAS).
In the opposite scenario, i.e. if the coverage is insufficient to achieve elimination ($px\le x_{c}$), the infection will continue to persist endemically, in the new long-term regime described by Eqs.~\ref{EE_SIR_vacc}. This \textit{post-vaccination} regime is characterized by the same susceptible proportion $(1/ \mathcal{R}_0)$ as in the pre-vaccination history, but by a different distribution of immune individuals between natural and vaccine-induced, and especially by a smaller prevalence and an increased inter-epidemic period with respect to the case where no vaccination campaign is enacted. The increase in $T$ straightforwardly follows from the reduced replenishment of the susceptible class due to vaccination. For example, considering again a measles-type parametrization ($\mathcal{R}_0 =15$, $1/\gamma =1 $ week, $\mu=(1/75)/year$), yielding a pre-vaccination inter-epidemic period of about $2$ years, a mass vaccination program with $x=0.9$ would increase the inter-epidemic period to a post-vaccination figure $T_V$ in excess of $10$ years.
The dynamic implications of the introduction of a mass vaccination for a measles-like infection are illustrated in Fig. \ref{SIR_VD_pre_postvacc_dynamics} for different vaccination programs (respective coverages: $x=0.80$, $x=0.90$ and $x=0.95$) initiated at time $t=20 years$ under the hypothesis that infection spread for $t<20$ was evolving in its pre-vaccination regime, characterized by an almost biennial inter-epidemic period. As expected for $x=0.95$ the infection is eliminated, since $x>x_{c}$. In the two remaining cases the infection remains endemic with a much lower prevalence but with an amazing increase in the inter-epidemic period. For example, for $x=0.90$ the inter-epidemic period initially increases to a level in excess of $20 years$ before gradually contracting and converging in the long-term to the above predicted figure of $T_V \approx 10 years$. The inflation in the inter-epidemic period has far reaching consequences since it implies that unvaccinated individuals who did not acquire infection early in life might do so at somewhat later ages, where they might be exposed at a much larger risk of serious disease from the infection \cite{AMay_1991,Hethcote_2000}. The increase in the average age at infection in the post-vaccination regime is the consequence in the decline in the individual risk of infection, as clear from the decline in the endemic FOI $\beta I_e$, which shifts from its pre-vaccination level of $\mu (\mathcal{R}_0 - 1)$ to $\mu (\mathcal{R}_0 (1-x) - 1)$. These phenomena are of importance, besides measles, for essentially all vaccine preventable infection, particularly for rubella \cite{AMay_1991,Hethcote_2000}.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{fig7.pdf}
\end{center}
\caption{The SIR model with vital dynamics: pre- and post- vaccination dynamics of the effective reproduction number $\mathcal{R}_E(t)$ (left panels) and the infective prevalence $I(t)$ (righ panels) under different programs of vaccination at birth initiated at time $t=20 years$ with coverage of $x=0.80$ (upper panels), $x=0.90$ (central panels) and $x=0.95$ (right panels) respectively. The demographic and epidemiological parameters are: $\mu = (1/75)year^{-1}$, $\gamma = 52 year^{-1}$, $\mathcal{R}_0 =15$.}%
\label{SIR_VD_pre_postvacc_dynamics}%
\end{figure}
\subsection{Return to susceptibility: SIS and SIRS models}
\label{SIS_SIRS_models}
If infection-acquired immunity is not lifelong, recovered subjects will eventually become again susceptible. A first possible scenario is the one where infective subjects directly return to susceptibility, yielding the so-called SIS model (Fig. \ref{Fig_SEIRS_compartments})
\begin{align}
S^{\prime} &= \mu (1-S) -\beta(t) I S + \gamma I, \label{SIS-S}\\
I^{\prime} &= \beta(t) I S - \mu I- \gamma I. \label{SIS-I}
\end{align}
Since $S+I=1$, one gets the scalar model
\begin{equation}\label{SIS}
I^{\prime} = (\mu+\gamma) I ( \mathcal{R}_0(t) (1- I) - 1),
\end{equation}
where
\begin{equation}
\mathcal{R}_0(t) = \frac{\beta(t)}{\mu+\gamma}
\end{equation}
and we still take $\beta(t)$ as a 1-year periodic function. Unsurprisingly, if $ \langle \mathcal{R}_0(t) \rangle \le 1 $ then the DFE $I = 0$ is globally attractive, whereas if $ \langle \mathcal{R}_0(t) \rangle > 1 $ the DFE is unstable, and $I(t)$ tends to a one-year periodic solution, i.e. the model predicts that the disease remains endemic via a series of annual epidemic outbreaks. For a constant transmission rate $\beta$ Eq.~\ref{SIS} is logistic, therefore implying a constant, globally attractive, endemic state:
\begin{equation}
I_{EE} = 1 - \frac{1}{\mathcal{R}_0},
\end{equation}
\noindent
where the susceptible fraction is still given by the inverse of $\mathcal{R}_0 $. Note that the establishment of a globally attractive EE also holds in absence of vital dynamics (i.e. for $\mu =0$, suggesting that the loss of immunity \textit{per se} is a further source of endemicity through rebuilding of the susceptible pool, as first noted by \cite{KMK}.
A second more general scenario occurs when return to susceptibility is buffered by a long sojourn time in the recovered class, yielding the SIRS endemic model (Fig. \ref{Fig_SEIRS_compartments})
\begin{align}
S^{\prime}&= \mu( 1- S) -\beta S I +\delta R, \label{eqSsirs}\\
I^{\prime}&= \beta I S -\gamma I - \mu I, \label{eqIsirs}\\
R^{\prime}&= \gamma I - \mu R -\delta R, \label{eqRsirs}
\end{align}
where $1/\delta$ is the average length of the immunity conferred by the disease. From $S=1-I-R$ one gets
\begin{align}
I^{\prime}&= (\mu+\gamma) I ( \mathcal{R}_0 (t)(1-I-R) -1), \label{eqIsirs1}\\
R^{\prime}&= \gamma I - (\mu + \delta) R, \label{eqRsirs1}
\end{align}
where
\begin{equation}\label{tvBRN} \mathcal{R}_0(t) = \frac{\beta(t)}{\mu+\gamma}. \end{equation}
Again, the infection ultimate fate, namely extinction vs endemicity, depends on whether
$ \langle \mathcal{R}_0(t)\rangle$ is smaller or bigger than one. Under constant $\beta$ , the threshold condition $ \mathcal{R}_0 >1 $ allows the existence of an endemic state:
\begin{equation}\label{valSee} S_{EE} = \frac{1}{\mathcal{R}_0}, I_{EE} = \left(1- \frac{1}{\mathcal{R}_0}\right)\frac{\mu+\delta}{\mu+\delta+\gamma},\end{equation}
which can be shown to be globally attractive by applying the same Dulac-Bendixon test used for the SIR endemic model.
Mass vaccination by a perfect vaccine can be introduced in SIS and SIRS models along the same route followed for the SIR model, with analogous results.
However, for infections with only temporary, or without, immunity, the hypothesis of a perfect vaccine is mostly unattainable. If the vaccine is \textit{imperfect}, several scenarios are possible i.e. vaccinated individuals might either (i) not produce any immune response, therefore remaining fully susceptible, or (ii) produce a weaker immune response. This weaker immune response might occur in the form of a temporary immunity which however typically wanes faster than natural one, eventually returning the individual to susceptibility, or in the form of reduced susceptibility, putting the individual at risk of \textit{breakthrough} infection, at some reduced force of infection $\lambda = \sigma \beta I$ with $\sigma<1$. For example, all these possibilities are known to occur for the current varicella vaccine \cite{Varicella_vaccine_imperfect}.
This opens a more complicate scenario which makes infection elimination more difficult. Indeed, as it can be demonstrated even for simple SIS models with vaccination at birth, there is a range $(x_{c}^{0},x_{c}^{*})$ of the vaccinated proportion $p$ where a bistability phenomenon arises, i.e. where the disease-free equilibrium is locally stable but it coexists with two endemic equilibria: one unstable and one locally stable. Therefore, infection elimination requires that $x>x_{c}^{*}$, whereas if $x<x_{c}^{0}$ then there is only one endemic steady state, coexisting with the DFE which is unstable. This whole scenario is termed as \text{backward bifurcation} \cite{Zaleta_Velasco_Hernandez}.
\subsection{Remarks on the wise use of the mean-field principle in theoretical epidemiology}
\label{stochastics_meanfield}
Thanks to the use of a few important deterministic models, theoretical epidemiology has become a key supporting tool to public health decision makers. These models, historically derived from classical physical chemistry and other deterministic applications of statistical mechanics, are, however, only approximations of the underlying real-world stochastic processes, obtained from the mean-field limits of stochastic birth and death processes analogous to those adopted in chemical kinetics \cite{gardiner}.
In this subsection we therefore concisely remind some well-known tools of nonequilibrium statistical physics \cite{gardiner, vank} with the aim to recall how deterministic epidemic models arise as limits on the underlying stochastic processes on the one hand, and their shortcomings, on the other hand.
For example, let us consider an epidemic (i.e. no vital dynamics) SIR infection process, with state variables $(X(t),Y(t),Z(t) = N -X(t)-Y(t))$. In this process there are two types of events: the \textit{infection}, responsible for the $S \rightarrow I$ transition with infinitesimal probability
\begin{equation}\label{ev1}
Prob\left( (X(t+dt),Y(t+dt)) = (X(t)-1,Y(t)+1 )\right) = \beta \frac{Y(t)}{N}X(t)dt,
\end{equation}
and \textit{recovery}, responsible for the $I \rightarrow R$ transition with infinitesimal probability
\begin{equation}\label{ev2}
Prob\left( (X(t+dt),Y(t+dt)) = (X(t),Y(t)-1 )\right) = \gamma Y(t)dt.
\end{equation}
For the SIS model (still without vital dynamics), since $X(t) = N -Y(t)$, the $S \rightarrow I$ transition has probability $\beta (Y/N)(N-Y)$.
The behavior of the realizations of these epidemic stochastic processes can be simulated exactly by means of the Gillespie algorithm \cite{gill}, whereas the probability density function of its state variables is described by a multivariate Master equation (ME) \cite{gardiner,vank}, which is bivariate for the epidemic SIR process, and univariate for the SIS process. For example, for the epidemic SIS model the ME reads
\begin{equation}\label{cmeSIS}
\frac{d}{dt}P(t,Y) = \beta \frac{Y-1}{N}\left(N-(Y-1)\right)P(t,Y-1)-\beta \frac{Y}{N}\left(N-Y\right)P (t,Y)+\gamma(Y+1)P(t,Y+1)-\gamma Y P(t,Y).
\end{equation}
Taking the average $\langle Y(t) \rangle = \sum Y P(t,Y)$ from Eq.~\ref{cmeSIS}, we get
\begin{equation}\label{meanY}
\frac{d}{dt}\langle Y(t) \rangle = (\beta -\gamma)\langle Y(t) \rangle - \beta \frac{1}{N}\langle Y^2(t) \rangle.
\end{equation}
Essentially the mean field hypothesis applied in theoretical population models stands, for large populations, in neglecting the correlations between the state variables and assuming that the average of the product of two state variables is approximately equal to the product of the averages \cite{mckanenewt}, for example in Eq.~\ref{meanY} above $ \langle Y^2(t) \rangle \approx (\langle Y(t) \rangle)^2 $.
However, due to the approximate nature of deterministic epidemic model and their impact of public health sciences, it is important to go beyond the mere application of the mean-field limit, and to understand under which conditions deterministic differential equations (whose state variables are continuous function of time) can safely be used in place of the originary "exact" stochastic formulation of the epidemic process (whose state variables are time-continuous stochastic processes taking integer non-negative values). Moreover, it is also important to have a feeling of the behavior of the system at least in the regime of small stochastic deviations from the deterministic behavior.
If the total population size is small, birth-death stochastic epidemic models must be used. On the other hand, when the size of the state variables becomes large (which is the case intuitively leading to the deterministic approximation), it is hard to simulate the process due to the crowding of events. In such a case, however, one can use a system-size approximations~\cite{gardiner,vank}. This methodology allows approximating, respectively, the solution of a multivariate ME with a multivariate Fokker-Plank equation (FPE) \cite{gardiner} (this approximate equation is sometime called Chemical FPE~\cite{gardiner}). In turn, from the obtained FPE one can derive a corresponding system of Stochastic Differential equations (SDEs) that approximate the dynamics of the state variables vectors. In physical chemistry it has been proposed a reverse equivalent approach, the Chemical Langevin equation (CLE)\cite{gill2000}, where first the approximating SDE system is obtained from the definition of the model, and only then the FPE is written. Namely, a spatially homogeneous system of particles/subjects with state variable $u$ characterized by $M$ "events" having probability $a_j(u)dt$ $j=1,\dots,M$ can be approximated by the following system of SDEs \cite{gill2000}
\begin{equation}\label{cle}
u_i^{\prime} = \sum_j \nu_{i,j}(a_j(u) + \sqrt{a_j(u)} \xi_j(t) ),
\end{equation}
where the $\xi_j(t)$ are $M$ uncorrelated white noises, and $\nu$ is the \textit{stechiometric} matrix where $\nu_{i,j} \in \{0,-1,+1\}$ depending on the changes induced by the $j$-th event into the $i$-th state variable. Note that here the original state vector, whose components were integer non-negative numbers, is approximated by a real vector.
For example, in the case of the SIR epidemic model, the CLE reads
\begin{align}
X^{\prime} &= -\beta \frac{Y}{N}X - \xi_1(t) \sqrt{\frac{Y}{N}X}, \label{CLE-X} \\
Y^{\prime} &= \beta \frac{Y}{N}X -\gamma Y + \xi_1(t) \sqrt{\frac{Y}{N}X}-\xi_2(t)\sqrt{\gamma Y}.\label{CLE-Y}
\end{align}
Since $N$ is constant, we can pass to the fractions $(S,I)$
\begin{align}
S^{\prime} &= -\beta SI - \xi_1(t) \frac{1}{\sqrt{N}}\sqrt{\beta SI}, \label{CLE-S} \\
I^{\prime} &= \beta SI -\gamma I + \xi_1(t) \frac{1}{\sqrt{N}}\sqrt{\beta SI}-\xi_2(t) \frac{1}{\sqrt{N}}\sqrt{\gamma I}.\label{CLE-I}
\end{align}
As a consequence, in the limit $N\rightarrow +\infty$ we obtain the deterministic model.
It is interesting to note that the use of the force of infection FOI=$\beta_1 Y$ leads, after passing to fractions, to a stochastic term $\xi_1(t) \sqrt{\beta_1 SI}$ which is not scaling as $1/\sqrt{N}$, and thus it does not disappear in the limit $N\rightarrow +\infty$. Moreover, note that an hypotetical model where the contact rate is decreasing to $0$ with the population size $N$, would yield the following trivial uncoupled model of the type $S^{\prime} = 0$, $I^{\prime}=-\gamma I$.
As stressed by a few authors among which Gillespie \cite{gill2000}, the accuracy of the approximation given by Eq.~\ref{cle} or by those derived by the system-size expansion \cite{gardiner,vank}, depends on the fulfillment of specific conditions that are \textit{time-varying} since they are state-variable dependent. We can roughly summarize these conditions as follows: for all times
$ a_j(u) dt $ must be sufficiently larger than $\sqrt{a_j(u) dt } $. This in turn implies that all state variables must be sufficiently large. Thus the above mentioned approximations require caution, since \textit{a rigori} they are invalid even when only one of the state variables becomes small.
For the specific case of the SIR model, note that in the initial epidemic phase if one sets $X(t)\approx X(0) \approx N$ the ensuing stochastic process for $Y(t)$ is linear and its mean is given by $E[Y] = Y(0) Exp((\beta S(0)-\gamma)t) $ as in the approximation of the deterministic SIR epidemic model. The corresponding variance is $Var[Y] = Y(0) (e^{(\beta S(0) - \gamma)t} -1 )/(\beta S(0) - \gamma)$. Of course, this approximation is valid for a relatively short time-interval. However, as Bartlett noted in his pioneering work on measles \cite{Bartlett_CCS}, the most critical difference between deterministic and stochastic models arises in the intervals between epidemics (for example during the undamped oscillations caused by periodic changes in the contact rate) when the prevalence can become so small that extinction of the infective population almost certainly occurs in the stochastic formulation \textit{also in presence of a large population} while deterministic models (e.g. the periodically forced SIR and SEIR models with vital dynamics) simply predict that system trajectories will be positive for all times. This is of relevance in presence of vaccination because stochasticity may cooperate with the further reduction in prevalence caused by vaccination in inducing infection elimination \cite{pej}.
The study of stochasticity-induced infection extinction led Bartlett to introduce \cite{Bartlett_CCS} the key concept of \textit{critical community size} (CCS). This is the (heuristically defined) threshold over which, in absence of vaccination, the infection is likely to persist, and below which extinction is very likely. Over the CCS it is therefore safe to "read" the output of a deterministic model as an average behavior. For example pre-vaccination measles data indicate a CCS in the region of $250 000$- $400 000$ \cite{KG,Bartlett_CCS}. The CCS is only weakly associated to the BRN, while it is strongly dependent on the duration of the infectious and latent periods. As for the latter, it was noted that simple stochastic models based on exponentially distributed durations, tend to severely overestimate the CCS \cite{KG}, while instead adequate predictions can be obtained by resorting to more realistic, e.g. bell-shaped, distributions of the latent and infectious periods \cite{KG}. On the other hand, the effect of space on the CCS is extremely complex \cite{pej}.
Stochastic epidemic models are characterized by a range of interesting phenomena not shown in their deterministic counterparts. The simplest is obviously stochastic extinction after importation, i.e. the fact that an infectious individual introduced in a fully susceptible population has a probability of recovering before being able to infect anyone (given by $1/\mathcal{R}_0$ ($1/\mathcal{R}_0^k$) \cite{pej}), in which case no epidemics will occur, unlike the deterministic model. Other stochastic effects are more subtle. For example, in the SIR model with vital dynamics, the stochastic fluctuations of the total population $N(t)$ around its deterministic equilibrium (which can be approximated by the CLE: $ N^{\prime} = \xi_b(t) \sqrt{\mu N} - \xi_d(t) \sqrt{\mu N}$), may transform the damped oscillations around the endemic equilibrium of the corresponding deterministic model in sustained oscillations\cite{Kuske}. This phenomenon, termed \textit{stochastic amplification}, is an instance of the more general phenomenon of \textit{coherence resonance} \cite{Kuske}. The theoretical possibility of strochastically inducing the disease eradication with a limited amount of vaccines via periodic delivery of vaccinations so that to induce resonances, and the related role of seasonal fluctuations of the contact rates, are invstigated in \cite{Meerson} by means of methodologies of the theory of fluctuation-induced population extinction. Further details on the stochastic theory of epidemics can be found in the complementary books \cite{tb,lja}.
\subsection{Space and beyond: mean-field metapopulations}
\label{metapop}
Everytime a new epidemic focus appears, the first concern of public health is to assess whether this isolated focus will remain localized or will spatially spread. This leads to a key missing dimension in the previous simple models, namely \textit{space}. The most straightforward way of modelling the geographical spread of
infections is to add spatial diffusion to the basic SIR and SEIR model \cite{Capasso,murray2,bailey}, leading to equations that are analogous to the reaction-diffusion equations of theoretical biochemistry \cite{murray1,murray2}. This approach, which is proved useful to understand some basic aspects of spatial epidemic behavior by using tools of reaction-diffusion theory, has obvious shortcomings. The first one is that human beings have patterns of movement dramatically different from that of particles moving under Fickian diffusion \cite{Gonzalez_Barabasi,Balcan_Vespignani}. This is a major complication leading to complex integro-differential equations where the diffusion part interacts with the reaction part, i.e. transmission \cite{mendez}. Moreover, human movement often includes rapid commuting between areas with highly clustered population, such as cities \cite{pej,pej2002}.
To cope with the above problems, population ecology and mathematical epidemiology have increasingly used the \textit{metapopulation}, or \textit{multi-patch} approach, where a large populations is split in a finite number of \textit{patches}, for example cities, where: i) each city obeys its own deterministic \textquotedblleft local \textquotedblright epidemic model, e.g. SIR-type; ii) local dynamics are linked through population fluxes, e.g. due to commuting for work. Commuting plays a fundamental role in the geograpical spread of infectious diseases in contemporary societies \cite{pej2002,pej}, by favouring the geographical spread and persistence of infections through the interlink between patches. The multi-patch approach was used in the pioneering work by Baroyan and Ravchev, who linked a large number of SIR epidemic models to simulate the spread of flu in URSS cities \cite{bailey}. Multi-patch epidemiological models have since then developed fast both on the theoretical and the applied side, e.g. for predicting the world-wide spread of pandemic influenza \cite{Vespignani_flu_2011,wang2012estimating,nah2016predicting,nah2016estimating}.
As for the mathematical viewpoint, multipatch epidemic models show the same spatial phenomena we may observe in traditional reaction-diffusion models, namely: i) emergent phenomena such as the onset of spatial heterogeneities by means of spatial instability of spatially homogeneous equilibrium points (Turing-like instabilites) or even of limit cycles \cite{lloydjmb,lloydmbs}; ii) the onset of traveling waves and solitons, representing the spatial displacement of epidemic outbreaks \cite{pej}; iii) the emergence of various spatio-temporal asynchronous/synchronous complex patterns \cite{pej,lloydeco}. To these classes of phenomena, one has to add those induced by parametric heterogeneities between patches \cite{pej}, which might mirror heterogeneities in the epidemiology (e.g. in social contacts), public health interventions (e.g. in vaccine uptake), demographics (e.g. population density and age-structure), as well as environmental (e.g. climate, temperature, etc).
As regards the onset of Turing-like patterns is concerned, let us consider the following system \cite{lloydjmb} for an infection with $m$ epidemiological states ($m=4$ for a SEIR system) spreading among $n$ interacting patches. If patches are isolated, the local spread is modeled by $n$ $m$-dimensional ODE systems of the form $u_j^{\prime} = f(u_j) $. In presence of linear population flows between patches one has the following interlinked dynamical systems
\begin{equation}\label{interpop}
u_j^{\prime} = f(u_j) + \sum_{i=1}^{n} c_{i,j} M u_i,
\end{equation}
where $u_i \in R^m$ are the state variables associated to patch $i$, $ U = (u_1,\dots,u_n) $ is the $m*n$ matrix of all state variables, $M$ is a $m*m$ diagonal matrix reporting the different mobility coefficients of individuals in the different epidemiological compartments (e.g. the mobility of infectious subjects might be reduced with respect to that of susceptible and removed \cite{bailey}), $\mathcal{C}$ is the $n*n$ \textit{connection} matrix whose elements are the transmission rates between individuals of different patches.
A \textit{spatially homogeneous} solution of Eq.~\ref{interpop} is one such that in all the patches the state vector is the same $ u_j = s(t) $, with $s(t) \in R^m$. Linearizing around the spatially homogeneous solution and applying some similarity transformation one gets the following $n$ independent systems in $C^m$ \cite{lloydjmb}
\begin{equation}\label{indep}
\psi^{\prime} = \left( Df(s(t)) + \lambda_h M \right)\psi, h =1,\dots,n,
\end{equation}
where $\lambda_h$ are the eigenvalues of the connection matrix $C$. Eq.~\ref{indep} is the discrete equivalent of the linearization equation arising in the study of the Turing bifurcation \cite{murray2}. Notably, this methodology can also be applied in the case of nonlinear population fluxes \cite{lloydmbs}.
Among multi-patch models, a fairly well-studied class is represented by the following \textit{multi-group, cross-coupled} SEIR model (\cite{lloydmbs} and refs therein) in a constant population
\begin{align}
S_i^{\prime} &= \mu(1-S_i) - S_i \sum_{j=1}^{n} \beta_{ij}I_j, \label{Scrosspop}\\
E_i^{\prime} &= S_i \sum_{j=1}^{n} \beta_{ij}I_j - (\mu + \alpha) E_i, \label{Ecrosspop}\\
I_i^{\prime} &= \alpha I_i - (\mu + \gamma)I_i, \label{Icrosspop}\\
R_i&= 1- S_i - E_i -I_i. \label{Rcrosspop}
\end{align}
\noindent Unlike Eq.~\ref{interpop} where all contacts are local, through inter-patch movements, in Eq.~\ref{Scrosspop} fluxes between patches are disregarded on the rationale that movements from own patch to other patches are short-lasting. The core of the description stands in the \textit{spatial transmission matrix} $\beta_{ij}$ assigning the per-capita rate at which a susceptible from patch $i$ is infected by an infective individual belonging to patch $j$ regardless of the patch where the at-risk contact occurred \cite{lloyd_may_1996,lloydmbs,pej}. For Eq.~\ref{Scrosspop} fairly general results are available, incuding a decoupling equivalent to those of Eq.~\ref{indep} \cite{lloydmbs}. In particular, Eq.~\ref{Scrosspop} was used to study the conditios under which the local (i.e., in each patch) oscillations of SIR and SEIR models become synchronised~\cite{lloyd_may_1996,lloydmbs}, as observed for measles in UK cities in the pre-vaccination era \cite{Earn_chaos_2000}. Notably, synchronization of unforced SIR and SEIR models arises at very low levels of patches connection, while seasonal forcing, besides inducing complicate behavior, allows maintenance of some phase differences in local dynamics~\cite{lloyd_may_1996}, and yields more plausible prevalence levels at the inter-epidemic intervals compared to the simple SIR and SEIR models.
Introduction of vaccination can have a complex impact on the spatio-temporal dynamics of multi-patch systems \cite{pej}. For example metapopulation models can explain the observed de-correlation observed in the time-series of UK major cities after the introduction of measles vaccination \cite{Bolker_Grenfell_pnas96,Earn_chaos_2000}.
\subsubsection{Large scale meta-population networks}
In very recent times, the increasing availability of data on both the spread of infectious diseases and human movements is allowing to finely model the dynamics of outbreaks on larger and larger scales, even world-wide \cite{Vespignani_flu_2011}. This has led to a number of investigations on the role of the topology of the networks of inter-patches contacts as resulting from population mobility \cite{vittoriaalex}. These analyses have shown that though the condition $\mathcal{R}_0>1 $ still guarantees that the epidemics is not self-extinguishing and that there are patches here and there in the network where the infection is circulating, in order to have global macroscopic spread a second threshold has to be overcome \cite{vittoriaalex,vittoriaalexjtb}. This second threshold, larger than the first one, depends both on population mobility and on the topology of network. Not suprisingly, if the network is heterogenous, i.e. its degree distribution $P(k)$ (which summarizes the probabiity for a node of the underlying graph of having $k$ links to other nodes of the networks) has a heavy tail, global epidemic spread is easier. However, for very large heterogeneous network structure the second threshold is close to the classical one, quite independent of population mobility. This can have profound implications as it implies that measures aimed at limiting international mobility to reduce infection spread might have little impact.
\subsubsection{Metapopulation models beyond space: social and age-specific heterogeneities}
Multi-group and metapopulation models are one of the most active and effective fields of investigation in the theoretical epidemiology of infectious diseases, as they \textit{latu sensu} represent a convenient framework for modeling a much wider class of epidemiological phenomena than geographical spread only. This has been done by straightforwardly extending the interpretation of patches to different types of internally homogeneous groups, such as social groups, age groups, etc, still relying on appropriate contact or transmission matrice, having elements $\beta_{i,j}$ reflecting heterogeneous contact patterns between social or age groups. Though the first works in the field were following a more mathematical vein, i.e. the need to remove the homogeneous mixing hypothesis, the public health modelling revolution described in section \ref{Mathepi} is much the legacy of this intuition. Indeed, the multigroup SIS model analogous to the SEIR model of Eqs.~\ref{Scrosspop}-\ref{Rcrosspop} \cite{lloydmbs}, first introduced by Lajmanovich and Yorke \cite{laj} to model the spread of an infection among societal groups with different social/sexual behavior, and later extended to model gonorrhea transmission in the US~\cite{Hethcote_1984}, has represented the departure point for the explosion of studies on the modeling of sexually transmitted infections that was subsequently triggered by the onset of the HIV/AIDS epidemics in the early eighties. Noteworthy extensions of Ref.~\cite{laj} to heterogeneous networks of contacts are available in \cite{pastor2001epidemic,olinky,don}.
On the other hand, in the field of common childhood infections, such as measles and pertussis, the large availability of age-specific infection data triggered the development of the \textquotedblleft standard \textquotedblright age-structured model \cite{Dietz_1981,AMay_1991,Hethcote_2000}. The \textquotedblleft standard \textquotedblright model is nowadays extensively used to investigate the nonlinear effects of mass vaccination programs, including the complicate patterns of increase in the average age at infection following the reduction in transmission, and the related perverse increase in the number of cases of serious disease, as it may be the case for rubella, measles and varicella \cite{Dietz_1981,AMay_1991,Hethcote_2000}. The key ingredient of the \textquotedblleft standard \textquotedblright model is represented by the age-specific transmission matrix of elements $\beta_{i,j}$, representing the per-capita transmission rates in contacts between individuals of different age groups. In a first phase these matrices were computed by resorting to simplifying hypotheses about contacts patterns between age groups, aimed to reduce the number of unknowns in the $\beta_{ij}$ matrix from $n^2$ to $n$, thereby making them identifiable from standard age-specific epidemiological data. These approaches include the so-called Who Acquires Infection From Whom (WAIFW) matrices (\cite{AMay_1991} and refs therein), and the \textit{proportionate} and \textit{preferred} mixing hypotheses (\cite{Hethcote_2000} and refs therein). Recently, considerable advances were made thanks to studies aimed to directly observe population contact patterns \cite{Mossong,wallinga,wang2013how,wang2013impact}, or to reconstruct them from official data \cite{Zagheni,Fumanelli}.
Apart the anagraphical age of subjects, two other chrono-biological variables impact on the in-population dynamics of infectious diseases. The first is the so called \textit{age since infection} \cite{thieme2003mathematics}, which is the time elapsed from the infection, a key factor modulating both the contact rate of infected subjects, the disease duration and the disease-related risk of death. Quite remarkably, this variable was first introduced in mathematical epidemiology by Kermack and McKendrik in their 1927 paper \cite{kermack1927contribution}, but later ignored for decades \cite{thieme2003mathematics}. The second is the \textit{age since vaccination} \cite{noi4,iannelli2005strain}, which is the time elapsed since the vaccine inoculation, and which modulates both the vaccine-induced immunity \cite{iannelli2005strain} and the occurrence of vaccine-related side effects \cite{noi4}. Note that if any of these temporal variables is modeled as a continuous variable, then the resulting model, due to the impact of these variables on the contact rate and other parameters, is not differential multi-patch but integro-differential, as in the seminal work \cite{kermack1927contribution}.
Finally, a number of recent re-assessments of the main future challenges for multi-group and metapopulation studies can be found in \cite{gianpi,Mick_Roberts,Metcalf,Many}.
\section{Basic concepts and methods in (non-behavioral) epidemiological modeling and vaccination: network models}\label{concep-net}
A real population is mix of heterogeneous individuals and the heterogeneity exists in several aspects, especially in contacts. The foundations of epidemiology and early epidemiological models were based on population wide random-mixing, which was well described in the last section. However, in practice, each individual has a finite set of contacts or individuals whom they can pass the infection. The ensemble of all such contacts forms a \textit{network}. Structural pattern of this contact network among individuals allows models to compute the epidemic dynamics at the population scale from the individual-level process of spreading infections.
In the recent years, overwhelming data explosion in human sciences shows complex and heterogeneous connectivity patterns in a wide range of biological and social systems. Network theory provides a natural gateway to understand all these complex patterns. Ideas from network theory have also inspired research in many fields to study interactions between elements and their dynamics in areas like computer science, system biology, social sciences, economic and statistical physics. However, epidemiological study has also used potentiality of network theory like any other scientific disciplines. The complex properties of networks have a profound impact on studying equilibrium and non-equilibrium phenomena in epidemic processes. Dated back to mid-eighties, the researchers have shown that understanding the spreading process of an infection have considerable overlap with the studies of network properties [151, 152]. The group of individuals and their connections in a population naturally defines a network in which disease spreads from one node (individual) to another distantly related node. So, insights from topological structure of the underlying network provides useful information that helps predicting spreading process including growth of infection at the first stage and subsequently its distribution over the entire contact network.
Although it has long been acknowledged that network theory is the key ingredient of epidemic modeling, recent abundance of data exemplifies the huge complexity in the spreading and persistence of infection in a population and thus calls for detailed theoretical understanding of the interplay between epidemic processes and networks. Previous works have shown that most real-world networks exhibit dynamic self-organization and are statistically heterogeneous -- typically all marks of complex systems \cite{albert2002statistical, baronchelli2013networks, boccaletti2006complex, caldarelli2007scale, cohen2010complex, costa2007characterization, dorogovtsev2002evolution, newman2010networks, newman2003structure}. A regular lattice rarely represents the real-world networks of relevance for epidemic spreading. In a real population, there are few individuals that may act as hubs (or `super-spreader'), whereas the majority of the population have very few interactions. Both social and infrastructure networks are organized in communities of tightly interconnected nodes of individuals \cite{girvan2002community}. Although randomness in the connection process of nodes is always present, organizing principles and correlations in the connectivity patterns define network structures that are deeply affecting the evolution and behavior of epidemic and contagion processes \cite{pastor2001epidemic, pastor2001aepidemic, moreno2002epidemic, kretzschmar1996measures}. Furthermore, the complex features of networks often find their signature in statistical distributions which are generally heavy-tailed, skewed, and varying over several orders of magnitude \cite{watts1998collective}. The evidence of large-scale fluctuations, clustering and communities characterizes the connectivity patterns of real-world systems; and this has prompted the need for mathematical approaches capable to deal with the inherent complexity of networks.
However, the connection between infectious disease epidemiology and the network theory goes far beyond the general discussion. Understanding network topology may help identifying disease transmission routs, which may help designing strategies to control the disease. For instance, `contact tracing' is a very powerful public health strategy during first phases of the spreading process. It relies on underlying transmission routes to control further spread without even any information on epidemiology of the infectious disease [167]. Thus studying network theory and how it can be applied to relate the infection spreading process may be a vital tool for understanding the disease and, designing control strategies. With a discussion of basic network metrics and topologies, here we review the development of current research at the interface of epidemiological studies and network theory including immunization to control the disease.
\subsection{Networks: types and topologies}
\subsubsection{Definitions and Notations }
Graph theory \cite{ore1962theory} can be used to mathematically formalize networks \cite{newman2003structure}. A graph is a collection of points, called \textit{vertices} (or \textit{nodes}), and a set of connections, called \textit{edges} (or \textit{links}). Edges indicate link between vertices, representing the presence of an interaction or relations between those vertices. Interaction can be bidirectional which defines \textit{undirected} networks, or it can be directional which defines \textit{directed} networks or \textit{digraphs}. We represent a graph of size $N$ (say, with $N$ vertices, and $L$ links) using $N \times N$ adjacency matrix $A$, with elements $a_{ij} = 1$ if an edge is connecting nodes $i$ and $j$ and zero otherwise. $A$ is symmetric matrix in undirected graphs, and asymmetric in directed graphs.\\
\noindent$\bullet$ \textbf{Degree and Degree distribution:} The \textit{degree} (or \textit{connectivity}) $k_i$ of vertex $i$ in an undirected network is the number of edges emanating from $i$, i.e. $k_i = \sum_j a_{ij}$. In the case of directed networks, we have both \textit{in-degree}, $k^{\mbox{in}}$, and \textit{out-degree}, $k^{\mbox{out}}$, as the number of edges that end in $i$ or start from $i$, respectively. The total degree, however, in directed network is defined by $k_i = k^{\mbox{in}}+k^{\mbox{out}}$. The \textit{degree distribution} $P(k)$ of a network is the probability that a randomly chosen vertex has degree $k$. In a finite network, it denotes the fraction of vertices with degree exactly equal to $k$. In case of directed networks, there are instead two different distributions, the \textit{out-degree} $P^{\mbox{out}} (k^{\mbox{out}})$ and the \textit{in-degree} $P^{\mbox{in}} (k^{\mbox{in}})$ distributions, though in-degree and out-degree of a given vertex might not be independent. It is also useful to consider the \textit{moments} of the degree distribution, $<k^n> = \sum_k k^n P(k)$. The first moment $<k> = 2L/N$ is called the \textit{average degree}, provides information about the density of the network. The degree distribution defines classes of networks with similar statistical features. A network is called \textit{sparse} if its number of links $L$ grows at most linearly with the network size $N$; otherwise, it is called \textit{dense}. In directed networks, since every edge contributes to one node in-degree and other node out-degree we have that $<k^{\mbox{in}}> = <k^{\mbox{out}}>$.\\
\noindent$\bullet$ \textbf{Connectedness, Shortest path length, and Diameter:}
A path $P_{i,n}$ is a sequence of different edges ${(i_j,i_{j+1})}, j = 0,...,n-1$, connecting vertices $i_0$ and $i_n$; the number of edges traversed, $n$, is also called the length of the path. A graph is \textit{connected} if there exists a path connecting any two vertices in the graph. A component $\mathcal{C}$ of a graph is defined as a connected subgraph. The \textit{shortest path} $l_{ij}$ between two nodes $i$ and $j$ is defined as the length of the shortest path (though may not be unique) joining $i$ and $j$, and the \textit{average shortest path length} $<l>$ is the average of the value of $l_{ij}$ over all pairs of vertices in the network. The \textit{diameter} of a network is the maximum value of all the pairwise shortest path lengths.\\
\noindent$\bullet$ \textbf{Degree correlations:} The \textit{degree correlation} between two vertices is defined as the conditional probability $P(k'|k)$ that an edge departing from a vertex of degree $k$ is connected to a vertex of degree $k'$ \cite{pastor2001epidemic}. A network is called \textit{uncorrelated} if this conditional probability is independent of the originating vertex $k$. In this case, $P(k'|k)$ can be simply estimated as the ratio between the number of edges pointing to vertices of degree $k'$, $k'P(k')N/2$, and the total number of edges, $L = <k>N/2$, to yield $P^{\mbox{un}}(k'|k) = k'P(k')/<k>$.
The most popular measure of degree correlation is \textit{Pearson correlation coefficient} $r$. Proposed by Mark Newman \cite{Newman_mixing,newman2002assortative}, the degree correlation coefficient is defined as
\begin{equation}
r = \sum_{_{jk}}\frac{jk(e_{jk}-q_jq_k)}{\sigma^2},
\end{equation}\label{degree-correlation-eq}
where \[\sigma^2 = \sum_kk^2q_k-\bigg[\sum_k kq_k\bigg]^2.\]
$e_{ij}$ is the degree correlation matrix that represents the probability of finding node with degrees $i$ and $j$ at the two ends of a randomly selected link, and $q_k$ is the probability that there is a degree-$k$ node at the end of the randomly selected link. The degree correlation varies between $-1\leq r \leq1$. For instance, $r > 0$ the network is assortative, for $r = 0$ the network is neutral and for $r < 0$ the network is disassortative. In reality, social networks are assortative, whereas technological and biological networks are disassortative \cite{newman2002assortative}. For example, for the scientific collaboration network we obtain $r = 0.13$, in line with its assortative nature; for the protein interaction network $r = - 0.04$, supporting its disassortative nature and for the power grid we have $r = 0$. Another related measure of correlations is the \textit{average degree} of the nearest neighbors of vertices of degree $k$, denoted by $\bar{k}_{nn}(k)$ which is formally defined as \cite{pastor2001epidemic} \[\bar{k}_{nn}(k) = \sum_{k'} k' P(k'|k).\]\\
\noindent$\bullet$ \textbf{Centrality and Betweenness:} The concept of \textit{centrality} encodes the relative importance of a node inside a network. Together with the \textit{degree} and the \textit{closeness of a node} (defined as the inverse of the average distance from all other nodes), another standard measure of node centrality is \textit{node betweenness}, which is obtained by counting the number of geodesics going through it. More precisely, the betweenness $b_i$ of a node $i$, sometimes referred to as $load$, is defined as
\begin{equation}
b_i = \sum_{j,k\in N, j\neq k} \frac{n_{jk}(i)}{n_{jk}},
\end{equation}
where $n_{jk}$ is the number of shortest paths connecting $j$ and $k$, while $n_{jk}(i)$ is the number of shortest paths connecting $j$ and $k$ and passing through $i$. Betweenness distributions have been investigated in \cite{molloy1995critical, molloy1998size, newman2001random, newman2002random, adamic2001search, goh2001universal, barthelemy2004betweenness, guimera2005worldwide}. Betweenness-betweenness correlations and betweenness-degree correlations have been studied respectively in \cite{ravasz2003hierarchical} and in \cite{vazquez2002large}. The concept of betweenness can also be extended to edges. The \textit{edge betweenness} is defined as the number of shortest paths between pairs of nodes that run through that edge \cite{newman2004finding}.\\
\noindent$\bullet$ \textbf{Clustering:} \textit{Clustering} refers to transitivity property of the network, i.e. the relative propensity of two nodes to be connected, provided that they share a common neighbor. The \textit{clustering coefficient} $C$ is defined as the ratio between the number of loops of length three in the network (i.e. triangles), and the number of connected triples (three nodes connected by two edges). An alternative definition is also available by Watts and Strogatz \cite{Strogatz2001}. A local measure $c_i$ of clustering is defined as the ratio between the actual number of edges among the neighbors of a vertex $i$, $e_i$, and its maximum possible value, measuring thus directly the probability that two neighbors of vertex $i$ are also neighbors of each other. The \textit{mean clustering} of the network $<c>$ is defined as the average of $c_i$ over all vertices in the network. The clustering spectrum $\bar{c}(k)$ is defined as the average clustering coefficient of the vertices of degree $k$ \cite{ravasz2003hierarchical, vazquez2002large}, satisfying $<c> = \sum_k P(k)\bar{c}(k)$.\\
\noindent$\bullet$ \textbf{Community Structure:} A \textit{community} or \textit{cluster} (or \textit{cohesive subgroup}) of a given network of vertices $N$ and set of edges $L$ is a subgraph $G'(N', L')$, whose nodes are tightly connected, i.e. cohesive. Since the structural cohesion of the nodes of $G'$ can be quantified in several different ways, there are different formal definitions of community structures.
A related terminology of community structure of a network is \textit{clique}. A \textit{clique} is a maximal complete subgraph of three or more nodes all of which are adjacent to each other, and such that no other nodes exist adjacent to all of them. This definition can be extended by weakening the requirement of adjacency into a requirement of reachability: a \textit{n-clique} is a maximal subgraph in which the largest geodesic distance between any two nodes is no greater than $n$. A detail description is given in \cite{boccaletti2006complex, fortunato2010community}.
\subsubsection{Type of networks}
Based on the qualitative and quantitative features of the degree distribution, the networks can be classified into a series of important types of networks, which are also useful for the epidemiological applications.\\
\noindent$\bullet$ \textbf{Random graph:} The simplest network model is the classical random graph model proposed by Erd{\"o}s and R{\'e}nyi (also named ER graph) \cite{erdHos1959random, gilbert1959random, solomonoff1951connectivity,erdHos1959random}. A random graph model $G_p(N)$ is constructed from a set of $N$ nodes in which each one of the $N(N-1)/2$ possible links is present with probability $p$. The degree distribution is given by a binomial form, which, in the limit of constant average degree (i.e. $p = <k>/N$) and large $N$ can be approximated by a Poisson distribution $P(k) = e^{-<k>} {<k>}/{{k!}}$ . The clustering coefficient is given by $<c> = p$, and the average shortest path length is $<l> \sim \log N/ \log<k>$ \cite{dorogovtsev2013evolution}. This model is therefore adequate in the case of networks governed only by stochasticity, although $G_p(N)$ tends to a regular graph for large N and constant $p$. Extensions of the basic model to allow for other degree distributions lead to the class of models known as \textit{generalized random graphs}, \textit{random graphs with arbitrary degree distributions} and the \textit{configuration model} \cite{newman2003properties}.\\
\noindent$\bullet$ \textbf{Small-world (SW) network:} A more sophisticated and tractable model of a network with small proximity and high clustering coefficient is the \textit{small-world network model} proposed by Watts and Strogatz (i.e. WS model) \cite{watts1998collective}. This network model is based on rewiring procedure of an ordered lattice. The initial configuration starts with a ring of $N$ vertices, each one of which symmetrically connected to its $2m$ nearest neighbors. This represents the model with a structure which has large clustering coefficient and large average shortest path length. Starting from it, a fraction $p$ of edges in the network are rewired, by visiting all $m$ clock-wise edges of each vertex and reconnecting them, with probability $p$, to a randomly chosen node. There is another version of the model proposed by Monasson \cite{monasson1999diffusion}, where a fraction $p$ of edges are added between randomly chosen pairs of vertices. The overall effect of the rewiring processes is to add long-range shortcuts, that, even for a small value of $p \sim N-1$, greatly reduce the average shortest path length, while preserving a large clustering for not very large values of $p$. This model, although better suited for social networks with high clustering coefficient, has a degree distribution and centrality measures decaying exponentially fast away from the average value. The small-world model thus generates homogeneous networks where the average of each metric is a typical value shared, with little variations, by all nodes of the network.\\
\noindent$\bullet$ \textbf{Scale-free (SF) network - static and dynamic:} In contrast to random graph, and small-world network models, real-world networks are structured in a hierarchy of nodes with a few nodes having very large connectivity (the hub nodes), while the vast majority of nodes have much smaller degrees. In precise, these heterogeneous networks exhibit heavy-tailed degree distributions often approximated by a power-law behavior of the form $P(k) \sim k^{-\zeta}$ , which implies a non-negligible probability of finding vertices with very large degree. The degree exponent $\zeta$ of many real-world networks takes a value between 2 and 3. In such cases, these network models are called \textit{scale-free}. Several variations of the classical random graph models have been proposed in order to generate networks with a power-law degree distribution. For example, \textit{configuration model} \cite{bender1978asymptotic, molloy1995critical, aiello2000random}, \textit{fitness model} by Caldarelli et al. \cite{caldarelli2002scale}, \textit{threshold graph model} by Masuda et al. \cite{masuda2004global} and \textit{gradient network models} by Toroczkai and Bassler \cite{toroczkai2004network}, where directed graphs induced by local gradients of a scalar field distributed on the nodes.
So far, we have not discussed evolution process of these scale-free networks, i.e. nodes and links do not change over time. There are many examples of real networks in which the structural changes are ruled by the dynamical evolution of the system. Most well-known and useful evolving network model is \textit{Barab{\'a}si-Albert model} \cite{barabasi2002evolution}. The Barab{\'a}si-Albert (BA) network is a model of network growth inspired to the formation of the World Wide Web and is based on two basic ingredients: i) \textit{growth} (of nodes) and ii) \textit{preferential attachment} (connection of vertices). A description of the growth precess and attachment function is discussed in \cite{boccaletti2014structure}. Variations in preferential attachment function produced several other evolving network models like \textit{Dorogovtsev-Mendes-Samukhin} (DMS) model (linear preferential attachment), \textit{Krapivsky} model (nonlinear attachment probability), etc.\cite{dorogovtsev2000structure}. The Barab{\'a}si-Albert (BA) model has been generalized further by the same authors, and also by others including certain characteristics of underlying growth process of nodes \cite{boccaletti2014structure}.\\
\noindent$\bullet$ \textbf{Weighted networks:} In the real world settings, many networks exhibit varied intensity of connections. For example, social network, where existence of strong and weak ties between individuals are seen \cite{marchiori2000harmony, latora2001efficient, krause2003compartments, latora2003economic}, metabolic reaction pathways, predator-prey interactions in food webs \cite{polis1998ecology, mccann1998weak}, different capabilities of transmitting electric signals in neural networks \cite{sporns2000connectivity}, unequal traffic on the Internet \cite{barrat2004architecture} or of the passengers in airline networks \cite{li2004statistical}. These types of complex systems can be explained with model networks in which each link carries a numerical value measuring the strength of the connection. Considering weights of the connections puts another dimension of complexity in the topology of weighted network. Similarly, we can also have networks with weighted nodes. Detail descriptions of the measuring and modeling weighted networks has been discussed in~\cite{boccaletti2006complex}.
There are also other type of networks, which are important to study real world phenomenon. These networks include \textit{spatial network} (whose nodes occupy a precise position in two or three-dimensional Euclidean space, and whose edges are real physical connections) \cite{li2014spatial,li2011dimension,sun2016pattern,wang2011evolution}, adaptive networks \cite{gross2008adaptive}, temporal or time-varying networks \cite{holme2011temporal,nicosia2014measuring,zhang2012towards} (for more details, please also see Section~\ref{section:digital_epidemiology}).
All the previous networks are single-layer. But there are many real world complex systems that can be conceived as multiple layers of such single networks. Most complex systems from physical, social, engineering, information and biological sciences include multiple subsystems and layers of connectivity, and they are often open, value-laden, directed, multilevel, multicomponent, reconfigurable systems of systems, and placed within unstable and changing environments. An example of multilayer networks conceived from age-dependent contact network is shown in Fig.~\ref{AgeMultilayer}. The recent development of \textit{multilayer networks} in network theory provides radical views of understanding these multilayer systems. For more information about single-layer and multilayer networks, please see the comparison in Table~\ref{multi}. Descriptions and examples of more different type spatial and multilayer networks are given in \cite{boccaletti2014structure}.
\begin{figure}[th]
\centering
\includegraphics[width=122mm,height=73mm]{fig8.pdf}
\caption{Schematic illustration of age-dependent contact networks into a multilayer system. Network with different color shades in (a) indicates network of individuals in the same age group, where network in gray region denotes connections across different age groups. Blue lines point to the same individuals from colored regions to gray region. (b) The same network can be conceived as multilayer networks.}
\label{AgeMultilayer}
\end{figure}
\begin{sidewaystable}%
\caption{Definitions, Measure and Expressions of network properties of single-layer and multilayer networks}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
&&\\
Definition & Single-layer networks & Multilayer networks\\
&&\\
\hline
&&\\
Notation& $\mathcal{G} = (\mathcal{N, L})$& $\mathcal{M} = (\mathcal{G, C})$\\
&&layers: $\mathcal{G} = \{\mathcal{G_{\alpha}}: \alpha \in \{1,\ldots, M\}\}$, $\mathcal{G_{\alpha} = (X_{\alpha},E_{\alpha}})$\\
&&links between layers: $\mathcal{C} = \{\mathcal{E_{\alpha\varphi} \subset X_{\alpha} \times X_{\varphi}}; \alpha \in \{1,\ldots, M\}, \alpha \neq \varphi\}$\footnote{$\alpha, \varphi$ denote layers in multilayer networks}\\
&&\\
\hline
&&\\
Degree \& Centrality °ree: $k_i = \sum_{j \in \mathcal{N}} a_{ij}$& degree: $\mathbf{k_i} = (k_i^{[1]}, \ldots k_i^{[M]})$, $k_i^{[\alpha]} = \sum_ja_{ij}^\alpha$\\
&betweenness: $b_i = \sum_{j,k\in N, j\neq k} \frac{n_{jk}(i)}{n_{jk}}$& overlapping degree: $o_i = \sum^M_{\alpha = 1} k_i^{[\alpha]} $\\
°ree correlation coefficient $c_i = \frac{1}{\lambda_I}\sum_{i\in \mathcal{N}}a_{ij}c_j$°ree correlation coefficient $c_i = (c_i^{[1]}, \ldots c_i^{[M]})$\\
&&\\
\hline
&&\\
Clustering coefficients&$c_i = \frac{2e_i}{k_i(k_i-1)}$&$C_\mathcal{M}(i) = \frac{2\sum_{\alpha=1}^M|\bar E_\alpha(i)|}{\sum_{\alpha=1}^M|\mathcal{N}_\alpha(i)|(|\mathcal{N}_\alpha(i)| -1)}$\\
&&\\
\hline
&&\\
Characteristic path length&$\frac{1}{N(N-1)}\sum_{i,j\in \mathcal{N}, i\neq j}d_{ij}$&$\frac{1}{N(N-1)}\sum_{u,v\in \mathcal{X_M}, u\neq v}d_{uv}$\\
&&\\
\hline
\end{tabular}
\end{center}
\label{multi}
\end{sidewaystable}%
\begin{table}
\caption{Epidemic thresholds for \textbf{SI, SIS} and \textbf{SIR} models over Erd{\"o}s and R{\'e}nyi (ER) and Scale-Free (SF) networks}\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
&&&\\
Model Type&Model expression& Characteristic time ($\tau_c$) & Spreading rate ($\lambda_c$)\\
&&&\\
\hline
&&&\\
SI&$\frac{di_k}{dt}=\beta_I [1- i_k(t)]k\theta_k(t)$&SF $(\zeta\geq 3)$: $\frac{<k>}{\beta_I(<k^2> - <k>)}$&SF $(\zeta > 3)$: 0 \\
&&SF $(\zeta < 3)$: 0&SF $(\zeta \leq 3)$: 0\\
&&ER:~~~$\frac{1}{\beta_I <k>}$&\\
&&&\\
\hline
&&&\\
SIS&$\frac{di_k}{dt}=\beta_I [1- i_k(t)]k\theta_k(t)-\gamma i_k(t)$&SF: $\frac{<k>}{\beta_I <k^2> - \gamma <k>)}$&$\frac{<k>}{<k^2>}$\\
&&ER:~~~$\frac{1}{\beta_I(<k>+1) - \gamma}$&$\frac{1}{<k>+1}$\\
&&&\\
\hline
&&&\\
SIR&$\frac{di_k}{dt}=\beta_I s_k(t)\theta_k(t)-\gamma i_k(t)$&$\frac{<k>}{\beta_I<k^2> - (\gamma+\beta_I)<k>)}$&$\frac{<k>}{<k^2>-<k>}$\\
&$s_k(t) = 1-i_k(t) - r_k(t)$&&\\
&&&\\
\hline
\end{tabular}\\
\end{center}
\scriptsize $i_k$: infected nodes with degree $k$.\\
\scriptsize $s_k$: susceptible nodes with degree $k$.\\
\scriptsize $r_k$: recovered or immune nodes with degree $k$. \\
\scriptsize $\theta_k$: fraction of the neighbors of node $k$ that are infected.\\
\scriptsize $\beta_I$: transmission rate.\\
\scriptsize $\gamma$: recovery rate.\\
\scriptsize $\zeta$: degree exponent SF network.\\
\label{DiseaseThresh}
\end{table}%
\subsection{Epidemic spread over networks}
Most infectious diseases such as influenza, SARS, HIV spread through human populations by physical contact between infective and susceptible individuals. The pattern of these disease-causing contacts forms a network, which has substantial influence on success of the spreading process. The traditional epidemic models such as classic \textbf{SIS} or \textbf{SIR} epidemic models do not explicitly incorporate the topological structure of the underlying contract network that facilitates the spread of a pathogen. Instead they assume that any individual can come into contact with any other individual (homogenous mixing hypothesis) and so all individuals have comparable number of contacts. It also assumes that probability of disease transmission is same across all individuals in the network. Both of these assumptions are limited in most of the diseases: in reality an individual can transmit a pathogen only to those it comes into contact with and also transmission occurs in varied probabilities. Hence pathogens spread can be assumed on complex contact networks.
Spreading epidemics on complex networks has been studied by many researchers including May and Lloyd \cite{may2001infection}. Most detail and systematical study is done by Newman \cite{newman2002spread}. Considering an SIR model with uniform recovery rate $\gamma$, i.e. where infected nodes become removed at rate $\gamma$ after infection, and infection rate $\beta_I$. Then the transmissibility $T$ is defined as the probability that the infection will be transmitted from an infected node to a connected susceptible neighbor before recovery takes place. For continuous-time dynamics, the transmissibility computed as \cite{newman2002spread}:
\begin{equation}
T = 1-\lim_{\delta t\rightarrow 0}(1-\beta_I\delta t)^{\gamma/\delta t} = 1-e^{-\gamma\beta_I}.
\end{equation}
In general, $\beta_I$ and $\gamma$ will vary between individuals, and assume initially that these two quantities are independent and identically distributed (iid) random variables chosen from some appropriate distributions $P(\beta_I)$ and $P(\gamma)$. Hence, transmission of the disease between two individuals is simply the average $T$ (i.e. homogeneous case) of $T_{ij}$ over the distributions $P(\beta_I)$ and $P(\gamma)$, which is
\begin{equation}
T =<T_{ij}>= 1-\int^{\infty}_0 P(\beta_I)P(\gamma) e^{-\gamma\beta_I} d\beta_I d\gamma.
\end{equation}
Computing threshold transmissibility for spreading the disease in a special non-degenerate case of SIR model has been discussed in \cite{boguna2002epidemic}.
Now, starting from a single infective individual, disease spreads across the contact networks and a outbreak occurs. The ultimate size of the outbreak would be precisely the size of the cluster of vertices that can be reached from the initial vertex by traversing only occupied edges (i.e., edge in the graph across which the disease is transmitted, say with average probability $T$). This is precisely a similar model equivalent to a bond percolation model with bond occupation probability $T$ on the graph representing the community. The connection between the spread of disease and percolation was in fact one of the original motivations for the percolation model itself. Below in the next subsection, we discuss how disease spread can be realized as the percolation problem on random graphs with arbitrary degree distributions discussing size of outbreaks, presence of an epidemic, transmissibility thresholds, etc.
\subsubsection{Percolation theory and spreading process}
Like mean-field method (mentioned in section 3), which is a useful tool to analyze the transmission dynamics in homogeneous mixing population, percolation theory also plays a significant role in prediction of epidemic or vaccination dynamics, such as emerging threshold and phase transition, in network population. One convenient universal approach to deal percolation problem with bond occupation probability $T$ on complex networks is considering \textit{generating functions} \cite{newman2002spread}. Generating functions and the problem of percolation in networks have been discussed in many earlier studies \cite{callaway2000network, cohen2000resilience, molloy1995critical, parshani2010epidemic}. We define generating functions $G_0(u)$ and $G_{1}(u)$ to describe a network as follows:
\begin{equation}
G_0(u) = \sum_{k=0}^\infty P(k)u^k,
\end{equation}
\begin{equation}
G_1(u) = \frac{G'_0(u)}{G'_0(1)},
\end{equation}
where $P(k)$ is the degree distribution of random graph $G$ with $N$ vertices. We can calculate back the degree distribution $P(k)$ and also all the higher moments
using $G_0(u)$ with
\begin{equation}
<k^n> = \sum_{k}k^nP(k) = \bigg[\bigg(u\frac{d}{du}\bigg)^nG_0(u)\bigg]_{u=1}.
\end{equation}
$G_{1}$ calculates the distribution of the degrees of vertices reached by following a randomly chosen edge. A modified set of generating functions is required to solve percolation problem with bond probability $T$: $G_0(u, T)$ and $G_{1}(u, T)$, where $G_i(u, T) = G_i(1+(u-1)T)$ \cite{van2012epidemics, van2009virus}. $G_i(u,T),~i=0,1$ represents the distributions of the number of \textit{occupied} edges attached to a vertex, as a function of the transmissibility $T$.
Outbreak size of a disease on a network precisely depends on the distribution of sizes of clusters of vertices together by occupied edges in the corresponding percolation model. Using the generation functions $G_0(u, T)$ and $G_{1}(u, T)$, we can find the mean outbreak size $<s>$ in closed form as
\begin{equation}
<s> = 1+\frac{G_0'(1;T)}{1-G_1'(1;T)}.
\end{equation}
So, given the values of generating function $G_i$, we can completely evaluate this expression to get the mean outbreak size for any value of $T$ and degree distribution. The outbreak size critically depends on the relation $TG'_1(1,T) = 1$. This point indicates the onset of an epidemic; it is the point at which the typical outbreak ceases to be confined to a finite number of individuals, and expands to fill significant fraction of the graph. This defines the critical transmissibility $T_c$, at which the critical transition occurs\\
\begin{equation}
T_c = \frac{1}{G_1'(1)} = \frac{\sum_kkP(k)}{\sum_kk(k-1)P(k)}= \frac{<k>}{<k^2> - <k>}.
\end{equation}
For $T>T_c$, we have an epidemic, or ``giant component'' in the language of percolation. This equation is valid for degree of uncorrelated networks which have no loops in the clusters. However, in the case of heterogeneous networks with degree distribution $P(k)\sim k^{-\zeta}$, it was found that the percolation threshold tends to zero for $\zeta<3$ in the limit of an infinite network size $N\rightarrow \infty$ (\cite{cohen2000resilience}). That is, an epidemic can spread in this network regardless of how small the infection probability and how quick is the recovery process. In general, the expansion of the generating functions around the nonzero solution yields the scaling behavior of the order parameter in the vicinity of the critical point: $P_G(T) \sim (T-T_c)^{{\beta_I}_{c}}$, where the critical exponent ${\beta_I}_{c}$ assumes following values
\begin{equation}
{\beta_I}_{c} = \left\{
\begin{array}{lll}
1/(3-\zeta)&\mbox{for} &\zeta<3,\\
1/(\zeta-3)&\mbox{for} &3< \zeta \leq 4,\\
1& \mbox{for}& \zeta \geq 4.
\end{array}
\right.
\end{equation}
For the case $\zeta = 3$, a stretched exponential form $P_G(T) \sim e^{1/p}$ is expected. For more details about disease thresholds, we provide a brief summary in Table ~\ref{DiseaseThresh}.
There are other approaches of modeling spreading process of infectious disease over networks such as \textit{Degree-based mean-field theory} (DBMF) and \textit{Individual-based mean-field theory} (IBMF). DBMF considers the dynamics of node of degree $k$ to be infected or remain susceptible, and assume that all the nodes with degree $k$ are statistically equivalent. The value of the epidemic threshold can be obtained by means of a linear stability analysis \cite{boguna2002epidemic, pastor2001epidemic}. It was the first approach to the study of the SIS model in complex networks \cite{pastor2001aepidemic}. Global stability of the disease-free state in such model, the role of seasonal oscillations and the coexistence of an endemic equilibrium state via multistability have been investigated in \cite{don}. In contrast, the state of the system in the disease model (such as SIS) in the IBMF framework is fully defined by a set of Bernoulli random variables $X_i(t) \in {0, 1}: X_i(t) = 0$ for a healthy, susceptible node and $X_i(t) = 1$ for an infected node; altogether it defines a $2^N$ Markov chain for a network with $N$ nodes \cite{van2012epidemics, van2009virus}, which specifies exactly the time evolution of the disease model. However, the Markov chain approach complicates analytical calculations of the model. A discussion on both DBMF and IBMF can be found in \cite{pastor2015epidemic}.
\subsubsection{Disease spreading in multilayer networks}
In many situations, disease spread can be modeled over \textit{interconnected networks} instead of a single-layer network (or called \textit{monoplex} network). For example, spreading of an infection amongst groups of communities, or even groups of individuals of different age profiles \ref{AgeMultilayer}. Studying the diffusion of pathogens over interconnected network or \textit{multilayer networks} has got very recent attention in epidemiology. Multilayer networks are made of multiple layers, where each layer is a monoplex network, where the same nodes can appear in multiple layers and nodes on different layers can be connected to each other. One important aspect of spreading infection across multilayer networks is that it can also spread from one layer to another. Generally, there are three possibilities for spreading process \cite{salehi2015spreading, min2014layer}: \textit{same-node, inter-layer}, when infection switches layer but remains on the same node, e.g., when an infected individual travels from one city to another city; \textit{other-node inter-layer}, when infection continues spreading to another node in another layer, e.g., spreading of infection between individuals in different age profiles through direct physical contact. In third type, \textit{other-node intra-layer}, when infection spreads across the same community. In the context of interacting spreading processes in multilayer networks, two types of thresholds have recently been introduced, called \textit{survival threshold} measuring if infection will survive and \textit{absolute-dominance threshold} indicating whether it can completely remove another competing process \cite{salehi2015spreading}.
Most of the works on modeling the dynamics of diffusion over multilayer networks have used epidemic models such as SIS, SIR \cite{min2014layer, magnani2011ml, wang2013effect, lee2014multiplex, berlingerio2013multidimensional, wasserman1994social, gao2014single, boccaletti2014structure, dickison2012epidemics, zhao2014multiple,wang2014epidemic}. In a similar approach with single-layer networks, the dynamics of epidemic spreading according to the SIS and SIR models over multilayer networks are described as a two- or three-state process, respectively. Thereafter, many extensions have been applied to SIR and SIS models. One of the most important extensions, Goldenberg et al. \cite{goldenberg2001talk} proposed a discrete-time version of the SIR model called Independent Cascade Model (ICM), where time proceeds in discrete time steps. In this model, each infected node $u$ at time $t$ can infect each of its neighbors. If the infection succeeds, then neighbor $v$ will become infected at step $t + 1$. ICM is often used in the literature on influence spreading. In \cite{salehi2015spreading}, the authors extended this model to analyze the dynamics of multiple cascades over multiplex networks.
Unlike a monoplex network, the infection may diffuse over inter- and intra-layer connections at different speeds over multiple networks, meaning that we may have different infection rates (i.e., transmissibilities) across the links of each layer and also the links between the layers. Therefore, most of the works on spreading processes over multilayer networks \cite{dickison2012epidemics, wang2011effects, qian2012diffusion, zhao2014multiple, marceau2011modeling, buono2014epidemics, funk2010interacting, wang2013effect, sanz2014dynamics} have extended epidemic-like models by considering different infection rates dependent on the types of the layers. A recent contribution in this context proposed a generalized epidemic mean-field (GEMF) framework, which was capable of modeling epidemic-like spreading processes with more complex states in multiplex network layers (compared to two or three states in the SIS and SIR models).
\subsubsection{Simulation-based models}
Simulation-based modelling, like analytical models described above, has similar appeal in epidemic theory \cite{eames2002modeling, eubank2004modelling, meyers2005network, read2003disease, wallinga1999perspective, watts1998collective, chao2010flute, merler2010role}. There are many good examples of simulation based network models of infectious diseases such as sexually transmitted diseases (STD) that highlight the importance of network structure (e.g., role of core groups or interconnected individuals with a large number of contacts) in disease transmission and persistence \cite{garnett1996sexually, ghani1997role, morris1997concurrent, potterat1999network, klovdahl2001networks, rothenberg2001net, mcelroy2003network, szendroi2004polynomial, doherty2005determinants}.
There are several simulation-based works that attempt to model airborne disease outbreaks \cite{halloran2002containing, cohen1997social, wallinga1999perspective}. Such models consider large population, like of order $\sim 10^6$ with certain specified social connections among individuals. The networks used are generated by computer simulation to conform to several observed demographic and social characteristics such as given age distribution, household size, public crowding. Variable activities in the entire day are also assigned to the individuals such as working place, public gathering, or even spending time at home. Different activities reflect the change in contact pattern in different time of the day. Children are assigned to school, day-care centre or play group where they interact (and therefore form connections) with other children. The simulation model puts emphasis particularly on transmission within households and family groups, which are probably the main routes of transmission for this disease. There are models that attempt similar tasks, using census data to determine interaction patterns estimating demographic and disease parameters \cite{eubank2004modelling}. Despite the number of approximations involved, the inherent stochasticity of such microsimulations allows a direct estimation of the variability between epidemics. For example, Meyers et al \cite{meyers2005network} generates a network model to characterize the spread of SARS in Vancouver, British Columbia. They have used census data of Vancouver and using computer simulations generated a plausible contact network that reflects an urban setting. They had chosen $N = 1000$ households at random from the Vancouver household size distribution, which yields approximately 2600 people. Household members were given ages according to the census data of Vancouver. Kids were then assigned to schools according to school and class size distributions to occupations according to (un)employment data, to hospitals as patients and caregivers according to hospital employment and bed data, and to other public places. Within each location then random connections were created among individuals with variable probabilities at households, hospitals and schools, workplaces, and other public places. They have also chosen the parameters of the Poisson and power law networks so that all three networks share the same epidemic thresholds. Conventional epidemiological models shows clear threshold effects, that is, the disease is expected to die out when transmission probability falls below a critical threshold level. Olinky and Stone \cite{olinky2004unexpected} have shown that epidemic propagation depends equally on the infection process and also on the network topology. Their recent analyses of infection processes on highly heterogeneous networks (e.g., scale-free networks) concludes that diseases spread and persist even for vanishingly small transmission probabilities.
However, network-based simulation models have certain pitfalls as well. As mentioned by Keeling and Eames \cite{keeling2005networks}, these models are limited to ascertain the sensitivity of the epidemiological results to the details of the network structure. For example, the study by Halloran et al \cite{halloran2002containing} may be limited to answer questions like whether the network is representative of an average American community, whether variation between communities will bias the results if large population sizes are considered and whether rare but epidemiologically important contact structures are missing from the network. It is difficult to answer such questions or gain an intuitive understanding of network structure if our experience is limited to simulations of sampled networks. However, a range of idealized networks and analytical tools have been developed that can reveal the elements of network structure, which are important determinants of epidemic dynamics.
\subsection{Vaccinations over networks}
Designing optimal vaccination strategies to prevent and control infection has been widely addressed in the epidemic modeling of infectious diseases. Vaccination over networks can be modeled as a site percolation problem \cite{albert2000error}. Each vaccinated node can be regarded as a site which is removed from the network. The goal of the vaccination process thus is to reduce the transmissibility and pass the percolation threshold, which leads to minimization of the number of infected individuals in the network. Thus implementing the SIR model and vaccination can be considered as a site - bond percolation model, and the vaccination is considered successful if the network is below the percolation threshold \cite{albert2000error,madar2004immunization}. However though, the epidemic threshold decreases with the standard deviation of the connectivity distribution in the heterogeneous networks \cite{anderson1992infectious, pastor2001epidemic, pastor2001aepidemic}. This feature is paradoxically amplified in SF networks that have diverging connectivity fluctuations. In fact, epidemic processes in SF networks do not possess, in the limit of an infinite network, an epidemic threshold below which diseases cannot produce a major epidemic outbreak or the inset of an endemic state. SF networks are, therefore, prone to the spreading and the persistence of infections, whatever virulence the infective agent might possess. In view of this, it becomes a major task to public health policy makers designing optimal vaccination strategies oriented to minimize the risk of epidemic outbreaks on networks such as SF networks. However, population with heterogeneous connected networks may take different paths to extinction and thus control of infection. A recent paper by Hindes and Schwartz (2016) discussed such issues on heterogeneous random networks under various configuration \cite{hindes2016epidemic}.
The simplest hypothetical way to experiment the effect of vaccination over networks is the random introduction of immune nodes in the network population, in order to get a \textit{uniform vaccination} density \cite{anderson1992infectious, heesterbeek2000mathematical}. Immune nodes cannot become infected and, thus, do not transmit the infection to their neighbors. In this case, for a fixed spreading rate $\beta_I$, the relevant control parameter is the immunity $x$, defined as the fraction of vaccinated nodes present in the network. At the mean-field level, the presence of uniform vaccination will effectively reduce the spreading rate $\beta_I$ by a factor $(1-x)$; i.e. the probability of finding and infecting a susceptible and non-immune node. In homogeneous networks, such as the Watts-Strogatz (WS) model, it is easy to show that in the case of a constant $\beta_I$, the critical vaccination rate $x_c$ is given by
\begin{equation}
x_c = 1-\frac{{\beta_I}_c}{\beta_I}.
\end{equation}
Thus, the critical vaccination that achieves eradication is related to the spreading rate and the epidemic threshold of the infection, which implies that the critical vaccination allows the complete eradication of the disease over the network.
However, in an heterogeneous population with a uniform vaccination scheme, it is necessary to vaccinate a fraction of the population larger than the estimate given by a simple (homogeneous) assumption \cite{anderson1992infectious}. In this case, it can be proved that optimal vaccination programs can eradicate the disease by vaccinating a smaller number of individuals. A straightforward way to reintroduce an intrinsic vaccination threshold in heterogeneous networks consists in using different fractions of vaccinated individuals according to their connectivity. Let us define $x_k$ as the fraction of immune individuals with a given connectivity $k$ and averaging $x_k$ over the various connectivity classes, we have the critical vaccination threshold as
\begin{equation}
x_c = \sum_{k>\beta_I^{-1}}\bigg(1-\frac{1}{\beta_I k}\bigg)P(k).
\end{equation}
Scale-free networks can be considered as a limiting case of heterogeneous systems and it is natural to look for specifically devised vaccination strategies. This however, does not work on SF network. Because of the absence of any epidemic threshold in SF networks, it becomes impossible to find any critical vaccination rate that ensures the eradication. In scale-free (SF) networks, the pervious equation takes the form
\begin{equation}
1-x_c = \frac{1}{\beta_I} \frac{<k>}{<k^2>}.
\end{equation}
This shows that only a complete vaccination $(x_c = 1)$ of the network works to eradicate the disease over scale free network with $<k^2> \rightarrow \infty$.
While uniform or proportional vaccination fails to protect contagion over scale-free networks due to its peculiar nature, it has been seen that scale-free networks are strongly affected by selective damage. For example, if a few of the most connected nodes are removed, the network suffers a dramatic reduction of its ability to carry information \cite{albert2000error, callaway2000network, cohen2000resilience}. With this idea, one can design a \textit{targeted vaccination} scheme to progressively make immune the most highly connected nodes, i.e., the ones more likely to spread the disease. In scale-free networks, it produces an significant increase of the network tolerance to infections at the price of a tiny fraction of immune individuals \cite{barabasi2009scale,pastor2002immunization,madar2004immunization}.
Suppose, a fraction $x$ of all nodes with connectivity $k>k_c$ are vaccinated in a network. Then $x = \sum_{k>k_c}P(k)$. At the same time, this also implies that all links emanating from vaccinated individuals can be considered as if they were removed. So, if we assume that the fraction of links is effectively removed from the network, the new connectivity distribution after the vaccination of a fraction $x$ of the most connected node is
\begin{equation}
P_x(k) = \sum_{q>k}^{k_c}P(q){q\choose{k}}(1-p)^kp^{q-k}.
\end{equation}
This gives, calculating the first two moments, the critical fraction of vaccination needed to eradicate the infection given by the relation:
\begin{equation}
\frac{<k^2>_{x_c}}{<k>_{x_c}}\equiv\frac{<k^2>_c}{<k>_c}[1-p(x_c)]+p(x_c) = \beta_I^{-1},
\end{equation}
where $<k>_c = \sum_{k_{min}}^{k_c}kP(k)$, $<k^2>_c = \sum_{k_{min}}^{k_c}k^2P(k)$. An explicit calculation for Barab{\'a}si-Albert network model, the approximate solution for the vaccination threshold is given by
\begin{equation}
x_c \sim \exp(-2/k_{min}\beta_I).
\end{equation}
This clearly indicates that the targeted vaccination program is extremely convenient in scale-free networks where the critical vaccination is exponentially small in a wide range of spreading rates $\beta_I$. Also in this case, the present result can be generalized for scale-free networks with arbitrary connectivity exponent $\zeta$.
A comparative analysis of targeted vaccination over SW and BA models exhibit contrastive results \cite{pastor2002immunization}. In the case of the SW networks, the behavior of the prevalence as a function of $x$ is equivalent in the uniform and targeted vaccination procedures. On the contrary, in the case of the BA networkss, we observe a drastic variation in the prevalence behavior, though it is sensitive to random choice of immune nodes. This however, confirms that targeted strategies do not have a particular efficiency in systems with limited heterogeneity, but are highly sensitive to the network with higher degree of heterogeneity such as scale-free networks. Targeted vaccination of a small fraction of the most connected nodes in scale-free networks shows significant impact on spread process of infection.
Another target-based strategy is \textit{acquaintance vaccination} that tries to target all of the most highly connected nodes for vaccination \cite{pastor2002immunization, gallos2007improving,madar2004immunization, liu2009common, cohen2003efficient}, but this strategy requires no knowledge of the node degrees or any other global knowledge. In this approach a random group of nodes are chosen and then a random set of their neighbors are selected for vaccination. The most highly connected nodes are far more likely to be in this group of neighbors. So immunizing this group results in targeting the most highly connected nodes, and it requires far less information about the network \cite{cohen2003efficient, christakis2010social, krieger2003focus}. Another variant of this strategy again calls for the random selection of nodes but instead asks for one of their neighbors with a higher degree, or at least more than a given threshold degree and immunizes them \cite{gallos2007improving}. These degree based strategies consistently require fewer nodes to be vaccinated and as such improve a network's chances against epidemic attacks \cite{cohen2003efficient,madar2004immunization}.
This however, is just a brief summary about vaccination on networks. For more details about mathematical and physics framework, please see next section.
\section{Non-behavioral epidemiological vaccination on networks}\label{sec:vac-net}
Due to the potential threat posed by the spreading of infectious disease, ample research has been devoted to the mitigation and prevention of epidemics. To date, one of the most popular and effective methods is network vaccination \cite{pastor2002immunization,barrat2008dynamical,pastor2015epidemic,barabasi2016network,bornholdt2003handbook,gao2011network,liu2016biologically}, where certain nodes in network are effectively immunized by a perfect vaccine, and are thus no longer able to transmit the disease to their neighbors \cite{cornforth2011erratic}. More precisely, given a fixed effevctive spreading rate $\beta_I/\gamma$ ($\beta_I$ and $\gamma$ are the infection rate and recovery rate respectively. Without loss of generality, $\gamma=1$ is fixed since it only affects the definition of the time scale of epidemic spreading.), the relevant control parameter is the immunity rate $x$, which is simply defined as the fraction of vaccinated nodes in a network. The challenge is how to arrive at a minimal value of $x$, which is the so-called vaccination threshold $x_c$, to reduce the spreading rate of the disease $\beta_I$ under the critical threshold ${\beta_I}_c$, such that the disease dies out completely \cite{wang2003epidemic,chakrabarti2008epidemic,restrepo2006characterizing}.
This section is devoted to the systematic review of such non-behavioral vaccination techniques, where networks are effectively immunized by means of a targeted vaccination of certain nodes. We will review research done on single-layer and multilayer networks, on static and adaptive networks, and theoretical as well as empirical results.
\subsection{Uniform (or random) vaccination}
The simplest vaccination strategy that one can conceive, the implementation of which requires no preparation or information at all, is uniform or random vaccination~\cite{anderson1992infectious,pastor2002immunization}. Here a fraction $x$ of nodes are randomly selected and then vaccinated. Accordingly, only the remaining $1-x$ fraction of nodes contributes to the spreading of the
disease. The effective number of neighbors that each
susceptible node possesses thus decreases from its degree $k$ to $k (1-x)$. If one considers the standard SIS model as an example, then after vaccination the evolution equation of the probability $\rho^I_k(t)$ that a node with degree $k$ is infected at time
$t$ is
\begin{equation}\label{}
\frac{d\rho^I_k(t)}{dt}=-\rho^I_k(t)+\beta_I k (1-x)[1-\rho^I_k(t)]\sum_{k'}P(k'|k)\rho^I_{k'}(t),
\end{equation}
where the first term defines the rate at which infected nodes of degree $k$ recover and become susceptible again, and in the second term $P(k'|k)$ refers to the conditional probability that an infected node with degree $k'$ points to the node with degree $k$.
It can be observed that in the mean-field model, where individuals interact randomly with one another, the uniform vaccination is equivalent to the simple decrease of the spreading rate of the disease from $\beta_I$ to $\beta_I (1-x)$. Since eradicating an epidemic by uniform vaccination requires
$\beta_I$ to be below its critical epidemic threshold ${\beta_I}_c$, we obtain \begin{equation}\label{x_c}
\beta_I(1-x_c)={\beta_I}_c,
\end{equation}
where $x_c$ is the critical vaccination threshold, above which the density of infected individuals in the stationary state is zero.
However, in predominantly homogeneous networks, such as random or small-world networks, ${\beta_I}_c=1/(\langle k\rangle+1)$ for the SIS model and ${\beta_I}_c=1/\langle k\rangle$ for the SIR model \cite{barrat2008dynamical,pastor2001epidemic}. Based on Eq.~\ref{x_c}, the vaccination threshold is
\begin{equation}\label{}
x_c=1-\frac{1}{\beta_I}\frac{1}{\langle k\rangle+1}
\end{equation}
for the SIS model, and
\begin{equation}\label{}
x_c=1-\frac{1}{\beta_I}\frac{1}{\langle k\rangle}
\end{equation}
for the SIR model. Thus, if the uniform vaccination level is larger than $x_c$, homogeneous
networks will be completely protected and no large epidemic outbreaks will be possible.
Conversely, the situation is much different for strongly heterogeneous networks, such as scale-free networks, where the epidemic threshold vanishes \cite{cohen2000resilience,cohen2001breakdown}. Here uniform vaccination will fail. In particular, the absence of an epidemic threshold $({\beta_I}_c=0)$ in the thermodynamic limit implies that no matter how one rescales
${\beta_I}\rightarrow {\beta_I} (1-x)$, the epidemic will not be stopped unless $x=1$.
Based on the fact that in heterogenous networks ${\beta_I}_c=\langle k\rangle/\langle k^2\rangle$ for the SIS model and ${\beta_I}_c=\langle k\rangle/(\langle k^2\rangle-\langle k\rangle)$ for the SIR model \cite{pastor2001epidemic,lloyd2001viruses,moreno2002epidemic}, and by using
Eq.~\ref{x_c}, the threshold of uniform vaccination is given by
\begin{equation}\label{}
x_c=1-\frac{1}{{\beta_I}}\frac{\langle k\rangle}{\langle k^2\rangle}
\end{equation}
for the SIS model and
\begin{equation}\label{}
x_c=1-\frac{1}{{\beta_I}}\frac{\langle k\rangle}{\langle k^2\rangle-\langle k\rangle}
\end{equation}
for the SIR model.
In a scale-free network with degree exponent $\zeta <3$, we have $\langle
k^2\rangle\rightarrow \infty$, which indicates $x_c\rightarrow 1$ for both the SIS and the SIR model \cite{pastor2002immunization,pastor2001epidemic}. In other words, a complete vaccination
of the network is required to stop an epidemic. As emphasized before, uniform
vaccination is largely inefficient for disease eradication in heterogenous
networks \cite{callaway2000network,cohen2000resilience}. This theoretical prediction is also consistent with the finding that
many infectious diseases require that 80\%-100\% of the
population be vaccinated to prevent an outbreak. For example, measles require 95\% of the population be vaccinated \cite{anderson1992infectious}, while for digital viruses the strategies
relying on uniform vaccination call for practically 100\% of the
computers to install the appropriate antivirus software
\cite{pastor2001epidemic,jeong2000large,cohen2000resilience}.
\subsection{Targeted vaccination}
\subsubsection{Degree-based targeted vaccination: theoretical prediction of the vaccination threshold}
The problems encountered with uniform vaccination are rooted in the
vanishing epidemic threshold. Therefore, to successfully eradicate a
disease in a heterogenous network, we must increase the epidemic
threshold. This requires us to modify the underlying contact network
to reduce the degree variance $\langle k^2\rangle$. Evidently, large-degree nodes
are known to be responsible for the large variance of heterogenous
networks. Thus, if we vaccinate only the nodes whose degree exceeds a
preselected threshold value $k_c$, it should be easy to decrease the variance and increase the
epidemic threshold \cite{pastor2002immunization,dezsHo2002halting,barabasi2016network}. To this aim, Pastor-Satorras and Vespignani have proposed a degree-based targeted vaccination strategy in which the most highly connected
nodes are progressively immunized, given that these are the ones that are most likely to spread the disease \cite{pastor2002immunization,chen2008finding,borgatti2005centrality,cohen2003efficient,dezsHo2002halting,eames2009epidemic,gallos2007improving,gao2011network,miller2007effective,schneider2011suppressing,vidondo2012finding,wang2009imperfect}.
The vaccination of a fraction $x$ of large-degree nodes can be regarded as the removal of nodes whose degree is larger than a certain value $k_c$. Accordingly, $x$ thus can be implicitly defined as
\begin{equation}\label{x}
x=\sum_{k=k_c+1}^\infty P(k).
\end{equation}
Moreover, the average $\langle k\rangle_{c}$ and the second moment $\langle k^2\rangle_{c}$ of the degree distribution with maximum degree $k_c$ of the network are
\begin{equation}\label{k}
\langle k\rangle_{c}=\sum_{k=k_{min}}^{k_c}kP(k),
\end{equation}
and
\begin{equation}\label{k2}
\langle k^2\rangle_{c}=\sum_{k=k_{min}}^{k_c}k^2P(k),
\end{equation}
where $k_{min}$ denotes the minimal degree of the node in a network.
However, the removal of large-degree nodes will modify the degree distribution of the remaining nodes, owing to the deletion of the links between removed and remaining nodes. The probability that the link connected to the removed nodes is also removed equals
the probability that the link points to a node whose degree is larger than $k_c$. We thus have\\
\begin{equation}\label{f}
f=\sum_{k=k_c+1}^\infty \frac{kP(k)}{\langle k\rangle},
\end{equation}
\\
and the degree distribution of the resulting network becomes~\cite{pastor2002immunization,cohen2001breakdown}\\
\begin{equation}\label{}
P'(k')=\sum_{k=k'}^\infty\left (\begin{array}{l} k\\k'\end{array}\right ) f^{k-k'}(1-f)^{k'}P(k).
\end{equation}
\\
Finally, the average degree $\langle k'\rangle$ and the second moment $\langle k'^2\rangle$ of the resulting network are \cite{pastor2002immunization,cohen2001breakdown}\\
\begin{equation}\label{k'}
\langle k'\rangle=\sum_{k'=k_{min}}^{k_c}k'P'(k')=(1-f)\langle k\rangle_{c},
\end{equation}
\\
and\\
\begin{equation}\label{k'2}
\langle k'^2\rangle=\sum_{k'=k_{min}}^{k_c}k'^2P'(k')=(1-f)^2\langle k^2\rangle_{c}+f(1-f)\langle k\rangle_{c},
\end{equation}
\\
respectively.
For the SIS model, the epidemic threshold of the resulting network can be calculated as\\
\begin{equation}\label{}
{\beta_I}'_c=\frac{\langle k'\rangle}{\langle k'^2\rangle}=\frac{(1-f)\langle k\rangle_{c}}{(1-f)^2\langle k^2\rangle_{c}+f(1-f)\langle k\rangle_{c}}.
\end{equation}
\\
Assuming $2<\zeta<3$ and combining Eqs.~\ref{k}-\ref{f}, we obtain~\cite{barabasi2016network}\\
\begin{equation}\label{}
{\beta_I}'_c=\bigg[\frac{3-\zeta}{\zeta-2}k_c^{3-\zeta}k_{min}^{\zeta-2}-\frac{3-\zeta}{\zeta-2}k_c^{5-2\zeta}k_{min}^{2\zeta-4}+k_c^{2-\zeta}k_{min}^{\zeta-2}\bigg]^{-1}.
\end{equation}
\\
A similar approach to the SIR model yields\\
\begin{equation}\label{}
{\beta_I}'_c=\bigg[\frac{3-\zeta}{\zeta-2}k_c^{3-\zeta}k_{min}^{\zeta-2}-\frac{3-\zeta}{\zeta-2}k_c^{5-2\zeta}k_{min}^{2\zeta-4}+k_c^{2-\zeta}k_{min}^{\zeta-2}-1\bigg]^{-1}.
\end{equation}
\\
In both the SIS and the SIR model, if $k_c \gg k_{min}$, we have \cite{barabasi2016network}\\
\begin{equation}\label{}
{\beta_I}_c\approx
\frac{\zeta-2}{3-\zeta}{k_{min}^{2-\zeta}}{k_c^{\zeta-3}}.
\end{equation}
\\
For here it follows that if more large-degree nodes are vaccinated, and thus $k_c$ becomes smaller, the epidemic threshold will become larger. By vaccinating an appropriate fraction of large-degree nodes, we enable ${\beta_I}_c$ to increase beyond the spreading rate ${\beta_I}$ of the disease. Accordingly, the network is then protected completely.
In order to assess the efficiency of degree-based targeted vaccination, it is useful to derive the explicit expression for the vaccination threshold in an uncorrelated network with a power-law degree distribution $P(k)=ck^{-\zeta}$ and $c \approx (\zeta-1)/k_{min}^{-\zeta +1}$, where $\zeta=3$ \cite{pastor2002immunization}. In this case, we obtain
$x=k_{min}^2k_c^{-2}$, $f=x^{1/2}$, $\langle k\rangle_{c}=2k_{min}$, and $\langle k^2\rangle_{c}=2k_{min}^2\textit{\textit{ln}}(x^{-1/2})$ based on Eqs.~\ref{x}-\ref{f}. The critical vaccination threshold $x_c$ needed to eradicate the disease is then
\begin{equation}\label{b}
\frac{\langle k'\rangle}{\langle k'^2\rangle}={\beta_I}.
\end{equation}
By inserting above values into Eq.~\ref{b} and combining Eqs.~\ref{k'}-\ref{k'2},
we obtain the approximate solution for the vaccination threshold in the case of targeted vaccination \cite{pastor2002immunization}
\begin{equation}\label{}
x_c\sim \emph{\emph{exp}}(-2/k_{min}{\beta_I}).
\end{equation}
This equation indicates that the degree-based targeted vaccination strategy can be very effective, giving a critical vaccination threshold that is exponentially small for a wide range of spreading rates.
This theoretical prediction can be tested numerically with simulations of the SIS model on a scale-free network. Results presented in Fig.~\ref{threshold} compare the efficiency of targeted vaccination and random vaccination, respectively. It can be observed that for random vaccination the disease prevalence decays very slowly and vanishes only in the $x\rightarrow 1$ limit. Conversely, for targeted vaccination there is a very sharp drop towards
the vaccination threshold (indicated by the arrow), above which the
system is infection-free. A linear regression from the largest
value of $x$ yields an approximate vaccination threshold $x_c
\simeq 0.16$ in this case, which proves that scale-free networks are
very sensitive to degree-based targeted vaccination even if a very small
fraction of large-degree nodes is selected for vaccination
\begin{figure}
\centering
\includegraphics[scale=0.5,trim=50 0 50 0]{fig9.pdf}
\caption{Reduced prevalence of disease $\rho_x/\rho_0$ for the SIS model on a scale-free network, as obtained with random (circles) and targeted (squares) vaccination at a fixed spreading rate ${\beta_I} = 0.25$, respectively. A linear extrapolation from the largest values of $x$ yields an estimation of the threshold $x_c\simeq 0.16$ for targeted vaccination.
Source: Adapted with permission from Ref.~\cite{pastor2002immunization}. Copyrighted by the American Physical Society.}\label{threshold}
\end{figure}
\subsubsection{Centrality-based targeted vaccination: definitions and simulation-based models}
The procedure of degree-based targeted vaccination is equivalent to altering the structure of the network. Namely, by vaccinating the large-degree nodes, we fragment the contact network, thus making it more difficult for the disease to reach the nodes in the other components. Based on this principle, it is possible to devise alternative vaccination strategies which take advantage of the heterogenous connectivity patterns in order to achieve a high level of tolerance towards infections~\cite{pastor2002immunization}. Accordingly, other centrality indexes, such as betweenness, closeness, eigenvalue, or PageRank, can be used to design targeted vaccination strategies~\cite{friedl2010critical,chen2008finding,hebert2013global,latora2007measure,borgatti2005centrality,schneider2011suppressing,ventresca2013evaluation,salathe2010dynamics,schneider2011suppressing,schneider2012inverse,bonacich1987power,restrepo2006characterizing,miller2007effective,page1999pagerank,yang2016immunization,yang2011control,du2016identifying}. In what follows, we will briefly survey these centrality indexes and their applications in targeted vaccination.
\emph{Betweenness centrality}. Betweenness centrality is defined as
the number of shortest paths between pairs of nodes that pass through a given node~\cite{freeman1978centrality,christley2005infection,holme2002attack}. More precisely, if $\sigma_{vw}$ is the total number of shortest paths from node $v$ to node
$w$ and $\sigma_{vw}(i)$ is the number of those shortest paths that pass through the node $i$, the betweenness of $i$ is given by $b_i=\sum_{v\neq w\neq i}\frac{\sigma_{vw}(i)}{\sigma_{vw}}$.
Since nodes with large betweenness centrality are often bridges between different communities in the network \cite{yu2010finding,salathe2010dynamics}, the vaccination of these nodes has a good potential to break up a large network into smaller parts.
Hence, if an epidemic starts in a particular component of the network, it cannot infect nodes in
other components. Additionally, betweenness centrality is an
effective measure to identify high risk individuals, as it determines the
volume of flow passing through each node \cite{borgatti2005centrality}. That is also one of the main reasons why largest betweenness-based vaccination is considered to be the most
effective targeted vaccination algorithm \cite{shams2014using}.
\emph{Random-walk centrality}. Betweenness is, in some sense, a
measure of the influence a node has over the spread of the epidemic
through the network. By counting shortest paths, betweenness-based vaccination simply assumes that epidemics spread only along those shortest paths, which, however, is inconsistent with most empirical observations. To relax this hypothesis, Newman proposed the random-walk centrality that involves the contributions from essentially all the paths between nodes, though it still gives more weight to short paths \cite{newman2005measure}. The random-walk centrality of a node $i$ is a measure based on random walks, counting how often the node $i$ is traversed by a random walk
between any pair of nodes $s$ and $v$
\begin{equation}\label{}
r_i=\sum_{s<v}I_{sv}(i),
\end{equation}
where $I_{sv}(i)=\frac{1}{2}\sum_jA_{ij}|T_{is}-T_{iv}-T_{js}+T_{jv}|$
for $i\neq s, v$, and $T_{is}$ is the element in the voltage matrix. This measure is particularly useful for finding high-centrality vertices that do not happen to lie on geodesic paths or on the paths formed by maximum-flow cut-sets, which are still important for the epidemic spreading \cite{newman2005measure}. Salath\'e and Jones have proven that vaccination strategies based on random-walk
centrality can result in the lowest number of infected cases if the
vaccination coverage is low \cite{salathe2010dynamics}.
\emph{Closeness centrality}. Closeness centrality is based on the
assumption that nodes with a short distance to other nodes can spread
the disease very effectively across the network \cite{borgatti2005centrality,freeman1978centrality}. It is defined as
\begin{equation}\label{}
c_i=\frac{1}{\sum_{j\neq i}l_{ij}},
\end{equation}
where $l_{ij}$ is the distance between nodes $i$ and $j$. This measure
gives a large centrality to nodes which have small shortest path
distance to the other nodes. Therefore, immunizing largest-closeness nodes not only vaccinates high risk individuals, but also postpones epidemic spreading through the network \cite{christley2005infection,freeman1978centrality}.
\emph{Eigenvector centrality}. Degree centrality awards one
``centrality point'' for every network neighbor a node has. In reality, however,
all the neighbors are not equivalent. Under many circumstances, a node's
importance in one network increases via building connections to other
nodes that are themselves important, which is the concept behind
eigenvector centrality \cite{shams2014using,bonacich1972factoring,miller2007effective,bonacich2007some}. Instead of awarding just one point for each neighbor, eigenvector centrality gives each node
a score proportional to the sum of the score of its neighbors. That
is, the eigenvector centrality $e_i$ of a node $i$ is proportional
to the sum of the eigenvector centrality of the nodes it is
connected to. It is defined as
\begin{equation}\label{}
e_i=\Lambda^{-1}\sum_j a_{ij}e_j,
\end{equation}
where $a_{ij}$ denotes the adjacency matrix of the network, $\Lambda$ is the largest eigenvalue and $e_i$ is the $i^{th}$
component of the eigenvector associated with $\Lambda$ of the
network.
The impact of eigenvector centrality on epidemic control has been
evaluated by \cite{bonacich1972factoring,tomovski2012simple}. Tomovski and Kocarev used nonlinear system stability analysis of the SIS model and proved that the
epidemic threshold equals to the reciprocal value of the largest
eigenvalue of the network adjacency matrix \cite{tomovski2012simple}, that is
\begin{equation}\label{}
{\beta_I}_c=\frac{1}{\Lambda}.
\end{equation}
Therefore, the risk of infection through networks can be reduced
by decreasing $\Lambda$. Along the same lines, Masuda defined $I_i$ as the decrement
of $\Lambda$ owing to the removal of node $i$ \cite{masuda2009immunization}
\begin{equation}\label{}
I_i=-\frac{\Delta \Lambda}{\Lambda}\approx\frac{e_i^2}{\sum_j e_j^2}.
\end{equation}
From here it is clear that the vaccination of a node with a large $e_i$ ($I_i$) will
significantly increase the epidemic threshold, and thus the epidemic will be effectively
suppressed.
\emph{PageRank centrality}. The PageRank centrality \cite{page1999pagerank},
introduced by Google for webpage ranking, is given by
\begin{equation}\label{}
p_i=c\sum_{j\in B_i}\frac{p_j}{N_j}+(1-c)1/N,
\end{equation}
where $i$ is a web page, $B_i$ represents the set of pages that point to $i$,
and $N_j$ denotes the number of pages that $j$ points to. Moreover, $c$ is a random jumping factor which is introduced by assuming that the web surfer
will browse the web pages along the links with probability $c$, and leave the current page and open a random page with probability $1-c$. Similar to eigenvector centrality, PageRank supposes that the importance of a node is determined by both the quantity and
the quality of the nodes that point to it. In terms of epidemic spreading, nodes with large PageRank centrality are more likely to be infected
or infect others \cite{miller2007effective}. Their
vaccination can thus eliminate a large number of potentially very effective disease transmission routes. Since nodes with large PageRank centrality have many low-degree neighbors, largest PageRank vaccination will immunize influential nodes whose vaccination choice can strongly protect their neighbors \cite{miller2007effective}.
To obtain a better understanding of all these proposed vaccination schemes, Shams et al. compared the efficiency of different centrality indexes in terms of the vaccination threshold $x_c$, and plotted it as a function of the average degree $\langle k\rangle$ for different network models \cite{shams2014using}. As shown in Fig.~\ref{gcvsk}, it can be observed that HP (for the meaning of the abbreviation please refer to the caption of Fig.~\ref{gcvsk}) delivers the best performance on all the networks, HB is second-best, while HE and HC deliver the worst performance. Despite of the weak performance of HD in small-world networks, the approach performs well on scale-free and ER random networks.
\begin{figure}[!htb]
\centering
\includegraphics[scale=0.4,trim=50 0 50 0]{fig10.pdf}
\caption{The critical threshold of vaccination $x_c$, as a function of the average degree $\langle k\rangle$ on different networks. The legend refers to the following vaccination strategies: targeted vaccination based on degree (HD), betweenness (HB), closeness (HC), eigenvalue (HE), and PageRank (HP). Source: Reprinted figure from Ref.~\cite{shams2014using}}
\label{gcvsk}\end{figure}
Moreover, we point out that targeted vaccination techniques can be improved by adaptive strategies, in which the centrality is recalculated for the network of unvaccinated nodes at each step of the vaccination process \cite{holme2002attack}. For example, degree-based vaccination with dynamic re-ranking entails an immediate update of node degrees, and then the nodes with the largest number of unvaccinated neighbors are vaccinated at each time step \cite{miller2007effective}. When a node has been vaccinated, the adaptive betweenness centrality is recalculated for the remaining network composed of unvaccinated nodes~\cite{holme2002attack,schneider2011suppressing,schneider2012inverse,miller2007effective}. Although such adaptive approaches are more efficient from the theoretical viewpoint, they usually require more information, and the complexity of the procedure might make a practical implementation difficult for all but very small populations \cite{schneider2012inverse}.
\subsection{Vaccination without global knowledge}
Although targeted strategies are in principle very effective, an important drawback in terms of practical applications in the real world is that they require complete knowledge of the network structure. Only so it is possible to identify and then vaccinate the most influential nodes as determined by the employed metric. However, the complete structure of the network is seldom, if ever, known, and often it is also not well-defined. In social networks, for example, the number of links an individual has depends strongly on the criteria based on which the network is constructed. To overcome the lack of global knowledge of the network structure, several different strategies have been proposed that require information only about the local structure of the network. The efficiency of these vaccination strategies can nevertheless be very similar to that of targeted vaccination, as reviewed in what follows.
\subsubsection{Acquaintance vaccination: theoretical prediction of the vaccination threshold}
The so-called acquaintance vaccination was first proposed by Cohen et al.~\cite{cohen2003efficient,madar2004immunization}, in which a fraction $p$ of nodes is
selected at random, with the constraint that each node is required to have at least one connection to another individual in the network. Subsequently, this neighbor, rather than the originally selected node, is vaccinated. The strategy thus requires no knowledge of the degree of nodes or other global knowledge about the network.
More precisely, the probability that a node with degree $k$ is selected for vaccination is $kP(k)/(N\langle
k\rangle)$. This quantifies the known fact that randomly selected
acquaintances have, on average, a higher degree than randomly
selected nodes \cite{feld1991your,newman2002assortative}. Based on percolation theory, Cohen et al. developed a theoretical framework to determine the critical fraction $p_c$ and vaccination threshold $x_c$. They followed a possible branch in the course of the epidemic, starting from a random link of the spanning
cluster. That is, they studied the possible spreading of the
epidemic by considering nodes that are not vaccinated, and are therefore susceptible to the epidemic and may become infected \cite{cohen2003efficient}. If $n_l(k)$ denotes the number of nodes with degree $k$ in some layer $l$ (hop-distance from the starting point), the number $n_{l+1}(k')$ of nodes with degree $k'$ that are susceptible in layer $l+1$ is given by\\
\begin{equation}\label{l+1}
n_{l+1}(k')=\sum_k n_l(k)(k-1)p(k'|k,s_k)p(s_{k'}|k',k,s_k),
\end{equation}
\\
where $k-1$ means that the node with degree $k$ in layer $l$ has $k-1$ new neighbors in layer $l+1$ (excluding the one through which we arrived). Moreover, $p(k'|k,s_k)$ is the probability of reaching a node with degree $k'$ by following a link from a susceptible node of degree $k$, while $s_k$ is the probability that a node with degree $k$ is susceptible. Lastly, $p(s_{k'}|k',k,s_k)$ is the probability that the reached node is also susceptible.
Based on Bayes' rule, it holds that\\
\begin{equation}\label{}
p(k'|k,s_k)=\frac{p(s_{k}|k,k')p(k'|k)}{p(s_{k}|k)}.
\end{equation}
\\
Furthermore, assuming that the network is uncorrelated, we have\\
\begin{equation}\label{}
\phi(k')=p(k'|k)=k'P(k')/\langle
k\rangle,
\end{equation}
\\
which is independent of $k$. The probability that the acquaintance is not selected in one particular vaccination attempt by a random node of degree $k$, is $1-1/(Nk)$, and in all $Np$ attempts, it reads\\
\begin{equation}\label{}
v_p(k)=(1-\frac{1}{Nk})^{Np}\approx e^{-p/k}.
\end{equation}
\\
However, if the neighbor's degree is not known, the average probability becomes $v_p=\langle v_p(k)\rangle$. Then the probability that a node with degree $k'$ is susceptible (not vaccinated) is\\
\begin{equation}\label{}
p(s_{k'}|k')=\langle v_p(k)\rangle^{k'},
\end{equation}
\\
if no other information on its neighbors exists. But when the degree of one neighbor is known to be $k'$, we have\\
\begin{equation}\label{}
p(s_{k}|k,k')=e^{-p/k'}\times\langle e^{-p/k}\rangle^{k-1}.
\end{equation}
\\
Since the fact that a neighbor with known
degree is vaccinated does not provide any further information about a node's probability of vaccination, it follows $p(s_{k}|k,k')=p(s_{k}|k,k',s_{k'})$.
Substituting above results in Eq.~\ref{l+1}, we obtain\\
\begin{equation}\label{l1}
n_{l+1}(k')=n_{l}(k')\sum_k\phi(k)(k-1)v_p^{k-2}e^{-2p/k}.
\end{equation}
\\
If the sum in the above equation is larger than 1, the branching
process will continue forever, which is referred to as the percolating phase. On the contrary, if it is smaller than 1, the vaccination is subcritical and the epidemic will be contained. Thus, we obtain a relation for $p_c$, which is\\
\begin{equation}\label{}
\sum_k\frac {P(k)k(k-1)}{\langle k\rangle}v_{p_c}^{k-2}e^{-2p_c/k}=1,
\end{equation}
\\
and the vaccination threshold is easily obtained
from the fraction of nodes which are not susceptible, namely\\
\begin{equation}\label{}
x_c=1-\sum_kP(k)p(s_{k}|k)=1-\sum_kP(k)v_{p_c}^k.
\end{equation}
\\
Results presented in Fig.~\ref{acquaintance vaccination} show the critical threshold that is required to eradicate a disease in a scale-free network. It can be observed that acquaintance
vaccination is more efficient than random vaccination for both high and low $\zeta$ values, corresponding to homogeneous and heterogenous networks, respectively.
\begin{figure}
\centering
\includegraphics[scale=0.4,trim=50 0 50 0]{fig11.pdf}
\caption{The critical vaccination threshold $x_c$ as a function of the degree exponent $\zeta$ of the contact network on which the disease spreads via the SIS model. The curves
correspond to two distinct vaccination strategies: uniform vaccination (top curve) and acquaintance vaccination (lower curve).
The continuous lines represent the analytical results while the
symbols represent simulation data for $N = 10^6$ and $k_{min} = 1$. Source: Adapted with permission from Ref.~\cite{cohen2003efficient}. Copyrighted by the American Physical Society.}\label{acquaintance vaccination}
\end{figure}
\subsubsection{Other strategies: definitions and simulation-based models}
In addition to acquaintance vaccination reviewed above, similarly efficient strategies can be devised based on local information about the structure of the network. An example is the so-called random walk vaccination strategy \cite{noh2004random}, in which a random walker diffuses through the network, and every node that it visits is vaccinated until a given fraction of the population is immune~\cite{ke2006immunization}. Given that a random walker visits a node with degree $k_i$ with the probability proportional to $k_i$, this strategy could lead to the same effectiveness as achieved by acquaintance vaccination \cite{pastor2015epidemic}. Another alternative to acquaintance vaccination is common acquaintance vaccination, where only the common neighbors of the randomly chosen nodes are selected for vaccination~\cite{liu2009common}. This may alleviate inefficiencies related to the fact that, of course, only very few of the randomly selected nodes and their neighbors will be the hubs of the network. Indeed, common acquaintance vaccination proves to be somewhat more effective than the simpler acquaintance vaccination.
The acquaintance vaccination strategy can also be improved by taking into account additional information about the local structure of the network. For example, if each node has the local knowledge about the degree of its nearest neighbors, and selects the neighbor with the degree to be vaccinated, then the efficiency in comparison to acquaintance vaccination is significantly improved \cite{holme2004efficient}. Similarly, the random walk vaccination strategy can also be improved via a bias favoring the exploration of large-degree nodes during the random walk process \cite{ke2006immunization,lee2009centrality}.
Along the same line, if even more local information is available, for example if nodes have the knowledge about the degree of their neighbors within short paths of length $D$, then the so-called $D$-steps vaccination strategy can be employed~\cite{gomez2006immunization,echenique2005distance}. In particular, for every node $i$, simply look for the largest-degree node within the distance $D$ and vaccinate it. In case several nodes within the distance have the same largest degree, then choose one of them uniformly at random.
Based on these considerations, in \cite{gomez2006immunization} detailed numerical simulations of four different vaccination schemes, including $D$-steps vaccination, degree-based targeted vaccination, random vaccination, and acquaintance vaccination, were performed and tested in terms of their efficiency. As shown in Fig.~\ref{dstep}, targeted vaccination expectedly produces the best results, and this irrespective of the interaction topology. Also understandably, the performance of random vaccination is not affected by the structure of the network. Turning to local algorithms, it is found that the vaccination scheme based on the covering algorithm performs
better than acquaintance vaccination, even for small
values of $D$. In fact, the $D$-steps vaccination is only outperformed by the targeted procedure and lies between the most efficient and the acquaintance vaccination scheme for all the values of $D$ and ${\beta_I}$. Moreover, from a practical point of view, the covering strategy could be a good policy since it balances the degree of
local knowledge and the efficiency of the vaccination. As most empirical network topologies are neither completely known nor completely unknown, this method thus allows us to fine-tune the value of $D$ on a case-by-case basis, i.e., according to the degree of
local knowledge of the network.
\begin{figure}
\centering
\includegraphics[scale=0.5,trim=50 0 50 0]{fig12.pdf}
\caption{Comparison of different vaccination strategies for the Internet router map. $R/R_{SIR}$ represents the ratio between the epidemic incidence ($R$) of the four proposed vaccination strategies and the epidemic incidence $(R_{SIR})$ without vaccination. The legend refers to the following vaccination strategies: $D$-steps vaccination (Local), targeted vaccination
($K_{max}$), random vaccination (Random) and single acquaintance vaccination (SAI). In each case, 1\% of the non-vaccinated nodes were initially infected at random. The distance considered in the local algorithm is: (a) $D = 1$; (b) $D = 2$; (c) $D = 3$; (d) $D = 5$. Source: Adapted with permission from Ref.~\cite{gomez2006immunization}. With permission of Springer.}
\label{dstep}\end{figure}
In addition, Gao et al. evaluated the relationship between the threshold $x_c$ of the aforementioned vaccination strategies and the power-law exponent of the degree distribution of the network, which is shown in Fig.~\ref{gcvspower}~\cite{gao2011network}. A low level of $x_c$ with a small variation indicates that the immunization strategy is robust. Results presented in Fig.~\ref{gcvspower} also show that the robustness of the $D$-steps vaccination strategy is close to that of targeted vaccination.
\begin{figure}
\centering
\includegraphics[scale=0.45,trim=50 0 50 0]{fig13.pdf}
\caption{The critical vaccination threshold $x_c$ as a function of
the power-law exponent of the degree distribution of the network. The legend refers to the following vaccination strategies: acquaintance vaccination (Random Neighbor and Max neighbor), $D$-steps vaccination (D-steps D=2 and D-steps D=3), node degree-based targeted vaccination
(Target) and node betweenness-based targeted vaccination (NodeBetweenness). Source: Adapted with permission from Ref.~\cite{gao2011network}. With permission of Springer.}
\label{gcvspower}\end{figure}
Vaccination plays an obviously important role in the study of epidemic transmission and prevention. Based on the above-reviewed research, we can conclude that the efficiency of vaccination depends significantly on the amount of information we have about the structure of the network, and whether this information concerns the global or the local structure. In the rare case information about the global structure is available, targeted vaccination works best. If only information about the local structure is available, then the $D$-steps vaccination seems to provide the optimal compromise between efficiency and flexibility. Table~\ref{table-vaccination} gives a systematic summary of the reviewed vaccination strategies, for a quick overview.
\begin{sidewaystable}\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}\centering{
\caption{Summary of non-behavioral epidemiological vaccination strategies, with details concerning vaccinated nodes, network models, and the required knowledge about the structure of the network.}\footnotesize \label{table-vaccination}
\begin{tabular}{|l|l|l|l|}
\hline
Vaccination strategy & Vaccinated nodes & Applied network models & Required knowledge of topology\\\hline
uniform (or random) vaccination & random chosen nodes & \tabincell{l}{ homogeneous networks \\ less heterogeneous networks} & no any need of network information \\\hline
targeted vaccination & nodes with high centrality\footnote{Centrality includes degree centrality, betweenness centrality, random-walk centrality, closeness centrality, eigenvector centrality and PageRank centrality.} & \tabincell{l}{heterogeneous networks \\networks with special centrality} & global knowledge of network topology\\\hline
acquaintance vaccination\footnote{Except for simple acquaintance vaccination, it also includes common acquaintance vaccination, random walk vaccination, D-steps vaccination strategy.} & random neighbor of random chosen node\footnote{Correspondingly, it also involves common neighbor of random chosen nodes, nodes visited by random walker, largest-degree node within the distance $D$ of random chosen node.} & \tabincell{l}{ homogeneous networks \\ heterogeneous networks} & local knowledge of network topology\\
\hline
\end{tabular}}
\end{sidewaystable}
\subsection{Vaccination on other types of networks}
Up to now, we have focused mainly on research concerning vaccination on traditional network models. This research provides vital insight in terms of the efficiency of vaccination under different circumstances, yet empirical networks usually have particular topology properties that require special care. This includes taking into account community structure~\cite{girvan2002community,newman2004finding,palla2005uncovering,newman2006modularity,liu2005epidemic,wu2008community}, changes in the network structure over time~\cite{gross2006epidemic,gross2008adaptive,gross2009adaptive,shaw2008fluctuating,marceau2010adaptive,shaw2010enhanced,gao2013modeling}, as well as multilayer properties that are typical for a large plethora of real-world networks~\cite{boccaletti2014structure,kivela2014multilayer,wang2016suppressing,wang2014asymmetrically,liu2015impacts,wang2014epidemic,gomez2013diffusion,cozzo2013contact,zhao2014multiple,zhao2014immunization,salehi2015spreading,zhao2016roin,gao2016competing,zhao2015finding}. In what follows, we will review vaccination programs that take into account the particular aspects of these properties on various types of networks.
\subsubsection{Vaccination on community networks}
Community structure is ubiquitous in a variety of real complex systems, such as human contact networks and social networks. The hallmark of this fact is that the connections among members
of the same community are much more common than connections among members of different communities \cite{girvan2002community,newman2004finding,palla2005uncovering,newman2006modularity,liu2005epidemic,wu2008community}. For networks with community structure, the weak ties that
connect a pair of nodes belonging to different communities, usually named the bridge nodes, therefore typically provide pathways for information and disease
to propagate from one community to the other. These nodes thus play a more important role in the spreading of disease than the nodes with no or fewer inter-community links \cite{onnela2007structure,zhao2010weak}. However, their importance
is not necessarily reflected by their degree centrality. In this regard, identifying
the bridge nodes in community networks is crucial in preventing
epidemic outbreaks~\cite{masuda2009immunization,gong2013efficient}. Identification of community bridges can be partially achieved by betweenness centrality
or random-walk centrality, since both locate nodes on the basis of
important paths of epidemic spreading. On a similar note, Chen et al. proposed a vaccination strategy based on an equal graph partitioning algorithm
to identify the minimum group that separates a network into several
clusters of approximately equal size \cite{chen2008finding}. The role of the
minimum separator group is actually similar to that of the bridge nodes
connecting different communities.
Salath{\'e} et al.~\cite{salathe2010dynamics} subsequently proposed a dedicated community bridge finder (CBF) algorithm that searches for the bridge nodes, thereby requiring only local structural
information about the structure of the network. Their research showed that such a method is more efficient than vaccination strategies targeting different kinds of hubs. The algorithm is based on the self-avoiding walk, which starts from a randomly chosen node
$v_0$, and then follows the procedure outlined in Fig.~\ref{CBF}. After $t$ steps ($t\geq 2$), the set of all the visited nodes is denoted by $\{v_{t'}\}$ for
$t'=0,1,\ldots,t$. Thus, $v_t$ is the node where the walker is located after $t$ steps, and $v_{t-1}$ is the node visited at step
$t-1$. The first process is to examine whether the node $v_t$ has a
link or several links to the nodes in the set $\{v_{t'}\}$, other
than the link between $v_t$ and $v_{t-1}$. If yes, the
self-avoiding walk proceeds towards step $t+1$; otherwise $v_{t-1}$
will be considered as a possible target of the bridge node. To determine
whether the node $v_{t-1}$ is one bridge node, two nodes are randomly
chosen among all the possible nodes that the walker could visit in
step $t+1$, i.e. two neighbors of the node $v_t$ are randomly chosen (except for the node $v_{t-1}$ due to the self-avoiding restriction of the walk). If there exists a path from any of the two chosen nodes back to any node in the set $\{v_{t'}\}$, then the node $v_{t-1}$ will not be a bridge node and the walker moves to node $v_{t+1}$. If there exists no path back to the set $\{v_{t'}\}$ from both chosen nodes, $v_{t-1}$ is
regarded as a bridge node that connects two communities and is vaccinated. Then a new self-avoiding walk starts and the above procedure is repeated until the desired vaccination ratio is attained. An important idea behind the CBF algorithm is actually that a community is formed by a circle of
close friends. Thus, when two randomly chosen neighbors of
$v_t$ cannot trace back to the community that $v_{t-1}$ belongs
to, the link between $v_{t-1}$ and $v_{t}$ is likely to be a bridge
between two communities, and the node $v_{t-1}$ is hence a bridge node.
It has been found that community networks typically exhibit a heterogeneous distribution
in the number of weak ties originating from the bridge nodes \cite{guimera2005functional,guimera2005worldwide}. Inspired by this finding, Gong et al. turned their attention to a particular kind of bridge nodes, named bridge hubs, which connect a community to many other communities, i.e., bridge hubs have a large number of weak ties~\cite{gong2013efficient}. They proposed a vaccination strategy named bridge-hub detector (BHD) to identify the bridge hubs. This algorithm extends the self-avoiding searching scheme to examining the overlap and the existence of links from all the neighbors of the last node back to the union of the friendship circles of all the nodes in the trail of the walk. A pair of nodes, a bridge node and a bridge hub, are
searched for vaccination via a self-avoiding walk. Compared with other local vaccination strategies such as acquaintance vaccination and CBF, this strategy is proven to be
more effective in preventing the epidemic~\cite{gong2013efficient}.
Also of note, Masuda recently proposed a vaccination strategy based on the eigenvector centrality for community-structured networks~\cite{masuda2009immunization}. In his work, a module based strategy technique is employed for measuring the contribution of each node to the weighted community network in order to preferentially vaccinate nodes that bridge important communities. The number of links that these nodes share gives the weight of
a link between any two communities. The influence of a node is thus related to the importance of the community together with its connectivity to other communities.
\begin{figure}
\centering
\includegraphics[scale=0.4,trim=50 0 50 0]{fig14.pdf}
\caption{A schematic presentation of the community bridge finder (CBF) algorithm. (a) A
random walker follows the path starting from $v_0$ to $v_1$ and $v_2$,
at which point it starts checking the connections from $v_2$ to $v_0$
and $v_1$. (b) Since there is more than one connection ($v_2-v_1$
and $v_2-v_0$), the walker turns to $v_3$. (c) Except for the obvious
$v_3-v_2$, there are no connections from $v_3$ to any of the
previously visited nodes, so $v_2$ is a potential target. (d) The
algorithm then picks two random neighbors of $v_3$ to check
connections with previously visited nodes and finds one (to $v_0$).
(e) Hence, $v_2$ is dismissed as a potential target, and the random
walker comes to $v_4$. Again, $v_4$ does not connect to any
previously visited node (except to $v_3$), and thus
$v_3$ is identified as a potential target.(f) Again, two random
neighboring nodes are picked to check connections with previously
visited nodes. Since no back connections can be found, $v_3$ is
identified as a target and vaccinated. Source: Reprinted figure from Ref. \cite{salathe2010dynamics}}\label{CBF}
\end{figure}
\subsubsection{Vaccination on adaptive networks}
From the epidemiological viewpoint, when an infectious disease occurs in a population, individuals are likely to adopt some self-protection measures to protect themselves from the disease. For example, susceptible people may break their contacts with infected partners, which can significantly alter the structure of the contact network, thus influencing the pathway of the epidemic spreading \cite{gross2006epidemic,shaw2008fluctuating,marceau2010adaptive,shaw2010enhanced,yang2012efficient,ruan2012epidemic,zhao2013efficient,zhou2012epidemic}. Both SIS and
SIR models have been studied on adaptive networks in which susceptible nodes rewire their links adaptively from infected neighbors towards other non-infected nodes~\cite{gross2006epidemic,shaw2008fluctuating}.
Such adaptation typically increases the epidemic threshold and
reduces the number of infectious cases, thus contributing favorably to the containment of epidemic spreading.
\begin{figure}
\centering
\includegraphics[scale=0.42,trim=50 0 50 0]{fig15.pdf}
\caption{The effect of IMR strategy and IMSI strategy versus the
starting time $T_0$ of quarantine for (a) $\Delta i_0(T_0)$, (b) $i_0$,
$i_{max}$, (c) $\Delta t(T_0)$ and (d) $r(T_0)$. Source: Reprinted figure from Ref.~\cite{yang2012efficient}}
\label{IMR}
\end{figure}
Moreover on the subject, Shaw et al. have studied vaccine control of disease spreading on an adaptive
network to model the specifics of disease avoidance behavior \cite{shaw2010enhanced}. Control
was implemented by adding a Poisson-distributed vaccination to
the susceptible individuals. Their research showed that vaccine control is much more effective
in adaptive networks than static networks due to the feedback
loop between the adaptive rewiring and the administration of the vaccine. On the other hand, Yang et al. have focused on the transient process rather than
the steady state, and found that strong community structure ($S$ nodes
community and $I$ nodes community, connected by $SI$ links) is
induced by the rewiring mechanism in the early stages of the epidemic
spreading \cite{yang2012efficient}. Subsequently, they proposed and examined various vaccination strategies that build on this fact, targeting in particular such $SI$ links. In particular, the vaccination of randomly selected $S$ bridge nodes from $SI$ links (IMSI) and the vaccination of randomly selected $I$ bridge nodes
from $SI$ links (ISSI) were considered. To evaluate the efficiency of IMSI and ISSI strategies, the vaccination of randomly selected $S$ nodes from the network (IMR) and the vaccination of randomly
selected $I$ nodes from the network (ISR) were also employed.
\begin{figure}
\centering
\includegraphics[scale=0.42,trim=50 0 50 0]{fig16.pdf}
\caption{The effect of ISR strategy and ISSI strategy versus the starting time $T_0$ of quarantine for (a) $\Delta i_0(T_0)$, (b) $i_0$, $i_{max}$, (c) $\Delta t(T_0)$ and (d) $r(T_0)$. Source: Reprinted figure from Ref.~\cite{yang2012efficient}}
\label{ISR}
\end{figure}
In terms of their results, in Figs.~\ref{IMR} and \ref{ISR}, $T_0$ denotes the starting time of
vaccination, $\Delta i_0(T_0)$ is the difference between the maximal
density of infected individuals $i_{max}(T_0)$ that can be reached after
vaccination and the density of infected individuals $i_0(T_0)$ at time $T_0$, i.e. $\Delta i_0(T_0)=i_{max}(T_0)-i_0(T_0)$. $\Delta t(T_0)$ represents the time interval of this process, and $r(T_0)$ denotes the total percentage of vaccinated nodes in $\Delta t(T_0)$.
In Fig.~\ref{IMR}, the impact of the IMR strategy is almost equal to
that of the IMSI strategy. Because most $S$ nodes are located on $SI$ links
when an epidemic prevails in an adaptive network, randomly vaccinating $S$
nodes can give rise to similar effects as vaccinating
$S$ nodes on $SI$ links. Nevertheless, for ISR and ISSI,
quarantining $I$ nodes on $SI$ links is significantly more effective than quarantining $I$ nodes randomly, since the former can
efficiently cut the pathways along which the disease could be invading the $S$ cluster
(Fig.~\ref{ISR}). Because of this fact, the ISSI strategy is more efficient in controlling the epidemic and has a larger optimal region. More importantly, the consideration of these different vaccination strategies reveals a counterintuitive conclusion: ``the
earlier the better'' does not necessarily hold for the implementation of vaccination measures. Optimal results can be expected more realistically if smart vaccination strategies are implemented when the community structure emerges after the initial rewiring of links that sets in due to an epidemic.
\subsubsection{Vaccination on multiplex networks}
\begin{figure}
\centering
\includegraphics[scale=0.5,trim=50 0 50 0]{fig17.pdf}
\caption{Schematic illustration of a multiplex configuration. (a) In a multiplex network, six nodes are connected through two different types of links, i.e., blue and black links. In this configuration, each node simultaneously belongs to two layers, and it is thus termed as a multiplex node. (b) Multiplex networks are composed of several, for example two, layers, each of which is composed by the same type of links. The term layer node indicates that the node is connected only through the one type of link in the corresponding layer. See main text for details with regards to the importance of this classification for vaccination strategies. Source: Reprinted figure from Ref.~\cite{zhao2016robustness}.}
\label{multi-layer}
\end{figure}
Multiplex networks, as a typical kind of multilayer network structure, can be regarded as the combination of several network layers, which contain the same nodes (or share at least some fraction of nodes) yet different intra-layer connections \cite{boccaletti2014structure,kivela2014multilayer,wang2016suppressing,wang2014asymmetrically,liu2015impacts,wang2014epidemic,peng2010models,gomez2013diffusion,cozzo2013contact,zhao2014multiple,zhao2014immunization,salehi2015spreading,zhao2016robustness,du2016analysis}. For each node, we term its counterpart in every layer as a `replica'. Along the above definition, may real-world social and engineering systems, such as online social networks \cite{min2014layer,hu2014conditions,zagenczyk2015multiplex,ma2015social,xie2014construction,shen2015novel,guo2014varibale,centola2010spread}, airport traffic networks \cite{hosseini2016review,du2016analysis,du2016physics}, biological metabolic networks \cite{de2016mapping,sporns2016modular}, and scientific collaboration and citation networks \cite{battiston2015emergence}, can be put into the framework of multiplex networks. Multiplex networks naturally shed new light into the research of epidemiology (see Refs. \cite{boccaletti2014structure,salehi2015spreading} for a comprehensive understanding). For example, the well known acquired immune deficiency syndrome (AIDS) usually propagates via three types of ways, namely sexual activity, blood, and breast milk. Different transmission routes could be mapped into different network layers which contain the same individuals yet different network topologies and dynamic properties \cite{zhao2014multiple}. Moreover, coupling methods from different layers will greatly affect the onset of disease and its outbreak threshold \cite{zhao2014multiple}. In what follows, we will focus on recent research concerning vaccination in multiplex networks.
To distinguish the node of a multiplex network and its replica in each network layer, Zhao et al. proposed the terminology of the multiplex node and the layer node~\cite{zhao2014immunization}. The former refers to the node which has neighbors in each layer via different intra-layer connections, like node 3 of Fig.~\ref{multi-layer}(a), whose neighbors are nodes $1$, $5$ and $4$ via blue and black links. While the latter is the partial case of a multiplex node and just involves the local connections of a node in one given layer. For example, if we only consider blue (black) links in Fig.~\ref{multi-layer}(b), node $3$ is the layer node of Layer-1 (Layer-2). Correspondingly, vaccination on multiplex networks could be classified into multiplex node-based vaccination and layer node-based vaccination. Multiplex node-based vaccination means that all the replicas of one node simultaneously possess immunity in all the layers, while layer node-based vaccination provides protection solely to nodes in a given network layer.
\begin{table}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\caption{Vaccinated probability of layer nodes and multiplex nodes for different vaccination strategies}\label{Vaccinated probability}
\begin{tabular}{|c|c|c|}
\hline
vaccination type \& probability & layer node-based vaccination & multiplex node-based vaccination\\
\hline
random vaccination & $x(k_{j_i})=x_i $, for $\forall j$& $x(\overrightarrow{k_j})=x$, for $\forall j$\\
\hline
targeted vaccination & $x(k_{j_i})=\left\{\begin{array}{ll} 1,\ \ \emph{\emph{if}}\ k_{j_i}>k_{ci}\\
f_i,\ \ \emph{\emph{if}}\ k_{j_i}=k_{ci}\\ 0,\ \ \emph{\emph{if}}\
k_{j_i}<k_{ci}\end{array}\right.$ & \tabincell{l}{$x(\overrightarrow{k_j})=\left\{\begin{array}{ll} 1,\ \ \emph{\emph{if}}\ K_j>K_c\\
f,\ \ \emph{\emph{if}}\ K_j=K_c\\ 0,\ \ \emph{\emph{if}}\
K_j<K_c\end{array}\right.$ \\ where $K_j=\sum\limits_{i=1}^m{\beta_I}_i{k_j}_i$} \\
\hline
acquaintance vaccination & \tabincell{l}{$x(k_{j_i})=1-$\\ $(1-q_i\sum\limits_{\overrightarrow{k'_j}}
\frac{k'_{j_i}p(\overrightarrow{k'_j})}{z_i}
\frac{1}{k'_{j_i}})^{k_{j_i}}$} & \tabincell{l}{$x(\overrightarrow{k_j})=1-$\\ $\prod_{i=1,\cdots,m}(1-\frac{q}{m}\sum\limits_{\overrightarrow{k'_j}}
\frac{k'_{j_i}p(\overrightarrow{k'_j})}{z_i}
\frac{1}{k'_{j_i}})^{k_{j_i}}$}\\
\hline
\end{tabular}\\
\scriptsize $\overrightarrow{k_j}=({k_j}_1,{k_j}_2,...,{k_j}_m)$: the degree expression of multiplex node $j$; ${k_j}_i$ refers to its degree in layer $i$.\\
\scriptsize $x(k_{j_i})$: the vaccinated probability of layer node $j$ in layer $i$ with degree $k_{j_i}$.\\
\scriptsize $x(\overrightarrow{k_j})$: the vaccinated probability of multiplex node $j$. \\
\scriptsize ${\beta_I}_i$: the infectious rate of the disease in layer $i$.\\
\scriptsize $k_{ci}$: the cutoff degree for vaccination of layer node in layer $i$.\\
\scriptsize $K_j$: the spreading degree of multiplex node $j$.\\
\scriptsize $K_c$: the cutoff spreading degree for vaccination of multiplex node.\\
\scriptsize $z_i$: the average degree of layer $i$.
\end{table}
Similar to the vaccination strategies on single-layer networks, Refs.~ \cite{zhao2014immunization,wang2015immunity,buono2015immunization,zuzek2015epidemic} proposed the corresponding vaccination strategies (including random vaccination, targeted vaccination, and acquaintance vaccination) on multiplex networks. Here, however, the aim of vaccination is specified to layer nodes and multiplex nodes. These vaccination strategies are determined by the vaccinated probability of layer nodes and multiplex nodes, which are summarized in Table~\ref{Vaccinated probability}
\emph{Multiplex node-based vaccination} refers to the case where a fraction
of multiplex nodes is vaccinated. If $x(\overrightarrow{k_j})$ is defined as the vaccinated probability
of multiplex node $j$, the generating function of the joint
degree distribution under multiplex node-based vaccination is
given by\\
\begin{equation}
G_0(\overrightarrow{u})=\sum\limits_{\overrightarrow{k_j}}
p(\overrightarrow{k_j})(1-x(\overrightarrow{k_j}))\prod\limits_{i=1}^{m}u_i^{{k_j}_i}.
\end{equation}
\\
The generating function of the remaining joint degree distribution by
following a randomly chosen link of layer $i$ is defined as\\
\begin{equation}
G_1^{(i)}(\overrightarrow{u})=\frac{1}{z_i}\frac{\partial}{\partial
u_i}G_0(\overrightarrow{u}).
\label{u-1}
\end{equation}
\\
Then, the probability $\upsilon_i$ that a multiplex node
connects to the infected cluster by following a random link of layer $i$ is given by the coupled self-consistency equation\\
\begin{equation}
\upsilon_i=G_1^{(i)}(\overrightarrow{1})-G_1^{(i)}(\overrightarrow{1-{\beta_I}
\upsilon}),
\label{u}\end{equation}
\\
where $\overrightarrow{1-{\beta_I} \upsilon}=(1-{\beta_I}_1 \upsilon_1,...,1-{\beta_I}_m
\upsilon_m)$. It is worth mentioning that the cluster of multiplex networks is defined as a set of connected multiplex nodes. A pair of multiplex nodes is regarded as having a connection if there exists at least one type of link between them.
Thus, the existence of an epidemic regime under multiplex node-based
vaccination just requires the largest eigenvalue $\Lambda$ of the
Jacobian matrix of Eq.\ref{u} at (0,0,...,0) to be larger than unity
\cite{min2014network}. Consequently, the critical vaccination threshold $x^{M}_c$ will
be the value of immunity $x^{M}=\sum p(\overrightarrow{k_j})x(\overrightarrow{k_j})$ satisfying $\Lambda=1$ in Eq. \ref{u}.
\emph{Layer node-based vaccination}, in contrast, aims to vaccinate layer nodes. To calculate the critical vaccination threshold, we just need to replace $G_0(\overrightarrow{u})$ by\\
\begin{equation}
G'_0(\overrightarrow{u})=\sum\limits_{\overrightarrow{k_j}}
p(\overrightarrow{k_j})\prod\limits_{i=1}^{m}(x_i({k_j}_i)+(1-x_i({k_j}_i))u_i^{{k_j}_i}),
\end{equation}
\\
in Eq.~\ref{u-1}, where $x(k_{j_i})$ is the vaccinated probability of layer node $j$ in layer $i$ with degree $k_{j_i}$. Since the vaccination of layer nodes is performed independently in each layer, the critical vaccination threshold becomes a vector $(x^{L}_1,...,x^{L}_m)_c$ which can be obtained when the immunity $x^{L}_i=\sum\limits_{{k_j}_i}p_i({k_j}_i)x_i({k_j}_i)$ is satisfying $\Lambda=1$ in Eq.~\ref{u}.
According to the above theoretical analysis, computer simulations have revealed that both multiplex node-based random vaccination and layer node-based random vaccination strategies are more effective in controlling disease spreading on multiplex ER networks~\cite{zhao2014immunization}. On the contrary, multiplex node-based targeted vaccination and layer node-based targeted vaccination strategies provide better protection on multiplex SF networks. As on single-layer networks, random vaccination strategies work well only in largely homogeneous networks, while targeted vaccination strategies are required for useful results on heterogeneous networks.
With regards to the efficiency of acquaintance vaccination in multiplex networks, we note that it is closely related to the correlation between network layers~\cite{wang2015immunity}. Interestingly, as the correlation coefficient increases, the vaccination threshold decreases for multiplex node-based acquaintance vaccination, but slowly increases under the layer node-based framework. Moreover, Buono et al. considered vaccination of only one layer and studied its effect on all the other layers while disregarding degree-degree correlations among them~\cite{buono2015immunization}. Though, relative to the case of no vaccination, the size of the epidemic is drastically reduced in the layer where the vaccination strategy is applied, the targeted strategy is still not as efficient as in single-layer networks. Thus, the selection of the vaccination strategy has a major effect on the layer where it is employed, but does not efficiently protect the individuals on other layers. Similar conclusions have also been reported in Refs.~\cite{vaidya2016modeling,zuzek2015epidemic}, thus corroborating the need to reconsider single-layer vaccination strategies when applied to multilayer networks.
\section{Rationale for studying behavior-disease dynamics} \label{sec:rationale}
As demonstrated by the foregoing sections of this review, epidemiological models that do not capture behavioral dynamics have been very successful over the past century, in terms of yielding rich theoretical opportunities for research as well as in terms of providing insights into epidemiological mechanisms and how to improve infection control. However, the impact of human behavior on disease dynamics is ubiquitous, and vaccinating behavior is no exception.
This section is concerned with explaining the rationale for studying and modeling coupled behavior-disease dynamics for vaccines, especially on networks. We start with a brief
summary of vaccine use in public health, discuss why vaccinating behavior may become an increasingly important part of disease control in coming decades, and then discuss
how models can help us to better understand and predict behavior-disease dynamics.
The twentieth century saw enormous progress in public health, and especially in terms of preventing and treating infectious diseases. For instance,
medical use of antibiotics began during World War II, and antibiotics have since changed how bacterial infections are treated and saved many lives. Improvements
in sanitation and hygiene are likewise credited with preventing many cases of cholera, typhoid fever, yellow fever, dysentery, tuberculosis, and malaria by providing
cleaner drinking water and living conditions and improved hygienic practice \cite{fink2011effect}. The use of vaccines, although already available for smallpox since the eighteenth century,
expanded considerably in the twentieth century as vaccines were invented for more infectious diseases, and were distributed to more people. Safe and effective vaccines
now exist for many of the most common infectious diseases that impose significant morbidity and mortality upon populations, such as smallpox, measles, pertussis,
influenza, hepatitis A and B, chickenpox, diphtheria, and others \cite{bonanni1999demographic}. Global smallpox eradication was achieved in large part due to the success of ring vaccination \cite{fenner1988smallpox}.
The use of vaccines has been estimated to save millions of children's lives per year in the 1990s alone \cite{bonanni1999demographic}. In the 2000s, special immunization activities (SIAs)--ambitious large-scale immunization campaigns in the world's poorest regions and especially in sub-Saharan Africa--reduced the number of deaths due to measles from an estimated 766,000 per year in 2000 to an estimated 164,000 per year
in 2008 \cite{dabbagh2009global}. Global eradication of measles in the coming decades is becoming seen as a real possibility for the first time in history \cite{levin2011global}.
Thus, although challenges still remain, it is becoming significantly easier to get vaccines administered to the people who need them. There are, however, important
exceptions. For instance, the challenges of administering vaccines in war zones have contributed to a large outbreak of polio in Syria as recently as 2015
\cite{cousins2015syrian}. At the same time in Pakistan, violence against WHO vaccinators has stymied efforts to combat polio \cite{ganapathiraju2015endgame}.
Despite this, global polio eradication is coming closer \cite{cochi2014global}.
While some countries with resource limitations struggle to contain vaccine-preventable infectious diseases, other comparatively wealthy countries are also struggling
with infectious disease control, albeit for different reasons. In countries such as the United Kingdom, Switzerland, and Germany, measles circulation in the 2000s and
2010s has been relatively widespread compared to previous decades, and has often re-established endemicity in some populations due to parental vaccine refusal or vaccine hesitancy on account
of unfounded concerns about the link between the measles-mumps-rubella (MMR) vaccine and autism \cite{de2001seroepidemiology, brown2010factors, larson2014understanding}.
Similarly, polio was on the cusp of global eradication and had been limited to northern Nigeria when an oral polio vaccine scare in 2003 caused resurgence of the disease and
spread of polio to countries like Syria and Pakistan that had previously eliminated the infection \cite{serpell2006parental}. These are only two examples where population resistance
to vaccines due to unfounded concerns has led to reduced vaccine coverage and resurgent infections, even in resource-rich countries where it is easy for children to
get vaccinated, if their parents want it.
Logistic, financial, technological, and administrative barriers to vaccination are likely to continue receding in the coming decades, enabling it to become easier to improve
vaccine coverage everywhere in the world. As this happens, we speculate that vaccine refusal and vaccine hesitancy such as recently observed in the case of MMR vaccine
will become an important barrier--and perhaps even the most important barrier--to global eradication of vaccine-preventable infections. Many factors contribute
to the decision to become vaccinated \cite{chapman1999predictors,sturm2005parental,brown2010factors}. Also, vaccine refusal is not a new phenomenon, and even the
first vaccine to be invented--the smallpox vaccine--was subject to considerable resistance (see Fig.~\ref{smallpox_cartoon}). However, growing herd immunity generated
by vaccine programs themselves can bring about the preconditions for vaccine refusal.
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{fig18.pdf}
\end{center}
\caption{Satirical cartoon by James Gillray capturing public fears of the smallpox vaccine in the 18th century, the first ``vaccine scare". Edward Jenner stands in the middle, administering the smallpox vaccine. Vaccinated persons grow cow parts out of various parts of their bodies (the smallpox vaccine of the time was based on the virus that causes cowpox, hence the fear among some that the vaccine would turn humans into cow-like hybrids). A painting of the Worshippers of the Golden Calf hangs in the background. Source: Wikimedia Commons.}
\label{smallpox_cartoon}
\end{figure}
Vaccine refusal is often construed in the context of ``free-riding'': individuals who do not vaccinate are consciously free-riding
on the herd immunity provided by those who do vaccinate. Debate on the usefulness of coupled behavior-disease models often centers on
whether individuals consciously free-ride when they make vaccinating decisions or not. However, it is not necessary for individuals to consciously free-ride in order for vaccine
refusal to be linked to growing herd immunity, or for behavior-disease models to be useful (although it is certainly possible that some individuals will consciously free-ride when
making vaccinating decisions \cite{ibuka2014free}). It is well-known that perceived infection risk and past exposure to infections are determinants of whether or not individuals
will choose to be vaccinated \cite{chapman1999predictors}. In particular, a high perceived infection risk and/or a past history of infection are positively associated with
decisions to accept vaccination. Therefore, lack of recent infection, such as caused by vaccine-generated herd immunity, can decrease the motivation to become vaccinated.
Parents who have never seen a case of measles infection leading to hospitalization will tend to under-estimate the risk and/or probability of measles infection. Similarly,
lack of a recent outbreak can cause carelessness or de-prioritization on the part of physicians, and therefore lower vaccine uptake \cite{swennen2001analysis}.
Whether or not non-vaccinators consciously free-ride is secondary in some respects. What matters is that vaccine-generated herd immunity brings
about lack of recent infections in a population, which can in turn cause vaccinating to become less of a priority, as well as creating fertile conditions for false rumours
about vaccine risk to spread. Vaccine programs are victims of their own success.
This is also echoed in observed behavior for pertussis and MMR vaccines. When declining vaccine coverage caused in partly by vaccine refusal leads to an outbreak, vaccine coverage responds by increasing during and after the outbreak \cite{goldstein1996effect, bauch2012evolutionary}. This vividly demonstrates the coupled nature
of disease dynamics and vaccinating choices, completely aside from whether we need to characterize this behavior as free-riding, or whether non-vaccinating
individuals are aware that is what they are doing. Individuals may not answer truthfully when surveyed about their motives for becoming vaccinated or not becoming
vaccinated, or more importantly they may answer very differently in surveys conducted during an outbreak, than during a time of elimination. However,
when we observe vaccinating behavior responding to changes in disease prevalence, as in the Disneyland, California measles outbreak of 2014 \cite{CDPH2016} and other examples,
we see the proof in the behavioral pudding that vaccinating behavior and disease dynamics are strongly linked to one another, particularly near the elimination threshold.
The Disneyland, California measles outbreak and similar outbreaks illustrate the important role played by social factors in determining individual vaccinating choices.
The influences of peer groups and medical professional opinions on vaccine decision-making are well-documented \cite{sturm2005parental,allen2010parental}.
Social contact networks can be mapped out using Bluetooth and other technologies (see Section \ref{section:digital_epidemiology} for more details) \cite{salathe2010high}. Moreover,
social influence has often been described as a contagious process in itself, as ideas and behaviors spread through networks \cite{christakis2007spread, campbell2013complex}.
Therefore, social processes can be very naturally modelled by defining processes on networks representing social contacts, and the methods of statistical physics concerned
with the properties of networks could therefore be very useful \cite{bansal2007individual}.
Because vaccine-generated herd immunity tends to create complacency toward vaccine programs in the way we have described above, we expect vaccine hesitancy and
outright vaccine refusal or vaccine ``scares" to become more common as eradication and elimination thresholds are approached for more vaccine-preventable infectious diseases. Moreover,
we have seen how the significant role of social context in influencing individual decision-making is very naturally represented using the network paradigm. The study
of coupled behavior-disease dynamics on networks could therefore help us address this potentially growing problem \cite{wang2015coupled}. In particular, mathematical models, often based on
methods borrowed from physics, can be a useful way of better understanding and predicting these coupled dynamics.
Human behavior is doubtless more difficult to mathematically model than the motion of a single billiard ball, for instance, or many of the slightly more complex systems studied
by physicists. However, at some levels of organization (in particular, the aggregate population level) it is often possible to develop simple models that have some predictive power.
For instance, mathematical models incorporating behavior-disease interactions have been fitted to time series data of infectious disease prevalence, and model selection approaches
such as Aikaike Information Criterion (AIC) have shown that models that include behavior can explain the data more effectively compared to models that neglect behavior, with little or no parsimony penalty
\cite{bauch2012evolutionary,he2013inferring,oraby2014influence}. Such models have also shown retrospective predictive power \cite{bauch2012evolutionary}. Other model fitting and data
analysis exercises without mechanistic representations of behavior have likewise shown the importance of accounting for behavior when explaining observed epidemic
patterns \cite{chowell2006transmission, bootsma2007effect}. Similarly, the results of experimental games and surveys have been useful
in the parameterization of coupled behavior-disease models \cite{galvani2007long,shim2012influence}. The advent of online social media
promises another source of data on both individual-level and population-level human behavior and this will be explored in greater depth in a later section.
Hence, there appears to be a strong public health and scientific rationale for studying behavior-disease interactions. The existing research shows that behavior and
disease are strongly coupled to another. This coupling is particularly important near the elimination threshold, hence vaccine refusal may become an increasingly
important barrier to achieving elimination and eradication, as logistic barriers recede into the distance. Moreover, models show promise to help us better understand
coupled behavior-disease dynamics, and comparing models with data have often shown that behavior is important for understanding observed epidemic patterns and
that behavioral models may have predictive power. Finally, these models can show interesting dynamics that are similar to those studied in physics--such as phase
transitions, oscillatory solutions, and spatio-temporal patterns--and therefore similar methodologies to those used in physics may be helpful in their study. In the next
section, we will introduce some of the basic concepts of behavioral modeling that have been applied to study coupled behavior-disease interactions.
\section{Basic concepts in behavioral modeling} \label{sec:behaviormodeling}
Much of the recent research on modelling the coupled dynamics of vaccinating behavior and disease dynamics has been conducted by investigators with
training in the natural sciences, including physics, applied mathematics and epidemiology. These investigators have borrowed concepts from the social
sciences in order to find behavioral models to couple with the existing infection transmission models they are more familiar with. These behavioral models
can be broken down into three main categories: phenomenological models, game theoretical models, and psychological models. However, we note
that the distinctions between these categories are not always well defined, and the delineations depends to some extent on subjective opinions.
We also note that our goal here is to characterize different ways of doing behavioral modelling, rather than different ways of modelling coupled
behavior-disease interactions \emph{per se} (see Refs.~\cite{funk2010modelling, perra2011towards} for classifications of coupled behavior-disease models).
\subsection{Phenomenological models}
Phenomenological approaches to modeling behavior mathematically describe the observed effects without positing mechanisms behind the observed
behavior. For instance, arguably the first model to capture the impact of adaptive human behavior on disease transmission used the mean-field
ordinary differential equations \cite{capasso1978generalization}
\begin{eqnarray}
\frac{dS}{dt} & = & -g(I)S, \nonumber \\
\frac{dI}{dt} & = & g(I) S, - \gamma I \\
\frac{dR}{dt} & = & \gamma I, \nonumber
\end{eqnarray}
where $S$ is the number of susceptible individuals, $I$ is the number of infectious individuals, $R$ is the number of recovered individuals, $\gamma$
is the per capita recovery rate, and $g(I)$ is the force of infection. The form of $g(I)$ was chosen to reflect the phenomenological impact of unspecified psychological effects. One of the functional forms explored in
Ref.~\cite{capasso1978generalization} included
\begin{equation}
g(I) = \frac{\beta I}{1 + \beta \delta I},
\end{equation}
which saturates at high prevalence of infection, $I$. Such saturation of the transmission rate at high levels of prevalence could occur if individuals reacted
to high prevalence by reducing their contact rate through hand-washing or other means, which is a plausible assumption for a sufficiently dangerous infectious
disease. We describe this model as phenomenological because, although it is motivated by psychological factors, the equations only describe the impact of
such psychological factors on the transmission rate without actually modelling specific psychological processes such as individual cognition or social learning.
Similarly, as noted in the previous section, a rapidly growing body of research uses the contagion metaphor to describe the transmission of ideas and behaviors through
social networks \cite{christakis2007spread,campbell2013complex}. We classify these approaches as phenomenological because they describe the effect of
behaviors and ideas being transmitted between individuals, without positing the mechanisms that might bring about this transfer. Contagion of ideas is
a metaphor, and caution must be exercised when applying metaphors to complex social phenomena. Vaccine scare and others panics undoubtedly share common
features with contagious processes, and using this metaphor can simplify and facilitate analysis of social processes. Moreover, the contagion metaphor
comes naturally to investigators experienced in modelling contagious diseases. However, the downside of such metaphor is that it runs the risk of being
too facile for some applications, by ignoring important subtleties of real-world social behavior \cite{alshamsi2015beyond}. For instance, some social science
literature distinguishes between descriptive and injunctive social norms \cite{cialdini1990focus}. Descriptive social norms are those which describe what other
individuals do, whereas injunctive social norms describe what other individuals approve or disapprove \cite{cialdini1990focus}. It is moreover possible to distinguish
between social learning and social norms. Social learning theory emphasizes the social aspects of cognitive processes: individuals do not learn many behaviors
by cogitating in isolation from others, but rather they learn through a more efficient process of imitating others \cite{bandura1963social}. An individual may use
social learning to choose among descriptive or injunctive social norms that seem to be most acceptable, or most successful. It is not immediately clear how
the contagion metaphor applies to these nuanced distinctions of how we learn ideas and behaviors from others. Indeed, empirical research seems to indicate
that the contagion metaphor (unsurprisingly) is not universally applicable \cite{alshamsi2015beyond}. Moreover, many of the usual criticisms of memetics also
apply to using the contagion metaphor for the spread of ideas \cite{atran2001trouble}.
However, in many contexts, these limitations of the contagion metaphor may not matter. Science works by adopting the simplest explanation that captures the
observed behavior (Occam's Razor) and there are some contexts in which the contagion metaphor may be perfectly adequate to explain the observations.
This is often the value of phenomenological models over mechanistic models. For instance, to prove that behavior influences the spread of infectious diseases,
it is sufficient to construct a phenomenological model that captures the impact of behavior and to show that it is superior to models that do not capture
behavior \cite{bauch2012evolutionary, he2013inferring}. And, precisely because phenomenological models are often simpler in structure, they may make
it easier to understand how simple mechanisms such as information spread through networks can give rise to complex emergent phenomena, such as
control of infection spread in a spatially structured population \cite{funk2009ZhenPNAS}. In other contexts, mechanistic models may be desirable, and
in the next two subsections we discuss two broad categories of mechanistic models: game theoretical models and psychological models.
\subsection{Game theoretical models}
Game theory is the formalization of strategic interactions between individuals in a group of two or more individuals \cite{von2007theory}. Game theory is
well exemplified by the Prisoner's Dilemma game (so called because it was originally explained as a decision process made by two prisoners being
interrogated in separate jail cells who must decide whether to confess their crime, or stick to their story of being innocent). Each player can choose to
either cooperate ('C') or defect ('D'), and receives payoffs depending on what s/he chooses, and what her/his opponent chooses (hence, the strategic
interaction of the game). If you, (the focal player) cooperates and your opponent also cooperates, then you receive \$3 for instance (Fig.~\ref{prisoners_dilemma}). But if you cooperate and your opponent defects, you get no money. On the other hand, if you defect and your
opponent cooperates, you get a huge \$10 payoff, but if your opponent also chooses to defect, you only get \$1. Your opponent has the
same payoff outcomes, and therefore the payoff matrix is called symmetric.
What should you--the focal player--do in this situation? If your opponent cooperates, then it is optimal for you to defect since you get \$10 instead
of \$3. If your opponent defects, then it is also optimal for you to defect, since you get \$1 instead of \$0. Therefore, regardless of what your
opponent does, you should choose `D', defect. Your opponent is thinking the same way, so s/he also chooses to defect. Therefore you both
defect and you get a payoff of \$1 each. This is a shame, since if you had both cooperated, you would each get \$3, but the selfishly optimizing
logic of maximizing personal gain in anticipation of what your opponent will do prevents this socially (Pareto) optimal outcome. The Defect-Defect
outcome is known as a Nash equilibrium, since neither player has an incentive to unilaterally change his or her strategy away from Defect and
therefore we expect the Nash equilibrium strategy choices to be stable.
\begin{figure}
\begin{center}
\includegraphics[width=0.25\textwidth, angle = 270]{fig19.pdf}
\end{center}
\caption{Payoff matrix describing the Prisoner's Dilemma, a classic example from game theory. Players must choose whether to Cooperate (C) or Defect (D) against their opponent, with the payoffs for taking these two different courses of action also dependent on what their opponent chooses. The money values are the payoff to the focal player, and the payoffs for the opponent are symmetrical hence payoffs for the opponent are the same, if they are treated as the focal player instead.}
\label{prisoners_dilemma}
\end{figure}
The Prisoner's Dilemma has been used to illustrate the clash between what is optimal for the individual and what is optimal for the group in
a variety of contexts, including vaccination \cite{may2000enhanced, bauch2003ZhenPNAS, bauch2004ZhenPNAS}. In the case of high levels of vaccine coverage,
non-vaccinators are the Defectors, who benefit from herd immunity generated by vaccinators without paying any real or perceived cost of
vaccination, while vaccinators are the Cooperators. However, this analogy only applies at high levels of vaccine coverage, since at low
levels of vaccine coverage it is optimal for individuals to accept at least some vaccination.
Game theoretical models of interactions between vaccinating behavior and disease dynamics typically define strategies such as Vaccinator
or Non-vaccinator, define payoffs for those strategies, and attempt to solve for a Nash equilibrium \cite{brito1991externalities,bauch2004ZhenPNAS,heal2005vaccination}
or the analogue from evolutionary biology, the evolutionarily stable state (ESS) \cite{smith1982evolution}. The approaches may also attempt to
determine whether the Nash equilibrium is convergently stable, which indicates whether players will evolve, over time, toward the Nash equilibrium.
However, classical game theory may not be well suited to describing real-world vaccinating behavior. It assumes that individuals are selfishly rational
optimizers, but in fact, effects like social learning, bounded rationality, and imperfect information are very important in vaccinating decisions
\cite{sturm2005parental, allen2010parental}, and this can impact predicted behavior \cite{bauch2012evolutionary}.
Research has explored the statistical physics of games such as the Prisoner's Dilemma or the Snowdrift Game played out on lattices \cite{nowak1992evolutionary,szabo1998evolutionary,hauert2004spatial,xia2015dynamic}, subsequently expanded to networks \cite{szabo2007evolutionary,tanimoto2007dilemma,dengentropy,deng2015Generalized,deng2014belief,deng2016novel,huang2015understanding,huang2015cooperative}.
Likewise, game theoretical concepts have been applied to vaccination decisions on networks
\cite{perisic2009social,fu2011ZhenPRSB, mbah2012impact, wells2013policy}.
However, the resemblance of this work with `game theory' gradually fades, as investigators use increasingly computational approaches
in their work (and less rigorous proof of Nash equilibria) as a result of including more realistic cognitive processes. Such cognitive processes
better reflect the importance of social influences and social norms, and the reality of imperfect information which makes individuals use `rules
of thumb' to make decisions, rather than assuming individuals can predict the future. In the end, the only resemblance to `game theory'
may be the use of utility functions where individuals make decisions (either non-strategic, explicitly strategic, or implicitly strategic) based on how
much utility (a measure of preference) they will receive for different actions such as vaccinating or not vaccinating
\cite{perisic2009social,fu2011ZhenPRSB,mbah2012impact, wells2013policy}. Hence, these more complex
game theoretically-inspired models blur the distinction between `game theoretical' models and `psychological' models,
the latter of which is the focus of the next subsection.
\subsection{Psychological models}\label{PsychologicalModels}
Some models of the interaction between vaccinating choices and disease dynamics invoke psychological theory in order to formulate
mechanisms underlying vaccinating decisions. An example of a model that uses social and psychological assumptions is the following SIR model modified
to include mechanistic modelling of vaccinating behavior according to an evolutionary game theoretical approach \cite{bauch2005ZhenPRSB,reluga2006evolving}:
\begin{eqnarray}
\frac{dS}{dt} & = & \mu (1 - x) - \beta S I - \mu, \nonumber \\
\frac{dI}{dt} & = & \beta S I - \gamma I - \mu S, \\
\frac{dx}{dt} & = & \hbar x (1 - x) (-1 + \omega I), \nonumber
\end{eqnarray}
where $S$ is the proportion of susceptible individuals, $I$ is the proportion of infectious individuals, $x$ is the proportion of individuals who adopt the vaccinator
strategy, $\mu$ is the per capita birth/death rate, $\beta$ is the transmission rate, $\gamma$ is the recovery rate, $\hbar$ is a rescaled parameter
that essentially captures the rate of social learning, and $\omega$ is a rescaled parameter that essentially captures the relative perceived risk of
the disease compared to the vaccine. In this model, individuals are not perfectly rational optimizers because they only switch strategies
through social learning ($\hbar$), after they have interacted with individuals playing a different strategy and moreover, they make the
`rule of thumb' assumption that their probability of eventually being infected depends on the current proportion of infectious persons, $I$,
rather than assuming that they could solve the SIR equations explicitly and figure out their chances of becoming eventually infected. This approach
can be straightforwardly extended to spatial settings, where individuals imitate successful strategies of their network neighbours \cite{fu2011ZhenPRSB}. A similar approach has also been adopted in models of coupled human-environment systems \cite{innes2013impact,barlow2014modelling}.
Other approaches incorporate specific theories from psychology and social psychology in order to formulate mechanisms in coupled behavior-disease
models. For example, random walk subjective expected utility (SEU) theory posits that individual decision-making can be modelled as a biased random
walk in a one-dimensional space of possible states with two decision outcomes on either end of the space. Individuals move toward one decision or the other depending
on the probability of certain outcomes of their decisions, and the subjective utility associated with those outcomes, as formulated through a decision
tree \cite{busemeyer1993decision}. Once a certain threshold has been crossed close to one end of the one dimensional space or the other, the individual
has finally decided on a course of action, and takes it. This has been applied to vaccinating decisions for influenza on a contact network, to explore
how decisions about influenza vaccination and decisions about contact precautions interact with one another through their mutual coupled influence on
disease dynamics \cite{andrews2015disease}.
Another relevant theory from psychology is prospect theory, which mathematically formalizes the phenomenon whereby humans tend to over-estimate
the probabilities of very rare events. Since vaccine adverse events and disease complications are rare events, the relevant of prospect theory to vaccinating
decisions is obvious. Prospect theory has been applied in mean-field models of coupled vaccinating behavior and disease dynamics for paediatric infectious
diseases, and has been found to significantly impact model predictions \cite{oraby2015bounded}.
A third example is the application of Dempster-Shafter theory, which is a general statistical framework (closely related to Bayesian statistics) for dealing
with decision-making under uncertainty \cite{shafer1976mathematical}. This theory has been incorporated into models of coupled dynamics of vaccinating
decisions and disease dynamics, where investigators have used the theory to explore how awareness of disease-related events and vaccine-related events
spreads through a group of socially connected individuals \cite{xia2014belief}.
\section{Behavior-vaccination dynamics in well-mixed (mean-field) populations}
\label{sec:beh-vac-mean}
The dynamics and control of vaccine-preventable infectious diseases under voluntary vaccination can be quite complex, because of the role of human decisions. These are in turn based on information and \textit{rumors} gathered on: i) the \textit{perceived} risks of getting the disease; ii) its spread; iii) the \textit{perceived} risks of being affected by vaccine-related side effects. This complexity becomes apparent in the modeling phase, due to the intrinsic difficulty of quantitatively representing human behavior, and is mirrored by the features of the resulting mean-field models, whose solutions are radically different from those of the basic model with mandatory vaccination illustrated in Section \ref{sec:compart-model}. Here we review the recent literature on the dynamic implications of the introduction of human behavior in mean-field SIR-like models for vaccine preventable infections under voluntary vaccination. The presentation broadly follows the categorization proposed in \cite{Funk_review2010} (also refer to Section~\ref{sec:behaviormodeling} for classification), contrasting \textit{phenomenological}, or \textit{behavior-implicit} models of vaccinating behavior, with \textit{behavior-explicit} ones, such as \textit{psychological} or \textit{game-theoretical} models. However, this categorization is not always effective from the modeling viewpoint since, under certain conditions, the various approaches can become mathematically indistinguishable. More mathematically focused reviews can be found in \cite{ourbook,Funk_review2010,Bauch_Galvani_2013}.
\subsection{\textit{Phenomenological} SIR models with information-dependent vaccine uptake}
\label{InfoDependentCoverageNoiTPB2007}
A general phenomenological framework is represented by the SIR model with \textit{information-dependent} vaccination introduced in \cite{noi1,noi2,noi3,noi8}. The underlying rationale is that the possibly complex behavioral rules parents follow when deciding whether to vaccinate or not their children will often translate into simple relationships between the individual probability to vaccinate, i.e. the \textit{vaccine demand}, and the gathered information on infection spread. These models therefore postulate relatively simple feedback rules whereby trends of infection (and of vaccine adverse effects, VAE) can affect the current vaccine uptake, eventually feeding-back on infection dynamics.
These ideas led to the following modified SIR model with vaccination by a perfect vaccine \cite{noi1} (note that for basic demographic and epidemiological quantities we use the notations adopted in Section \ref{sec:compart-model})
\begin{align}
S^{\prime}&=\mu\left( 1-x(M)\right) -\mu S-\beta(t)SI, \label{General_SI_S}\\
I^{\prime}&=I(\beta\left( t\right) S-(\mu+\gamma)) \label{General_SI_I},
\end{align}
where the vaccine demand $x(.)$ is an increasing function of a new variable, $M$, which summarizes the information available on current and/or past disease trend, used e.g. to formulate expectations about future risks. Here we considered uniquely vaccination of newborns, however models considering 'delayed' vaccination strategies targeted also to later ages, i.e. of the type
\begin{equation}
S^{\prime} = \mu(1-S) -\psi(M) S - \beta(t) SI,
\end{equation}
where $\phi(M)$ is an increasing function of $M$, have a qualitative behavior that is similar to the one of the model \ref{General_SI_S}-\ref{General_SI_I} \cite{noi1}.
Focusing on the perceived risk of disease as the driving force of vaccination decision, $M$ might be any continuous and increasing function $g$ of the \textit{current} infection prevalence or incidence. For linear $g$ it would hold $M=h(t)\beta(t)SI$ (with $h(t)>0$) or $M=k(t)I$ (with $k(t)>0$). Namely, the latter expression defines the perceived risk of serious disease as the product of the perceived risk of infection, proportional to infection prevalence $I$, times a constant perceived risk of serious disease given infection.
More realistically $M$ also depends on past values of state variables, due e.g. to time-delays in the reporting/acquisition of information, and mainly to the fact that parents might also include their experience of past disease trends, etc. In this case, $M$ can be modeled as follows
\begin{equation}
M(t)=\int\limits_{-\infty}^{t}g(I(\tau))K(t-\tau)d\tau,
\label{eqM}%
\end{equation}
where the \textit{delaying kernel} $K$ \cite{Mac} is a probability density function. Besides the trivial case $K(t)=\delta(t)$ (yielding back the unlagged case), the literature has focused on kernels allowing reduction to ordinary differential equations\cite{noi1,noi2,noi3,noi8, Reluga1}, in particular on the \textit{exponentially fading memory} kernel $K(t)=a\exp(-at)$, which describe an exponential decay of the "`memory"' of agents, with characteristic memory lenght $T=1/a$ \cite{Mac}. Another kernel investigated in \cite{noi8} is the \textit{acquisition-fading} kernel $K(t) = a_1a_2(a_2-a_1)^{-1} (e^{-a_1 t}-e^{-a_2 t} )$. This unimodal kernel is null at $t=0$, and it mimics not only the exponential decay of memory, but also an initial phase of information acquisition.
Finally, function $x$ can be written as \cite{noi1,noi3,noi8}:
\begin{equation}
x(M)=x_{0}+x_{1}(M)\ \ , 0<x_{0}<1,\label{pM}%
\end{equation}
where $x_{0}$ is a non-null baseline constant coverage, mirroring awareness of disease severity also in absence of spread, and $x_{1}(M)$ is a sufficiently regular increasing function, with $x(M)\le 1$.
For the sake of simplicity, from now on we shall assume that the information index $M$ only depends on (current or past values of) $I$, the so called \textit{prevalence-dependent} case \cite{Funk_review2010,Geoffard97}. In the current case this assumption implies that epochs of increasing prevalence would rapidly promote an increase in vaccine uptake in children (and vice-versa).
Two main substantive questions arise about model of Eqs.~\ref{General_SI_S}-\ref{pM}, namely how prevalence-dependent vaccination affects infection control on the one hand, and, on the other hand, how it might affect the dynamical pattern of infection and vaccination, e.g. by triggering oscillations. From now on we always assume that in absence of vaccination the infection is endemic, i.e. $\mathcal{R}_0>1$.
\subsubsection{Implications of information-dependent vaccination: elimination \textquotedblleft mission impossible\textquotedblright}
The basic SIR model with constant mandatory vaccination $x$ at birth shows a simple threshold behavior governed by the \textit{vaccine reproduction number} (VRN) $\mathcal{R}_{V}=(1-x)\mathcal{R}_0$ such that if $\mathcal{R}_{V}\le 1$ (i.e. $x\ge x_{c}$) then the infection can be eliminated, otherwise the disease remains endemic. This simple result no longer holds under prevalence-dependent vaccination which makes elimination impossible. This is a general fact holding regardless of the form of the information index $M$, of the memory kernel $K(.)$ and of the nature of the transmission rate $\beta(t)$. This may be inferred by investigating the local as well as the global stability of the disease-free equilibrium (DFE): $DFE=(1-x_{0},0,0)$ of model in Eqs.~\ref{General_SI_S}-\ref{pM}, yielding the following global eradication condition
\begin{equation}
B=\frac{1-x_{0}}{\mu+\gamma}\frac{1}{\theta}\int\limits_{0}^{\theta}\beta
(u)du \le 1\label{C_GAS_DFE},%
\end{equation}
(where $B$ represents the average VRN when the vaccine uptake is set to the baseline coverage $x_{0}$) while if $B>1$ the DFE is unstable. For constant $\beta$ the condition $B\le1$ is equivalent to the condition $x_{0}>x_c$, in other words elimination can only be achieved if the baseline vaccine uptake lies above the elimination threshold. Actually, it is a documented fact that western countries have been able to maintain very high coverages - i.e. above the estimated elimination threshold - against e.g. polio and diphtheria for decades. This however has been possible under situations where these vaccinations were essentially mandatory. Whether similar results could be maintained under fully voluntary vaccination regimes seems to be quite remote, given that scenarios where prevalence is very small would also make the perceived reward of vaccination to vanish, so that people would start escaping from vaccination.
\subsubsection{Memory-triggered oscillations}
\label{delay_oscillations}
Given that infection elimination is ruled out, the question turns into the types of dynamics that might be triggered by prevalence-dependent vaccination.
For $\mathcal{R}_{0}(1-x_{0})>1$ and constant $\beta$, the system of Eqs.~ \ref{General_SI_S}-\ref{pM} has a unique endemic equilibrium
$EE=(S_{e},I_{e},M_{e})$.
In the unlagged case the system becomes 2-dimensional and it is easy to prove that the endemic state is globally asymptotically stable (GAS), so that only damped oscillations are possible \cite{noi1}. Instead, in the delayed case, stable oscillations - via Hopf bifurcation of the endemic state - already appear under the simplest pattern of delay, namely the exponentially fading memory. In particular, assuming as
key parameter the inverse of the average memory/delay lenght $a=1/T$, and taking for simplicity $g(I)=kI$, it holds that \cite{noi1} if and only if%
\begin{equation}
\left( \beta I_{e}+\mu\right) ^{2}-\beta I_{e}\mu kx_{1}^{\prime}%
(M_{e})+2\left( \beta I_{e}+\mu\right) \sqrt{\beta I_{e}(\gamma+\mu
)}<0,\label{Cond_Hopf}%
\end{equation}
there is a range $[ a_{1},a_{2}]$ for $a$ such that: i) for $a\in(a_{1},a_{2})$ Yabucovich oscillations \cite{yf3} occur via Hopf bifurcations at $a_1$ and $a_2$; ii) for
$a\notin\left[ a_{1},a_{2}\right]$ the endemic equilibrium EE is locally stable. The global stability of EE, suggested by numerical sumulations, is nontrivial and it has been investigated in \cite{mbsbbdonlac} by means of the Li-Muldowney geometric theorem, which extends the Dulac-Bendixon theorem.
Note that Eq.~\ref{Cond_Hopf} shows that sustained oscillations can appear only if $x_{1}^{\prime}(M_{e})$ is sufficiently steep, that is only if the behavioral response of vaccine uptake to changes in the perceived risks is sufficiently intense. In this case oscillations occur in an appropriate intermediate window of the average delay $1/a$. These results continue to hold for more realistic patterns of delay, such as acquisition-fading and Erlangian kernels \cite{noi1,noi2,noi3,noi4,noi8}, for generic increasing $g(I)$ functions \cite{noi3}, in presence of disease-related mortality and a non-constant population \cite{noi3}, and when the incidence of VAE is taken as the key determinant of the vaccine demand \cite{noi4}. Finally, as shown in \cite{buonomo2013modeling}, oscillations in behavior-depending vaccination models are also observed in absence of memory mechanisms but in presence of disease-intrinsic delays, such as in the SEIR model (the latent period being, of course, equivalent to an exponentially distributed delay in the onset of infectivity).
\subsubsection{Dynamic pattern for a measles-like infection}
\label{Dynamics_pheno_model_delay}
We illustrate some of the above described dynamic oscillatory patterns, under the parameter values adopted in section \ref{SIR_vaccination}. As for behavioral rules we choose $g(S,I)=I$, and a Michaelis-Menten-type \cite{murray1} coverage function $ x_{1}(M)=(1-x_{0}-\varepsilon) D M/(D M+1)$ where $D>0$ is a shape parameter tuning the reactivity of $x_{1}$, and $1-x_{0}-\varepsilon$ represents the \textit{ maximal coverage } arising for large $D M$ values. Behavioral parameters are chosen in the oscillatory region, with the average delay $T=1/a$ set to 4 months, $x_{0}=0.75$, $\varepsilon=0.01$ and $D=15\times 10^4$. The selected values of $\varepsilon$ and $D$ allow, in principle, the vaccine uptake to reach values close to $100\%$ during epochs of high perceived risk.
The model is initialized at time $t=0$ at the endemic state determined by a constant coverage set at the baseline level $x_0$. Fig.~\ref{TPB} reports the transient (left-hand side) and long-term (righ-hand side) time paths of the ERN $R_E(t)=\mathcal{R}_0 S(t)$ (top), of the infective prevalence (medium), and of the overall vaccine uptake $x(M)=x_0+x_{1}(M)$ (bottom), jointly with its time average. State variables converge to a stable limit cycle in about 150 years, with a long-term inter-epidemic period around $12$ years, i.e. about 2.5 times the pseudo-period of the SIR model with constant coverage $x_{0}=0.75$. The most striking fact is that though the vaccine uptake $x(M)$ reaches levels as high as $96\%$ during epochs of high perceived risk, this is totally insufficient for elimination, as witnessed by the average long-term coverage, which remains below $80\%$, i.e. well below the critical coverage $x_{c}\approx 0.93$.
This oscillatory nature appears to be intrinsic to SIR systems with delayed prevalence-dependent vaccine demand: epochs of increasing prevalence will yield, after a certain period of time, an increase in the \textit{demand} for vaccines, which will in turn reduce infection prevalence, therefore eventually feeding-back on the vaccine uptake and so on, as first noted in \cite{Geoffard97}.
\begin{figure}[t]
\centering \includegraphics[width=0.7\textwidth] {fig20.pdf}
\caption{Dynamics of model in Eqs.~\ref{General_SI_S}-\ref{pM} for $g(S,I)=I$ under an exponentially fading kernel ($T=4$ months) and a Michaelis-Menten-type prevalence-dependent component of vaccine uptake ($D=500$). Transient (left-hand side) and long-term (righ-hand side) paths of: (top) effective reproduction number $R_E=R_0 S$; (middle) infective prevalence $I$ (normalized to its equilibrium value); and (bottom) overall vaccine uptake $x(M)$ (compared to its time average given by the flat line in the right-bottom graph). $R_0$ is equal to $15$. Source: Reprinted figure from Ref.~\cite{noi8} }
\label{TPB}
\end{figure}
\subsubsection{Further extensions and remarks on phenomenological models}
The information-dependent framework is a flexible one from several standpoints. As mentioned above, a model analogous to the one presented here has been applied to investigate the effects of VAE \cite{noi4} which, in the current western situation where incidence of most infections is persistently low, appear to represent the most important component of the vaccine demand. In particular, in \cite{noi4} the key determinant of the vaccine demand was the information on the incidence of VAE. Role of VAE as vaccination determinants will be considered more in detail in the \textit{behavior-explicit} models of next subsection. However, many other applications are possible. For example the framework in~\cite{noi1} was also used to consider the possibility that parents adopt the strategy of delaying the age at vaccination of their children, later extended in a more general behavior-explicit setting in \cite{BhattaBauch2010}.
More important, the phenomenological model of Eqs.~\ref{General_SI_S}-\ref{eqM} represents a general form in that, as we have remarked, also \textit{behavior-explicit} (e.g., game theoretic) formulations for vaccine uptake eventually collapse into the same structure, once the vaccine uptake function resulting from the adopted formulation is set within the framework of a dynamic epidemiological model \cite{Reluga1, Geoffard97}. For example, game-theoretic approaches (discussed in section \ref{game-theoretic models}) often yield step-wise \textit{best response} behavioral functions. The simplest example for the vaccination problem would be \textit{not to vaccinate} with probability one when the risk of infection is below a certain threshold, \textit{to vaccinate} above that threshold, and to vaccinate with a certain probability when the risk is exactly at threshold. However, a number of reasons (e.g., bounded rationality) suggest to approximate the best response function by a corresponding \textit{smoothed responses} \cite{Xu_Cressmann_2016}. The shape of smooth response function is typically analogous to that of function $x_{1}(.)$ used here.
\subsection{\textit{Behavior-explicit} psychological models of vaccinating behavior}
\label{ImitationBasedVaccination}
Most available psychological models of vaccinating behavior in previous sections relied on what we will term \textit{Imitation Game Dynamics} (IGD), an important class of models developed within \textit{Evolutionary Game Dynamics Theory} (EGDT)\cite{nowak:2006a,nowak:2006b,Hofbauer98}, which is a powerful approach to modeling behavior by setting ideas from game theory into a population dynamics framework \cite{nowak:2006a,nowak:2006b,Hofbauer98}. \textit{Imitation} represent a main avenue through which the different strategies enacted by players can spread within a population, through mutual learning between individuals \cite{Bauch2}. The first application of IGD to investigate vaccinating behavior and its feedback on infection dynamics and control was developed in a seminal paper by Bauch \cite{Bauch2}, later extended \cite{noi5} by a more general formulation which will prove useful to link the class of imitation-based models with the phenomenological models of previous section.
\subsubsection{Imitation dynamics}
IGD is usually presented using the framework of game theory, both in its economic and in bioevolutionary standpoints. However, we note that at its very core there is again a mass-action diffusion process based on statistical physics \cite{Hofbauer98}. Let us consider a binary decision problem, i.e. one for which only two, mutually exclusive, strategies are played: strategy $V$ and $N$ (these might for example be \textit{to Vaccinate}, and \textit{Not to Vaccinate}). Let us also assume that individuals can switch from the strategy they are currently playing to the alternative one by \textit{learning and imitating} after social encounters with subjects that are playing the other strategy. Considering a population of constant size, and indicating with $y_V$ and $y_N$ the fractions of the $N$ and $V$ sub-populations ($y_V +y_N=1$), leads to the following mean-field \textit{double infection} model:\\
\begin{align}
y_V^{\prime} =& - K_{VN}y_N y_V + K_{NV}y_N y_V \label{sV},\\
y_N^{\prime} =& - K_{NV}y_N y_V + K_{VN}y_N y_V \label{sN},
\end{align}
\noindent where the coefficients $K_{NV}, K_{VN}$ have a clear "epidemiological" interpretations as the \textit{strategy-specific} transmission rates
following social contacts with individuals playing the other strategy. Following EGDT \cite{Hofbauer98}, these coefficients are proportional to the perceived benefit, or \textit{payoff}, arising from strategy switching.
Since $y_N = 1 - y_V $, it follows
\begin{equation}\label{igdeq}y_V^{\prime} = (K_{NV}- K_{VN})y_V(1-y_V). \end{equation}
Note that if $K_{NV}- K_{VN}$ is strictly positive (negative) in all the state space, strategy $V$ ($N$) will eventually prevail, by spreading to the entire population with a logistic-type dynamics, as is intuitive given that the quantity $K_{NV}-K_{VN}>0$ must be proportional to the perceived \textit{net payoff gain} when switching from strategy $N$ to $V$. More interesting, dynamic situations arise when the net payoff $K_{NV}-K_{VN}$ has non-constant sign, which happens when it is made dependent on other state variables of the system, as is the case for vaccination games \cite{Bauch2,noi5,BhattaBauch2010,BauchBhattaPLOSCB2012, oraby2014influence, OrabyBauch2014_bounded_rationality}.\\
\subsubsection{Basic SIR models with imitation-based vaccinating behavior}
\label{VAE_prevalence_models}
Unlike previous subsection where the vaccine uptake at birth $x$ was taken as a "static" function of $I$ (and/or of $S$), we assume that $x$ is a state variable determined by the switches of parents between strategy $N$ (not to vaccinate) and $V$ (vaccinate) according to the above described IGD. Therefore, $x(t)$ actually represents the proportion of parents favorable to vaccination at time $t$, which is taken as a proxy for the actual emerging vaccine uptake. Setting $y_V = x,$ yields the following family of mean-field SIR models with vaccination \cite{noi5,noibeta}
\begin{align}
S^{\prime} & =\mu(1-x)-\mu S-\beta SI\label{sirvs},\\
I^{\prime} & =\beta SI-(\mu+\gamma)I\label{sirvi},\\
x^{\prime} & =\kappa_{1} \Delta E x(1-x) \label{sirvp},%
\end{align}
where the term $K_{NV}-K_{VN}$ has been represented by the product between the scale coefficient $\kappa_{1}$, tuning the speed of the imitation process, and the net expected \textit{payoff gain} of vaccination, $\Delta E$. The latter can in turn be defined as the difference $e_{V}-e_{N}$ between the \textit{vaccinator payoff} and the \textit{non vaccinator payoff}, or alternatively as $C_{V}-C_{N}$ where $C_{i}=-e_{i}$ represents the corresponding \textit{cost}.
Note preliminarly that irrespective of the specific form of $\Delta E(t)$, the family of models in Eqs.~\ref{sirvs}-\ref{sirvp}, always has the following three equilibria: (i) a disease-free state with no vaccinators $A=(1,0,0)$, which is unstable due to the above assumption that $\mathcal{R}_0>1$; (ii) a \textit {pure-vaccinator} disease-free equilibrium $B=(0,0,1)$ where everyone is vaccinated; (iii) the pre-vaccination endemic equilibrium $C=\left(
S_{SIR}, I_{SIR},x_{SIR})=(\mathcal{R}_0^{-1},\mu(1-\mathcal{R}_{0}^{-1})/(\mu+\gamma),0\right)$. On the other hand, the stability of $B$ and $C$ as well as the existence of further equilibria (induced by the behavioral component) depend on the chosen form of the perceived payoff gain $\Delta E(t)$.
\subsubsection{The baseline SIR model with imitation-based vaccinating behavior}
\label{Bauch_2005}
The seminal study \cite{Bauch2}, motivated by need to understand the causes and dynamics of \textit{vaccine scares}, such as the pertussis or the MMR vaccine scares,
rests on two main hypotheses, namely: (a) a prevalence-dependent cost of not vaccinating (see Fig. 21), of the form $C_N=r_{I}m I(t)$, where $m I(t)$ is the current perceived risk of infection (taken as an estimate of the actual FOI $\lambda=\beta I$), and $r_{I}>0$ is the perceived (conditional) risk of serious disease given infection; and (b) a constant cost of vaccinating $C_V = r_{V}>0$, representing the perceived risk of suffering a VAE per single vaccination. Hence
\begin{equation}
\Delta E(t)=r_{I}mI(t)-r_{V}=r_{V}\left( \vartheta I(t)-1\right),
\end{equation}
where $\vartheta=mr_{I}/r_{V}$ is a measure of the
\textit {relative cost} of disease with respect to that of vaccination. Based on these hypotheses, a number of substantive results were shown \cite{Bauch2}. First, the pure-vaccinator equilibrium $B$ is unstable, reflecting the fact that in a wholly vaccinated population, prevalence is equal to zero and therefore the payoff of vaccination is strictly negative ($\Delta E=-r_{V}$), removing incentive to vaccinate, so that vaccination will start declining. Second, there is a \textit{partial vaccinator} endemic state $D$, that is an endemic state with positive vaccine uptake $D=\left(\mathcal{R}_0^{-1},\vartheta^{-1},\widehat{x}\right)$, where
$\widehat{x}= \left(1+\gamma/\mu\right)\left(I_{SIR}-\vartheta^{-1} \right).$ The equilibrium coverage $\widehat{x}$ is meaningful only for values of the behavioral parameters ensuring that $I_D<I_{SIR}$, i.e. such that the corresponding endemic prevalence is lower than the prevalence observed in absence of any vaccination. This in turn requires $\vartheta > I_{SIR}^{-1}$, meaning that $D$ exists only for sufficiently high levels of the relative cost of disease. Finally, at $\vartheta=\vartheta_{0}=I_{SIR}^{-1}$ there is a transcritical bifurcation between $C$ and $D$. In words, given that endemic infection for which at some stage a vaccine becomes available, then: (i) if the relative cost of the infection is low (i.e. the perceived cost of vaccination is high with respect to infection) then the vaccine would not be adopted due to lack of incentive; (ii) if the relative cost of infection is high then the vaccine would spread in the population, therefore destabilizing the pre-vaccination endemic state $C$, and sustaining the appearance of the new, locally stable, endemic state $D$ with positive vaccination. In turn, the stability of $D$ depends on $\kappa_{1} (\vartheta-\vartheta_{0})$, showing in particular that appropriately large values of the imitation coefficient can yield sustained oscillations by Hopf bifurcation of $D$, indicating that fast social learning is a source of oscillations.
Previous results add robust evidence, based on a behavior explicit model, to those reported in section \ref{InfoDependentCoverageNoiTPB2007}, indicating that prevalence-dependent vaccinating behavior represents a serious threat for programs aimed to infection elimination.
\begin{figure}[t]
\centering \includegraphics[width=0.7\textwidth] {fig21.pdf}
\caption{Trends in pertussis vaccine uptake (dashed line) and case notifications (solid) in England and Wales 1967-2003 motivating the idea of prevalence-dependence in vaccine uptake: (a) absolute levels, (b) with vaccine uptake represented as deviations from a linear least-squares
fit. Note in particular that when coverage went abruptly down due to vaccine scare (mid seventies), large epidemics restarted, possibly uprising the perceived risk of infection, and eventually restoring vaccine uptake at high levels. Nonetheless symptoms of prevalence-dependence persist, though at a lower scale, even after the overall relapse, as evidenced in (b). Moreover, the initial (pre-1975) persistently high coverage, bringing incidence to low and declining levels, might have sharply increased the number of VAE, eventually triggering the vaccine scare. Source: Reprinted figure from Ref.~\cite{Bauch2}.}
\label{Fig1_Bauch2005}
\end{figure}
\subsubsection{An alternative model with myopic perception of the cost of vaccine adverse events}
\label{IGD_with_myopic_perception_of_VAE}
The hypothesis made in previous subsection that the perceived risk of vaccinating is constant, implies that the public correctly evaluates this risk, as the ratio between total reported VAEs for a given infection, and the total number of vaccinations for that infection per unit of time. However, due to the current high levels of herd immunity in modern industrialized countries, the perceived risk of serious disease from many infections is currently very low, while the large number of vaccinations administered unavoidably yields a steady flow of VAEs \cite{CDC_VAE}. The VAE burden might then make the perceived risk of suffering vaccine side effects (VSE) higher than risks perceived from infection.
Therefore, in \cite{noi5} it was assumed that the public \textit {myopically} evaluates the risk of VSEs by available information on the total number of VAE, which is proportional to actual vaccine uptake $x$. In particular, it was investigated the case $C_{V} = \alpha x$, defining the perceived cost of vaccination as the product of the (perceived) probability of being vaccinated, times a perceived probability of suffering VSE given vaccination. Similarly, $C_{N}$ is taken as an increasing function of infection prevalence: $C_{N}=h_{1}(I(t))$. Note that the case $h_{1}(0)>0$ is a realistic one as it might represent a scenario of local elimination where however infection re-emergence (e.g. by importation of cases) is feared.
The resulting dynamic equation for $x(t)$ is
\begin{equation}
x^{\prime}=\kappa\left( h(I)-x\right) x\left( 1-x\right), \label{general}%
\end{equation}
where $\kappa =\kappa_{1} \alpha$, and $h_1(\cdot)=h(\cdot)/\alpha$.
Interestingly, if imitation dynamics is fast compared to infection dynamics, i.e. $\kappa>>(\mu+\gamma)$, then a quasi-stead state approximation yields
\begin{equation}
x(t)\approx\min\left( h\left( I(t)\right) ,1\right), \label{spert}%
\end{equation}
i.e. the information-dependent model with current information (i.e. Eqs~\ref{General_SI_S}-\ref{pM}) presented in the previous subsection is recovered. In other words, the phenomenological model \cite{noi1} is a special case of the imitation-based case when the imitation process is fast.
Comparing with the baseline model of previous section \ref{Bauch_2005}, we note that the alternative model of Eqs. \ref{sirvs}-\ref{general} also has the equilibria $A$ (unstable), $B$ (unstable) and $C$. Unlike section \ref{Bauch_2005} \cite{Bauch2}, the pre-vaccination endemic equilibrium $C$ is always unstable, reflecting the fact that introduction of a vaccine will always be successful, due to the low perceived cost of vaccination at low coverage levels.
Moreover, two further equilibria are induced by the inclusion of behavior i.e., a disease-free equilibrium $E$ with positive vaccine uptake (\textit{disease free partial vaccinator} equilibrium) $E=(1-h(0),0,h(0))$, and the partial vaccinator endemic state $D$
\begin{equation}
D=\left( \mathcal{R}_{0}^{-1},I_{e},h\left( I_{e}\right) \right),
\end{equation}
where $\mathcal{R}_{0}=\beta/(\mu+\gamma)$ is the basic reproduction number of the infection, and $I_{e}$ is the unique solution of the equation
\begin{equation}
h(I)= 1-\mathcal{R}_{0}^{-1} -\frac{\mu+\gamma}{\mu}I . \label{equi}%
\end{equation}
Eq. \ref{spert} alone can explain the behavior of the disease-free equilibrium $E$. Indeed, from $x^{\prime} \ge \kappa(h(0)-x)x(1-x)$ it follows that for large times $x(t)\ge min(h(0),1)$ and in turn: if $h(0)\ge x_{c}$ then $x(t)\rightarrow 1$, i.e. $E$ is globally attractive. Moreover, linearizing at $E$ it follows that if $h(0)<x_{c}$ then $E$ is unstable. Therefore, in case of myopic evaluation of VAE elimination turns out to be possible, but only if the perceived cost of disease associated with infection re-emergence is so large to bring the vaccine uptake in excess of the elimination threshold.
As all equilibria are independent of $\kappa$, it is natural to choose
$\kappa$ as a bifurcation parameter. Not surprisingly, the dynamics around the partial vaccinator endemic state $D$ depends on the steepness of function $h(I)$ at $D$. Namely, it exists a threshold:
$$ th = \frac{(\mu+\beta I_e)\left( (\mu+\beta I_e) + 2 \sqrt{\beta I_e(\mu+\gamma)} \right)}{\beta I_e \mu} $$
such that i) if $h^{\prime}\left(I_e\right)<th$ then equilibrium $D$ is locally stable irrespective of the imitation speed $\kappa$; ii) if $h^{\prime}\left(I_e\right)>th$ then there is a range of values $(\kappa_1,\kappa_2)$ for the imitation speed $\kappa$ such that for $\kappa \in [\kappa_1,\kappa_2]$ and the system's orbits $x(t)=\left(S(t),I(t),x(t)\right)$ are oscillatory (i.e. recurrent epidemics are predicted) in the sense of Yabucovich \cite{yf3}, which onset via Hopf bifurcations at $\kappa_1$ and at $\kappa_2$.
Note that unlike the model reported in section \ref{Bauch_2005} where sustained oscillations around the endemic
state were triggered by large values of the imitation coefficient $\kappa$ \cite{Bauch2}, in the present model oscillations occur in an intermediate window of values of $\kappa$. This suggests that both slow and fast imitation might be stabilizing forces, quite similarly to what is reported in section \ref{delay_oscillations} for the phenomenological model. This is not surprising since $x(t)$ depends on $I(t)$ via a differential equation, i.e. $x(t)$ "follows" $h(I(t))$ with an appropriate delay.
Possible dynamics of the system for some parametric constellations promoting sustained oscillations are illustrated in Fig.~\ref{Fig3_AlbertoPiero_JTB2011} for a measles-like infection. A noticeable fact is that large levels of the imitation coefficient $\kappa$, besides triggering very long inter-epidemic periods, can promote long-lasting epochs where the vaccine uptake $x$ lies above the elimination threshold $x_{c}$.
\begin{figure}[t]
\centering \includegraphics[width=0.75\textwidth] {fig22.pdf}
\caption{Oscillatory dynamics of the SIR model with vaccination decisions governed by Eq.~\ref{general} for a linear $h(I)$ of the form: $h(I)= \theta I, \theta>0$. The graphs report the dynamics of the vaccinating proportion $x(t)$ (top) and of infection prevalence $I(t)$ (bottom), for different values of the rescaled imitation coefficient $\kappa$: $\kappa=0.0005 day^{-1}$ (left), $\kappa=0.002 day^{-1}$ (centre), $\kappa=0.0035 day^{-1}$ (right). The system is initialized from the pre-vaccination endemic levels of variables $S,I$ and from $x(0)=0.95$. Demographic and epidemiological parameters are: $\mu = (1/75)year^{-1}$, $\gamma = 1/7 day^{-1}$, $\mathcal{R}_0 =10$, $= 15,000$, $\kappa=0.002 day^{-1}$. Source: Reprinted figure from Ref.~\cite{noi5}. With permission from Elsevier.}
\label{Fig3_AlbertoPiero_JTB2011}
\end{figure}
\paragraph{Modeling irrational behaviors}
A positive aspect of the imitation game-based seminal models \cite{Bauch2,noi5,noibeta} is that they parsimoniously describe, by means of a classical tool of game theory, the complex social learning underlying the parental decision to vaccinate their children. Moreover, broadly speaking, the imitation process represent a first deviation from full rational behavior. However, models
\cite{Bauch2,noi5,noibeta} are based on an important but implicit assumption, i.e. that the "force" driving the vaccine non-acceptance is para-rational. Indeed either it is constant (and fully dictated by a rational thought, as in \cite{Bauch2}) or it depends on the information on vaccine side effects (as in \cite{noi5,noibeta}). In reality in doing this the models \cite{Bauch2,noi5,noibeta} miss the fully irrational behavior underlying the periods of vaccine scares.
In \cite{BauchBhattaPLOSCB2012} Bauch and Bhattacharyya go an important step further by assuming that the "anti-vaccination force" (which depends on the vaccine risk perception) is an empirical function mimicking the rumours-driven sudden rise, stay and decay of vaccine scare. As far as the "force" favoring the vaccine uptake (i.e. the feedback from disease prevalence) they assume that it is proportional to the time-series of notified cases. The resulting dynamics for the fraction of vaccinated newborns $x(t)$ is thus of the type:
\begin{equation}\label{BM} x^{\prime}= \kappa x(1-x)( u L(t) -\omega(t) ), \end{equation}
where $L(t)$ is an interpolation of notified cases, $u$ is a proportionality constant and $\omega(t)$ is an appropriate unimodal function. Interestingly, the authors consider also other simplified models,\\
\noindent a) absence of social learning:
\begin{equation}
x(t) = u L(t) -\omega(t);
\end{equation}
b) absence of disease driven feedback:
\begin{equation}
x(t) = 1 -\omega(t);
\end{equation}
c) absence of pro-vaccination "force":
\begin{equation}
x^{\prime}= \kappa x(1-x)( -\omega(t) );
\end{equation}
d) explanatory model with force pro-vaccine depending on the state variable $I$ (as in \cite{Bauch2}):
\begin{equation}\label{BI}
x^{\prime}= \kappa x(1-x)( I(t) -\omega(t) ),
\end{equation}
complemented by an SIR model with vital dynamics, of course.
They applied the various models, and various models of the function $\omega(t)$, to available data concerning the whole cells pertussis vaccine scare and to data on the MMR vaccine scare. These multiple comparison not only were done in order to find the best fitting and predicting model, but also in order to disentangle the effect of each component: the social learning, the kind of curve $\omega(t)$ modelling the vaccine scare; the type of disease-related feedback. They obtained the following interesting results: i) the best fit to available data were obtained by the model of Eq.~\ref{BI} including both the disease-related feedback and the social learning mechanism; ii) this model also produced good predictions, up to 10 years in the case of pertussis data; iii) the model of Eq.~\ref{BM} performed better than the simplified models.
\subsubsection{Delaying vaccination as a further strategy}
A wide empirical evidence suggests that parents often adopt the strategy of delaying their children's \textit{age at vaccination} against common vaccine preventable infections (\cite{BhattaBauch2010} and references therein), mostly because they fear that the immune system of a newborn is still too weak to adequately tolerate immunization. Bhattacharyya and Bauch \cite{BhattaBauch2010} proposed a minimal 3-strategy model, with two possible vaccination rounds, the first at birth (age $0$), and the second one at a fixed subsequent age $a_1$ set to four years \cite{BhattaBauch2010}, always by a \textit{perfect} vaccine. This is remindful of the two-rounds MMR (measles-mumps-rubella) vaccination, with an early first dose and a second one offered during the childhood to reach unvaccinated or non-successfully vaccinated children. Parents can opt between the following strategies: i) \textit{timely vaccinator} ($V$) i.e. vaccinate at the first round; ii) \textit{delayer} ($Del$) i.e. vaccinate at the second round; ii) \textit{Non vaccinators} ($N$). The three corresponding fractions of the population are denoted as $x_{V}$, $x_{Del}$ and $x_{N}=1-x_V-x_{Del}$. The adopted model is a two-age classes meta-population version of the SIR endemic model, with exponential transitions between age groups and inhomogeneous age-specific mixing (see section \ref{metapop}).
Individual learn and switch between strategies according to an IGD driven by the constant perceived risks of VAE from vaccination at ages $0$ and $a_1$ respectively, denoted by $r_{V,1}< r_{V,2}$, and by the direct prevalence-dependent perceived risk of disease in the two age groups \cite{Bauch2}. The latter depends on the corresponding risks $\eta_i$ ($i=1,2$) of acquiring infection in age groups $i=1,2$, that are modeled as follows $\eta_i = \chi_i I_i/(\rho_1 +I_1)$, namely as increasing and saturating functions of the (perceived) group-specific infective prevalence $I_i$ ($i=1,2$) to mirror two stylized facts, namely that (i) social contacts relevant for transmission are largely assortative i.e., tend to mostly occur with individuals of the same age group \cite{Mossong}, and (ii) agents form their perceptions of risk according to simple heuristics. As for payoffs: (i) the timely vaccinator payoff only includes the vaccination cost at age zero, i.e. $C_V = -r_{V,1}$; (ii) the delayer payoff includes the sum of the cost of acquiring disease in period 1, plus the cost of vaccination at age 2, which arises only if he/she escaped infection during period 1, i.e. $C_{Del}=-r_I \eta_1 - r_V (1-\eta_1)$, (iii) the non-vaccinator payoff is $C_{N}=-r_I \eta_1 - r_I \eta_2 (1-\eta_1)$.
The resulting dynamics of state variables $x_V$ and $x_{Del}$ (the dynamics of $x_N$ is given by $x_{N}=1-x_V-x_{Del}$) obey
\begin{align}
x_{V}^{\prime} & = \kappa_{v} ( \Delta E_{V,Del} x_{V}x_{Del} + x_{V} (1-x_{V}-x_{Del}) \Delta E_{V,N}), \label{p1_timely_vacc}\\
x_{Del}^{\prime} & = \kappa_{Del} ( x_{Del}(1-x_{V}-x_{Del}) \Delta E_{Del,N} - x_{Del} x_{V} \Delta E_{Del,V} ), \label{p2_delayer}%
\end{align}
\noindent where (similarly to the previously illustrated IGD-based models) the quantities $\Delta E_{i,j}=$ $ e_{i}-e_{j}=$ $C_{j}-C_{i}$ $(i,j = V,Del,N)$ represent the pairwise perceived payoff gain resulting from social encounters between individuals belonging to group $i$ and group $j$, resulting from the difference of the related costs. Quite reasonably Eqs. \ref{p1_timely_vacc}-\ref{p2_delayer} postulate a simplified mixing pattern between players of various strategies (compared to the mixing relevant for infection transmission), as indeed parents of children eligible for vaccination represent a restricted subgroup of the population in age group 2. The full model is obtained by adding to Eqs. \ref{p1_timely_vacc}-\ref{p2_delayer} the equations for susceptible and infective individuals in age groups 1,2.
The qualitative behavior of the model is more complicated than the baseline model, although basic intuitions from section \ref{Bauch_2005} explain well the main results. For example, in addition to the unstable \textit{infection free-all vaccinators} equilibrium, there is an unstable endemic \textit{pure delayer} equilibrium with $I_2=0$. This further confirms the impossibility of disease elimination under prevalence-dependent cost of infection.
Several mutually exclusive other locally stable endemic states exist, including the pre-vaccination equilibrium and various \textit{mixed} states with sub-optimal vaccine uptake combining different proportions of individuals following the different strategies. Interesting oscillatory regimens were observed in the simulations: individuals switch between strategies depending on the pattern of prevalence, with frequently observed anti-phase oscillations between strategies $V$ and $Del$, e.g. epochs of high prevalence incentivate timely vaccination, which in turn reduce prevalence, thereby promoting the delayer attitude, which in turn favors a restart of prevalence in age group 1, and so on. However, inclusion of the third strategy has a stabilizing effect with respect to the basic ($V$-$N$) strategy of section \ref{Bauch_2005} \cite{Bauch2}. The model also offers some insight on the role of the delayer strategy during epochs of vaccine scare.
\subsubsection{Dynamical effects of social norms}
\label{social_norms}
Previous models assuming \textit{prevalence-dependent} vaccinating behavior predict that elimination is impossible since vaccine uptake eventually sets around some equilibrium levels below the critical threshold $x_{c}$. However, in western countries vaccination coverage for certain diseases such as poliomyelitis and dyptheria attained and maintained very high levels, ensuring persistent local elimination, even in countries with voluntary vaccination. This suggests that further behavioral mechanisms are at work. A number of psychological theories relevant for health decisions such as the theory of planned behavior \cite{Ajzen} and its extensions, have stressed the importance of social pressure as a determinant of individual behavior. In particular, our own perceptions of what persons relevant to us might think we ought to do are relevant - the so-called \textit{injunctive social norms} (ISNs) \cite{Cialdini_Trost}
might affect vaccinating behavior in a number of ways, i.e. by building self-legitimation for non-vaccinators, by enforcing their cohesion, or by increasing the \textit{conformism} of vaccinators as a majority.
Oraby et al. \cite{oraby2014influence} included ISNs into the IGD model of section \ref{Bauch_2005} by assuming, in line with the game-theoretical work by \cite{Helbing_Johansson_2010}, that ISNs add a new component to payoffs which is proportional to \textit{group pressure}, i.e. to how many other people in the population are also playing that strategy. The perceived payoffs for nonvaccinators and vaccinators become: $C_N=-r_i m I(t) + \delta_0 (1-x)$, and $C_V = -r_V + \delta_0 x$, where $\delta_0$ tunes the effects of social pressure on payoffs, taken as symmetric. Note that social pressure has the potential to mitigate, or even to sign-reverse, the costs of the two strategies.
After some rescaling, this yields the following equation for the vaccinators proportion $x$
\begin{equation}\label{sirvp_eq_social_norms}
x^{\prime} =\kappa_{1} x(1-x) (I- \omega - \delta (1-2x) ),
\end{equation}
\noindent which, differently from section \ref{Bauch_2005}, emphasizes the relative cost of vaccination and the relative strength of social pressure with respect to the cost of disease, given by $\omega=r_V/(r_i m I)$ and $\delta=\delta_0/(r_i m I)$ respectively.
With respect to the model in section \ref{Bauch_2005} considering only the direct costs of vaccination or not, the inclusion of ISNs yields a far richer dynamics. First, besides the four basic equilibria $A,B,C,D$ observed in section \ref{Bauch_2005}, there is a fifth equilibrium $E$ induced by social pressure, where $S_E=1-x_E$, $I_E=0$, $x_E=2^{-1} (1+\omega/ \delta)$, with $x_E>1/2$. However, this \textit{disease free partial vaccinator} equilibrium, which exists only if $\delta < \omega$ (i.e. if the strength of group pressure is smaller than the relative cost of vaccination) is always unstable.
Second, the pre-vaccination endemic state $C=\left(S_{SIR}, I_{SIR},x_{SIR}=0\right)$ is locally stable if $I_{SIR} <\omega +\delta$, i.e. if the perceived cost of disease is small. Third, if $\delta > \omega$ ($\delta < \omega$), i.e. if the strength of group pressure is larger (smaller) than the cost of vaccination, then the pure vaccinator equilibrium $B$ is locally stable (unstable).
As a consequence of the above two stability conditions, both $B$ and $C$ can coexist in their stability ranges, i.e. the system exibits bistability and its mathematical behavior will depend on the initial conditions (see Fig. \ref{Fig2_Oraby_2014}). Bistability phenomena also involve the \textit{partial vaccinator} endemic state $D$ (see Fig. \ref{Fig2_Oraby_2014}). Finally, oscillations due to destabilization of $D$ are possible, as in section \ref{Bauch_2005}.
\begin{figure}
\centering \includegraphics[width=0.5\textwidth] {fig23.pdf}
\caption{Bistability of equilibria in the model with imitation dynamics and ISNs: regions of the $(\delta,m)$ parameters plane for the stability of the equilibria $B$ (disease-free full vaccinator equilibrium), $C$ (endemic pre-vaccination equilibrium), and $D$ (partial vaccinator endemic state). Equilibrium $B$ is stable in the hatched region below the thick black diagonal line, $C$ is stable in the light grey region, $D$ is stable in the dark grey region on the left. Details on other behavioral parameters and basic demographic and epidemiologic parameters are reported in \cite{oraby2014influence}. Source: Reprinted figure from Ref.~\cite{oraby2014influence}.}
\label{Fig2_Oraby_2014.pdf}
\end{figure}
Overall, including ISNs widely expands the spectrum of qualitative mathematical behaviors w.r.t. the baseline case where payoffs only include direct costs of the alternative strategies. In particular, ISNs can mitigate the main perverse consequence of prevalence-dependence, namely the "elimination impossible" result, by allowing the pure vaccinator equilibrium to become stable at large levels of social pressure. Other effects are more controversial, as social pressures can induce bistability phenomena in a large portion of the parameter space, with the well-known amplification of uncertainity due to both the imprecise measurement of initial state and and the equilibrium switch that can be induced by extrinsic stochasticity. Thus, depending on the context, social norms can either support or hinder vaccination goals. Finally, inclusion of ISNs allows, under a few further assumptions, an excellent fit of prevalence and coverage data of pertussis in the
|
UK since the seventies, including the epoch of the pertussis vaccine scare and the subsequent epoch of coverage relapse (see Fig. \ref{Fig1_Oraby_2014}).
\begin{figure}
\centering \includegraphics[width=0.7\textwidth] {fig24.pdf}
\caption{Empirical and modelled (a,c,e) pertussis vaccine coverage and (b,d,f ) pertussis incidence in the UK, (a,b) from 1971 to 1988 without injunctive social
norms, (c,d) from 1967 to 2010 without injunctive social norms and (e,f ) from 1967 to 2010 with injunctive social norms. The solid line is the empirical data and the dashed line is the best-fitting model. Source: Reprinted figure from Ref.~\cite{oraby2014influence} }
\label{Fig1_Oraby_2014.pdf}
\end{figure}
\subsubsection{Implications of bounded rationality in vaccination decisions}
\label{sec:bounded_rational_vaccination}
Vaccinating (or not) is a typical example of \textit{decision under uncertainty} conditions, for which bounded rationality approaches, such as Kahneman and Tversky \textit{prospect theory} \cite{Kahneman_Tversky_1979} (see section \ref{PsychologicalModels}) typically work better than classical rational calculus. Oraby and Bauch \cite{OrabyBauch2014_bounded_rationality} have investigated the dynamic implications of bounded rationality in vaccinating behavior by an SIR model in which vaccination decisions are formulated according to the ideas of prospect theory.
The chosen coupled behavior-disease model is formulated as an IGD, i.e. parents still learn about other's strategies through imitation, also including the effects of ISNs (see the previous subsection) and a non-perfect vaccine. The $x$-equation for the proportion of parents choosing strategy $V$ is formulated as
\begin{equation}\label{sirvp_eq_bounded_rationality}
x^{\prime} =\kappa_{1} x(1-x) F(\pi_{V|N}(I,x)) - F(\pi_{N|V}(I,x)),
\end{equation}
where $F(.)$ is a suitable probability distribution function, and $\pi_{V|N}(I,x)$ (alt. $\pi_{N|V}(I,x)$) is the net reward that a parent playing stategy $N$ (alt. $V$) perceives to gain from switching to strategy $V$ (alt. $N$), given current conditions on prevalence and vaccine uptake $I,x$. The map $F$ transforms net rewards to a probability scale, so that the term $\Delta_F = F(\pi_{V|N}(I,x)) - F(\pi_{N|V}(I,x))$, replacing the \textit{sure} perceived payoff gain of previous models, represents the \textit{probability gain} arising from the switch between strategies. The perceived net rewards $\pi_{V|N}(I,x)$, $\pi_{N|V}(I,x)$ are defined as
\begin{align}
\pi_{V|N}(I,x) & = \pi_{N}(V) - \pi_{N}(N) + (\delta_{V}x - \delta_{N} (1-x) ), \label{pi_from_N_to_V}\\
\pi_{N|V}(I,x) & = \pi_{V}(N) - \pi_{V}(V) + (\delta_{N}(1-x) - \delta_{V} x). \label{pi_from_V_to_N}%
\end{align}
In Eqs. \ref{pi_from_N_to_V}-\ref{pi_from_V_to_N}, besides the social norm components (analogous to those described in the previous section but taken asymmetric for sake of generality), the true core is represented by the differences $\Delta \pi_{V|N}=\pi_{N}(V) - \pi_{N}(N)$, and
$\Delta \pi_{N|V} = \pi_{V}(N) - \pi_{V}(V)$, where the quantities $\pi_{N}(V), \pi_{N}(N), \pi_{V}(N), \pi_{V}(V)$ represent the expected utilities arising from each of the four possible \textit{prospects} faced by agents. In \cite{OrabyBauch2014_bounded_rationality} it is hypothesized that agents classify the possible realizations of infection in a four item scale $r_{i,I}$ ($i=1,..,4$) i.e. (i) \textit{mild} infection, (ii) \textit{moderate}, (iii) \textit{morbid}, up to (iv) \textit{death} due to disease/vaccine adverse events (VAE), scaled to the base-case of absence of infection, with corresponding probabilities $p(r_{i,I})$. Analogous classification $r_{i,VAE}$ ($i=1,..,4$) is done for VAE. At each realization it is attached a utility evaluation $U_I (r_{i,I})$ as well as a weight $W(x(r_{i,I}))$ to the corresponding probability. The various prospects are defined consequently. For example, agents currently playing $N$ who continue to play $N$ can only experience the consequences of infection, so their prospect includes the utility outcomes $U_I (r_{i,I})$ with expected utility $\pi_{N}(N)=\sum U_I (r_{i,I})W(x(r_{i,I}))$. On the other hand agents currently playing $N$ who consider to switch to $V$ will include in their prospect both the utility outcomes from VAE, $U_{VAE} (r_{i,VAE})$, as well as those due to infection, as a consequence of vaccine imperfection, and so on.
The key step then lies in assigning the utility evaluations and corresponding weights, according to the principles of bounded rationality, i.e. accounting for the two different possible parents' attitudes, namely \textit {risk averting} vs \textit {risk seeking}, that classically cause departures from full rationality,
depending on a number of parameters reflecting \textit{boundedly rational} behavior. This is done by appropriate parametric specifications of $U$ and $W$ \cite{OrabyBauch2014_bounded_rationality} including \textit{perfect rationality } as a sub-case.
The resulting SIR vaccination model based on Eq. \ref{sirvp_eq_bounded_rationality} and Eqs. \ref{pi_from_N_to_V}-\ref{pi_from_V_to_N} is quite richer compared to section \ref{social_norms} as it includes, besides bounded rationality of vaccination decisions, also vaccine imperfection and asymmetric effects of social pressure. In particular, the underlying perfect rationality model is the extension of the model of previous section accounting for vaccine imperfection and asymmetric social pressure.
The ensuing equilibrium structure differs from that in \ref{social_norms} due to the appearance of a sixth endemic state $F$, which is a \textit{pure vaccinator} endemic state forced by vaccine imperfection. In particular (i) the \textit{disease free partial vaccinator} equilibrium $E$ remains always unstable, as in \ref{social_norms}, (ii) the pure vaccinator, disease-free equilibrium $B$ is stable for infections that are not not highly transmissible, if the
vaccinators social pressure is relatively strong compared to the vaccine cost, while the \textit{pure vaccinator endemic state} $F$ is stable if and only if vaccinator pressure is large and the vaccine efficacy is low.
To sum up, the effects of bounded rationality are mostly dynamical. Indeed, while the dynamics observed in the underlying rational model are much alike those in section \ref{social_norms}, with bistability regions having simple forms, the inclusion of bounded rationality yields a zoo of dramatic dynamical changes, which include complicate nonlinear shapes of the stability regions, and steady oscillations about the pure vaccinator disease-free equilibrium $B$. In addition, boundedly rational behavior makes it harder to eliminate infection, because it can offset the beneficial effects of the social pressure of the vaccinators group.
\subsubsection{Modeling public communication on disease and vaccine risks}
\label{public_communication_PONE2012 }
A parsimonious explanation of some observed features of vaccination systems can be obtained by removing an unrealistic feature of models based on IGD, namely that switching between strategies is entirely determined by learning from other agents (i.e., parents of children to be vaccinated) during social encounters. Indeed, the pure-imitation model poorly represents the process of information provision in modern countries because, even when a given vaccination is fully voluntary. Public health systems will nonetheless keep the role of main suppliers of the relevant information on risks associated to diseases and vaccines, as suggested by a body of evidence \cite{noibeta}. In this subsection we report an extension of the basic imitation-based framework of sections \ref{Bauch_2005}-\ref{IGD_with_myopic_perception_of_VAE}, which accounts also for perceptions based on information (about risks) supplied by the public health systems \cite{noibeta}. In this new framework, remindful of Bass classic model for information diffusion \cite{Bass}, switches between $N$ and $V$ strategies is determined by the balance between private information, exchanged through inter-personal communication between parents of children eligible for vaccination during their social contacts, and public information, communicated by the public health authorities through media and related channels. This allows to amend the equations for the dynamics of switch between the two strategies as follows
\begin{align}
y_V^{\prime} =& G(t)y_N- K_{VN}y_N y_V + k_{NV}y_N Y_V, \label{sVaw}\\
y_N^{\prime} =& -G(t)y_N- k_{NV}y_N Y_V + K_{VN}y_N y_V, \label{sNaw}
\end{align}
where $G(t)$ represents the rate of transition from strategy $N$ to strategy $V$ induced by mediatic information (which is independent of the level of individuals' social activity), yielding
\begin{equation}\label{igeq}y_V^{\prime} = G(t)(1-y_V) + (K_{NV}- K_{VN})y_V(1-y_V), \end{equation}
where $K_{NV} = \kappa \theta(I)$ and $K_{VN}=\kappa \alpha(x)$. In the simplest case one can assume that $G(t)$ is a strictly positive constant, i.e. $G(t)=k_{G}>0$. The underlying idea is that the information provided by public health systems aims to convey a different perception of risks related to disease and vaccination compared to what might happen in private social encounters. In particular, a "`wise"' public communication aims to convince that (i) vaccines are highly safe, with a very low, constant, risk of VAE, and that (ii) risks associated with the disease are prevalence-independent.
Setting $x=y_V$ and simplifying Eq. \ref{igeq} yields the following dynamic equation for the proportion favourable to strategy $V$
\begin{equation}\label{p_equation_G_model}
x^{\prime} = \kappa (1-x) \left( \left( \theta(I)-\alpha(x)\right) x + \psi \right),
\end{equation}
where parameter $\psi=G/\kappa$ summarizes the effectiveness of the public \textit{effort} (information, education, availability of vaccination infrastructures, subsides to vaccination staff, etc) in affecting perceptions on vaccines and disease.
The properties of the ensuing behavior-disease model arising by coupling Eq. \ref{p_equation_G_model} with Eqs. \ref{sirvs}-\ref{sirvi} depend in a critical manner
on parameter $\psi$. Briefly, there exist appropriate thresholds $\psi_c < \psi_1$ such that
\begin{itemize}
\item For high levels of public effort ($\psi \ge\psi_1$) only the pure vaccinator equilibrium $B$ exists and it is globally asymptotically stable (GAS).
\item For intermediate levels of public effort (say $\psi_c <\psi < \psi_1$) the pure vaccinator equilibrium is unstable, but there is a \textit{disease free partial vaccinator} equilibrium $E$, with vaccine uptake above the elimination threshold, which is GAS.
\item For low levels of public effort ($\psi \le \psi_c$) the disease free partial vaccinator equilibrium $E$ exists but is unstable, and the \textit{partial vaccinator} endemic state $D$ appears ensuring endemicity, though it is not necessarily stable i.e. oscillations may occur.
\end{itemize}
In a control perspective, if the infection is endemic at $D$ in a scenario where vaccination is voluntary with public intervention absent, it is possible to increase the equilibrium coverage by increasing the public effort in providing information about benefits of vaccination. Suitable further increases in public effort can allow the equilibrium vaccine uptake to expand until the endemic state $D$ disappears by exchanging its stability with the disease free partial vaccinator $E$, thus achieving elimination. Further increases in $\psi$ can push $E$ to collapse on the pure vaccinator equilibrium $B$.
These results indicate that public intervention can completely mitigate the negative implications of prevalence-dependent, pure-imitation models (sections \ref{Bauch_2005}-\ref{IGD_with_myopic_perception_of_VAE}), particularly removing the impossibility to eliminate the infection. A simple calibration to Italian data on measles coverage shows that this model explains the recent Italian story, characterized by a fast increase in measles vaccine uptake following substantial efforts by the public health system, largely better than models including imitation only \cite{noibeta}.
\subsection{\textit{Behavior-explicit} game-theoretic models of vaccinating behavior}
\label{game-theoretic models}
The first application of classical game theory and epidemiological modeling to investigate the implications of vaccination decisions dealt with the conflict between group interest and self-interest in relation to smallpox vaccination upon reintroduction following a terroristic attack \cite{Bauch0}. A companion paper \cite{Bauch1} applied the same approach to mass vaccination against common vaccine-preventable childhood infections under voluntary vaccination. The latter work gave much momentum to subsequent research in the field. In line with the previous subsections, this subsection reviews this seminal work and a few subsequent fundamental contributions focusing on childhood vaccination \cite{Bauch1,Reluga1,RelugaGalvani,Shim_JTB_2012}. The flavor of the presentation is slightly different however, as the focus of these approaches, though framed within SIR models for vaccine-preventable infections, is more on classic game theoretic questions, namely existence of Nash equilibria for individuals' vaccine uptake and comparisons with the level which is optimum for the community as a whole, rather than on dynamic properties.
\subsubsection{The baseline equilibrium model}
\label{Earn_Bauch_2004}
The work in \cite{Bauch1} integrates ideas from the endemic SIR model with vaccination at birth (see section~\ref{SIR_vaccination}) into a game theoretical framework to investigate the interplay between individual vaccination decisions and the population-level dynamics of infection in a voluntary vaccination scenario. Game theory is the natural framework given that the vaccination strategy of each specific agent (i.e. parents), whose sum defines the vaccine coverage at the population level, is influenced by other parents' decisions in the sense that the corresponding payoff will depend on the strategies played by all other players.
Letting the individual strategy be represented by the parents' probability to vaccinate $x$, authors then seek a \textit{convergently stable Nash equilibrium} (CSNE) solution $x^*$, which is expected to be observed in a real population of players.
The key determinants of individual strategies are the perceived \textit {relative cost of vaccination} $r$, defined as the ratio between the perceived risk of VAE ($r_V$) and the perceived risk of disease given infection ($r_I$), and the probability $\pi_x$ of experiencing infection under a given vaccination coverage $x$. For the latter it is assumed that parents correctly know the underlying epidemic model and estimate $\pi_x$ by the corresponding lifetime probability of infection at endemic equilibrium for an unvaccinated individual, given by
\begin{equation}
\pi_x = 1-\frac{1}{(1-x)\mathcal{R}_0}.
\end{equation}
Note that the scenario of full absence of vaccinators ($x=0$) yields $\pi_0=x_{c}$.
The authors show that there is always a unique CSNE solution $x^*$. For high levels of the relative perceived risk of vaccination, namely for $r>\pi_0$, i.e. for $r>x_{c}$, the CNSE corresponds to the situation where no one vaccinates ($x^*=0$), i.e. the pure non-vaccinator strategy. On the other hand, if $r<\pi_0$, i.e. $r<x_{c}$, the CNSE solution predicts a strictly positive probability to vaccinate $x^*$. The condition $r<\pi_0$ implies
\begin{equation}
(1-r)\mathcal{R}_0 <1.
\end{equation}
Under the above constraint, the CSNE is
\begin{equation}
x^* = 1 - \frac{1}{(1-r)\mathcal{R}_0},
\end{equation}
which is such that
\begin{equation}
x^*(r)< x_{c}
\end{equation}
(unless $r=0$). In other words, under voluntary vaccination elimination turns out to be impossible if individual only take their self-interest into account (Fig. \ref{Figs1_2_BauchEarn2004}, left panel). Moreover, the case where $r$ exceeds the elimination threshold ($r>x_{c}$), a fact possibly observable during periods of vaccine scare, the CNSE predicts absence of any vaccinator at all. A key point stressed in \cite{Bauch1} is that when the perceived relative risk switches from an initial level $r^0<x_{c}$ to another one $r^{'}$ greater than $x_{c}$, then the resulting payoff gain is increasing with $r^{'}$ and it has no plateau. In other words, the vaccine scare can go unlimited. In \cite{Bauch1} it is also considered the effect of measures induced by the public health systems to counteract the vaccine scare, i.e. a reduction of the relative risk from a value in excess of $x_{c}$ down to another value below $x_{c}$. The authors obtained that the incentive to vaccinate dimishes as the vaccine coverage approaches to its Nash equilibrium. Summarizing, a vaccine scare easily induces a substantial drop in vaccination, whereas restoring the pre-scare coverage levels is difficult. These findings are illustrated in the right panel of Fig.~\ref{Figs1_2_BauchEarn2004}.
\begin{figure}
\centering \includegraphics[width=0.9\textwidth] {fig25.pdf}
\caption{The baseline equilibrium model of the vaccination game. Left panel: patterns of the vaccination coverage $x^*$ predicted by the CSNE as a function of the relative risk $r=r_V / r_I$ of vaccination for different values of the BRN $\mathcal{R}_{0}$. The dashed horizontal lines drawn at the level of the critical coverage $x_c=x^* (r=0)$ (for the different levels of $\mathcal{R}_{0}$) show the impossibility to eliminate the infection in a regime where individuals vaccinate according to self-interest driven by direct perceived risks. Right panel: impact of a vaccine scare as represented by a large increase of the relative risk of vaccination from a very low level $r^0<<x_{c}$ to a new level $r^{'}$, for different levels of the BRN $\mathcal{R}_{0}$. The graph reports the trend of the payoff gain $\Delta E$ and of the change $\Delta P$ in vaccine uptake for different levels of $r^{'}$. For moderate levels of $r^{'}$, namely $r^{'}<x_{c}$, the impact is moderate but becomes dramatic when $r^{'}$ exceeds $x_{c}$ causing the CSNE coverage dropping to zero. Source: Reprinted figure from Ref.~\cite{Bauch1}.}
\label{Figs1_2_BauchEarn2004}
\end{figure}
\subsubsection{Best individual vs community vaccination response in a heterogeneous population}
\label{Reluga_Bauch_2006}
In \cite{Reluga1} a general game-theoretical approach was adopted, aimed to relate the population vaccine demand to individual decisions in a behaviorally heterogeneous population sharing a common information `signal' about the dynamic spread of an SIR-type infection. In their model the individuals choose between strategy $V$ (vaccinate their children) and $N$ (not to vaccinate) based on perceived benefits and costs of vaccinating, in turn based on the information signal $\sigma$ carrying information about risks of acquiring the infection. This information signal has the same meaning (and form) of the information index $M$ in the phenomenological models of section \ref{InfoDependentCoverageNoiTPB2007}. The population is assumed to be behaviorally heterogeneous, and its internal structure is described by a single real variable $y$. Individuals having type $y$ choose strategy $V$ (to vaccinate) with probability $\chi(y,\sigma)$ and strategy $N$ with probability $1-\chi(y,\sigma)$. Benefits of vaccination are summarized by the utility function of strategy $V$ which depends on $\sigma$ and $y$: $U_V(y,\sigma)$, and is increasing in $\sigma$. On the other hand, the utility of vaccine refusal (strategy $N$) is assumed to also depend on the population average vaccine uptake
\begin{equation}
\bar{\chi}=\int_y \chi(y,\sigma)dy,
\end{equation}
i.e. $U_N(y,\sigma,\bar{\chi})$, where $U_N$ is decreasing in $\bar{\chi}$, mirroring a free-riding effect (compare this assumption with sections \ref{IGD_with_myopic_perception_of_VAE} - \ref{public_communication_PONE2012 } where the perceived cost of vaccination was increasing in the average vaccine uptake).
By taking the individual's expected utility
\begin{equation}
U(y,\sigma,\bar{\chi}) = \chi(y,\sigma )U_V(y,\sigma)+ (1-\chi(y,\sigma))U_N(y,\sigma,\bar{\chi}),
\end{equation}\label{Utility_Reluga200}
one notes that the resulting individual's optimal strategy will necessarily depend on the strategies adopted by other individuals in the population i.e., the above problem defines a game.
The search for the \textit{best response strategy} $\chi^*$ for the previous problem has to be framed into the theory of set-valued maps \cite{aubin}, yielding
\begin{equation}
\chi^*(y,\sigma,\bar{\chi})= \begin{cases} 0 &\mbox{if } U_V(y,\sigma)<U_N(y,\sigma,\bar{\chi}), \\
1 &\mbox{if } U_V(y,\sigma)>U_N(y,\sigma,\bar{\chi}), \\
[0,1] &\mbox{if } U_V(y,\sigma)=U_N(y,\sigma,\bar{\chi}). \end{cases}
\end{equation}
In order to completely characterize the Nash equilibrium, it is needed to fulfill a self-consistency condition concerning the average vaccine uptake at equilibrium:
\begin{equation}
\bar{\chi^*} \in \int \chi^*(y,\sigma,\bar{\chi^*})dy,
\end{equation}
i.e.
\begin{equation}
\int \inf \chi^*(y,\sigma,\bar{\chi^*} ) dy< \bar{\chi^*}<\int \sup \chi^*(y,\sigma,\bar{\chi^*} )dy .
\end{equation}
This condition ensures that there exists a Nash equilibrium and that the population's average response is the same for all Nash equilibria.
On the other hand, at the population level, a benevolent social planner will seek the optimal community strategy by maximizing the \textit{community utility}
\begin{equation}
\widehat{U}(\chi(.)) = \int \left( U(y,\sigma,\bar{\chi}) \chi(y,\sigma )U_N(y,\sigma)+ (1-\chi(y;\sigma))U_V(y,\sigma,\bar{\chi}) \right) dy .
\end{equation}
By elementary functional analysis it is easy to verify that the above individual-wise optimum $\chi^*$ and the community-wise optimum $\chi^{C}$ are such that for all $y$ and given $\sigma$
\begin{equation}
\chi^*(y,\sigma)<\chi^{C}(y,\sigma).
\end{equation}
In other words, the Nash equilibrium for individuals will always be characterized by "less vaccination" than the corresponding community-based equilibrium, unless the Nash equilibrium is universal vaccination, confirming in a very general manner a long-standing result of the literature, namely that the pursuit of optimal self-interest leads to vaccine uptakes that are systematically lower than those that are optimal at community level.
Note that both the optimal $\chi^*$ and its average value $ \bar{\chi}^*$ depend implicitly on the information signal $\sigma$, which is a function of the state variables. For example in \cite{Reluga1} it is assumed that $\sigma$ is an increasing and saturating function of infection prevalence $I$: $\sigma(t)=\beta I / (\mu + \beta I)$. Note also that $\bar{\chi^*}$ results to be an increasing function of prevalence if the signal $\sigma$ has this property.
The implications of the above described approach for infection transmission and control are then investigated by setting the resulting vaccination strategies within the framework of an SIR model with vaccination. Three main cases are considered: (a) the vaccine coverage $x$ is equal to the average Nash equilibrium response $ x(t) = \bar{\chi^*}(I)$; (b) the vaccine coverage adapts dynamically to the differences between its departures from the average Nash response, according to the equation $ x^{\prime}(t) = \alpha(\widehat{\chi^*}(I) - x)$; (c) the vaccine coverage adapts dynamically according to an IGD where the switch rate from $N$ to $V$ strategy is determined by the payoff gain of vaccination (which, as it can be easily seen, is constant) and the inverse switch is determined by the perceived vaccine cost (which depends on $\sigma(I)$).
Unsurprisingly, model (a) (compare with the phenomenological model in section \ref{InfoDependentCoverageNoiTPB2007}) leads to a unique locally stable endemic equilibrium with partial vaccination. Similarly in model (b), where $x$ 'follows' $\bar{\chi^*}$ with an exponential delay, the endemic equilibrium can be unstabilized by varying $\alpha$, yielding sustained oscillations. This confirms the already stated fact that phenomenological and game-theoretic coupled behavior-infection models are often indistinguishable when the game-theoretic solution is simply set, heuristically, within the framework of a dyamic epidemiological model.
\subsubsection{Markov decision processes show that the interaction between vaccine-believers and vaccine-skeptics yields sub-optimal measles vaccine uptake}
\label{Markov_decision_processes_measles}
Within game-theoretic approaches several efforts \cite{Reluga2,RelugaGalvani,Shim_JTB_2012} have been devoted to the investigation of the population level implications, of individual choices determined within the framework of Markov decision processes theory (MDPT). An attempt of systematization of the subject in reference to vaccination games was developed in \cite{RelugaGalvani}. The framework rests on three interacting building blocks, namely a model for individual decisions, a model for changes at the individual scale
($\mathcal{I}$-scale), and a mean-field epidemiological model for describing the dynamics at the population scale ($\mathcal{P}$-scale). Individuals handle uncertainty by maximizing their (inter-temporal) expected utility, on the assumption that they are able to predict - in probabilistic terms - their future evolution in terms of both their future states $\mathcal{Y}(t)$ and the ensuing utility flow, that follow from taking a given strategy (i.e., vaccinating a child at a certain age, or not vaccinating him), or action, $A(t)$. Path $\mathcal{Y}(t)$ is a stochastic process which depends on the adopted strategy as well as on the strategies that have been adopted by other individuals, that is the average strategy at the population level. For example, for those not vaccinating the future utility path depends on the risk of catching infection which under homogeneous mixing depends on the overall coverage prevailing in the population. The $\mathcal{P}$-scale dynamics of infection is specified by an appropriate mean-field model, e.g. an SIR model, where the actual coverage is determined by the average of individuals' strategies. Finally, the $\mathcal{I}$-scale dynamics is specified by taking a suitable continuous-time Markov chain embedded in the $\mathcal{P}$-scale model. In \cite{Shim_JTB_2012} this framework was applied to investigate the consequences of a typical phenomenon in current vaccination systems, namely the coexistence of well-identified and stable groups valuing differently the costs arising from infection and vaccination.
\begin{figure}
\centering \includegraphics[width=0.5\textwidth] {fig26.pdf}
\caption{Pattern of the best Nash response in terms of vaccine uptake among vaccine skeptics, drawn as a function of the cost of vaccination relative to the cost of infection ($C_{V,1}/C_{I,1}$) for different levels of the BRN $\mathcal{R}_{0}$ consistent with measles infection ($\mathcal{R}_{0}=13,15,17)$, and for a proportion of skeptics ($q_{1}=0.30$, showing the dramatic fall of the optimal vaccine uptake when the relative perceived risk exceeds a threshold level. Source: Reprinted figure from Ref.~ \cite{Shim_JTB_2012}. With permission from Elsevier.}
\label{Fig3A_Shim_2012}
\end{figure}
The simplest scenario, introduced in the static game-theory framework in \cite{Manfredi}, is represented by the conflict between two groups, namely vaccine vaccine \textit{skeptics} (group 1) and \textit{believers} (group 2), with different asymmetric perceived costs of infection and VAE: $C_{V,1}>C_{V,2}, C_{I,1}<C_{I,2}$. Let $q_{n}$ ($n=1,2, q_{1}+q_{2}=1$) denote the relative size of each group (assumed to be constant), $\phi_{n}$ the vaccination \textit{strategy} adopted by individuals in group $n$ and $\overline{\phi}_{n}$ the average coverage prevailing in group $n$. The $\mathcal{P}$-scale model is represented by the following simple two groups' SIRV model with a class vaccinated at birth by a perfect vaccine, in a constant population, where the two groups differ only in the average coverage:
\begin{align}
S_{n}^{\prime}&= \mu( 1- \overline{\phi}_{n})q_{n} -\beta S_{n} (I_{1}+_I{2}), \label{SIRV_Shim2012_eqS_k}\\
I_{n}^{\prime}&= \beta S_{n} (I_{1} + I_{2}) -(\gamma +\mu)I_{n}, \label{SIRV_Shim2012_eqI_k}\\
V_{n}^{\prime}&= \mu \overline{\phi}_{n}q_{n} - \mu V_{n}, \label{SIRV_Shim2012_eqV_k}
\end{align}
\noindent where $S_{n},I_{n},V_{n}$ respectively represent the susceptible, infective, removed and vaccinated proportions in each group, with the removed compartment obeying $R_{n}=q_{n}-(S_{n} + I_{n} + V_{n})$. The rates determining the various state transitions in Eqs. \ref{SIRV_Shim2012_eqS_k}-\ref{SIRV_Shim2012_eqV_k} determine the associated Markov chain for state transition at the $\mathcal{I}$-scale, under the further assumption that agents estimate the FOI faced by their children by its endemic equilibrium value
$\lambda=\lambda_{e} = \beta (I_{1,e} + I_{2,e})$, which agents know from the knowledge of the $\mathcal{P}$-scale model. It holds
$\lambda=\mu (\mathcal{R}_{V}-1)$ where $\mathcal{R}_{V}=\mathcal{R}_{0}(1-\overline{\phi}_{1}q_{1}-\overline{\phi}_{2}q_{2} )$ is the vaccine reproduction number (section \ref{SIR_vaccination}), meaning that individuals correctly perceive the long-term risk as a function of the overall coverage prevailing in the community. The corresponding system for the individual's state probabilities has the form $\mathcal{Y}_{n}^{\prime} = \mathcal{Q}_{n} \mathcal{Y}_{n}$, driven by the $4 \times 4$ transition matrix $\mathcal{Q}_{n}$:
$$
\mathcal{Q}_{n} =\begin{pmatrix}
-(\lambda+\mu)&0&0&0\\
0&\lambda & -(\gamma+\mu) &0 \\
0& 0 & -\mu &0\\
0& 0 &0 &-\mu \\
\end{pmatrix}
$$
\noindent with initial probability distribution, actually representing the probabilities that the generic individual occupies at the moment of birth any of the four possible epidemiological states, given by the vector $\mathcal{Y}_{n}(0)=[1-\overline{\phi}_{n}, 0, \overline{\phi}_{n}, 0]$. Note that the transition matrix is the same for both groups, since perceptions about epidemiological and demographic hazards are assumed to be the same, so that difference in $\mathcal{I}$-scale dynamics is only due to the decision to vaccinate or not made at birth. Given the adopted individual strategy $\phi$, and the ensuing $\mathcal{I}$-scale path $\mathcal{Y}_{n} (t)$, individuals belonging to group $n$ have a (discounted) utility flow
\begin{equation}
U_{n} = F_{n} \mathcal{X}_{n}(0) + \int_{0}^{\infty} e^{-rt}f_{n}^{T} \mathcal{X}_{n}(t)dt, \label{Shim2012_Utility_flow_k} %
\end{equation}
\noindent where vector $f_{n}=[0, 0, -C_{V,n}, 0]^T, (n=1,2)$ includes the costs associated with occupancy of various states at any time, while
$F_{n}=[0, -C_{I,n}, 0, 0]^T$ includes the costs associated with transitions occurring at age $0$, which occur only in the event of vaccination. Integrating Eq. \ref{Shim2012_Utility_flow_k} yields $U_{n} = (F_{n}^{T} + f_{n}^{T}(rI-Q_{n})) \mathcal{X}_{n}(0) \label{Shim2012_Utility_k}$, which expands as
\begin{equation}
U_{n} = -\phi_{n}C_{V,n} - (1-\phi_{n})C_{I,n} (r+\mu+\gamma)^{-1}\frac{\lambda}{r+\mu+\lambda}. \label{Shim2012_Utility_k}%
\end{equation}
\\
The mathematical analysis shows that the Nash strategy, i.e. the \textit{best response curve} $\widehat{\phi}_{n}$ arising from the maximization of expected utility of Eq. \ref{Shim2012_Utility_k}, has the following form: (i) not to vaccinate ($\widehat{\phi}_{n}=0$) for levels of the perceived (equilibrium) FOI $\lambda$ strictly below a group-specific threshold $\lambda_{\mathcal{T},n}$, (ii) to always vaccinate ($\widehat{\phi}_{n}=1$) for values of of $\lambda$ strictly above $\lambda_{\mathcal{T},n}$, and (iii) to choose some (unspecified) intermediate level $\widetilde{\phi}_{n}, 0<\widetilde{\phi}_{n}<1$ for $\
|
in Q' \subseteq Q$ and $y \in Q$, which is a contradiction. We conclude that $(\theta(Z_{1}), \theta(Z_{2}))$ is an IDL-congruence.
Conversely, we assume $(\theta(Z_{1}), \theta(Z_{2}))$ is an IDL-congruence. Suppose that $(Z_{1},Z_{2})$ is not a $T_{\mathcal{M}}$-closed set of $\mathcal{F}_{\mathcal{M}}$. Then there exist $P \in Z_{1}$, $Q \in \mathcal{X}(\textbf{A})$ and $R \in \mathcal{X}(\textbf{B})$ such that $(R,Q) \in \mathcal{D}(P)$ and $(R,Q) \notin Z_{2} \times Z_{1}$. If $R \notin Z_{2}$, then since $Z_{2}$ is a closed set of $\mathcal{X}(\textbf{B})$ there exist $b,c \in B$ such that $b \in R$, $c \notin R$ and $(b \wedge c, b) \in \theta(Z_{2})$. Let us consider ${\rm{Fig}}_{\textbf{B}}(R \cup \{c\})$. Since $R \in T_{\mathcal{M}}^{1}(P,Q)$, then $i({\rm{Fig}}_{\textbf{B}}(R \cup \{c\}), P) \nsubseteq Q$, i.e., there exists $x \in A$ such that $x \notin Q$ and $p \leq i(r \wedge c, z)$, for some $r \in R$ and $p \in P$. Hence $i(r \wedge c, z) \in P$ and by Proposition \ref{propo_1}, $i(r \wedge b \wedge c, z) \in P$. On the other hand, since $(\theta(Z_{1}), \theta(Z_{2}))$ is a congruence, we obtain that $(i(r \wedge b \wedge c, z), i(r \wedge b, z)) \in \theta(Z_{1})$ and $i(r \wedge b, z) \in P$. So, $i(R,P) \subseteq Q$, $r \wedge b \in R$ and by Lemma \ref{lem_1} we have $z \in Q$, which is a contradiction. If $Q \notin Z_{1}$, then there exist $x,y \in A$ such that $x \in Q$, $y \notin Q$ and $(x \wedge y, x) \in \theta(Z_{1})$. Let us consider $I={\rm{Idg}}_{\textbf{A}}(Q^{c} \cup \{y\})$. Observe that $I \cap i(R,P) \neq \emptyset$, because otherwise from the Prime Filter Theorem, there would exists $H \in \mathcal{X}(\textbf{A})$ such that $i(R,P) \subseteq H$, $H \subseteq Q$ and $x \notin H$ which is absurd since $Q$ is minimal. Thus, there exist $a \in A$ such that $a \leq q \vee x$ and $p \leq i(r,a)$, for some $q \notin Q$, $r \in R$ and $p \in P$. So, by Proposition \ref{propo_1}, $p \leq i(r,a) \leq i(r, q \vee x)$ and $i(p, q \vee x) \in P$. Therefore, since $(\theta(Z_{1}), \theta(Z_{2}))$ is a congruence, it follows that $(i(r, q \vee (x \wedge y)), i(r, q \vee x)) \in \theta(Z_{1})$. Hence $i(r, q \vee (x \wedge y)) \in P$. Since $i(R,P) \subseteq Q$ and $r \in R$, then by Lemma \ref{lem_1} we get that $q \vee (x \wedge y) \in Q$, which is a contradiction because $Q$ is prime. Then $(Z_{1},Z_{2})$ is a $T_{\mathcal{M}}$-closed set.
\end{proof}
Let $\{\mathcal{M}_{k}\}_{k\in K}$ be a family of FIDL-modules, with $\mathcal{M}_{k}=\langle \textbf{A}_{k}, \textbf{B}_{k},f_{k},i_{k}\rangle$. Then
\begin{equation*}
\underset{k\in K}{\prod}\mathcal{M}_{k} = \left\langle \underset{k\in K}{\prod}{\bf{A}_{k}}, \underset{k\in K}{\prod}{\bf{B}_{k}}, f, i \right\rangle
\end{equation*}
has a FIDL-module structure, where $f(a,b)(k)=f_{k}(a(k),b(k))$ and $i(b,a)(k)=i_{k}(b(k),a(k))$, for every $k \in K$. Let $\pi^{\bf{A}}_{k}: \underset{k\in K}{\prod}{\bf{A}_{k}} \rightarrow \bf{A}_{k}$ and $\pi^{\bf{B}}_{k}:\underset{k\in K}{\prod}{\bf{B}_{k}} \rightarrow \bf{B}_{k}$ be the projection homomorphisms. Note that the pair $(\pi^{\bf{A}}_{k}, \pi^{\bf{B}}_{k})$ is a FIDL-homomorphism, for every $k \in K$. It is no hard to see that $\underset{k\in K}{\prod}\mathcal{M}_{k}$ together with the family $\{(\pi^{\bf{A}}_{k}, \pi^{\bf{B}}_{k})\}_{k \in K}$ is in fact the categorical product of $\{\mathcal{M}_{k}\}_{k\in K}$.
Let $(\alpha,\gamma)$ be a FIDL-homomorphism. We say that $(\alpha,\gamma)$ is a 1-1 FIDL-homomorphism if $\alpha$ and $\gamma$ are 1-1, and similarly, we say that $(\alpha,\gamma)$ is a onto FIDL-homomorphism if $\alpha$ and $\gamma$ are onto.
If $\mathcal{M}$ is a FIDL-module, then we introduce the following concepts:
\begin{itemize}
\item We will say that $\mathcal{M}$ is a {\it{subdirect product}} of a family $\{\mathcal{M}_{k}\}_{k\in K}$ of FIDL-modules, if there exists a 1-1 FIDL-homomorphism
\begin{equation*}
(\alpha, \gamma) \colon \mathcal{M} \to \underset{k\in K}{\prod}\mathcal{M}_{k}
\end{equation*}
such that $(\pi^{\bf{A}}_{k} \alpha, \pi^{\bf{B}}_{k} \gamma)$ is an onto FIDL-homomorphism, for every $k\in K$
|
.
\item We will say that $\mathcal{M}$ is \emph{subdirectly irreducible} if for every family of FIDL-modules $\{\mathcal{M}_{k}\}_{k\in K}$ and 1-1 FIDL-homomorphism
\begin{equation*}
(\alpha, \gamma) \colon \mathcal{M} \to \underset{k\in K}{\prod}\mathcal{M}_{k}
\end{equation*}
there exists a $k\in K$ such that $(\pi^{\bf{A}}_{k} \alpha, \pi^{\bf{B}}_{k} \gamma)$ is an isomorphism of FIDL-modules.
\item We will say that $\mathcal{M}$ is \emph{simple} if the lattice of the FIDL-congruences has only two elements.
\end{itemize}
The following result is immediate from Theorem \ref{Characterization of congruences}.
\begin{corol} \label{subdirectly irreducible DLFI-modules}
Let $\mathcal{M}$ be a FIDL-module. Then $\mathcal{M}$ is subdirectly irreducible if and only if $\mathcal{M}$ is trivial or there exists a minimal non-trivial FIDL-congruence in $\mathcal{M}$.
\end{corol}
If $\mathcal{U}$ is an Urquhart space, then from Theorem \ref{Characterization of congruences} it is clear that $\mathcal{C}_{s}(\mathcal{U})$ is an algebraic lattice. So, if $Z_{1} \times Z_{2} \subseteq X \times Y$, let ${\rm{cl}}_{\mathcal{C}_{s}}(Z_{1},Z_{2})$ be the smallest element of $\mathcal{C}_{s}(\mathcal{U})$ which contains $Z_{1} \times Z_{2}$. Let $(x,y) \in X \times Y$. If there is no place to confusion, we write ${\rm{cl}}_{\mathcal{C}_{s}}(x,y)$ instead of ${\rm{cl}}_{\mathcal{C}_{s}}(\{x\},\{y\})$.
\begin{prop} \label{Simple algebras}
Let $\mathcal{M}$ be a FIDL-module and $\mathcal{F}_{\mathcal{M}}$ be the Urquhart space associated of $\mathcal{M}$. Then $\mathcal{M}$ is simple if and only if ${\rm{cl}}_{\mathcal{C}_{s}}(P,Q) = \mathcal{X}(\textbf{A}) \times \mathcal{X}(\textbf{B})$, for every $ (P,Q) \in \mathcal{X}(\textbf{A}) \times \mathcal{X}(\textbf{B})$.
\end{prop}
\begin{proof}
Since $\mathcal{M}$ is simple if and only $Con(\mathcal{M})=\{(\Delta^{\mathbf{A}}, \Delta^{\mathbf{B}}), (\nabla^{\mathbf{A}},\nabla^{\mathbf{B}})\}$, then by Theorem \ref{Characterization of congruences} this is equivalent to $\mathcal{C}_{s}(\mathcal{F}_{\mathcal{M}})=\{(\emptyset,\emptyset),(\mathcal{X}(\textbf{A}),\mathcal{X}(\textbf{B}))\}$ and the result follows.
\end{proof}
\begin{theorem} \label{Subdirectly irreducible algebras}
Let $\mathcal{M}$ be a FIDL-module and $\mathcal{F}_{\mathcal{M}}$ be the Urquhart space associated of $\mathcal{M}$. Then $\mathcal{M}$ is subdirectly irreducible but no simple if and only if the set
\begin{equation*}
\mathcal{J} = \{ (P,Q) \in \mathcal{X}(\textbf{A}) \times \mathcal{X}(\textbf{B}) \colon {\rm{cl}}_{\mathcal{C}_{s}}(P,Q)=( \mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B})) \}
\end{equation*}
is a non-empty open set distinct from $(\mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B}))$.
\end{theorem}
\begin{proof}
Let us assume that $\mathcal{M}$ is subdirectly irreducible. Then $Con(\mathcal{M})-\{(\Delta^{\mathbf{A}},\Delta^{\mathbf{B}})\}$ has a minimum element. From Theorem \ref{Characterization of congruences}, $\mathcal{C}_{s}(\mathcal{F}_{\mathcal{M}})-(\mathcal{X}(\textbf{A}),\mathcal{X}(\textbf{B}))$ has a maximum element. Let $(Z_{1},Z_{2})$ be such an element. Then $Z_{1}$ and $Z_{2}$ are non-empty. We prove that $\mathcal{J}=(Z_{1},Z_{2})-(\mathcal{X}(\textbf{A}),\mathcal{X}(\textbf{B}))$. On the one hand, if $(P,Q)\notin (Z_{1},Z_{2})$, then $(Z_{1},Z_{2}) \subseteq (Z_{1},Z_{2}) \cup {\rm{cl}}_{\mathcal{C}_{s}}(P,Q)$. So it must be that ${\rm{cl}}_{\mathcal{C}_{s}}(P,Q)=( \mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B}))$, because if it is not the case, then $(Z_{1},Z_{2})$ it would not be the maximum of $\mathcal{C}_{s}(\mathcal{F}_{\mathcal{M}})-(\mathcal{X}(\textbf{A}),\mathcal{X}(\textbf{B}))$, which is a contradiction. On the other hand, if $(P,Q) \in \mathcal{J} \cap (Z_{1},Z_{2})$, then ${\rm{cl}}_{\mathcal{C}_{s}}(P,Q)=( \mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B}))=(Z_{1},Z_{2})$, which is absurd from assumption. We conclude the proof by noticing that if $\mathcal{J}$ is a non-empty open set distinct from $(\mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B}))$, then it is easy to see that $\mathcal{J}-(\mathcal{X}(\textbf{A}), \mathcal{X}(\textbf{B}))$ is the maximum of $\mathcal{C}_{f}(\mathcal{F}_{\mathcal{M}})-(\mathcal{X}(\textbf{A}),\mathcal{X}(\textbf{B}))$. Then the result is an immediate consequence of Theorem \ref{Characterization of congruences}.
\end{proof}
|
\section{Introduction}
Modern, deep, uniform, large-area photometric surveys have shown unambiguously that the
Milky Way outer halo contains accretion-derived substructure
(e.g., Ibata et al. 2001, Newberg et al. 2002, Majewski et al. 2003, Rocha-Pinto et al. 2003, 2004,
Vivas et al. 2004, Conn et al. 2005a). These
structures of known or assumed remnants of satellite accretion,
have long-lived, coherent tidal features that can be used to
model the Galactic gravitational potential (e.g., Ibata et al. 2001, Law et al. 2005), as well as the
characteristics of the original satellite (e.g., Johnston et al. 1999).
However, such modeling studies have been limited by the meager
available kinematical data over large angles along the tidal features.
While radial-velocity programs have just recently begun to address this problem
for the few known Galactic tidal tails (Sagittarius - Majewski et al. 2004, Vivas et al. 2005,
and the Monoceros ring - Crane et al. 2003), no systematic survey has begun
to address the transverse (tangential)
velocities (i.e., absolute proper motions).
Without this information, dynamical models remain
poorly constrained, and therefore limited to describing merely a range of
possible events, rather than an accurate description of the real event.
The work described here is a proper-motion survey that provides
high precision (1 to 3 mas/yr per well-measured
star) absolute and relative proper motions down to a magnitude of
$V \sim 19$, and for a few selected fields down to
$V \sim 21$, in $\sim 50$ lines of sight
in the Selected Areas (SA) designed by Kapteyn for Galactic structure
studies in 1906. Current proper-motion surveys do not achieve this
precision at a similar magnitude limit and thus have limited capability to detect
and characterize distant halo substructure. To date there are only a few
similarly deep, precise, pencil-beam-type proper-motion data
sets that are primarily centered on globular clusters or dwarf spheroidals,
and even fewer that are focused on Galactic field stars
(e.g., Chiu 1980, Majewski 1992, Guo et al. 1993, Dinescu et al. 2002).
Obviously, ground-based studies of this
type are limited to parts of the sky where suitable first epoch astrometric data
exist. Here we exploit a unique cache of nearly century old, deep photographic
plates having good scale, as well as other plate material collected in Kapteyn fields.
We intend to complement the new proper motion data with radial velocities, distances and
metallicity estimates from our own photometric and spectroscopic work as well
as from overlapping surveys such as 2MASS, SDSS, QUEST and RAVE.
With this survey we aim to (1) determine the extent and orbital motion of the
highly obscured Monoceros ring-like structure above and below the
Galactic plane, and explore its possible relation to other low latitude structures,
such as the Canis Major (CMa) overdensity,
(2) characterize the transverse motion of the Sagittarius tidal streams,
(3) search for and characterize additional substructures in the halo of the
Milky Way and (4) determine the kinematical properties of numerous thick
disk and halo stars in our fields as a function of Galactic position.
Future papers will include detailed kinematical analyses for specific
regions, while in this paper
we characterize the survey and show its potential in two
specific regions where tidal streams have already been identified from other
sources. In the following Section we will describe the survey in detail. In Section 3
we show results in a few SA fields, and a brief summary is presented in Section 4.
\section{Survey Description}
\subsection{The Collection of Photographic Plates}
This survey is made possible by the visionary, now century-old Milky Way
survey introduced by Kapteyn in 1906. Kapteyn devised the {\it Plan of
Selected Areas} as a means to systematically study the Milky Way.
The {\it Plan}, as
originally envisioned, involved photometry, astrometry and spectroscopy
of stars in 206 SAs collected at numerous observatories around the world
and focused on characterizing the ``sidereal world"; the perceived importance of this
grand effort in the early 20th century prompted the creation of
International Astronomical Union (IAU) Commission 32: Selected Areas,
as well as the Subcommittee on Selected Areas of IAU Commission 33:
Structure and Dynamics of the Galactic System. Among the important early
contributions to the SA program was the Carnegie Institution's systematic
analysis of stellar photometry from Mount Wilson 60-inch plates of the
139 northern accessible SAs by Sears, Kapteyn and van Rhijn (1930).
The photographic plates used in this analysis were taken by Fath and
Babcock with the 60-inch telescope between 1909 and 1912.
For 54 near-equatorial fields there exist deliberately matched
(in area, approximate plate scale and depth) photographic plates taken with the
Las Campanas Du Pont 2.5-m telescope by S. Majewski between 1996 and 1998.
This collection of photographic plates provides the opportunity for an
unprecedentedly deep, high-precision proper-motion survey that takes advantage of the excellent plate scale ($10.92\arcsec$/mm for the 2.5-m Du Pont,
and $27.12\arcsec$/mm for the 60-inch Mt. Wilson)
of the images taken in both epochs
which span a $\sim 90$ year baseline. Each field of view is
$40\arcmin\times40\arcmin$. Unfortunately, because the old 60-inch plates go only as deep as
$V \sim 19$ for blue objects, there are very few galaxies to determine
the correction to absolute proper motion in a $40\arcmin\times40\arcmin$
field of view, and with those that are available being at
the plate limiting magnitude, and therefore
yielding rather poor centroids. Primarily for this reason, we have included in our
proper-motion determinations the first
Palomar Observatory Sky Survey (POSS-I)
plates as measured by both Space Telescope Science Institute (STScI,
the Digitized Sky Survey - DSS) and by USNO. The POSS-I plates were
taken in the early fifties
with the Oschin Schmidt telescope and have a plate scale of $67.2\arcsec$/mm.
While the scale and the digitization of the POSS-I plates (see Section 2.3)
are much poorer than those of the other two sets of plates, they offer
a 40-year baseline with the DuPont plates, and extend the proper
motion limiting magnitude to $V \sim 20-21$.
Minimally, each field has an early epoch Mt. Wilson plate with two offset images,
one from a 60 min exposure and one from a 3 min exposure. Many fields have a second early Mt. Wilson plate,
sometimes with two exposures, sometimes with one. For each field
there are two recent-epoch Du Pont plates taken in the
blue (IIIa-J + GG385) and visual (IIIa-F + GG495) passbands.
These plates also contain a pair of offset exposures of about 60 min and 3 min integration.
The intermediate epoch
POSS-I plates were taken in the blue (103a-O, no filter) and red
(103a-E + RP2444) passbands, with typical exposure times of 10 and 50 minutes
respectively.
Finally, a handful of fields have KPNO Mayall 4-m prime focus photographic
plates (plate scale $18.6\arcsec$/mm) taken in the mid seventies by A. Sandage,
and mid nineties by S. Majewski.
The modern plates
were taken in the blue (IIIa-J+GG385) and visual (IIIa-F+GG495) passbands,
while the Sandage plates were taken in the blue (IIa-O+GG385/GG3), visual
(IIa-D+GG495/GG11) and red (127-04+RG610) passbands.
\subsection{Area Coverage}
In Figure 1 we show the location of the centers of the
SA fields (open circles) on the sky in an Aitoff
projection. The filled circles represent the fields that also
have Mayall 4-m plates.
The survey samples three declination zones ($0\arcdeg$ and
$\pm15\arcdeg$) and
the full range in right ascension, except for low ($|b| < 25\arcdeg$)
Galactic latitude zones. The Galactic plane is represented with a grey
dash-dot line. There is one field, SA 29, which is at higher declination,
and which has only 4-m plates. We are including this field because
it falls within the QSO catalog determined from Data Release 3 (DR3)
of the Sloan Digital Sky Survey (SDSS) area (grey area in Fig.\ 1).
Although, for our astrometric
reductions, we make use of the photometry from DR4, Figure 1 highlights
the approximate SDSS footprint corresponding to the QSO catalog.
Thus it can be seen that
for a good number of fields QSOs have already been identified that can be
used in setting the absolute proper-motion reference frame.
\begin{figure}[htb]
\includegraphics[scale=1.00,angle=-90,clip=true]{f1.eps}
\caption{Distribution of the SA field centers (open circles) in equatorial coordinates. Filled circles show the fields that have additional 4-m plates. The continuous line is the approximate orbital plane of Sgr, the dash-dot line is the Galactic plane.
The grey area shows the SDSS coverage as given by the catalog of QSOs (Schneider et al. 2005). The cross-hatched bands represent the areas where the Monoceros
structure was mapped. The globular cluster Pal 12, Sagittarius's core and
the Canis Major dwarf galaxy candidate are also marked.}
\end{figure}
We have indicated the location of the Sagittarius dwarf galaxy's (Sgr) center,
and that of globular cluster Pal 12, which is believed to have belonged to Sgr
according to proper-motion data (Dinescu et al. 2000),
surrounding field photometry (Mart\'{i}nez-Delgado et al. 2002)
and chemical abundances (Cohen 2004).
The continuous dark line
shows Sgr's most recent orbit determination (Dinescu et al. 2005a). The orbit
is roughly indicative of Sgr's tidal streams that can be as wide
as $10\arcdeg$, as Pal 12's location for instance suggests.
It is apparent that Sgr's southern, trailing arm, which is
closer in distance to the Sun ($\sim 20-30$ kpc, Majewski et al. 2003)
than much of the northern, leading arm,
will be sampled in perhaps six SA fields (depening on the true stream width)
from RA = $0^h$ to $5^h$. In this paper
we present results for SA 93 and 94.
The second most well-known, putative halo substructure is the Monoceros stream or ring (``Mon";
Newberg et al. 2002, Ibata et al. 2003), a low Galactic-latitude structure
which is represented here by the two cross-hatched
bands above and below the Galactic plane. This area is drawn
only approximately as based on
observations from Yanny et al. (2003), Ibata et al. (2003), Rocha-Pinto et al.
(2003),
Crane et al. (2003), Conn et al. (2005a), Martin et al. (2006).
The stellar overdensity discovered by Martin et al. (2004) in the constellation
of Canis Major and subsequently analyzed by other groups, may be a
distinct structure (but cf. Rocha-Pinto et al. 2006), and it has been suggested to be the core of
the satellite that is responsible for the Monoceros stream (see, e.g.
Pe\~{n}arrubia et al. 2005). We have marked the location of this
structure (CMa) and we note that Conn et al. (2005b) claim, based on
radial velocities, to have detected the Monoceros ring behind the
CMa structure as well. At first glance, between six and twelve SA
fields are likely to sample the Monoceros ring. Fields SA 71 and 72 are
more problematical because both Sgr and Mon are expected in these areas
(see Fig. 1). This paper presents results in SA 96, 100 and 101
that fall within/close to the Mon region.
Another equatorial survey that overlaps some of our proper-motion survey is the
QUEST survey (QUasar Equatorial Survey Team), which
identifies RR Lyrae variable stars along the celestial equator (Vivas et al. 2004). Their
most recent results (Duffau et al. 2006) indicate the discovery
of a new halo structure in the constellation of Virgo that is not
related to either Sgr or Mon (see also Juri\'{c} et al. 2006).
Two of our SA fields, 103 and 104 are in the
area indicated by QUEST to sample the Virgo structure.
Provided the depth of the plates allows it, we may be able to
determine an absolute proper motion for this structure
in the near future.
\subsection{Photographic Plate Measurements}
\subsubsection{Du Pont 2.5-m, Mt. Wilson 60-inch and Mayall 4-m Scans}
For each field, we start by fully digitizing one Du Pont, one 60-inch and one 4-m
plate. These initial coarse scans serve to build input lists for
high-resolution scans.
All of the scans, except for those of the Mayall 4-m plates
in SA 29, 71, 94 and 118,
were done with the Yale PDS Microdensitometer.
The coarse scans of most of the 4-m plates were done with the University
of Virginia's microdensitometer.
The size of the field is defined by the 10 inch $\times$ 10 inch Du Pont plate size, which corresponds to
$40\arcmin\times40\arcmin$.
For the 60-inch plate in each field,
we digitize the same size area that matches the Du Pont field. This corresponds to
a 10-cm box located at the center of the Mt. Wilson plate.
Stellar
images on the 60-inch plates are affected by coma, and outside this region
they are practically unusable for astrometric purposes.
Based on these coarse scans, an input list of objects is determined
using the software package SExtractor (Bertin \& Arnouts 1996),
for each epoch. Long and short exposures on each plate are
separated into two different lists. Then, each exposure on each plate is
measured in a fine raster, object-by-object mode, with a pixel size of
$12.7\mu$m ($0.138\arcsec$) for the Du Pont plates, and $10\mu$m
($0.275\arcsec$) for the 60-inch plates.
The input lists for the 4-m plates were made from the coarse
scans done at UVa, and using the software package FOCAS
(Valdes 1982)\footnote[1]{See also Valdes's 1993 FOCAS User's Guide,
an NOAO document available at
ftp://iraf.noao.edu/iraf/docs/focas/focasguide.ps.Z}. These 4-m input catalogs
prepared earlier (Dinescu et al. 2002) were used to measure the 4-m plates
at Yale in a fine raster mode with a pixel size of $10\mu$m ($0.186\arcsec$).
The objects' positions, instrumental magnitudes, and other object parameters
were derived from the fine raster scans using the
Yale 2D Gaussian centering routines (Lee \& van Altena 1983).
As is customary, a set of five to eight stars well-distributed over the plate
are repeatedly measured during the scan in order to monitor and correct
for thermal drifts during the scan.
From coordinate transformations of same-epoch, same-telescope
plates we obtain,
for well-measured stars, a centering precision of $1.2\mu$m (13 mas) per
single measurement, per star for the Du Pont plates. For the 60-inch
plates this number is $3.3\mu$m (90 mas), while for the 4-m plates,
it is $1.3\mu$m (24 mas).
\subsubsection{POSS-I Scans}
There are two readily available scans of the POSS-I plates:
those of the red plates done with a PDS machine at STScI,
widely known as the DSS, and those of both blue and red plates done at
US Naval Observatory (USNO), Flagstaff Station
with the Precision Measuring Machine (PMM, see Monet et al. 2003 for
its description).
The DSS scans are retrieved directly from the
web\footnote[2]{http://archive.stsci.edu/cgi-bin/dss-form}
as the area in question is smaller than a degree on a side.
The resolution of the DSS scans is $25\mu$m/pix ($1.7\arcsec$/pix).
Sections of the PMM scans were kindly made available to us by S. Levine at
USNO-Flagstaff. These scan sections are centered on the SA fields, and
cover $40\arcmin\times40\arcmin$.
The PMM is an 8-bit, fast measuring
machine that uses a CCD detector to take ``footprint'' images
of the photographic plate. The POSS-I plates were scanned at a resolution of
$13\mu$m/pix ($0.9\arcsec$/pixel). Each Schmidt plate is covered
by some 588 exposures with a field of view of $20\arcmin\times15\arcmin$ each.
The CCD footprints are assembled together in the
subsequent software by using an offset,
to provide the entire digitized sky (see details in Monet et al. 2003).
The ``stitching'' of the CCD footprints however is not perfect,
and thus position-dependent systematics are introduced
(see Section 2.5). Without access to the individual PMM footprints, a
method is required to correct the systematics in the assembled scans.
We have made use of the DSS scans, which were produced by a traditional
PDS measuring machine, to correct the PMM positions (see Section 2.5).
The SA field scans from DSS and USNO, for each POSS-I plate
are processed as follows:
Objects are detected with SExtractor and then re-centered with the
Yale centering routines.
Coordinate transformations between overlapping plates for
the DSS measurements indicate
a centering precision of $2.1\mu$m (141 mas) per single measurement, per star.
Similarly, for the USNO scans we obtain $\sim 2.6\mu$m (174 mas) for both
the red and blue plates. Based on these values, if we use only the
Du Pont and the POSS-I plates (a $\sim 40$-year baseline),
we can obtain a proper-motion uncertainty of $\sim 2$ mas/yr per
well-measured star.
\subsection{Photometry and Spectroscopy}
Here we will only briefly mention our campaign to obtain
photometry and spectroscopy in the SA fields; more details of
these observations will be presented in subsequent papers.
All SA fields have $UBV$ CCD photometry taken with
with the SITe\#1 2048$^{2}$ CCD on the Swope 1-m
at Las Campanas Observatory on the nights of UT 1997 December 23-31,
1998 June 19-26, and 1998 December 15-16. The CCD field of view
covers $\sim 30\%$ ($22\arcmin\times22\arcmin$, $0.697\arcsec$/pixel)
of the astrometric field. Exposure times were 120,
200, and 900 seconds for the Johnson $V, B,$ and $U$ filters,
respectively, yielding data shallower by 1 to 1.5 magnitudes than the
typical Du Pont plates, for red stars, and by $\sim 0.3$ magnitudes for
blue stars. Short exposures (5, 7, and 40-45 seconds in
$V, B,$ and $U$) were also taken to obtain photometry for the bright stars
in these fields.
For the astrometric reductions, we need $BV$ colors for all stars
to map out the color terms. These were obtained by calibrating
photographic instrumental magnitudes for each blue and visual
Du Pont plate with CCD magnitudes.
The CCD magnitudes used in this process are not calibrated to
the standard Johnson system because at the time the astrometric reductions
were done, the CCD photometry had not been calibrated yet. Nevertheless,
the CCD photometry helped to linearize the photographic magnitudes.
To date, the reduced CCD data cover only a
handful of fields. Therefore, for the rest of the fields where
there is coverage with SDSS, we have used the SDSS
$(g-r)$ colors in the astrometric reductions.
In Figure 2 (left panel) we show the photographic, CCD-calibrated $B-V$ colors
versus the SDSS $(g-r)$ colors in SA 100. Dark symbols are stars, and
red symbols are galaxies.
The good color correlation justifies our use of the $(g-r)$
colors in the astrometry. The right panel shows the
relationship between the SDSS $r$ magnitude and the photographic
$V$ magnitude. This shows that the Du Pont plates reach a limiting magnitude of
$r \sim 21$.
\begin{figure}[htb]
\includegraphics[scale=0.80,clip=true]{f2.eps}
\caption{SDSS $(g-r)$ colors as a function of photographic $(B-V)$
colors (left panel) and SDSS $r$ magnitudes as a function of photographic $V$
magnitudes (right panel). The photographic magnitudes are derived from the
scan of a Du Pont plate calibrated to CCD photometry. The $B,V$ magnitudes are
not calibrated to the standard Johnson system. Dark symbols represent the stars,
while red symbols represent the galaxies.}
\end{figure}
For the fields where we have detected structure in the proper-motion
distribution as well as in the corresponding
SDSS color-magnitude diagram (CMD), we have started an observing
program with $Hydra$ on the 3.5-m $WIYN$ telescope
in order to measure radial velocities.
So far, preliminary radial velocities have been obtained in SA 71 and SA 96.
These results will be presented elsewhere.
\subsection{Astrometry}
The Du Pont plates were precorrected for differential refraction
(third-order refraction theory, Taff 1981) and for distortion
(see details in Dinescu et al. 2000). The distortion coefficients
applied here were determined from the coordinate transformations of
eight visual and seven blue plates into the Second Naval Observatory
CCD Astrograph Catalog (UCAC2, Zacharias et al. 2004).
The distortion coefficient for the visual plates is
$(-6.55 \pm 0.09) 10^{-8}$ mm$^{-2}$, and for the blue plates
is $(-7.02 \pm 0.06) 10^{-8}$ mm$^{-2}$. Between 100 and 200 stars in
common with UCAC2 in the various SA fields were used to
model the coordinate transformations. The center of distortion
was assumed to coincide with the tangent point. The tangent point
was determined by minimizing quadratic terms in the
transformations of plate coordinates into the UCAC2 positions
(see e.g., Guo et al. 1993). The cubic distortion coefficients
determined here, although
slightly smaller than that determined by Cudworth \& Rees (1991), are
in agreement within quoted uncertainties. The center of
distortion is determined with a precision of $\sim 0.5$ mm, or 5.5 arcsec.
The Mayall 4-m plates were precorrected for distortion using
the coefficients from Chiu (1976). The center of distortion is
determined similarly to the process used for the Du Pont plates, i.e.,
by minimizing quadratic terms from a coordinate transformation into UCAC2.
This gives a distortion center known no better that $\sim 0.05$ mm (1 arcsec).
We have not precorrected the 4-m plates for differential refraction,
as this effect is much smaller than that of distortion;
for the Du Pont plates these effects are similar in size (Dinescu et al. 2000).
The positions determined from the DSS scans of the POSS-I plates were used
to correct the positions determined from the PMM scans that are affected
by the assemblage of multiple CCD footprints (see Section 2.3).
In Figure 3
we show the residuals from a coordinate transformation between
PMM measurements and DSS measurements as a function of position.
The transformation includes up to fourth-order terms. The top two panels
show the residuals for the red plate (i.e., same red plate, two
different measurements, PMM into DSS) and the bottom two for
the blue plate (i.e., blue PMM measurements
into red DSS measurements) as labeled.
The right panels show the residuals
after the correction was applied. The correction for each object is derived
by taking a local average (of some 20 neighbors) from a 2-d map of the residuals
shown in Fig. 3.
\begin{figure}[htb]
\includegraphics[scale=0.90,clip=true]{f3.eps}
\caption{Residuals from a transformation of the PMM measurements into the DSS
measurements. The first two rows are for PMM measurements of a red plate
transformed into DSS measurements of the same red plate. The bottom two rows
are for PMM measurements of a blue plate transformed into DSS measurements
of a red plate. Left panels show the residuals before a 2-d correction
was applied, and right
panels after the correction was applied (see text).}
\end{figure}
It is well known that astrometry from the wide-field POSS-I plates,
which were taken with a Schmidt telescope, is strongly affected
by position and magnitude-dependent systematics (e.g., Morrison et al.
2001 and references therein). These are particularly large and
difficult to model when an entire plate of $6.5\arcdeg\times6.5\arcdeg$
is being mapped into astrometric catalogs. For our purposes however,
we are using only small regions on a plate, which can be mapped into one another
and into an external system, such as the Du Pont plates for example,
by using up to third order polynomials. As for magnitude dependent
systematics (i.e., the magnitude equation) ,
these are minimized by using stars within a relatively
narrow magnitude range,
toward the faint end of the plate limit: $V\sim 15$ to 19.
The old 60-inch plates were not precorrected in any manner
as they are the least well-understood.
We are aware that coma is present from the shape of
the images on the plates, and that it affects the astrometry
from coordinate transformations
of long into short exposure. In the following Section we describe
the way in which these plates were incorporated into the
proper-motion determinations.
\subsection{Proper-Motion Determinations}
The lists of coordinates from the 60-inch, POSS-I and 4-m
plates are matched with the list from the Du Pont plates.
The matching radius is 2 arcsec. This basically limits our catalog to objects
with proper-motions smaller than $\sim 50$ mas/yr.
Although stars with proper motions larger than this value can be recovered
from the current lists of positions for all measured objects on each plate,
it is not our immediate goal to do so.
The proper-motions are determined differentially, by adopting
one plate as the master plate, into which all the others are mapped.
The master plate is chosen from the Du Pont plates, which are
best understood and modeled. During long exposures, guiding-induced
magnitude-dependent systematics (also known as the magnitude equation,
see e.g., Majewski 1992, Guo et al 1993, Girard et al. 1998)
affect practically all photographic material.
However, the Du Pont plates have a short exposure, and we have
assumed that this exposure is not strongly affected by the magnitude equation.
Therefore we have transformed the long exposures into one short exposure,
and the magnitude-dependent trend of the residuals was used to
correct the long-exposure measurements. This essentially corrects
the bright stars ($V \le 16$), over which the trend is apparent.
One of the magnitude-equation corrected Du Pont long-exposures is then chosen
as the master plate. The remaining plates are then transformed into
the master plate using up to fourth order
polynomial coordinate transformations and linear color terms.
These high-order geometric terms are present most likely due to our
inability to accurately determine the center of distortion (see Section 2.5).
Preliminary proper motions are calculated based on the Du Pont and
POSS-I plate measures, and, for a handful of fields, from Du Pont
and 4-m plates. The transformation of POSS-I plates into the Du Pont system
occasionally requires third-order polynomials in the coordinates,
and linear color terms.
The derived preliminary proper motions allow propagation
back in time to the epoch of the 60-inch plates, and
thus the 60-inch plates are tied to the system of the Du Pont
master plate. These transformations include third order polynomials
in the coordinates,
coma terms and linear color terms.
New proper motions are then calculated from the entire
set of plates and one more iteration is performed to obtain the final
values. The reference system that is used to determine the
plate transformations consists of faint stars, varying in number
from a couple hundred to a couple thousand, depending upon the
Galactic latitude of the field and the depth of the corresponding plates. The resulting
proper motions are relative; the correction from relative to absolute
proper motions is derived from the offset defined by the mean proper motion of QSOs and galaxies
in the field.
We note that the Du Pont and 60-inch short exposures
provide measurements of stars
as bright as some of the faintest $Hipparcos$ stars ($V\sim 8$).
However, we will not rely on any of the stars at the bright end
to determine the correction to absolute proper motions, because
we believe our proper-motions at the bright end ($V < 15$) are
affected by unaccounted for magnitude-dependent biases.
The proper motions of galaxies and the QSOs are also used at each
iteration to check magnitude and color-dependent systematics.
In some of the areas that have 4-m plates, galaxies were used to
correct small trends with color and magnitude left in the
first-iteration set of proper motions.
QSOs and galaxies are selected from the SDSS (Schneider et al. (2005) and
DR4 classification). According to the SDSS documentation, the
galaxy classification is quite reliable down to $r = 21$.
For areas that do not overlap with SDSS we have selected galaxies from
a visual inspection of the deepest Du Pont plate.
We calculate a proper motion for each object that has at least three measurements
separated in time by at least $\sim 40$ years ($\sim 20$ years for
fields that include 4-m plates). The proper motion is calculated for
each object from a linear least-squares fit of positions as a function of
plate epoch. The formal proper-motion uncertainty is given by the
scatter about this best-fit line. Measurements that differ
by more than $0.2\arcsec$ from the best-fit line are excluded.
Objects that have only three measurements should however be considered
with caution because they may have unrealistically small formal uncertainties.
\subsection{Proper-Motion Uncertainties}
There are two ways to estimate externally the proper-motion
uncertainties:
(1) by direct comparison with another high-quality
proper-motion catalog and (2) from the proper-motion scatter of
objects that have no or negligible intrinsic proper-motion dispersion.
For the first test, we compare our proper motions with those in the Munn et al. (2004)
catalog in five SA fields. We remind the reader that the Munn et al. (2004)
proper-motion catalog was made by combining USNO-B (Monet et al. 2003)
with the SDSS (DR1). Munn et al. (2004) have used SDSS galaxies to correct
for position-dependent proper-motion
systematics and to place the proper motions on an absolute reference frame.
In Figure 4 we show proper-motion differences (i.e.,
our relative proper motions minus proper motions from Munn et al. 2004)
as a function of magnitude. Units for proper motions are mas/yr
throughout the paper. The left
panels show proper-motion differences along right ascension (RA), and the
right panel along declination (Dec). It is apparent that, in all fields, there
is a larger scatter in the RA proper-motion differences, than in
Dec. Proper-motion differences plotted as a function of
positions and colors show that the larger scatter in RA is not due to
systematics related to these quantities.
Our proper motions for galaxies and QSOs
(i.e., objects with no intrinsic proper-motion dispersion)
do not indicate that our measurements are consistently poorer in the RA
direction than in Dec (see below). Interestingly,
the histogram of QSO proper motions in Munn et al. (2004, their Figs 1 and 3)
shows that the RA proper-motion dispersion is larger than that in
Dec, in agreement with our findings.
From coordinate transformations of the modern Du Pont plates
directly into SDSS positions in some thirteen SAs,
we do not find indications that
the positional error in RA is larger than that in declination.
This leads one to conclude that the positional precision
in the USNO-B catalog is poorer in RA than in declination.
Indeed, Figure 1 in Munn et al. (2004), which shows the distribution of
QSOs' proper motions in RA and Dec for both
the USNO-B catalog and the new Munn et al. (2004) catalog,
indicates that the scatter in RA is larger than that in Dec
in both catalogs.
\begin{figure}
\includegraphics[scale=0.90,clip=true]{f4.eps}
\caption{Proper-motion differences (ours minus those in the Munn et al. 2004 catalog) as
a function of magnitude in five SAs. The left panels show
the RA direction, right panels the Dec direction.}
\end{figure}
From our comparison with the Munn et al. (2004) catalog,
we obtain a scatter of $\sim 3.8$ mas/yr in RA and $\sim 2.8$ mas/yr
in Dec , for $r < 18$.
Munn et al. (2004) quote uncertainties of
3.6 and 2.8 mas/yr in RA and Dec respectively for the
same magnitude range.
This indicates that our uncertainties are substantially smaller.
The fact that we see in the
proper-motion differences (Fig.\ 4) the larger scatter in RA than
in Dec which is characteristic of the Munn et al. (2004) catalog,
implies that the errors in the latter catalog dominate the scatter in
the proper-motion differences.
Our second test uses galaxies, QSOs,
stars in known streams or in clusters, i.e., objects for which the
proper-motion dispersion reflects only the measurement uncertainty.
In practice, galaxies have larger proper-motion uncertainties than
stars due to poor centering of their fuzzy, low-gradient image profiles, and
there are very few QSOs in each SA field. Thus these
uncertainty estimates should also be viewed as conservative numbers.
In Figure 5 we show relative proper motions of galaxies (open circles)
and QSOs (filled red circles) as a function of magnitude
for the same SA fields shown in Fig. 4 (left panel is for RA, right panel for
Dec). Clearly SA 94 is better
measured, and this is because it includes 4-m plates.
\begin{figure}
\includegraphics[scale=0.90,clip=true]{f5.eps}
\caption{Relative proper motions of galaxies (open circles) and QSOs
(filled red circles) as a function of magnitude for five SAs. Left panels show
the RA direction, right panels the Dec direction.}
\end{figure}
Figure 5 indicates that the proper-motion uncertainty varies considerably
with magnitude, and there may be slight
variations from field to field. These variations
are primarily due to the image quality of the old POSS I plates.
For a typical field, we obtain between 3 and 5 mas/yr for
reasonably well-measured galaxies, while for fields that
have 4-m plates, we obtain 2 to 3 mas/yr. Most fields have of the order of
100 galaxies, therefore the uncertainty in the correction to absolute
proper motion is $\le 0.5$ mas/yr. For the QSOs, it is more difficult
to reliably estimate uncertainties, due to small number statistics and
the fact that most QSOs are toward the faint limit of the survey.
However, for SA 94, we obtain between 1 and 2 mas/yr for well-measured
($r \le 19$) images.
Proper-motion uncertainty estimates as derived from
``known'' tidal structures will be given in the following Section, where
we present the tidal tail results.
\begin{figure}[htb]
\includegraphics[scale=0.60,angle=-90,clip=true]{f6.eps}
\caption{Correction to absolute proper motion as a function of magnitude
as given by QSOs (red circles), galaxies (green circles), NPM1 stars (black filled circles),
UCAC2 stars (open circles) and Tycho2 stars (filled triangles).
As in the previous two figures, the left panel shows
the RA direction, the right panel shows the Dec direction.}
\end{figure}
Finally, in Figure 6 we show the correction to absolute
|
proper
motion as given by a number of calibrating objects for SA 101.
Most of the fields show similar characteristics.
Toward the faint end, the galaxies (green symbols) and QSOs (red
symbols) dominate. At the bright end we show the correction to absolute proper
motion as given by proper-motion differences of
Tycho2 (Hog et al. 2000) stars (filled triangles),
UCAC2 stars (open
circles), and NPM1 (Lick Northern Proper-Motion Program, Klemola et al. 1987)
stars (filled circles). For all of these three catalogs,
the stars matched with our survey are at the faint limit of those catalogs,
so that these are stars with the largest measurement errors in these other
catalogs. Indeed, the proper-motion scatter is between 4 and 6 mas/yr,
poorer than that of our measured galaxies. This is one reason
why we have not used Tycho2, UCAC2 and NPM1 stars to calibrate to
absolute proper motions. The second reason is due to obvious offsets
between the zero point as determined by galaxies and QSOs in our survey,
and that determined based on brighter stars in these external catalogs. This
is due to residual magnitude-dependent systematics in either or
both the listed catalogs and
in our survey. Since the magnitude range of galaxies and QSOs better matches
that of our survey stars, using these to establish the abolute proper motion
correction minimizes
magnitude-dependent systematics.
We will use both galaxies and QSOs in the determination of the
absolute proper-motion correction.
One more question remains: Why not use stars in the Munn et al. (2004)
catalog as astrometric zero-point calibrators? From comparisons of the zero point as determined
by galaxies and by the Munn et al. (2004) stars, we find
differences between 0.2 and 5 mas/yr. The scatter in the difference is
2 mas/yr.
The absolute proper-motion calibration in Munn et al. (2004) is also
based on SDSS-selected galaxies, and their first epoch material is in our
time series.
Nevertheless, differences appear in the
two independently determined catalogs. We feel our proper motions
are more reliable for two reasons: (1) the modern epoch consists of precise
measurements of the high quality Du Pont plates, and (2)
the old epoch, i.e., the POSS-I scans were re-reduced here in a very careful
manner to minimize position-dependent systematics (see Section 2.5),
while the Munn et al. (2004) proper motions necessarily had to
rely on the batch-reduced PMM scans of the POSS-I plates for
nearly $10^{9}$ objects.
For these reasons, as well as because of the effect seen in Figure 4, we will use
our internal zero-point calibration based on galaxies and QSOs.
\subsection{Completeness}
\begin{figure}
\includegraphics[scale=0.60,clip=true]{f7.eps}
\caption{Distribution of objects as a function of magnitude
for SA 107 and SA 94. The SDSS catalog is represented with a red line,
our survey with a black line. The hatched area is
our survey, only for objects measured on the old 60-inch
plates. }
\end{figure}
In Figure 7 we show the distribution of objects detected
as a function of magnitude
in two fields, SA 107 a typical field in our survey, and SA 94, a
deep field that includes 4-m plates. The SDSS distribution is represented
with a red line, our survey with a black line, and the hatched area is
that fraction of our
stars that were measured on the old 60-inch plates.
For a typical survey field, our catalog is nearly complete to $ r = 19 $, and
this magnitude limit varies slightly from field to field depending
on the deepness of the POSS-I plates. Figure 7 shows that the SDSS
distribution and our catalog distribution follow each other closely.
A difference between the SDSS and our catalog counts is however
present between $r = 17$ and 19. This difference indicates that
SDSS has 6 to $9\%$ more counts than our catalog.
For these two fields, we have checked by hand
the objects that appear in SDSS and do not appear in our catalog.
We found that the absence of these objects from our catalog is
due to two reasons: (1) from its construction (Section 2.6)
our catalog misses high (greater than $\sim 50$ mas/yr)
proper-motion stars, and (2) SDSS has a non-negligible
number of spurious detections near bright stars that, upon inspecting the
SDSS images, appear to be located on diffraction spikes.
We note that the latter objects have
SDSS flags that qualify them as real, primary detections
(i.e., flag ``GOOD'').
To correctly quantify the contribution of each of
these effects is not a trivial matter.
If we assume however that the SDSS represents
the true counts, we then obtain a conservative estimate of our
completeness limit which is greater than $90\%$ to $ r= 19$. The limiting
magnitude at $50\%$ completeness is $r \sim 19.5 $. For fields that include 4-m plates, the
catalog is near-complete (greater than $90\%$)
to $r \sim 21 $ and reaches a limiting magnitude
of $ r= 22 $ at $50\%$ completeness.
\section{Results}
In what follows we show proper-motion results in SA fields containing
some currently known halo substructures. Quantitative analyses that involve
absolute proper motions combined with other information and
with models will be presented elsewhere.
\subsection{Sagittarius Tidal Tails}
Based on Hess diagrams, Newberg et al. (2002) have identified
overdensities of F-colored stars related to debris from the
Sagittarius dwarf in the regions labeled S167-54-21.5 (their Fig.\ 7),
and S341+57-22.5 (their Fig.\ 5).
The latter region includes our SA 105 and 106.
We find however very little evidence of clumpiness in the proper-motion diagram
because Sgr's main sequence turnoff is at very faint magnitudes ($r = 22.3$).
A small number of Sgr red clump stars at $r \sim 19.5$ may be present in
SA 105 and 106, but these are at the faint end of our survey and
will have large proper-motion uncertainties.
This SDSS region samples distant parts of
the leading tidal tail of Sgr (see Fig. 1 and e.g., Majewski et al. 2003).
Thus we no longer consider these fields here.
The SDSS region S167-54-21.5 includes
SA 93 at $(l,b) = (154.2\arcdeg,-58.1\arcdeg)$
and SA 94 at $(l,b) = (175.3\arcdeg,-49.2\arcdeg)$, and it samples
the trailing tidal tail of Sgr, which is generally closer to the Sun than the
leading trail (e.g., Majewski et al. 2003).
Yanny et al. (2003, 2004), Majewski et al. (2004) and Law et al. (2005)
confirm with radial velocities
the presence of Sgr debris in this region.
In Figure 8 we show the SDSS CMDs and relative proper-motion
distributions for SA 94, 93 and 107. The middle and
right hand panels show proper motions: the $x$ direction corresponds to
proper motions along RA, and the $y$ direction to Dec.
Each row represents one
field. SA 107 located at $(l,b) = (5.7\arcdeg,+41.3\arcdeg)$
was chosen for comparison with the other two fields, and
it is roughly symmetrically placed in the Galaxy with SA 94.
The reddening is relatively low in all fields: for SA 93 it is $E(B-V) \sim 0.03$,
for SA 94 it is $0.09$, and for SA 107 it is 0.11 (Schlegel, Finkbeiner, \& Davis 1998).
The CMDs show objects in the SDSS that were matched with our proper-motion
survey. Galaxies according to the SDSS classification were eliminated.
The magnitudes and colors are not dereddened. For SA 94, where we have better statistics
for the QSOs than in the other fields, we highlight the
QSOs with red symbols so that their proper-motion distribution can be compared to
that of stars.
The middle panels show the proper-motion distribution of blue stars
($ 0.2 < g-r < 0.8$) chosen to represent the
turnoff of thick disk and halo stars, and the right panels that of red stars
($1.2 < g-r < 1.7$) chosen to represent more nearby, disk dwarf stars.
Sgr's turnoff is visible in the CMDs of SA 94 and 93, at $r \sim 21 $.
SA 107 is also located in the
SDSS region S6+41-20, which Newberg et al. (2002)
find is most consistent with a smooth halo/thick disk population.
Because of its direction toward the inner Galaxy
and the fact that the
scale length of the thick disk is 3-4 kpc (e.g., Juri\'{c} et al. 2006),
SA 107 samples more thick
disk stars than SA 94. The proper-motion distributions
show the intrinsic dispersion convolved with the proper-motion
measurement uncertainty, which increases with magnitude.
\begin{figure}
\includegraphics[scale=0.95,clip=true]{f8.eps}
\caption{CMDs (left panels) and relative proper-motion diagrams
(middle and right panels) for stars and QSOs in
SA 94, 93 and 107. The middle and right panels show the proper motions for blue
$(0.2 < (g-r) < 0.8)$ and red $(1.2 < (g-r) < 1.7)$ stars respectively.
In the top row, QSOs are highlighted with red symbols for comparison
with stars' distributions.
The bottom row shows SA 107, where only every
other star was plotted to mimic a lower density field.}
\end{figure}
By comparing the Figure 8 proper-motion distribution of presumably nearby red stars
in all fields, it is
clear that SA 94 is a much better measured field --- a
result of having 4-m plates in its time series data.
The proper-motion uncertainties for SA 93 and SA 107 are however comparable
to one another.
The proper-motion distribution in SA 107 shows a clear kinematical distinction
between blue stars --- i.e., distant thick disk and halo stars near the main sequence turnoff ---
and red, nearby dwarf stars.
The fourth row shows the same diagrams as in the third row, but with only half the stars
in SA 107 plotted so as to approximately match the number of
stars in SA 94 in the appropriate magnitude range.
It is evident that the blue stars in SA 94 and 93 show more concentrated clumping
in proper-motions due to the presence of a population with a dispersion much
tighter than that expected for random thick-disk/halo stars (i.e., SA 107).
This identified proper-motion clumpiness together with a sudden overdensity of stars
at a particular magnitude in the CMD is a clear signature of distinct substructure in the halo,
presumably dominated by Sgr tidal debris.
In SA 94, the best measured field,
the proper-motion scatter of the clumped, blue stars is
$\sim 1.8 - 2.0$ mas/yr.
This number is determined primarily by faint stars ($r = 21$) and
reflects our proper-motion uncertainty.
In this contribution we present the mean absolute proper motion of candidate
Sgr debris in SA 94 and 93.
In SA 94, the correction to absolute proper motion is given by the error-weighted
mean of two determinations: that with respect to QSOs and that with
respect to galaxies.
The former determination is the average of nine QSOs and it
is $\mu_{\alpha}cos\delta = -0.08 \pm 0.32$ mas/yr and $\mu_{\delta} = -0.07 \pm 0.73$ mas/yr.
The latter includes 885 galaxies with magnitudes between
$r= 15$ and 21 that
have measurements on at least four plates and proper-motion values less than 20 mas/yr
in both coordinates. Their average proper motion
is $\mu_{\alpha}cos\delta = -0.39 \pm 0.18$ mas/yr and $\mu_{\delta} = -0.01 \pm 0.19$ mas/yr.
The error-weighted average of these two determinations is: $\mu_{\alpha}cos\delta = -0.32 \pm 0.16$ mas/yr
and $\mu_{\delta} = 0.00 \pm 0.18$ mas/yr.
Candidate Sgr stream stars were selected based on the SDSS CMD (Fig.\ 8)
to belong to the turnoff region. Only stars with measurements on at least four plates and
with $r$ magnitudes brighter than 21.7 were included, to avoid stars with
highly uncertain proper motions. Furthermore, stars with proper motion
values larger than 15 mas/yr are eliminated as outliers. In this way,
the sample of Sgr candidates consists of 156 objects. To determine their mean and dispersion,
we have used the probability-plot method (Hamaker 1978)
using the inner $80\%$ of the proper-motion distribution.
By doing so we aim to eliminate the minor contribution of distant
halo stars to the estimate of the mean
proper-motion of the candidates. The Besancon model (Robin et al. 2003) predicts 21
halo stars in the CMD region and proper-motion range used to select
Sgr debris candidates. Therefore the chosen contamination fraction of $20\%$ seems to
slightly overestimate the Besancon model output.
We find the mean relative proper-motion of Sgr debris candidates in SA 94 is
$\mu_{\alpha}cos\delta = -0.18 \pm 0.32$ mas/yr and
$\mu_{\delta} = -2.02 \pm 0.28$ mas/yr.
By using fractions between $60\%$ and $90\%$, the estimate of the mean
did not change within its formal uncertainty.
The mean absolute proper motion for candidate Sgr debris in SA 94
is therefore $\mu_{\alpha}cos\delta = 0.14 \pm 0.36$ mas/yr and
$\mu_{\delta} = -2.02 \pm 0.33$ mas/yr.
For SA 93 we also present a preliminary value for Sgr stream candidates.
We plan to improve both determinations for SA 93 and 94 by obtaining
radial velocity data that will help establish membership to the stream.
For SA 93, the correction to absolute proper motion
is given by galaxies and QSOs together, rather than two
independent measurements as
was done for SA 94. This is because
there are only 2 QSOs measured on this field, one of which is a galaxy as well.
Proceeding in a similar manner as for SA 94, we obtain the correction to
absolute proper motion
as given by the mean of 222 extragalactic objects:
$\mu_{\alpha}cos\delta = -1.96 \pm 0.40$ mas/yr and $\mu_{\delta} = 2.62 \pm 0.43$ mas/yr.
Since there are far fewer Sgr candidates in SA 93 than in SA 94,
we have made use of all
stars within $r=14$ to 20 and $(g-r) = 0.0$ to 0.9 to determine the mean motion,
since some subgiant and red clump stars may also be present besides the
turnoff stars. In this case the halo contamination is much more important
than in SA 94, and probability plots will not give an accurate result.
We have thus simply selected the
candidates by defining a very conservative proper-motion cut: a circular region
of radius $\sim 2$ mas/yr that encompasses the proper-motion clump
seen in Figure 8. The mean is thus based on 63 stars, and it is:
$\mu_{\alpha}cos\delta = -2.54\pm 0.18$ mas/yr and
$\mu_{\delta} = 0.30 \pm 0.24$ mas/yr.
The uncertainties are derived from the scatter given by the 63 Sgr
candidates. We caution
that this uncertainty is formally low because it does not account for the
uncertainty in choosing Sgr stream members. Finally, the absolute proper motion
for Sgr candidates in SA 93 is:
$\mu_{\alpha}cos\delta = -0.58 \pm 0.47$ mas/yr and
$\mu_{\delta} = -2.32 \pm 0.50$ mas/yr. We regard this number as
preliminary, and with uncertainties possibly underestimated.
In both SA 94 and 93 determinations the membership to the stream is the
major source of error.
Majewski et al. (2006) have shown that by measuring the proper motion of
Sgr debris, especially along the trailing tail, one can
determine the rotation velocity of the Local Standard of Rest (LSR),
whereas Law et al. (2005) and
Johnston et al. (2005) show how the dynamics of the Sgr arms are also affected by the
flattening, $q$, of the Galactic potential.
In Figures 9 and 10 we show proper-motion predictions by Law et al. (2005) for Sgr tidal debris
in the direction of SA 93 and SA 94 compared to the observed results found here
in these two fields. The proper motions are shown as a function of
the longitude along Sgr's orbit (Majewski et al. 2003); $\Lambda = 0^{\circ}$
corresponds to the main body of Sgr and $\Lambda$ increases in the trailing direction.
In Figure 9, the adopted velocity of the LSR is 220 km/s,
while the flattening of the halo $q$, has three values corresponding to
prolate (top), spherical (middle) and oblate (bottom) halos.
In Figure 10, the adopted flattening of the halo is $q = 0.9$, and
the velocity of the LSR varies from 260 km/s to 180 km/s, as indicated
in each panel. The general agreement between our proper motion results and the model,
which was constrained
solely from the distribution and radial velocities of M giants
(Majewski et al. 2003, 2004) and a given Galactic potential
(i.e., no proper-motion data), is remarkable. This correspondence reinforces
the assumption that we are
measuring proper motions in Sgr's tidal stream.
We note that the proper-motion
range shown in Figs. 9 and 10 is only $\sim 3$ mas/yr, a typical value for the
proper-motion uncertainty per well-measured star in catalogs such as Tycho2 (Hog et al. 2000),
SPM 3 (Girard et al. 2004).
To better illustrate this, we include here the mean absolute proper motion of
red field stars (see also Fig. 8). In SA 94, we obtain $\mu_{\alpha}cos\delta = 2.80 \pm 0.57$ mas/yr and
$\mu_{\delta} = -3.22 \pm 0.46$ mas/yr, and in SA 93, $\mu_{\alpha}cos\delta = 3.99 \pm 0.88$ mas/yr
and $\mu_{\delta} = -0.94 \pm 0.80$ mas/yr.
Thus, along RA the mean
proper-motions of red field stars in both areas lie outside the
proper-motion range shown in Figs. 9 and 10.
\begin{figure}
\includegraphics[scale=0.90,clip=true]{f9.eps}
\caption{Proper motions along Sgr's southern trailing tail as predicted
by the Law et al. (2005) models (color symbols). The dark symbols with
$1$-$\sigma$ error bars show our preliminary results in SA 93
($\Lambda = 103^{\circ}$) and SA 94 ($\Lambda = 116^{\circ}$). The LSR velocity adopted
for this model is 220 km/s, while the flattening of the halo $q$ varies
as specified in each panel. The colored dots represent N-body model particles stripped from Sgr
since its last apoGalacticon, i.e. present orbit, ({\it yellow symbols}), during the previous orbit ({\it magenta}),
and two orbits ago ({\it cyan}); this color scheme matches that used in Law et al. (2005)}
\end{figure}
\begin{figure}
\includegraphics[scale=0.90,clip=true]{f10.eps}
\caption{Similar to Figure 9. The flattening of the halo adopted
for this model is 0.9, while the velocity of the LSR varies as specified
in each panel. Color representation is as in Fig.\ 9 and Law et al. (2005).}
\end{figure}
While our preliminary proper motions in only two SA fields do not yet lend discriminatory
power between the Galactic models shown, with improved proper motion samples and the inclusion
of additional fields at other $\Lambda$ (e.g., SA 116 and 117; see Fig.\ 1) we expect to be able
to more rigorously address this issue in the near future.
\subsection{The Monoceros Structure}
In Figure 11 we show similar plots to those in Figure 8 for the fields
SA 96 at $(l,b) = (198.3\arcdeg,-26.0\arcdeg)$,
SA 100 at $(l,b) = (227.6\arcdeg,+26.7\arcdeg)$,
and SA 101 at $(l,b) = (239.0\arcdeg,+39.9\arcdeg)$.
These areas sample regions in the Monoceros structure
at the anticenter and across the Galactic plane.
For SA 96, the data for the CMD are not in the available SDSS data releases.
We have obtained the data set from B. Yanny, and the photometry is
dereddened for this field. Reddening is however rather low for all three
of these fields: $E(B-V) = 0.07$ for SA 96, $E(B-V) = 0.04$
for SA 100 and $E(B-V) = 0.04$ for SA 100.
Since the data for SA 96 are dereddened, we have shifted blueward
the color ranges
for selecting blue and red stars, by 0.1 mag compared to the other fields
(Fig. 11).
Newberg et al. (2002) designate the region containing SA 96 as
S200-24-19.8, for which their Fig. 15 shows a clear main sequence turnoff at $g = 19.8$.
SA 100 is located only $3\arcdeg$ away from the eastern edge of
S223+20-19.4, which also displays a clear main sequence with the
turnoff at $g\sim 19.4$ (see Fig. 12 in Newberg et al. 2002).
At $b \sim 40\arcdeg$, SA 101 is farther away from the Monoceros ring (see also Fig. 1).
The majority of the stars measured in our survey are however
brighter than these turnoffs and the corresponding main sequence
stars that were studied by SDSS and assigned to the Monoceros structure.
\begin{figure}
\includegraphics[scale=0.95,clip=true]{f11.eps}
\caption{Same as Fig. 8, only for SA 96, 100 and 101. The red symbols
in SA 96 are stars with radial velocities measured by Yanny et al. (2003, 2004)
in the turnoff of the Monoceros structure.}
\end{figure}
Yanny et al. (2003, 2004) measured radial velocities of candidate
turnoff stars in the Monoceros ring to demonstrate that their dispersion is
indicative of a kinematically cold stream (but cf. Crane et al. 2003).
In Fig. 11, top row, the red
symbols are the stars in our SA 96 field that have radial
velocities measured by
Yanny et al. (2003, 2004). These stars are at the faint limit
of our survey, therefore their proper motions are quite uncertain.
Since the mean radial velocity of the candidate Monoceros stars
overlaps with that of the thick disk (Yanny et al. 2003, 2004),
it is also possible that some
of the stars are indeed thick disk stars, i.e., have
a larger proper-motion dispersion that that of a cold stream.
The tight clump seen in the proper-motion diagram of blue stars in SA 96
is comprised of stars with $r = 15$ to 19. Are these only thick disk/halo
stars? Comparing SA 96 with SA 100 which is located at the same
latitude as SA 96, only above the plane, and only $\sim 30\arcdeg$
away in longitude from SA 96, we sample
very similar parts of the major Galactic components. However, there is a
clear difference in the number of blue stars in the two regions.
A careful inspection of Fig. 15 in Newberg et al. (2002)
which corresponds to SA 96 suggests that
multiple main sequences and turnoffs may be present, with the faintest
one in SDSS being the most distinct.
To better illustrate the proper-motion clumpiness in SA 96,
we show a zoomed-in proper-motion diagram of the blue
stars (as defined in Fig. 11) for SA 96 (left panels) and SA 100
(right panels) in Figure 12. The middle and bottom panels show
the proper motions as a function of magnitude.
The stellar excess as well as the proper-motion tightness in SA 96
are obvious when compared to SA 100.
\begin{figure}
\includegraphics[scale=0.90,clip=true]{f12.eps}
\caption{Relative proper-motion distributions and proper motions as a
function of magnitude for blue stars in SA 96 (left panels)
and in SA 100 (right panels).}
\end{figure}
The stellar excess is also apparent when comparing the proper-motion
distributions in the three fields with those predicted by the
Besancon Galactic model (Robin et al. 2003) which obviously contains
only the major Galactic components.
In Figure 13 we show the absolute proper-motion distribution
along one coordinate (RA, for example)
in all three fields as determined from the
Besancon Galactic model (Robin et al. 2003) (top panels), and from our
data (bottom panels). The filter system of the Besancon models is the
CFHT MEGACAM system which is very close to the SDSS system. At any
rate, we are interested only in relative comparisons between fields,
rather than a direct comparison between data and model counts.
The model proper motions were convolved with a 1 mas/yr
proper-motion uncertainty to approximately match the errors of the
observed proper motions.
\begin{figure}
\plotone{f13.eps}
\caption{Absolute proper-motion distribution along RA as given by
the Besancon Galactic model (top) and by our data (bottom). Blue and red
stars as selected in Fig. 11 are shown here in the left and right hand panels respectively. The magnitude range of the selected stars is $r=14$ to 19.}
\end{figure}
The distributions were constructed for blue and red stars
as defined above (see also Fig. 11) and in the magnitude range $r = 14$ to 19.
SA 96 is represented with a red
line, SA 100 with a green line and SA 101 with a blue line.
Rather than comparing directly the distributions given by the
data with those given by the model for each field, we proceed to do a
relative comparison as follows. We compare pairs of fields, i.e.
take ratios of the distributions as given separately
by the data and the model for
SA 100 and SA 101, and for SA 96 and SA 100. These ratios are readily apparent
from the plot as given by the areas under each curve. Alternatively,
one can use the ratios of the peaks for pairs of fields.
By doing so, for the red stars it is apparent that the observations
are in good agreement with the model predictions.
For the blue stars, the ratios of the proper-motion distributions for
SA 100 and SA 101 as given by the model and the observations are in reasonably good agreement.
However, this is not the case for the ratios for
SA 96 and SA 100. Clearly the data show that SA 96 has an excess of stars when compared to the model predictions.
We also note here the good agreement between model and data for the
displacement in absolute proper motion between fields as the Galactic plane
is crossed.
One interpretation for the excess counts in SA 96 is that
in this region we see multiple, wrapped streams from the
Monoceros structure, with the most distant one detected by SDSS at
$\sim 13$ kpc. Our preliminary absolute proper motion for candidate
Mon stars in SA 96 indicates a thick-disk-like motion in agreement
with the recently-modeled orbit of the Mon system
by Pe\~{n}arrubia et al. (2005).
Another possible interpretation
is that the excess stars in SA 96 are part of the Canis Major
overdensity, under the assumption that this overdensity is
the core of a disrupted dwarf galaxy.
The nature of this overdensity is strongly debated
in the current literature. Other interpretations besides the dwarf galaxy
hypothesis are: the overdensity is the
warp of the Galactic disk (e.g., Momany et al. 2006), a spiral
arm in the warped disk (e.g., Carraro et al. 2005), and the
periphery of a system centered in the region of the Argo
constellation (Rocha-Pinto et al.
2006). Currently, no interpretation of the reported CMa overdensity is clearly proved or
widely accepted. If CMa proves to be any of the above interpretations
other than the dwarf galaxy one, then the excess in SA 96 is very unlikely to be
related to the CMa overdensity, primarily because of its location.
We therefore explore the possible connection between SA 96 and the CMa
overdensity interpreted as a dwarf galaxy.
In a recent wide-area study of the distribution of red clump stars
from 2MASS, Bellazini et al. (2006) map out the overdensity in
Canis Major, as well as the Galactic warp. From their study, CMa stands out
as a distinct overdensity in the outer regions of the Galactic disk.
More importantly, because it is so nearby, CMa covers a large portion of the
sky, and the geometry is such that at lower longitudes, toward the anticenter,
CMa is closer to the Sun ($d_{\odot} \sim 6$ kpc) than its core,
which is supposedly at $(l, b) = (240\arcdeg, -8\arcdeg), d_{\odot} \sim 8$ kpc
(Bellazzini et al. 2006, Martinez-Delgado et al. 2005).
Two figures in Bellazzini et al. (2006) support the notion that SA 96 may be
on the outskirts of the reported CMa overdensity: (a) their Figure 9 (top panel), which shows the
excess of surface density of CMa and indicates that CMa is
extended away from the Galactic plane at longitudes closer to the Galactic
Anticenter; and (b) Figure 13 (bottom panel) of Bellazini et al., which shows
a Sgr-type galaxy placed at the distance and location of CMa.
Indeed its putative extension is impressive, and SA 96 would lie at the
tidal radius of such a system.
Another indication that the system may be elongated in the direction of
Galactic rotation and away from the plane is the motion of CMa as
measured by Dinescu et al. (2005b): the $\Theta$ velocity component is
$188 \pm 15$ km/s, while the $W$ component is $-49 \pm 15$ km/s.
Radial velocities of our proper motion stars in the field of SA 96 should help clarify this
issue and hopefully lend further clues to the nature of Mon and CMa structures.
We proceed now to estimate the proper-motion uncertainty by assuming that
the blue stars in the proper-motion clump seen in SA 96 belong to
a kinematically cold system, and therefore their proper-motion dispersion
reflects measurement errors only.
In the proper-motion diagram of SA 96,
in Fig. 12 there appear to be two overlapping proper-motion clumps with different dispersion:
one less tightly clumped, centered at
$(\mu_x, \mu_y) \sim (0,0)$ mas/yr and a ``radius'' of $\sim 4$ mas/yr,
and the other more tightly clumped centered at
$(\mu_x, \mu_y) \sim (-1,+2)$ mas/yr and a ``radius'' of $\sim 2$ mas/yr.
Selecting stars within the radius of the less tight clump, and within $r= 14$ to 18,
we obtain a proper-motion scatter of 1.6 mas/yr. Similarly, for the tighter clump
we obtain a proper-motion scatter of $\sim 1$ mas/yr. These numbers are only
approximate estimates to illustrate the precision of the proper motions in this work.
A rigourous kinematical understanding and therefore a better separation
of these two clumps is beyond the scope of this paper. This will be addressed in future
work where radial velocities will be considered also.
One more cautionary note should be made:
the population of the
less tight proper-motion clump seems to appear as well in SA 100 (Fig. 12), and may
therefore be representative of distant halo/thick disk stars, i.e., a population with a
non-negligeable intrinsic proper-motion dispersion.
\section{Summary}
We describe our ground-based, photographic
absolute proper-motion survey along three near-equatorial declination
zones. Proper motions are derived in $40\arcmin\times40\arcmin$ fields
from a collection of 2.5-m Du Pont, 4-m Mayall, 60-inch Mt. Wilson and
POSS-I plates. The time baseline varies between 40 and 85 years.
We have demonstrated that we obtain proper-motion uncertainties
between $\sim 1$ and 3 mas/yr per star down to a magnitude of 19,
and for a few fields down to 21. The typical uncertainty in the
correction to absolute proper motions as given by galaxies and QSOs is
between 0.2 and 0.8 mas/yr.
The described proper-motion survey is complemented by our own ongoing
radial-velocity and photometric follow-up programs
as well as by current surveys such as SDSS, QUEST, etc.
The characteristics of this proper-motion survey make it suitable to address
many topics related to both the main Galactic components, and the
tidal features seen in the halo and associated with streams from disrupted
satellites.
In this contribution, we present preliminary results in the southern
trailing tidal tail of Sgr and in the Monoceros ring region.
We thank Stephen Levine from USNO-Flagstaff for making available the
POSS-I USNO scans.
SRM is grateful to the Carnegie Observatories and its former director Augustus Oemler
for a Carnegie Visiting Associateship that made possible the collection of the Du Pont plates
used in this survey.
The financial support from NSF grants AST-0406884
and AST-0407207 for this research is acknowledged.
This publication makes uses of SDSS data products.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society,
and the Higher Education Funding Council for England. The SDSS Web Site
is http://www.sdss.org/.
|
\section{Introduction}
\label{sec:introduction}
Stochastic variational inference is framed as maximizing a global\footnote{The evidence lower bound is locally optimized with respect to local variational parameters.} variational parameter $\mathbf{\Lambda}$, which is the natural parameter of a conjugate exponential distribution \cite{JMLR:hoffman13a}.
In this framework, stochastic gradient steps are taken along the natural gradient \cite{Amari_1998} to optimize for $\mathbf{\Lambda}$.
A pleasing property of stochastic variational inference on a conjugate exponential distribution and approximation $q(\mathbf{\Lambda})$ is that
the gradient is automatically rescaled so that a unit-length step size will minimize it.
For a general Bayesian network, where the global variational parameters are subdivided to parameterize different factors $q_i$ in the network's variational approximation, the picture is less clear.
Hoffman \emph{et al.}'s appendix suggests a stochastic updating scheme like that of the global version \cite{JMLR:hoffman13a}.
We show here that the problem is more subtle in the general case, as component-wise noisy natural gradients
can tightly couple variational parameters,
and following the default recipe can sometimes lead to a scheme that ``diverges'' beyond recovery!
These remarks are of particular value to the Xbox recommender system,
which uses stochastic variational inference in a Bayesian network on ``worldwide'' scale \cite{xbox-www, PaquetK13}.
Some of the results presented in Sec.~\ref{sec:bmf} are preliminary investigations
that were done when designing the system in 2012.
\section{Variational Bayes}
A Bayesian network between the variables $\mathbf{X} = \{ \mathbf{x}_j \}$ defines the conditional dependency structure between them through their joint probability
$
p(\mathbf{X}) = \prod_j p(\mathbf{x}_j | \mathrm{pa}_j)
$.
Following Fig.~\ref{fig:bayesnet},
let $\mathrm{pa}(j)$ be the set of indexes of parents of random variable(s) $\mathbf{x}_j$;
for notational convenience we let $\mathrm{pa}_j \stackrel{\text{\tiny def}}{=} \{ \mathbf{x}_k : k \in \mathrm{pa}(j) \}$ denote the parent variables.
The variables in the network can be hidden or observed, $\mathbf{X} = \{ \mathbf{X}^\mathrm{h}, \mathbf{X}^\mathrm{o} \}$.
Variational Bayes (VB) approximates the posterior $p(\mathbf{X}^\mathrm{h} | \mathbf{X}^\mathrm{o})$ with $q(\mathbf{X}^\mathrm{h})$,
by maximizing the evidence lower bound
\[
\mathcal{L}[q] \stackrel{\text{\tiny def}}{=} \int q(\mathbf{X}^\mathrm{h}) \log \frac{p(\mathbf{X})}{q(\mathbf{X}^\mathrm{h})} \, \mathrm{d} \mathbf{X}^\mathrm{h}
\le \log p(\mathbf{X}^\mathrm{o}) \ .
\]
If $i$ indexes the hidden variables, we factorize the approximation with
\[
q(\mathbf{X}^\mathrm{h}) = \prod_i q_i(\mathbf{x}_i) \ .
\]
Let $\mathbb{E}_{j \neq i}$
indicate the expectation taken over $\prod_{j \neq i} q_j(\mathbf{x}_j)$.
The bound can be maximized in a component-wise fashion by iteratively setting
each $q_i(\mathbf{x}_i)$ to the maximum
\begin{equation} \label{eq:vbupdate}
\log q_i^*(\mathbf{x}_i) = \mathbb{E}_{j \neq i} \Big[ \log p(\mathbf{x}_i | \mathrm{pa}_i) \Big]
+ \sum_{k \in \mathrm{ch}(i)} \mathbb{E}_{j \neq i} \Big[ \log p(\mathbf{x}_k | \mathrm{pa}_k) \Big] + \mathop{\rm const} \ .
\end{equation}
In many practical networks there are some $\mathbf{x}_i$ for which number of children $N_i \stackrel{\text{\tiny def}}{=} |\mathrm{ch}(i)|$ is large.
In \cite{JMLR:hoffman13a}, $\mathbf{x}_i$ is a topic-vocabulary distribution from which millions of documents are generated.
In Sec.~\ref{sec:bmf} and \cite{xbox-www, PaquetK13} the interaction is bilinear,
where user $\mathbf{x}_i$ and item $\mathbf{x}_j$ variables are combined to represent a user's affinity to an item.
Rather than summing over all $\mathrm{ch}(i)$ for each update in (\ref{eq:vbupdate}), we aim to stochastically approximate the expectations.
It alleviates two problems: firstly, the sum contains many terms; secondly, the update depends on some $q(\mathbf{x}_j)$ which will be re-estimated, and the expense of fully estimating $q(\mathbf{x}_i)$ is lost as it too will be re-estimated.
\begin{figure}[t]
\begin{minipage}[b]{0.6\linewidth}
\begin{center}
\begin{tikzpicture}[bend angle=45,>=latex]
\tikzstyle{obs} = [ circle, thick, draw = black!100, fill = blue!20, minimum size = 6mm ]
\tikzstyle{lat} = [ circle, thick, draw=black!100, fill = red!0, minimum size = 6mm ]
\tikzstyle{par} = [ circle, draw, fill = black!100, minimum width = 3pt, inner sep = 0pt]
\tikzstyle{parents} = [ rectangle, thick, draw=green!50, fill = green!0, minimum size = 6mm ]
\tikzstyle{children} = [ rectangle, thick, draw=red!40, fill = red!0, minimum size = 6mm]
\tikzstyle{copar} = [ rectangle, thick, draw=blue!40, fill = blue!0, minimum size = 6mm]
\tikzstyle{every label} = [black!100]
\begin{scope}[node distance = 1.5cm and 1.5cm,rounded corners=4pt]
\node [lat] (xi) {$\mathbf{x}_i$};
\node [lat] (xpar1) [above of = xi] { }
edge [post] (xi);
\node [lat] (xpar2) [right of = xpar1] { }
edge [post] (xi);
\node [parents] (xpar3) [left of = xpar1] { $ \cdots \mathrm{pa}_i \cdots $ };
\node [lat] (xch1) [below of = xi] { }
edge [pre] (xi);
\node [lat] (xch2) [right of = xch1] {$\mathbf{x}_k$ }
edge [pre] (xi);
\node [children] (xch3) [left of = xch1] { $\cdots \mathrm{ch}_i \cdots $ };
\node [lat] (xcp1) [right of = xi] { }
edge [post] (xch2);
\node [lat] (xcp2) [right of = xcp1] { }
edge [post] (xch2);
\node [copar] (copar) [right of = xcp2] { $\cdots \mathrm{cp}_i \cdots$ };
\begin{pgfonlayer}{background}
\filldraw [line width = 1pt, draw=green!50, fill=green!10]
($(xpar3.north west) + (0.4cm, 0.25cm)$) rectangle ($(xpar2.south east) + (0.3cm, -0.3cm)$);
\filldraw [line width = 1pt, draw=red!40, fill=red!10]
($(xch3.north west) + (0.4cm, 0.3cm)$) rectangle ($(xch2.south east) + (0.3cm, -0.3cm)$);
\filldraw [line width = 1pt, draw=blue!40, fill=blue!10]
($(xcp1.north west) + (-0.3cm, 0.3cm)$) rectangle ($(copar.south east) + (-0.4cm, -0.25cm)$);
\end{pgfonlayer}
\end{scope}
\end{tikzpicture}
\end{center}
\end{minipage}
\begin{minipage}[b]{0.39\linewidth}
\caption{A Bayesian network, indicating $\mathbf{x}_i$'s Markov blanket. The parents of $\mathbf{x}_i$ are $\mathrm{pa}_i$ and its
children $\mathbf{x}_k \in \mathrm{ch}_i$. For a compact notation we also write $k \in \mathrm{ch}(i)$ as the index set of children, where it is clear from context. Each child $k$ has parents $\mathbf{x}_i$ and $\mathrm{cp}_i$ (the co-parents with $\mathbf{x}_i$).
The form of our notation loosely matches Winn and Bishop's in \cite{Winn_Bishop_2005}, as Alg.~\ref{alg:stochastic}
can be interpreted as \emph{``stochastic variational message passing''}.
}
\label{fig:bayesnet}
\end{minipage}
\vspace{-5pt}
\end{figure}
\subsection{Conditionally conjugate models}
The updates in (\ref{eq:vbupdate}) are straightforward when the Bayesian network is conditionally conjugate; that is, when the distribution of $\mathbf{x}_i$, conditioned on $\mathrm{pa}_i$, is (a) drawn from an exponential family, and (b) is conjugate with respect to the distribution of $\mathrm{pa}_i$.
We define the exponential family as
\begin{equation} \label{eq:exponential}
\log p(\mathbf{x}_i | \mathrm{pa}_i) = \bm{\eta}_i(\mathrm{pa}_i)^T \phi_i(\mathbf{x}_i) + f_i(\mathbf{x}_i) + g_i(\mathrm{pa}_i)
\end{equation}
where $\bm{\eta}_i(\mathrm{pa}_i)$ is the natural parameter vector, $\phi_i(\mathbf{x}_i)$ forms the sufficient
statistics, and $g_i(\mathrm{pa}_i)$ defines the normalizing constant through
$
g_i(\mathrm{pa}_i) = - \log \int \exp \{ \bm{\eta}_i(\mathrm{pa}_i)^T \phi_i(\mathbf{x}_i) + f_i(\mathbf{x}_i) \} \, \mathrm{d} \mathbf{x}_i
$.
We can view (\ref{eq:exponential}) as a ``prior'' over $\mathbf{x}_i$.
Now consider a node $\mathbf{x}_k \in \mathrm{ch}_i$ in Fig.~\ref{fig:bayesnet}. We subdivide $\mathrm{pa}_k$, the parents of $\mathbf{x}_k$, into $\mathbf{x}_i$ and its co-parents $\mathrm{cp}_i$:
\[
\log p(\mathbf{x}_k | \mathbf{x}_i, \mathrm{cp}_i) = \bm{\eta}_{k}(\mathbf{x}_i, \mathrm{cp}_i)^T \phi_k(\mathbf{x}_k) + f(\mathbf{x}_k) + g(\mathbf{x}_i, \mathrm{cp}_i) \ .
\]
We can view this as a contribution to the ``likelihood'' of $\mathbf{x}_i$. We include the co-parents as they are part of $\mathbf{x}_i$'s Markov blanket.
Through conjugacy, $p(\mathbf{x}_i | \mathrm{pa}_i)$ and $p(\mathbf{x}_k | \mathbf{x}_i, \mathrm{cp}_i)$ have the same functional form with respect to $\mathbf{x}_i$, so that we can
rewrite $p(\mathbf{x}_k | \mathbf{x}_i, \mathrm{cp}_i)$ in terms of the sufficient statistics $\phi_i(\mathbf{x}_i)$ by defining some function $\bm{\eta}_{ki}$ with
\[
\log p(\mathbf{x}_k | \mathbf{x}_i, \mathrm{cp}_i) = \bm{\eta}_{ki}(\mathbf{x}_k, \mathrm{cp}_i)^T \phi_i(\mathbf{x}_i) + h(\mathbf{x}_k, \mathrm{cp}_i) \ .
\]
We furthermore parameterize the $q(\mathbf{x}_i)$ distributions in terms of their natural parameters. To distinguish them, we denote their natural parameters by $\bm{\lambda}_i$, and define $\mathbf{\Lambda} \stackrel{\text{\tiny def}}{=} [\bm{\lambda}_1, \ldots, \bm{\lambda}_I ]$:
\begin{equation} \label{eq:q}
\log q_i(\mathbf{x}_i | \bm{\lambda}_i) = \bm{\lambda}_i^T \phi_i(\mathbf{x}_i) + f_i(\mathbf{x}_i) + \tilde{g}_i(\bm{\lambda}_i) \ .
\end{equation}
\subsection{Variational Bayes updates and their stochastic version}
Returning to (\ref{eq:vbupdate}), we can write
\begin{equation}
\log q_i^*(\mathbf{x}_i) = \mathbb{E}_{j \neq i} \Bigg[ \bm{\eta}_i(\mathrm{pa}_i)
+ \sum_{k \in \mathrm{ch}(i)} \bm{\eta}_{ki}(\mathbf{x}_k, \
|
mathrm{cp}_i) \Bigg]^T \phi_i(\mathbf{x}_i)
+ f_i(\mathbf{x}_i) + \mathop{\rm const} \ , \label{eq:vbconjugate}
\end{equation}
from which we can directly read off the updated natural parameter $\bm{\lambda}_i^*$ through (\ref{eq:q}).
Notice now that $\bm{\eta}_i$ is a multi-linear function of the random variables $\mathrm{pa}_i$, i.e.~it is linear in \emph{each} parent random variable.
In the same way $\bm{\eta}_{ki}$ is a multi-linear function of the random variables $\mathbf{x}_k$ and $\mathrm{cp}_i$. Furthermore, $q$ factorizes over these variables (except where they are observed, of course). We can therefore reparameterize (\ref{eq:vbconjugate}) in terms of expectations over $q_j$, $i \neq j$ with
\begin{align*}
\mathbb{E}_{j \neq i} \Big[ \bm{\eta}_i(\mathrm{pa}_i) \Big] & =
\widetilde{\bm{\eta}}_i \left( \Big\{ \mathbb{E}_{j} \Big[ \phi_j(\mathbf{x}_j) \Big] \Big\}_{j \in \mathrm{pa}(i)} \right)
\stackrel{\text{\tiny def}}{=} \widetilde{\bm{\eta}}_i \\
\mathbb{E}_{j \neq i} \Big[ \bm{\eta}_{ki}(\mathbf{x}_k, \mathrm{cp}_i) \Big] & =
\widetilde{\bm{\eta}}_{ki} \left( \mathbb{E}_{k} \Big[ \phi_k(\mathbf{x}_k) \Big], \, \Big\{ \mathbb{E}_{j} \Big[ \phi_j(\mathbf{x}_j) \Big] \Big\}_{j \in \mathrm{cp}(i)} \right)
\stackrel{\text{\tiny def}}{=} \widetilde{\bm{\eta}}_{ki} \ .
\end{align*}
\begin{wrapfigure}[12]{L}[0pt]{0pt}
\noindent\begin{minipage}[t]{0.51\textwidth}
\vspace{-14pt}
\begin{algorithm}[H]
\caption{Stochastic Variational Bayes}
\label{alg:stochastic}
\begin{algorithmic}[1]
\FOR{$t=1$ to $t_{\max}$ or convergence}
\STATE $\rho_t = (t + \tau)^{-\kappa}$
\FOR{each hidden $\mathbf{x}_i$}
\STATE $\mathcal{C} \leftarrow C$ random nodes from $\mathrm{ch}_i$
\STATE $\bm{\lambda}_i^{\mathrm{temp}} \leftarrow \widetilde{\bm{\eta}}_i + \frac{N_i}{C} \sum_{k \in \mathcal{C}} \widetilde{\bm{\eta}}_{ki}$
\STATE \emph{option (a):} $\bm{\lambda}_i \leftarrow (1 - \rho_t ) \bm{\lambda}_i + \rho_t \bm{\lambda}_i^{\mathrm{temp}}$
\ENDFOR
\STATE \emph{option (b):} $\mathbf{\Lambda} \leftarrow (1 - \rho_t ) \mathbf{\Lambda} + \rho_t \mathbf{\Lambda}^{\mathrm{temp}}$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\end{minipage}
\end{wrapfigure}
This is a key ingredient of algorithms like variational message passing \cite{Winn_Bishop_2005}.
(When $\mathbf{x}_k$ is observed, $\phi_k(\mathbf{x}_k)$ is kept as is, as it is averaged over a delta function.)
The variational update in (\ref{eq:vbconjugate}) becomes
\begin{equation} \label{eq:vb-update}
\bm{\lambda}_i^{*} = \widetilde{\bm{\eta}}_i + \sum_{k \in \mathrm{ch}(i)} \widetilde{\bm{\eta}}_{ki} \ .
\end{equation}
The update is a step along the natural gradient \cite{Amari_1998}, equivalent to setting the gradient to zero by solving for the zero of the derivative of $\mathcal{L}$ with respect to $\bm{\lambda}_i$.
In particular, (\ref{eq:vb-update}) updates $\bm{\lambda}_i$ from its old value to $\bm{\lambda}_i^{*}$ using a \emph{step of unit length}
along the natural gradient.
Sec.~\ref{sec:component-grad} derives the gradient $\nabla_{\bm{\lambda}_i} \mathcal{L}$,
and Sec.~\ref{sec:component-natural-grad} states its natural form $\widehat{\nabla}_{\bm{\lambda}_i} \mathcal{L}$.
When $N_i \stackrel{\text{\tiny def}}{=} |\mathrm{ch}(i)|$ is large, not all the child nodes might be accessed in reasonable time.
Furthermore, when $q(\mathbf{x}_i)$ is re-estimated, the (previous) large computation is discarded and recomputed.
We may alternatively consider a subsample of nodes from $\mathrm{ch}(i)$ to determine the sufficient statistics.
By placing a uniform distribution $\widetilde{p}_i$ on the atoms $\widetilde{\bm{\eta}}_{ki}$, the update from (\ref{eq:vb-update}) is equivalent to
$
\bm{\lambda}_i = \widetilde{\bm{\eta}}_i + N_i \, \mathbb{E}_{\widetilde{p}_i} [ \widetilde{\bm{\eta}} ]
$.
This expectation can be estimated in many ways.
Let set $\mathcal{C}$ be a sample of $C$ children from $\mathrm{ch}_i$ without replacement and let
\begin{equation} \label{eq:stochasticupdate}
\bm{\lambda}_i^{\mathrm{temp}} = \widetilde{\bm{\eta}}_i + \frac{N_i}{C} \sum_{k \in \mathcal{C}} \widetilde{\bm{\eta}}_{ki} \ .
\end{equation}
Taking expectations gives $\bm{\lambda}_i^{*} = \mathbb{E}[ \bm{\lambda}_i^{\mathrm{temp}} ] = \widetilde{\bm{\eta}}_i + N_i \mathbb{E}_{\widetilde{p}_i} [ \widetilde{\bm{\eta}} ]$.
With $\rho_t \to 0$, $\sum_{t=1}^{\infty} \rho_t = \infty$ and $\sum_{t=1}^{\infty} \rho_t^2 < \infty$, these stochastic
natural gradients are used in Alg.~\ref{alg:stochastic}, which is a stochastic version of variational message passing.
In Alg.~\ref{alg:stochastic}, scalar $\kappa \in (\frac{1}{2}, 1]$ is a forgetting rate, while delay $\tau \ge 0$ discounts early iterations more.
There are two options in Alg.~\ref{alg:stochastic}. For option \emph{(a)},
the mean value of the parameters of $q(\mathbf{X}^{\mathrm{h}})$ is periodic in $I$, the number of factors in $q$, and convergence to a local optimum can also be guaranteed for $I$-dependent mean values \cite{Kushner_2003}. Option \emph{(b)} is the update scheme given in \cite{JMLR:hoffman13a}.
\section{Bayesian matrix factorization} \label{sec:bmf}
To illustrate a general Bayesian network,
we factorize a sparse matrix of a subsample of a million entries in the Netflix data set ($M = 4805$ users and $N = 16015$ items). Each entry $r_{mn}$ is user $m$'s rating of movie $n$ on a five-star rating scale. For illustrative purposes, consider a Gaussian bilinear ratings model
\[
p(r_{mn} | \mathbf{u}_m, \mathbf{v}_n) = \mathcal{N}( r_{mn} \, ; \, \mathbf{u}_m^T \mathbf{v}_n , \, 1)
\]
for user parameter vector $\mathbf{u}_m \in \mathbb{R}^K$ and item trait vector $\mathbf{v}_n \in \mathbb{R}^K$.
We place a factorized prior $\mathcal{N}(u_{mk} \, ; \, 0, 1)$ on each of the entries of $\mathbf{u}_m$ and $\mathbf{v}_n$.
We choose a fully factorized Gaussian approximation $q(\mathbf{U}) = \prod_m \prod_k q(u_{mk})$, with a similar approximation for $q(\mathbf{V})$.
The VB update for $q(u_{mk})$ therefore incorporates $2K-1$ co-parents due to the inner product.
With the Gaussian's natural parameters being its precision and mean-times-precision, it is
\begin{equation} \label{eq:bmf}
\bm{\lambda}_{mk}^{\mathrm{temp}} =
\begin{pmatrix}
\mathsf{prec} \\
\mathsf{mean} \cdot \mathsf{prec}
\end{pmatrix}
=
\begin{pmatrix} 1 \\ 0 \end{pmatrix}
+ \frac{N_{m}}{C}\sum_{n \in \mathcal{C}}
\begin{pmatrix}
\mathop{\sf var}_q[v_{nk}] + \mathbb{E}_q[v_{nk}]^2 \\
\mathbb{E}_q[v_{nk}] (r_{mn} - \sum_{k' \neq k} \mathbb{E}_q[u_{mk'}] \mathbb{E}_q[v_{nk'}] )
\end{pmatrix} \ .
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.49\textwidth]{ComponentFull.pdf}
\includegraphics[width=0.49\textwidth]{HoffmanFull.pdf} \\
\includegraphics[width=0.49\textwidth]{Component.pdf}
\includegraphics[width=0.49\textwidth]{Hoffman.pdf} \\
\includegraphics[width=0.49\textwidth]{ComponentZoom.pdf}
\includegraphics[width=0.49\textwidth]{HoffmanZoom.pdf}
\end{center}
\caption{Convergence of ${\cal L}[q]$ with $\rho_t = t^{-0.6}$. Alg.~\ref{alg:stochastic}'s option \emph{(a)} is shown in the left
column; option \emph{(b)} is shown in the right column. The x-axes are on a logarithmic scale.
The \emph{global} stochastic gradient is not in its natural form,
and the effect of a large variance in the gradient estimate
and overshooting with too large step sizes of $\rho_t \in (0, 1]$ is clearly visible for small $C$.
Note that $r_{mn}$'s can be revisited over multiple loops in Alg.~\ref{alg:stochastic}.
Different magnifications of the same two convergence plots for options \emph{(a)} and \emph{(b)} are shown in the three rows of graphs.
}
\label{fig:convergence}
\vspace{-18pt}
\end{figure}
Fig.~\ref{fig:convergence} shows ${\cal L}[q]$ as a function of the number of times that individual ratings (observed nodes) $r_{mn}$ are accessed or queried (using $K=5$).
The value of the bound is shown for the use of at most $C = 1, \ldots, 20$ children when estimating the gradient of each random variable with (\ref{eq:stochasticupdate}) and (\ref{eq:bmf}). Both options \emph{(a)} and \emph{(b)} in Alg.~\ref{alg:stochastic} ``diverge'' in a numerically unrecoverable way when $C$ is small.
This is due to the global gradient not being in its natural form, and using a step size of $\rho_t \in (0, 1]$ that is too big, overshooting with too large gradient steps.
Full VB, shown in black in Fig.~\ref{fig:convergence}, implicitly uses $\rho_t = 1$ in (\ref{eq:vb-update}).
As the stochastic natural gradient depends on other $\bm{\lambda}_j$, much smaller initial step sizes are required to not ``overshoot''.
The variance of the gradient is simply too large compared to $\rho_t$.
Fig.~\ref{fig:convergence} illustrates this problem for \emph{general Bayesian networks};
see especially the \emph{top left} figure.
In practice, we can overcome this problem overcome by starting with sufficiently small initial step sizes $\rho_t \ll 1$.
For $C = 1$ in option \emph{(a)} this was starting from $\rho_1 = \frac{1}{512}$,
and $\rho_1 = \frac{1}{64}$ for option \emph{(b)}. In \cite{xbox-www, PaquetK13} the value of $C$ varied depending on a user or item's usage,
and there $\rho_t = 1$ was fixed for the first ten iterations before slowly decreasing it.
Have we thrown the baby (well-scaled steps) out with the bath water (exact gradients)?
Maybe some. As shown by this short note, it is still an open question.
\renewcommand\section{\subsubsection}
\section*{Acknowledgment}
To the anonymous reviewer who pointed out that the Fisher information matrix of $q(\mathbf{X}^\mathrm{h} | \mathbf{\Lambda})$ block-diagonal, having
the Fisher information matrices of $q(\mathbf{x}_i | \bm{\lambda}_i)$ along its diagonal: thank you!
\small
|
\section{Introduction}
Continued fractions have always held great fascination, for both aesthetic reasons and practical purposes. Among the many clever properties of periodic continued fractions, Legendre found how to obtain the representation of an integer $N$ as the sum of two squares, in his own words, "{\em sans aucun t\^atonnement}" from the continued fraction expansion of $\sqrt N$ when the period is odd \cite{legendre}.
In particular, this property holds for any prime $p$ congruent $1$ modulo $4$, \cite{legendre,sierp}.
As a kind of counterpart to Legendre's finding, this paper shows how to obtain a factor of a composite $N$
directly from the continued fraction expansion of $\sqrt N$ when the period is even.
In particular, this is certainly possible when at least one prime factor of $N$ is congruent $3$ modulo $4$.
\noindent
Based on this result, derived from peculiar properties of the continued fraction convergents, and on an adaptation of Shanks' infrastructural machinery, a factoring algorithm is proposed whose complexity depends on the accuracy of the evaluation of certain integrals of Dirichlet's.
%
To these aims, the paper is organized as follows. Section 2 summarizes the properties of the continued fraction expansion of $\sqrt N$.
In Section 3, some new properties of the convergents are proved,
and Shanks' infrastructural method is revisited and applied
to a sequence of quadratic forms generated from the convergent of the continued fraction expansion of $\sqrt N$. Section 4 discusses the factorization of composite numbers $N$
when the period of the continued fraction expansion of $\sqrt N$ is even.
Lastly, Section 5 briefly reports some conclusions.
\section{Preliminaries}
A regular continued fraction is an expression of the form
\begin{equation}
\label{cf}
\displaystyle a_0+\frac{1}{\displaystyle a_1+\frac{1}{\displaystyle a_2+\frac{1}{\displaystyle a_3+ \cdots}}} ~~,
\end{equation}
%
where $a_0$, $a_1$, $a_2, \ldots, a_i, \ldots$ is a sequence, possibly infinite,
of positive integers.
A convergent of a continued fraction is a sequence of fractions $\frac{A_m}{B_m}$,
each of which is obtained by truncating the continued fraction at the $m$-th term.
The fraction $\frac{A_m}{B_m}$ is called the $m$-th convergent \cite{dave,hardy,perron}.
The first few initial terms of the convergent of (\ref{cf}) are
$$ \frac{A_0}{B_0}=\frac{a_0}{1},~~ \frac{A_1}{B_1}=\frac{a_0 a_1 +1}{a_1}, ~~
\frac{A_2}{B_2}=\frac{a_0 a_1 a_2 +a_0+a_2}{a_1a_2+1}, \ldots ~~~. $$
Numerators and denominators of each $m$-th convergent satisfy the second-order
recurrences
\begin{equation}
\label{mainrecur}
\left\{ \begin{array}{ll}
A_m = a_m A_{m-1} + A_{m-2} & ~~,~~ A_0=a_0, ~~A_1=a_0 a_1 +1 \\
B_m = a_m B_{m-1} + B_{m-2} & ~~,~~ B_0=1, ~~B_1 = a_1\\
\end{array} \right. \hspace{5mm} , ~~\mbox{$ \forall ~ m \geq 2$;}
\end{equation}
further, we have \cite[p.85]{dave} the relationships
\begin{equation}
\label{fund}
A_mB_{m-1}-A_{m-1}B_m=(-1)^{m-1}
\end{equation}
\begin{equation}
\label{fund1}
A_mB_{m-2}-A_{m-2}B_m=(-1)^{m-2}a_m ~~.
\end{equation}
Equation (\ref{fund}) shows that numerator and denominator of any $m$-th convergent are relatively prime. \\
A continued fraction is said to be definitively periodic,
with period $\tau$, if, starting from a finite $n_o$, a fixed pattern
$a_1'$, $a_2', \ldots, a_{\tau}'$ repeats indefinitely.
Lagrange showed that any definitively periodic continued fraction represents
a positive number of the form $a+b\sqrt{N}$, $a,b \in \mathbb Q$, i.e. an element of $\mathbb F=\mathbb Q(\sqrt N)$, and conversely
any such positive number is represented by a definitively periodic continued fraction
\cite{dave,sierp}. The maximal order of $\mathbb F$ is denoted $\mathfrak O_{\mathbb F}$.
Let $\mathcal G(\mathbb F/\mathbb Q)=\{\iota, \sigma\}$ be the Galois group of $\mathbb F$ over $\mathbb Q$,
where $\iota$ denotes the group identity, and the action of the automorphism $\sigma$,
called conjugation, is defined as $\sigma(a+b\sqrt{N}) =a-b\sqrt{N}$.
The field norm $\mathbf N_{\mathbb F}(\mathfrak a)$ of $\mathfrak a \in \mathbb F$
is defined to be $\mathbf N_{\mathbb F}(\mathfrak a)=\mathfrak a \sigma(\mathfrak a)$. \\
In the continued fraction expansion of $\sqrt{N}$, the period of length $\tau$ begins
immediately after the first term $a_0$, and consists of a
palindromic part formed by $\tau-1$ terms $~~a_1,a_2, \ldots, a_2, a_1,~$
followed by $2a_0$.
Periodic continued fractions of this sort are conventionally written in the form
\begin{equation}
\label{sqrtN0}
\sqrt{N} = \left[ a_0 , \overline{a_1,a_2, \ldots, a_2, a_1, 2a_0} \right] ~~,
\end{equation}
where the over-lined part is the period.
Note that the period of the irrational $ \frac{a_0+\sqrt{N}}{N-a_0^2}$ starts immediately without anti-period; in this case,
the continued fraction is called purely periodic and is denoted
$ \left[ \overline{a_1,a_2, \ldots, a_2, a_1, 2a_0} \right]$.
\noindent
In Carr's book \cite[p.70-71]{carr} we find a good collection
of properties of the continued fraction expansion of $\sqrt{N}$, which are summarized in the
following, with the addition of some properties taken from \cite{dave,sierp,riesel}:
\begin{enumerate}
\item Let $c_n$ and $r_n$ be the elements of two sequences of positive integers defined by
the relation
$$ \frac{\sqrt{N}+c_n}{r_n}=a_{n+1}+\frac{r_{n+1}}{\sqrt{N}+c_{n+1}} $$
with $c_0= \left\lfloor \sqrt N \right\rfloor$, and $ r_0 =N-a_0^2$;
the elements of the sequence $a_1, a_2, \ldots , a_n \ldots$
are thus obtained as the integer parts of the left-side fraction
\begin{equation}
\label{approxN}
a_{n+1} = \left\lfloor \frac{\sqrt{N}+c_n}{r_n} \right\rfloor ~~.
\end{equation}
\item Let $a_0= \lfloor \sqrt{N} \rfloor$ be initially computed, and set $c_0 = a_0$, $r_0 =N-a_0^2$,
then sequences $\{ c_n \}_{n \geq 0}$ and $\{ r_n \}_{n \geq 0}$ are produced by the
recursions
\begin{equation}
\label{contfrac}
a_{m+1} = \left\lfloor \frac{a_0+c_m}{r_m} \right\rfloor \hspace{5mm},\hspace{5mm} c_{m+1}=a_{m+1} r_m -c_m
\hspace{5mm},\hspace{5mm} r_{m+1} = \frac{N-c_{m+1}^2}{r_m} ~~.
\end{equation}
These recursive equations, together with (\ref{approxN}), allow us to compute the sequence
$\{a_m\}_{m\geq 1}$ using rational arithmetical operations; however,
the iterations may be stopped when $a_m = 2 a_0$, having completed a period.
\item The $n$-th convergent to $\sqrt{N}$ can be recursively computed as
\begin{equation}
\label{approx10}
\frac{A_n}{B_n} = \frac{a_n A_{n-1}+ A_{n-2}}{a_n B_{n-1}+ B_{n-2}} ~~~n \geq 1~~,
\end{equation}
with initial conditions $A_{-1}=1$, $B_{-1}=0$, $A_{0}=a_0$, and $B_{0}=1$.
\item The sequence of ratios $ \frac{A_n}{B_n}$ assumes the limit value $\sqrt{N}$ as $n$ goes to infinity,
due to the inequality
$$ \left| \frac{A_n}{B_n} -\sqrt N \right| \leq \frac{1}{B_n B_{n+1}} ~~, $$
since $A_n$ and $B_n$ go to infinity along with $n$. Furthermore, $\frac{A_n}{B_n} <\sqrt N$,
if $n$ is even, and $\frac{A_n}{B_n} >\sqrt N$ if $n$ is odd \cite[p.132]{hardy}. Therefore, any convergent
of even index is smaller than any convergent of odd index.
\item The true value of $\sqrt{N}$ is the value which (\ref{approx10}) becomes when the "approximated"
quotient $a_n$, as defined in (\ref{approxN}), is substituted with the complete
quotient $\frac{\sqrt{N}+c_{n-1}}{r_{n-1}}$. This gives
$$ \sqrt{N}= \frac{(\sqrt{N}+c_{n-1}) A_{n-1}+r_{n-1} A_{n-2}}{(\sqrt{N}+c_{n-1}) B_{n-1}+r_{n-1} B_{n-2}}~. $$
\item The value $c_0=a_0$ is the greatest value that $c_n$ may assume.
No $a_n$ or $r_n$ can be greater than $2a_0$. \\
If $r_n=1$ then $a_{n+1}=a_0$.
For all $n$ greater than $0$, we have $a_0-c_n < r_n$.
\item The first complete quotient that is repeated is $\frac{\sqrt{N} +c_0}{r_0}$,
and $a_1$, $r_0$, and $c_0$ commence each cycle of repeated terms.
\item Through the first period (or cycle) of length $\tau$, the elements
$a_{\tau-j}$, $r_{\tau-j-2}$, and $c_{\tau-j-1}$ are respectively equal to
$a_j$, $r_j$, and $c_j$.
\item The period length cannot be greater than $2a_0^2$.
This bound is very loose and was tightened by Kraitchik \cite[p.95]{steuding},
who showed that $\tau$ is upper bounded by
\begin{equation}
\label{boundP}
0.72 \sqrt{N} \ln N \hspace{6mm} N > 7 ~~.
\end{equation}
However, the period length has irregular behavior as a function of $N$, because it may assume any
value from $1$, when $N=M^2+1$,
to values close to the order $O( \sqrt{N} \ln N )$ \cite{sierp}.
\item The element $\mathfrak c_m=A_m+B_m \sqrt N \in \mathfrak O_{\mathbb F}$
is associated to the $m$-th convergent.
\end{enumerate}
\noindent
Numerators and denominators of the convergents satisfy interesting relations \cite[p.92-95]{perron}
\begin{equation}
\label{fact5}
\left\{ \begin{array}{l}
A_0 A_{\tau-1} + A_{\tau-2} - N B_{\tau-1} =0 \\
A_1 A_{\tau-2} + A_0 A_{\tau-3} - N (B_1 B_{\tau-2}+B_0 B_{\tau-3}) =0 \\
A_j A_{\tau-j-1} + A_{j-1} A_{\tau-j-2} - N (B_j B_{\tau-j-1}+B_{j-1} B_{\tau-j-2}) =0
\hspace{10mm} 3\leq j \leq \tau-3 ~~.\\
\end{array} \right.
\end{equation}
\noindent
Besides these properties, the following equations, \cite[p.329-332]{sierp}, are used in the proofs:
\begin{equation}
\label{fact0}
\left\{ \begin{array}{l}
A_{\tau} = 2 a_0 A_{\tau-1} +A_{\tau-2} \\
B_{\tau} = 2 a_0 B_{\tau-1} +B_{\tau-2} \\
\end{array} \right.
\end{equation}
\begin{equation}
\label{fact1}
\left\{ \begin{array}{l}
A_{\tau} B_{\tau-1} - A_{\tau-1} B_{\tau} = (-1)^{\tau-1} \\
A_{\tau-1} B_{\tau-2} - A_{\tau-2} B_{\tau-1} = (-1)^{\tau-2} \\
A_{\tau} B_{\tau-2} - A_{\tau-2} B_{\tau} = 2 a_0 (-1)^{\tau}\\
\end{array} \right.
\end{equation}
\begin{equation}
\label{fact2}
\left\{ \begin{array}{l}
A_{\tau-2} = - a_0 A_{\tau-1} + N B_{\tau-1} \\
B_{\tau-2} = A_{\tau-1} - a_0 B_{\tau-1} \\
\end{array} \right.
\end{equation}
\begin{equation}
\label{fact3}
\left\{ \begin{array}{l}
A_{\tau} = a_0 A_{\tau-1} + N B_{\tau-1} \\
B_{\tau} = A_{\tau-1} + a_0 B_{\tau-1} \\
\end{array} \right.
\end{equation}
\begin{remark}
\label{remark1}
The smallest positive solution of Pell's equation $x^2-Ny^2=(\pm 1)$ is $\mathfrak c_{\tau-1}$,
whenever a solution exists.
If $\{ 1, \sqrt N \}$ is an integral basis of $\mathbb F$, then $\mathfrak c_{\tau-1}$ coincides with
the fundamental positive unit $\epsilon_0$ of $\mathbb F$. If $\{ 1, \frac{1+\sqrt N}{2} \}$
is an integral basis of $\mathbb F$, then $\mathfrak c_{\tau-1}$ may be either
$\epsilon_0$ or $\epsilon_0^3$.
An easy way to check whether $\mathfrak c_{\tau-1}=\epsilon_0^3$ is
to solve in $\mathbb Q$ the equation $(x+ y \sqrt N)^3=A_{\tau-1}+B_{\tau-1}\sqrt N$,
which is equivalent to verifying whether some solution of the following Diophantine equation
is a rational number with $2$ at denominator
$$ \begin{array}{l}
64 x^9-48 A_{\tau-1}x^6+ (27 N B_{\tau-1}^2-15 A_{\tau-1}^2) x^3- A_{\tau-1}^3=0 \\
\end{array} ~~.
$$
If a rational solution $x_o$ of this equation exists, the corresponding $y_o$ can be computed as
$y_o= \sqrt{\frac{A_{\tau-1}-x_o^3}{3 N x_o}}$.
\end{remark}
\noindent
The following proposition describes how to move from one period to another.
\begin{proposition}
\label{prop1}
The sequence $\{ \mathfrak c_m \}_{m \geq 0} $ satisfies the relation
\begin{equation}
\label{fact5per}
\mathfrak c_{m+k\tau} = \mathfrak c_m \mathfrak c_{\tau-1}^k ~~\forall~m, k \in \mathbb Z~~.
\end{equation}
\end{proposition}
\begin{proof}
The two dependencies, with respect to $m$ and $k$, are disposed of separately. \\
The claimed equality is trivial for $m=k=0$, and fixing $k=1$, equation (\ref{fact3}) allows us to write
$ \mathfrak c_\tau =a_{0}\mathfrak c_{\tau-1}+\sqrt N \mathfrak c_{\tau-1} =
(a_0+ \sqrt N) \mathfrak c_{\tau-1}= (A_0+B_0 \sqrt N) \mathfrak c_{\tau-1}$.
Then, by the recurrences (\ref{mainrecur}) and the periodicity of the $a_i$s, we can write
$$ \mathfrak c_{\tau+1} = a_1 \mathfrak c_{\tau} +\mathfrak c_{\tau-1} =
a_1 (A_0+B_0 \sqrt N) \mathfrak c_{\tau-1} + \mathfrak c_{\tau-1} = \mathfrak c_{1} \mathfrak c_{\tau-1} ~~.$$
Clearly, we can iterate by using the recurrences (\ref{mainrecur}) and the symmetry of the $a_i$s to
obtain the relation
$ \mathfrak c_{\tau+m} = \mathfrak c_{m} \mathfrak c_{\tau-1}$,
which shows that multiplication by $ \mathfrak c_{\tau-1}$ is equivalent to a translation by $\tau$.
The conclusion is immediate by iterating on $k$.
\end{proof}
\section{Convergents and quadratic forms }
\label{convprop}
Let $\Delta_m=A_m^2-N B_m^2$ denote the field norm of
$\mathfrak c_m=A_m + \sqrt{N} B_m \in \mathfrak O_{\mathbb F}$.
Several properties of convergents are better described considering, besides the sequence
$\mathbf \Delta = \{\Delta_m\}_{m \geq 0}$, a second sequence
$\mathbf \Omega = \{ \Omega_m = A_m A_{m-1}-N B_m B_{m-1} \}_{m \geq 1}$.
Using (\ref{fund}), the following relation can be shown
\begin{equation}
\label{dnorm}
\Omega_{m+1}^2-\Delta_m \Delta_{m+1}=N ~~\forall m \geq 0~~.
\end{equation}
The elements of the sequences $\mathbf \Delta$ and $\mathbf \Omega$ satisfy the recurrent relations
\begin{equation}
\label{Deltarecur}
\left\{
\begin{array}{ll}
\Delta_{m+1} = a_{m+1}^2 \Delta_{m} + 2 a_{m+1} \Omega_{m}+ \Delta_{m-1} \\
\Omega_{m+1} = \Omega_{m} + a_{m+1} \Delta_{m} \\
\end{array} \right. ~~~~~~m \geq 1
\end{equation}
with initial conditions $\Delta_0=a_0^2-N$, $\Delta_1=(1+a_0a_1)^2-N a_1^2$ and
$\Omega_1=(1+a_0a_1) a_0-N a_1$. Using (\ref{Deltarecur}), it is immediate to see that
$c_{m+1}= |\Omega_m|$ and $r_{m+1}=|\Delta_m|$. \\
Introducing the matrix
\begin{equation}
\label{Deltarecur2}
T(a_{m}) = \left[ \begin{array}{ccc}
a_m^2 & a_m & 1 \\
2a_m & 1 & 0 \\
1 & 0 & 0
\end{array} \right] ~~,
\end{equation}
and defining the column vector $\mathbf \Lambda_m=[\Delta_m,2\Omega_{m},\Delta_{m-1} ]^T$, the
equation (\ref{Deltarecur}) can be written as
\begin{equation}
\label{Deltarecur3}
\mathbf \Lambda_{m+1}= T(a_{m+1}) \mathbf \Lambda_{m} ~~~~ \forall~ m \geq 1 ~~.
\end{equation}
Iterating this relation, we have
\begin{equation}
\label{recurn3}
\mathbf \Lambda_{m+n}=T(a_{m+n}) T(a_{m+n-1}) \cdots T(a_{m+2}) T(a_{m+1}) \mathbf \Lambda_{m} =
T_{(m,n)} \mathbf \Lambda_{m} ~~~~ \forall~ m,n \geq 1 ~~,
\end{equation}
where $T_{(m,n)}=\prod_{j=m+1}^{m+n}T(a_{j})$ is a matrix
that only depends on the sequence of coefficients $a_t$.
Furthermore, from (\ref{dnorm}) we may derive the relation
$$ \Omega_{m+1}^2-\Omega_{m}^2 = \Delta_m (\Delta_{m+1}-\Delta_{m-1}) ~~ \forall~ m \geq 1 , $$
which allows us to write equation (\ref{Deltarecur}) as
\begin{equation}
\label{Deltarecur1}
\left\{
\begin{array}{ll}
\Delta_{m+1} = \Delta_{m-1} + a_{m+1} ( \Omega_{m+1} +\Omega_m) \\
\Omega_{m+1} = \Omega_m + a_{m+1} \Delta_m ~\\
\end{array} \right. ~~ \forall~ m \geq 1~~.
\end{equation}
\begin{definition}
\label{seqqf}
Let $\mathbf \Upsilon$ be the sequence of quadratic forms
$\mathbf f_m(x,y)=\Delta_m x^2+ 2 \Omega_m x y + \Delta_{m-1}y^2$, $m \geq 1$,
defined by means of the sequences $\mathbf \Delta$ and $\mathbf \Omega$.
\end{definition}
\noindent
Note that it may sometimes be convenient to denote a quadratic form simply
with the triple of coefficients, i.e. the $3$-dimensional vector $\mathbf \Lambda_m$; further,
due to equation (\ref{dnorm}), all quadratic forms in $\mathbf \Upsilon$
have the same discriminant $4N$.
\begin{remark}
The absolute values of $\Delta_m$ and $\Omega_m$ are bounded as
$$ |\Delta_m| < 2 \frac{1}{a_{m+1}} \sqrt{N} \leq 2 \sqrt{N} \hspace{7mm} , \hspace{7mm}
|\Omega_m| < \sqrt{N} ~~~~\forall~ m\geq 1~~ .$$
The bound $2 \sqrt{N}$ for $\Delta_m$ is well known, \cite[Theorem 171, p.140]{hardy},
and can be slightly tightened considering the following chain of inequalities
$$ |A_m^2-N B_m^2| = B_m^2 \left|\frac{A_m}{B_m}-\sqrt{N} \right|(\frac{A_m}{B_m}+\sqrt{N} )\leq
\frac{B_m}{B_{m+1}} \left|\frac{A_m}{B_{m}}-\sqrt{N}+2 \sqrt{N}\right| $$
$$ \leq \frac{B_m}{B_{m+1}} \left|\frac{A_m}{B_{m}}-\sqrt{N}\right|+
2 \sqrt{N}\frac{B_m}{B_{m+1}} \leq \frac{1}{B_{m+1}^2} +
2 \frac{B_m}{a_{m+1}B_m+B_{m-1}} \sqrt{N} $$
$$ = 2 \frac{1}{a_{m+1}}\sqrt{N} + \frac{1}{B_{m+1}^2} -
2 \sqrt{N}\frac{B_{m-1}}{a_{m+1}(a_{m+1}B_m+B_{m-1})} < 2 \frac{1}{a_{m+1}}\sqrt{N}~~. $$
The bound for $ |\Omega_m|$ is an immediate consequence of equation (\ref{dnorm}), we have
$\Delta_m \Delta_{m+1} <0$ since the signs in the sequence $\mathbf \Delta$ alternate; consequently
$$ \Omega_m^2 =N+\Delta_m \Delta_{m+1}< N ~~, $$
thus taking the positive square root of both sides, the inequality
$|\Omega_m| < \sqrt{N}$ is obtained.
\end{remark}
\subsection{Periodicity and Symmetry}
The sequences $\mathbf \Delta$ and $\mathbf \Omega$ are periodic in the same way as the sequence of coefficients $a_m$,
although their periods are even, and may be $\tau$ or $2\tau$ depending on whether $\tau$ is even or odd.
Further, within a period, there exist interesting symmetries.
\begin{theorem}[Periodicity of $\mathbf \Delta$]
\label{per2}
Starting with $m=1$, the sequence $\mathbf \Delta = \{ \Delta_m \}_{m \geq 0}$ is periodic with period $\tau$
or $2\tau$ depending on whether $\tau$ is even or odd. The elements of the first block
$\{ \Delta_m \}_{m=0}^{\tau} \subset \mathbf \Delta$ satisfy the symmetry relation
$\Delta_m=(-1)^{\tau}\Delta_{\tau-m-2}$, $\forall ~0 \leq m \leq \tau-2$.
\end{theorem}
\begin{proof} The period of the sequence $\mathbf \Delta$ is $\tau$ or $2\tau$, as a consequence of
equation (\ref{fact5per}), because the norm of $A_{\tau-1} + \sqrt{N} B_{\tau-1}$ is $(-1)^\tau$. \\
The symmetry of the sequence $\mathbf \Delta$ within the $\tau$ elements of the first period follows from
the relations
\begin{equation}
\label{fact4}
\left\{ \begin{array}{l}
A_{\tau-m-2} = (-1)^{m-1} A_{\tau-1} A_m + (-1)^{m} N B_{\tau-1} B_m \\
B_{\tau-m-2} = (-1)^{m} A_{\tau-1} B_m + (-1)^{m-1} B_{\tau-1} A_m \\
\end{array} \right. ~~,~~ 0\leq m \leq \tau-2~,
\end{equation}
which are proved using the recurrences (\ref{mainrecur}) together with
(\ref{fact2}) and (\ref{fact3}) backwards \cite[p.329-330]{sierp};
the transformation defined by (\ref{fact4}) is identified by the matrix
\begin{equation}
\label{keymatrix}
M_{\tau-1} = \left[ \begin{array}{cc}
- A_{\tau-1} & N B_{\tau-1} \\
- B_{\tau-1} & ~~ A_{\tau-1} \\
\end{array} \right] ~~.
\end{equation}
We have
$$ \left\{ \begin{array}{lcl} A_{\tau-m-2}^2-N B_{\tau-m-2}^2 &=&
(A_{\tau-1} A_m - N B_{\tau-1} B_m)^2-N(-A_{\tau-1} B_m + B_{\tau-1}A_m)^2 \\
&=& (A_{m}^2-N B_{m}^2)(A_{\tau-1}^2-N B_{\tau-1}^2) = (-1)^{\tau} (A_{m}^2-N B_{m}^2)
\end{array} \right. $$
that is $\Delta_{\tau-m-2}= (-1)^{\tau}\Delta_m $. Actually, equation (\ref{fact4})
can be written in the form
\begin{equation}
\label{factfund}
A_{\tau-m-2}+\sqrt{N} B_{\tau-m-2} = (-1)^{m-1} (A_{\tau-1}+\sqrt{N} B_{\tau-1})
(A_{m}-\sqrt{N} B_{m})
\end{equation}
or more compactly as
$\mathfrak c_{\tau-m-2} = (-1)^{m-1} \mathfrak c_{\tau-1} \sigma(\mathfrak c_m)$.
\end{proof}
\begin{theorem}[Periodicity of $\mathbf \Omega$]
\label{per3}
The sequence $\mathbf \Omega= \{ \Omega_m \}_{m \geq 1}$ is periodic of period $\tau$
or $2\tau$ depending on whether $\tau$ is even or odd.
The elements of the first block $\{ \Omega_m \}_{m=1}^{\tau} \subset \mathbf \Omega$ satisfy the symmetry relation \\
$\Omega_{\tau-m-1}=(-1)^{\tau+1}\Omega_{m}$, $\forall~m \leq \tau-2$.
\end{theorem}
\begin{proof}
The periodicity of the sequence $\mathbf \Omega$ follows from the property
expressed by equation (\ref{fact5per}), noting that
$$ \Omega_j = \frac{1}{2} \left( (A_{j} + \sqrt{N} B_{j})((A_{j-1} - \sqrt{N} B_{j-1})+
(A_{j} - \sqrt{N} B_{j})((A_{j-1} + \sqrt{N} B_{j-1}) \right) ~~. $$
The symmetry property of the sequence $\mathbf \Omega$ within a period follows from
(\ref{fact4}) in the same way as the sequence $\mathbf \Delta$;
we have
$$ \begin{array}{rl} A_{\tau-1-j}A_{(\tau-1)-j-1}-NB_{\tau-1-j}B_{(\tau-1)-j-1}= &
- (A_{\tau-1} A_j - N B_{\tau-1} B_j)(A_{\tau-1} A_{j-1} - N B_{\tau-1} B_{j-1}) \\
& ~~~~~~ + N(A_{\tau-1} B_j - B_{\tau-1} A_j)(A_{\tau-1} B_{j-1} - B_{\tau-1} A_{j-1}) \\
= & - (A_{\tau-1}^2-N B_{\tau-1}^2)(A_jA_{j-1} -N B_jB_{j-1})
\end{array} $$
that is, $\Omega_{\tau-j-1}=(-1)^{\tau+1}\Omega_{j}$.
\end{proof}
\noindent
The two quadratic forms $ \mathbf f_{n}(x,y)=\Delta_n x^2+ 2 \Omega_n x y + \Delta_{n-1}y^2$ and
$ \mathbf f_{\tau-1-n}(x,y)=\Delta_{n-1} x^2-2 \Omega_n x y + \Delta_{n}y^2$ are associated respectively
to the positions $n$ and $\tau-1-n$, as a consequence of the symmetries of the sequences
$\mathbf \Delta$ and $\mathbf \Omega$ shown by Theorems \ref{per2} and \ref{per3},
within the first block of length $\tau$ in $\mathbf \Upsilon$.
It should be noted that $\mathbf f_{m}(x,y)$ and $\mathbf f_{\tau-1-m}(x,y)$ are improperly equivalent.
\paragraph{Key matrix.}
Clearly, the column vectors $\mathbf \Lambda_m$ and $\mathbf \Lambda_{\tau-m-1}$ are transformed
one into the other by an involutory matrix $J$ of determinant $1$
$$ \left[ \begin{array}{c}
\Delta_{m-1} \\
- 2\Omega_{m} \\
\Delta_{m}
\end{array} \right] =
\left[ \begin{array}{ccc}
0 & 0 & 1 \\
0 & -1 & 0 \\
1 & 0 & 0
\end{array} \right]
\left[ \begin{array}{c}
\Delta_{m} \\
2\Omega_{m} \\
\Delta_{m-1}
\end{array} \right] ~~.
$$
Using the matrices $T(a_n)$ and equation (\ref{recurn3}), and
applying to $ \mathbf \Lambda_{m}$ the sequence of matrices
$T(a_{m+1})$, $T(a_{m+2}), \ldots $, $T(a_{\tau-1-m})$ in reverse order,
we obtain $\mathbf \Lambda_{\tau-1-m}$
\begin{equation}
\label{key1}
\mathbf \Lambda_{\tau-1-m} = T(a_{\tau-1-m}) \cdots T(a_{m+1}) \mathbf \Lambda_{m}
~~ \Rightarrow ~~ \mathbf \Lambda_{m} = J T(a_{\tau-1-m}) \cdots T(a_{m+1}) \mathbf \Lambda_{m} ~~.
\end{equation}
Assuming $\tau$ even, this equation implies
that $\mathbf \Lambda_m$ is an eigenvector of eigenvalue $1$ of the matrix
$$ E_m =J T(a_{\tau-1-m}) \cdots T(a_{m+1}) = J T(a_{m+1}) T(a_{m})
\cdots T(a_{\frac{\tau}{2}-1}) T(a_{\frac{\tau}{2}})T(a_{\frac{\tau}{2}+1}) \cdots T(a_{m+1}) $$
since $T(a_{\tau-1-n})=T(a_{n+1})$ by the symmetry of the sequence $\{ a_n \}_{n=1}^{\tau-1}$.
Observing that $JT(a_{m})J=T(a_{m})^{-1}$ and $J^2=I$, we have
\begin{equation}
\label{eqforD}
\begin{array}{lcl}
E_m &= &(J T(a_{n+2})J) (J T(a_{n+3}) J)J\cdots
\cdots (JT(a_{\frac{\tau}{2}-1})J)J T(a_{\frac{\tau}{2}})T(a_{\frac{\tau}{2}-1}) \cdots T(a_{n+2}) \\
&= & T(a_{n+2})^{-1} \cdots T(a_{\frac{\tau}{2}-1})^{-1} J T(a_{\frac{\tau}{2}})T(a_{\frac{\tau}{2}-1})
\cdots T(a_{n+2}) \\
&= & (T(a_{\frac{\tau}{2}-1}) \cdots T(a_{n+2}))^{-1} J T(a_{\frac{\tau}{2}})(T(a_{\frac{\tau}{2}-1})
\cdots T(a_{n+2}) ) ~~.\\
\end{array}
\end{equation}
It follows that the matrix $E_m$ has the same characteristic polynomial
$z^3-z^2-z+1$ as $J T(a_{\frac{\tau}{2}})$, i.e. $E_m$ has
eigenvalue $-1$ with multiplicity $1$, and eigenvalue $1$ with geometric multiplicity $2$.
\vspace{1mm}
\noindent
Assuming $\tau$ odd, the symmetries of the sequences $\{ a_n \}_{n=1}^{\tau-1}$, $\{ \Delta_n \}_{n=1}^{\tau-1}$, and
$\{ \Omega_n \}_{n=1}^{\tau-1}$, refer to an even number $\tau -1$ of terms, and equation (\ref{eqforD})
is written as
\begin{equation}
\label{eqforDodd}
\begin{array}{lcl}
D_n &= &(J T(a_{n+2})J) (J T(a_{n+3}) J)J\cdots
\cdots (JT(a_{\frac{\tau-3}{2}})J)J T(a_{\frac{\tau-3}{2}}) \cdots T(a_{n+2}) \\
&= & T(a_{n+2})^{-1} \cdots T(a_{\frac{\tau-3}{2}})^{-1} J T(a_{\frac{\tau-3}{2}})
\cdots T(a_{n+2}) \\
&= & (T(a_{\frac{\tau-3}{2}}) \cdots T(a_{n+2}))^{-1} J (T(a_{\frac{\tau-3}{2}})
\cdots T(a_{n+2}) ) ~~. \\
\end{array}
\end{equation}
It follows that the matrix $D_n$ has the same characteristic polynomial
$z^3+z^2-z-1$ of $J $, i.e. $D_n$ has
eigenvalue $1$ with multiplicity $1$, and eigenvalue $-1$ with geometric multiplicity $2$.
\begin{theorem}
\label{uniquqf}
The correspondence $m \leftrightarrow \mathbf \Lambda_m$ is one-to-one for $1 \leq m \leq \tau$,
i.e. all quadratic forms $\mathbf f_m(x,y)$ within a period are distinct.
\end{theorem}
\begin{proof}
The proof is by contradiction.
Suppose, contrary to the theorem's claim, that
$\mathbf \Lambda_{n_1}=\mathbf \Lambda_{n_2}=\mathbf X$ for some $n_1 < n_2$, then
equation (\ref{recurn3}) implies the existence of a matrix $P_{n_2n_1}= \prod_{j=n_1+1}^{n_2}T(a_j)$
such that $\mathbf \Lambda_{n_2}=P_{n_2n_1}\mathbf \Lambda_{n_1}$. Thus $\mathbf X$ must be
an eigenvector, for the eigenvalue $1$, of the non-negative (positive whenever $n_2-n_1 \geq 2$)
matrix $P_{n_2n_1}$ which is the product of non-negative matrices. \\
If $n_2 =n_1+1$, it is direct to compute the characteristic polynomial $p(x)$ of $P_{n_2n_1}=T(a_{n_1})$
$$ p(x) = x^3-(a_{n_1}^2+1) x^2-(a_{n_1}^2+1) x+1 ~~, $$
which is a $3$-degree reciprocal polynomial which has a single root $-1$, and the remaining roots
are certainly different from $1$, because $a_{n_1} \neq 0$; thus, in this case, $\mathbf X$ cannot exist.\\
To prove in general that $\mathbf X$ does not exist, we observe that any $P_{n_2n_1}$ has a reciprocal
characeristic polynomial $q(x)$ of degree $3$, because we have
$$ q(x)= \det\left(\lambda I_3 -\prod_{j=n_1+1}^{n_2}T(a_j) \right) = \det\left(\lambda I_3 -J\prod_{j=n_1+1}^{n_2}T(a_j)J \right)= \det\left (\lambda I_3 -\prod_{j=n_1+1}^{n_2}T(a_j)^{-1} \right) ~~,$$
$$ q(x)= \det\left(\lambda I_3 -\prod_{j=n_1+1}^{n_2}T(a_j) \right) = \det\left(\lambda I_3 -\left(\prod_{j=n_1+1}^{n_2}T(a_j) \right)^{-1} \right) ~~,$$
where the last equality is justified by \cite[Theorem 1.3.20, p.53]{horn}.
The reciprocal polynomial $q(x)$ has an eigenvalue equal to either $-1$ or $1$. If the eigenvalue is $-1$, which occurs when $n_2-n_1$ is odd,
the eigenvector $\mathbf X$ does not exist.
If the eigenvalue is $1$, which occurs when $n_2-n_1$ is even, there is a second eigenvector for the same eigenvalue, because we have
$$ J\mathbf \Lambda_{n_2}=JP_{n_2n_1}\mathbf \Lambda_{n_1} = JP_{n_2n_1} J \cdot J \mathbf \Lambda_{n_1} = \left(\prod_{j=n_1+1}^{n_2}T(a_j)\right)^{-1}J \mathbf \Lambda_{n_1}
~~\Rightarrow~~ \left( \prod_{j=n_1+1}^{n_2}T(a_j) \right) J\mathbf \Lambda_{n_2}= J\mathbf \Lambda_{n_2}. $$
Then, $\mathbf X$ and $J\mathbf X$ should be distinct eigenvectors (because $\Omega_{n_2} \neq 0$ for every $n_2$) of the same eigenvalue $1$ of multiplicity one, which is impossible. \\
|
In conclusion, the eigenvector $\mathbf X$ of eigenvalue $1$ does not exist, so
$m \leftrightarrow \mathbf \Lambda_m^T$ is a one-to-one mapping within each period.
\end {proof}
\subsection{Odd period}
Legendre describes in \cite[p.59-60]{legendre} a constructive method for computing the representation of a positive
(square-free) $N$ as the sum of two squares by means of the continued fraction expansion of $\sqrt N$.
This result is stated as a theorem with a different proof from that of Legendre \cite[p.60]{legendre}.
\begin{theorem}
\label{serret}
Let $N$ be a positive integer such that the continued fraction expansion of $\sqrt N$ has odd period $\tau$.
The representation of $N=x^2+y^2$ is given by $x= \Delta_{\frac{\tau-1}{2}}$ and
$y=\Omega_{\frac{\tau-1}{2}}$.
\end{theorem}
\noindent
{\sc Proof}.
Since $\tau$ is odd, by the anti-symmetry in the sequence $\{ \Delta_n \}_{n=0}^{\tau-2}$, we have
$\Delta_{\frac{\tau-1}{2}-1}=-\Delta_{\frac{\tau-1}{2}}$,
so that the quadratic form $\Delta_{\frac{\tau-1}{2}} X^2+2\Omega_{\frac{\tau-1}{2}} XY+
\Delta_{\frac{\tau-3}{2}}Y^2$ has discriminant
$4 \Delta_{\frac{\tau-1}{2}}^2+4\Omega_{\frac{\tau-1}{2}}^2=4N$, which shows the assertion.
\begin{flushright} $\Box$ \end{flushright}
\subsection{Even period}
\label{subsect2}
Let $N$ be a square-free composite integer such that the continued fraction of $\sqrt{N}$ has even period.
We say that $\mathfrak c_{\tau-1}=A_{\tau-1}+B_{\tau-1}\sqrt{N} $ splits $N$ whenever $A_{\tau-1}+1$ and $A_{\tau-1}-1$ are divisible
by proper factors, say $m_1$ and $m_2$, of $N=m_1m_2$, respectively.
\begin{lemma}
\label{sym3a}
If the period $\tau$ of the continued fraction expansion of $\sqrt{N}$ is even, we have
$$ \Delta_{\tau} = \Delta_{\tau-2} ~~~~\mbox{and}~~~~ \Omega_{\tau}=- \Omega_{\tau-1} $$
with $\Omega_{\tau-1}=-a_0$.
\end{lemma}
\begin{proof}
Since $\Delta_{\tau-1}=1$, we have $\Omega_{\tau-1}^2-\Delta_{\tau-2}=N$, thus
$\Omega_{\tau-1}=-\sqrt{N+\Delta_{\tau-2}}$ ~because $\tau-1$ is odd. Considering
the Taylor series around the origin for the square root, we have
$$ \Omega_{\tau-1}=-\sqrt{N+\Delta_{\tau-2}} = -\sqrt N \left( 1 - \frac{\Delta_{\tau-2}}{2N} + \frac{\Delta_{\tau-2}^2}{8N^2} + \cdots \right)
=-\left\lfloor \sqrt{N} \right\rfloor =-a_0 ~~. $$
Using equation (\ref{Deltarecur}) with $m=\tau-1$ we have
$$ \Delta_{\tau} = \Delta_{\tau-2} +a_{\tau} \left( 2 \Omega_{\tau-1} + a_{\tau}\Delta_{\tau-1} \right) =
\Delta_{\tau-2} ~~. $$
Thus, equation (\ref{Deltarecur1}) finally gives
$ \Omega_{\tau}=- \Omega_{\tau-1}$.
\end{proof}
\begin{lemma}
\label{sym3}
Let $\tau$ be even, and define the integer $\gamma \in \mathfrak O_{\mathbb F}$ by the product
$$ \gamma= \prod_{m=1}^\tau \left(\sqrt N +(-1)^{m} \Omega_m \right) ~~,$$
then $ \frac{\gamma}{\sigma(\gamma)}=\mathfrak c_{\tau-1}=A_{\tau-1}+B_{\tau-1}\sqrt{N}$.
\end{lemma}
\begin{proof}
The norm of $\frac{\gamma}{\sigma(\gamma)}$ is patently $1$, then it remains to prove
that $ \frac{\gamma}{\sigma(\gamma)} \in \mathfrak O_{\mathbb F}$, which in turn is true if
$\mbox{Tr} \left(\frac{\gamma}{\sigma(\gamma)} \right) \in\mathfrak O_{\mathbb F}$.
We have
\begin{equation}
\label{unit3}
\mbox{Tr}\left(\frac{\gamma}{\sigma(\gamma)} \right) = \frac{\gamma}{\sigma(\gamma)} +\frac{\sigma(\gamma)}{\gamma} = \frac{\gamma^2+\sigma(\gamma)^2}{\gamma\sigma(\gamma)} = \frac{[\gamma+\sigma(\gamma)]^2}{\gamma\sigma(\gamma)} -2 ~~.
\end{equation}
Observing that $\gamma\sigma(\gamma)= \prod_{m=1}^\tau (\Omega_m^2-N)=\prod_{m=1}^\tau (\Delta_m \Delta_{m+1}) = \prod_{m=1}^\tau \Delta_m^2$ by the periodicity of the sequence
$\{ \Delta_m \}_{m \geq 1}$, it follows that $\gamma\sigma(\gamma)$ is a perfect square, then equation (\ref{unit3}) can be written as
$$ \mbox{Tr}\left(\frac{\gamma}{\sigma(\gamma)} \right)=\left(\frac{\gamma+\sigma(\gamma)}{ \prod_{m=1}^\tau (-1)^m \Delta_m}\right)^2 -2 ~~,$$
where the factor $(-1)^m$ has been introduced at the denominator in order to have all positive factors.
This last equation shows that the trace is certainly integral if
$$ \frac{\gamma}{ \prod_{m=1}^\tau (-1)^m \Delta_m} = \prod_{m=1}^\tau \left(\frac{\sqrt N +(-1)^{m} \Omega_m}{ (-1)^m \Delta_m}\right) $$
is integral, a property that is implied by an argument of Cohen's \cite[Proposition 5.7.3, p.267]{cohen},
or using the following identity
$$ \frac{\sqrt N +(-1)^{m}\Omega_m}{ (-1)^m \Delta_m}=\frac{A_{m-1}-B_{m-1} \sqrt N}{A_m-B_m \sqrt N} ~,$$
which is obtained by rationalizing its right-side member, and taking into account the definitions of $\Delta_m$, $\Omega_m$,
and the fundamental property (\ref{fund}). Thus we can write
\begin{equation}
\label{unitnorm}
\frac{\gamma}{ \prod_{m=0}^{\tau-1} (-1)^{m-1} \Delta_m} = \prod_{m=0}^{\tau-1} \frac{A_{m-1}-B_{m-1} \sqrt N}{A_m-B_m \sqrt N} =
\frac{A_{-1}-B_{-1} \sqrt N}{A_{\tau-1}-B_{\tau-1} \sqrt N} =A_{\tau-1}+B_{\tau-1} \sqrt N ~~.
\end{equation}
which shows the claimed property.
\end{proof}
\noindent
The strict connection between the continued fraction expansion of $\sqrt{N}$ and the factorization
of $N$ is proved using the matrix $M_{\tau-1}$ defined in equation (\ref{keymatrix}).
Note that the matrix $M_{\tau-1}$ is involutory, or neg-involutory, since its square
is either plus or minus the identity matrix $I_2$, i.e. $M_{\tau-1}^2=(-1)^{\tau} I_2$.
If $\tau$ is even, the eigenvalues of matrix $M_{\tau-1}$ are $\pm 1$, and $M_{\tau-1}$
is involutory.
If $\tau$ is odd, the eigenvalues are $\pm i$, and $M_{\tau-1}$ is neg-involutory.
\begin{theorem}
\label{locfactor}
If the period $\tau$ of the continued fraction expansion of $\sqrt{N}$ is even, the element
$\mathfrak c_{\tau-1}$ in $\mathbb Q(\sqrt N)$ splits $4N$, and a factor of $N$ is located at
positions $\frac{\tau-2}{2}+j\tau$, $j=0,1, \ldots$, in the sequence
$\mathbf \Delta=\{\mathfrak c_{m} \sigma(\mathfrak c_{m}) \}_{m \geq 1}$.
\end{theorem}
\noindent
\begin{proof}
It is sufficient to consider $j=0$ due to the periodicity of $\mathbf \Delta$.
Since $\tau$ is even, $M_{\tau-1} $ is involutory and has eigenvalues $\pm 1$ with
corresponding eigenvectors
$$ \mathbf X^{(h)}= \left[ \frac{A_{\tau-1}-(-1)^{h}}{d}, \frac{B_{\tau-1}}{d} \right]^T ~~~~\mbox{with}~~~~
d=\gcd \{A_{\tau-1}-(-1)^{h}, B_{\tau-1}\} ~~~~h=0,1 ~~. $$
Considering equation (\ref{fact4}) written as
$$ \left[ \begin{array}{c} A_{\tau-j-2} \\ B_{\tau-j-2} \end{array} \right] =
(-1)^{j-1} M_{\tau-1}
\left[ \begin{array}{c} A_{j} \\ B_{j} \end{array} \right] ~~, $$
we see that $\mathbf Y^{(j-1)}=[ A_{j} , B_{j} ]^T$ is an eigenvector of $M_{\tau-1}$, of
eigenvalue $(-1)^{j-1}$ if and only if
$j$ satisfies the condition $\tau-j-2=j$, that is $j=\frac{\tau-2}{2}=\tau_0$.
From the comparison of $\mathbf X^{(h)}$ and $\mathbf Y^{(j-1)}$, we have
\begin{equation}
\label{keycond}
A_{\tau_0} = \frac{A_{\tau-1}-(-1)^{\tau_0-1}}{d} \hspace{10mm}
B_{\tau_0} = \frac{B_{\tau-1}}{d} ~~,
\end{equation}
where the equalities are fully motivated because $\gcd \{A_{\tau_0},~ B_{\tau_0}\}=1$. A direct computation yields
\begin{equation}
\label{keyfact}
\Delta_{\tau_0} = \frac{(A_{\tau-1}-(-1)^{\tau_0-1})^2- N B_{\tau-1}^2}{d^2} =
2\frac{(-1)^{\tau_0-1}A_{\tau-1}-1}{d^2} ~~,
\end{equation}
which can be written as $A_{\tau_0}^2-N B_{\tau_0}^2 = 2(-1)^{\tau_0-1} \frac{A_{\tau_0}}{d}$; dividing this equality by
$ 2 \frac{A_{\tau_0}}{d}$ we have
$$ \frac{dA_{\tau_0}}{2} -N \frac{1}{\frac{2A_{\tau_0}}{d}} B_{\tau_0}^2 = (-1)^{\tau_0-1} ~~.$$
Noting that $\gcd\{A_{\tau_0},~B_{\tau_0} \}=1$, it follows that $\frac{2A_{\tau_0}}{d}$ is certainly a divisor of $4N$, i.e. $ \Delta_{\tau_0} |4N$.
\end{proof}
\begin{example}
Consider $N=3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 =21945$ ,
the period of the continued fraction of $\sqrt{21945}$ is $10$, and is fully shown in the
following table for the sequences $\mathbf \Delta$ and $\mathbf \Omega$
\begin{center}
\begin{tabular}{|c|c|r|} \hline
$j$ & $\Delta_j$ & $\Omega_j$ \\ \hline
$-1$ & 1 & \\
0 & -41 & 148 \\
1 & 64 & -139 \\
2 & -129 & 117 \\
3 & 16 & .141 \\ \hline
4 & -21 & 147 \\ \hline
5 & 16 & .147 \\
6 & -129 & 141 \\
7 & 64 & -117 \\
8 & -41 & 139 \\ \hline
9 & 1 & -148 \\
10 & -41 & 148 \\ \hline
\end{tabular}
\end{center}
In position $j=\frac{\tau-2 }{2}=4$ we find $21$, a factor of $N$, as expected.
The same factor $21$ can be found by considering the fundamental unit
$ \mathfrak c_9 = 3004586089+20282284 \sqrt{21945}$,
in fact we have $3004586089-1 =2^3\cdot (3\cdot 7)\cdot 4229^2$, and the second factor
$5\cdot 11\cdot 19$ may be obtained from $3004586089+1=2\cdot (5\cdot 11^3\cdot 19) \cdot109^2$.
\end{example}
\section{Factorization}
Gauss recognized that the factoring problem was important, although very difficult,
\begin{quotation}
\noindent
{\em $\ldots$ Problema, numeros primos a compositis dignoscendi, hosque in factores
suos primos resolvendi, ad gravissima ac utilissima totius arithmeticae pertinere, et
geometrarum tum veterum tum recentiorum industriam ac sagacitatem occupavisse,
tam notum est, ut de hac re copiose loqui superfluum foret. $\ldots$ }
\hfill {\scriptsize \sc C. F. Gauss [{\em Disquisitiones Arithmeticae} Art. 329]}
\end{quotation}
and, in spite of much efforts, various different approaches, and the increased importance due to the large
number of cryptographic applications, no satisfactorily factoring method has yet been found.
\noindent
Many factorizations make use of the regular continued fraction expansion of
$\sqrt N$, combined with the idea of using quadratic forms
\cite{gauss,riesel}. The infrastructure method, proposed by Shanks \cite{shanks}, considers the subset
$\mathbf \Psi =\{\mathbf f_m(x,y) \}_{ 1 \leq m \leq \tau-1}$ in the periodic sequence
$\Upsilon =\left\{ \mathbf f_m(x,y) \right\}_{m \geq 1}^\infty$ of reduced principal quadratic forms.
It should be remarked that the forms
$\mathbf f_m(x,y)=\Delta_mx^2+2\Omega_m x y + \Delta_{m-1}y^2$ in $\Upsilon$
are reduced following a different convention from that commonly adopted \cite{buell}.
\begin{definition}
A real quadratic form $\mathbf f(x,y)=ax^2 +2bx y +cy^2$ of discriminant $4N$ is said to be reduced
if, defining $\kappa=\min \{ |a|, |c| \}$, $b$ is the sole integer such that
$\sqrt N- |b|<\kappa <\sqrt N+|b| $, with the sign of $b$ chosen opposite to the sign of $a$.
\end{definition}
\begin{definition}
The distance between $\mathbf f_{m+1}(x,y)$ and $\mathbf f_{m}(x,y)$ is defined to be
\begin{equation}
\label{defdist}
d(\mathbf f_{m+1}, \mathbf f_{m}) =\frac{1}{2} \ln \left( \frac{\sqrt{N}+(.1)^m\Omega_m }{\sqrt{N}-(.1)^m\Omega_m} \right) ~~.
\end{equation}
The distance between two quadratic forms $\mathbf f_{m}(x,y)$ and $\mathbf f_n(x,y)$,
with $m > n$, is defined to be the sum
\begin{equation}
\label{defdist1}
d(\mathbf f_{m}, \mathbf f_{n}) = \sum_{j=n}^{m-1} d(\mathbf f_{j+1}, \mathbf f_{j}) ~~.
\end{equation}
\end{definition}
\noindent
Taking the above definitions, Shanks showed
that, by the Gauss composition law of quadratic forms with the same determinant, followed by reduction,
the set $\mathbf \Psi$ equipped with the distance
$d(\mathbf f_{m+1}, \mathbf f_{m})$ modulo $R=\ln \mathfrak c_{\tau-1}$
resembles a cyclic group, with $\mathfrak c_{\tau-1}$ playing the role of identity.
Composition, followed by reduction, permits big steps within $\mathbf \Psi$, thus
two operators were further defined \cite[p.259]{cohen} to allow small steps, precisely
\begin{enumerate}
\item One-step forward:
The operator $\rho^+$ that transforms one reduced quadratic form into the next in the sequence
$\mathbf \Upsilon$, is defined as
$$ \rho^+([a,2b,c]) = [\frac{b_1^2-N}{a},2b_1, a] ~~, $$
where $b_1$ is $2b_1= [2b \bmod (2a)] +2ka$ with $k$ chosen in such a way
that $-|a| < b_1 < |a|$.
\item One-step backward:
The operator $\rho^-$ that transforms a reduced quadratic form into the immediately preceding quadratic form
in the sequence $\mathbf \Upsilon$ is defined as
$$ \rho^-([a,2b,c]) = [c,2b_1, \frac{b_1^2-N}{c}] ~~, $$
where $b_1$ is $2b_1=[ -2b \bmod (2c)] +2kc$ with $k$ chosen such that $-|c| < b_1 < |c|$.
\end{enumerate}
\noindent
The infrastructure machinery has been used to compute the fundamental unit, the regulator, and
the class number, with complexity smaller than $O(\sqrt N)$, although not of polynomial complexity
in $N$.
From a different perspective, by Theorem \ref{locfactor}, in many cases a factor of $N$ is exactly
positioned in the middle of a period of the sequence $\mathbf \Delta$. Therefore, instead of trying
to find special quadratic forms randomly located in $\mathbf \Psi$ (the principal genus),
or some ambigue form in some non-principal
genus, we may try to localize the position of some factor of $N$ within a period whose length $\tau$
is unknown.
Then, it is shown that, by extending the infrastructure machinery to the whole sequence $\Upsilon$,
some factors of $N$ can be computed with a complexity substantially bounded by the complexity
required to evaluate an integral of Dirichlet's at a given accuracy: the more precise the evaluation
of the integral, the less complex the factorization; at the limit, it is of polynomial complexity;
clearly, to be more accurate in the integral evaluation, greater complexity is required.
To pursue this idea, we quickly review and adapt the previous definitions of the infrastructure
components to the new task.
Let us recall that the quadratic forms $\mathbf f_{m}(x,y)$ are primitive,
i.e. $\gcd \{\Delta_m, 2\Omega_m , \Delta_{m-1} \}=1$, and at least one between $|\Delta_m|$
and $|\Delta_{m-1}|$ is less than $\sqrt N$ and $0 < |\Omega_m| < \sqrt N$. Further, since
$\mathfrak c_{\tau-1}$ is either equal to the positive fundamental unit of
$\mathbb F=\mathbb Q(\sqrt N)$ or equal to its cube,
the regulator of $\mathfrak O_{\mathbb F}$ is either $R_{\mathbb F} = \ln\mathfrak c_{\tau-1}$,
or $R_{\mathbb F} = \frac{1}{3} \ln\mathfrak c_{\tau-1}$.
The following observations are instrumental to motivate the procedure:
\begin{enumerate}
\item The sign of $\Delta_{m-1}$ is the same as that of $\Omega_m$, which is opposite to that of $\Delta_m$,
thus in the sequence $\mathbf{\Upsilon}$ the two triples of signs $(-,+,+)$ and $(+,-,-)$ alternate.
\item The distance of $\mathbf f_{m}(x,y)$ from the beginning of $\mathbf \Upsilon$ is defined by
referring to a hypothetical quadratic form properly selected, i.e.
$ \mathbf f_{0}(x,y)= \mathbf f_{\tau}(x,y)= \mathbf f_{0}(x,y)= \Delta_0 x^2-2 \sqrt{N-\Delta_0} x y+y^2$, which is located before
$ \mathbf f_{1}(x,y)$, that is
$d(\mathbf f_{m}, \mathbf f_{0})$ is given by (\ref{defdist1}) if $m \leq \tau$, and by
$d(\mathbf f_{m}, \mathbf f_{0}) =d(\mathbf f_{m \bmod \tau}, \mathbf f_{0}) +kR_{\mathbb F}$ if $k \tau \leq m < (k+1) \tau$.
\item Let "$\bullet$" denote the form composition $\mathbf f_{m}(x,y) \bullet \mathbf f_{n}(x,y) $ in $\mathbf \Upsilon$,
that is the Gauss composition \cite{cohen} of $\mathbf f_{m}(x,y)$ and $\mathbf f_{n}(x,y)$
followed by a reduction performed with the minimum number of steps, ending with a reduced form whose
triple of signs is $(-,+,+)$ if $m$ and $n$ have the same parity, and $(+,-,-)$ otherwise.
This distance defined by (\ref{defdist}) holds in $\mathbf \Upsilon$ with good approximation, and is
compatible with the "$\bullet$" operation, that is we have
$$ \mathbf f_{m,n}(x,y) = \mathbf f_{m}(x,y) \bullet \mathbf f_{n}(x,y) \Rightarrow
d(\mathbf f_{m,n}, \mathbf f_{0}) \approx d(\mathbf f_{m}, \mathbf f_{0})+ d(\mathbf f_{n}, \mathbf f_{0}) ~~.$$
It is remarked that the error affecting this distance estimation is of order $O(\ln N)$
as shown by Schoof in \cite{schoof0}.
\item Shanks \cite{shanks} observed that, within the first period, the composition law "$\bullet$"
induces a structure similar to a cyclic group for the addition of distances modulo the regulator.
\item Between the elements of $\mathbf{\Upsilon}$ the distance is nearly maintained by the giant-steps,
and is rigorously maintained by the baby-steps.
\end{enumerate}
\begin{theorem}
\label{mainreg}
The distance $d(\mathbf f_{\tau}, \mathbf f_{0} )$ is exactly equal to
$\ln \mathfrak c_{\tau-1}$, i.e. this distance $d(\mathbf f_{\tau}, \mathbf f_{0} )$ is either the regulator $R_{\mathbb F}$ or $3R_{\mathbb F}$.
\end{theorem}
\begin{proof}
The distance between $\mathbf f_{\tau}$ and $\mathbf f_{0}$ is the summation
$$ d(\mathbf f_{\tau}, \mathbf f_{0}) = \sum_{j=0}^{\tau-1} d(\mathbf f_{j+1}, \mathbf f_{j}) =
\sum_{j=0}^{\tau-1} \frac{1}{2} \ln \left( \sum_{j=0}^{\tau-1} \frac{\sqrt{N}+(-1)^{j}\Omega_j }{\sqrt{N}-(-1)^j\Omega_j} \right) =
\frac{1}{2} \ln \left( \prod_{j=0}^{\tau-1} \frac{\sqrt{N}+(-1)^{j}\Omega_j }{\sqrt{N}-(-1)^j\Omega_j} \right)
~~. $$
Recalling that $N-\Omega_j^2=-\Delta_j \Delta_{j-1} >0$, and taking into account the periodicity of the sequence
$\mathbf \Delta$, the last expression can be written with rational denominator as
$$ \frac{1}{2} \ln \left( \prod_{j=0}^{\tau-1} \frac{(\sqrt{N}+(-1)^j\Omega_j )^2}{-\Delta_j \Delta_{j-1}} \right)
= \frac{1}{2} \ln \left( \prod_{j=0}^{\tau-1} \frac{(\sqrt{N}+(-1)^j\Omega_j )^2}{\Delta_j^2} \right) =
\ln \left( \prod_{j=0}^{\tau-1} \frac{\sqrt{N}+(-1)^j\Omega_j}{(-1)^{j-1}\Delta_j} \right) ~~.$$
The conclusion follows from Lemma \ref{sym3},
showing that the product
$\prod_{j=0}^{\tau-1} \frac{\sqrt{N}+(-1)^j \Omega_j}{(-1)^{j-1}\Delta_j}$, which has field norm one
and is an element
of the order $\mathfrak O_{\mathbb F}$, is actually the unit $\mathfrak c_{\tau-1}$ by equation (\ref{unitnorm}). The connection between $\ln \mathfrak c_{\tau-1}$ and the regulator is
motivated by Remark \ref{remark1}.
\end{proof}
\noindent
Since Theorem \ref{locfactor} guarantees that, when $\tau$ is even, a factor of $N$ is located in the positions
$\frac{\tau-2}{2}+k\tau$ of the sequence $\Upsilon$, Shanks method
allows us to find such a factor, if $\ln(\mathfrak c_{\tau-1})$, or an odd multiple of it, is
exactly known. Now, a formula of Dirichlet's gives the product
\begin{equation}
\label{dirichlet}
h_{\mathbb F} R_{\mathbb F} = \frac{\sqrt{D}}{2} L(1, \chi) =
- \sum_{n=1}^{\lfloor \frac{D-1}{2} \rfloor} \jacobi{D}{n} \ln\left( \sin \frac{n \pi}{D} \right)
\end{equation}
where $R_{\mathbb F}$ is the regulator, $ L(1, \chi)$ is a Dedekind $L$-function, $D=N$ if $N\equiv 1 \bmod 4$ or
$D=4N$ otherwise, and character $\chi$ is the Jacobi symbol in this case.
If the product $h_{\mathbb F} R_{\mathbb F}$ is known exactly (computed), for example using equation (\ref{dirichlet}),
the distance from the beginning of the sequence where the quadratic form can be found
$[1, 2\Omega_{\tau-1}, \Delta_{\tau-2}]$ is known.
Since this distance is an integer multiple of the regulator, and our target
is to find a quadratic form that is located in the middle of some period, then
\begin{enumerate}
\item if $h_{\mathbb F}$ is odd, a factor of $N$ is found in the position at distance
$\frac{h_{\mathbb F} R_{\mathbb F}}{2}$, or $3\frac{h_{\mathbb F} R_{\mathbb F}}{2}$, from the beginning;
\item If $h_{\mathbb F}$ is even, in a position at distance $\frac{h_{\mathbb F} R_{\mathbb F}}{2}$, or
$3\frac{h_{\mathbb F} R_{\mathbb F}}{2}$ the quadratic form $[1, 2\Omega_{\tau-1}, \Delta_{\tau-2}]$
is found, (which reveals a posteriori that $h_{\mathbb F}$ is even);
in this case, the procedure can be repeated with target the position at distance
$\frac{h_{\mathbb F} R_{\mathbb F}}{4}$, or $3\frac{h_{\mathbb F} R_{\mathbb F}}{4}$, again, either a factor of $N$
is found or $h_{\mathbb F}$ is found to be a multiple of $4$.
Clearly the process can be iterated $\ell$ times until
$\frac{h_{\mathbb F} R_{\mathbb F}}{2^\ell}$ is an odd multiple of $R_{\mathbb F}$, and a factor of $N$ is found.
\end{enumerate}
\noindent
When the factor $m_1$ of $N$ is found, the second factor is
$m_2=\frac{N}{m_1}$, thus the procedure can be iterated to find all factors of $N$.
Mimicking Shank's infrastructure, giant-steps are performed to get close to
forms at distance $\frac{k R_{\mathbb F}}{2}$, or $3\frac{k R_{\mathbb F}}{2}$, for some $1 \leq k \leq h_{\mathbb F}$, then
baby-steps are performed to get the exact position.
\section{Conclusions}
It has been shown that the complexity of factoring a composite number $N$ is upper bounded
by the complexity of evaluating, at a certain degree of accuracy, the product
$h_{\mathbb F} R_{\mathbb F}$, as defined by Dirichlet using the $~L(1,\chi_N)$ function,
and also that is not necessary to know $h_{\mathbb F}$ and $R_{\mathbb F}$ separately.
The more precise the evaluation of the product $h_{\mathbb F} R_{\mathbb F}$, the less complex the
factoring $N$; at most the complexity could be polynomial in $N$.
Of the many expressions for the product $ h_{\mathbb F} R_{\mathbb F}$ as a function of $N$,
one is particularly convenient \cite[p.262]{cohen}
\begin{equation}
\label{eqhR}
h_{\mathbb F} R_{\mathbb F} = \frac{1}{2} \sum_{x \geq 1} \jacobi{N}{x} \left(\frac{\sqrt N}{x} \mbox{erfc} \left(x \sqrt{\frac{\pi}{N}} \right)+ E_1\left(\frac{\pi x^2}{N}\right)\right) ~~ ,
\end{equation}
where the complementary error function $\mbox{erfc}(x)$, and the exponential integral function $E_1(x)$,
can be closely approximated \cite[p.297-299]{hand}
$$ \mbox{erfc}(z) = \frac{2}{\sqrt \pi} \int_z^\infty e^{-t^2} dt = 1- \mbox{erf}(z) = 1- \frac{2}{\sqrt \pi} \sum_{n=0}^\infty \frac{(-1)^nz^{2n+1}}{n!(2n+1)} $$
$$ E_1(z)= \int_1^\infty \frac{e^{-t z}}{t} dt= -\gamma-ln(z) - \sum_{n=1}^\infty \frac{(-1)^n z^n}{n \cdot n!} ~~. $$
\noindent
Then, as a last observation,
according to \cite[Proposition 5.6.11, p.262-263]{cohen} the series in equation (\ref{eqhR}) converges exponentially.
A few numerical experiments, with small numbers $N <10^{10}$, in truth very small in comparison to
$10^{200}$, indicate that an approximation of the order $O((\ln N)^2)$ is achievable by taking
$O((\ln N)^2)$ terms in the series; it remains to prove that this balance
holds for every $N$. Then, arguably, a fast (possibly polynomial)
algorithm for factoring is achievable simply by combining results of Dirichlet, Shanks,
and the above observations, which were suggested by Legendre's finding that continued fractions
permit the representation of primes as the sum of two squares to be explicitly computed.
|
\section{Introduction}
Perturbative calculations in lattice gauge theories (for a review, see \cite{CapitaniPR}) are of interest from several points of view.
Firstly, they are needed to determine the $\Lambda_{LAT}$ parameter of QCD in the
lattice regularization and its relation to the respective value $\Lambda_{QCD}$ in the continuum theory.
Secondly, every lattice action defines a specific regularization scheme,
and thus one needs a complete set of renormalization computations
in order for the results obtained in Monte Carlo simulations
be understood properly. Perturbation theory is required to
establish the connection of the matrix elements computed on a lattice
with their values in the continuum theory \cite{Braun}, \cite{Hagler}.
In this connection, it should be emphasized that the use of one-loop perturbative renormalization constants gives rise
to large systematic uncertainties in lattice calculations of the momenta of hadronic structure functions
\cite{Hagler} and respective two-loop computations are needed.
Thirdly, perturbative calculations provide the only
possibility for an analytical control over the continuum limit in QCD.
One can also mention anomalies, proof of renormalizability,
Symanzik improvement program and other fields of application
of lattice perturbation theory.
Here we consider one- and two-loop diagrams with Wilson ($r=1$) fermions
at zero external momenta \cite{Kawai}.
We outline the Burgio-Caracciolo-Pelissetto (BCP) method \cite{Burgio} of calculations of
one-loop integrals and describe the respective computer algorithm \cite{Rogalyov}.
This algorithm allows to compute the fermionic propagator in the coordinate
representation and, therefore, to extend the L{\" u}scher-Weisz (LW) method \cite{Luscher}
to the fermionic case; such extension is presented in Section~4.
\subsection{Notation}
We use the following designations: $\tilde n$ stands for the set $n_1,n_2,n_3,n_4$;
$\ x=(x_1,x_2,x_3,x_4)$, where $x_\mu$ are
integer-valued coordinates of an infinite four-dimensional lattice
$\Lambda=\{x:\ x_\mu\in Z\!\!\!Z\}$; we also need the lattice $\Lambda' = \Lambda \backslash \{0\}$ with removed site x=(0,0,0,0);
\begin{equation}
|x| = |x_1| + |x_2| + |x_3| + |x_4|, \qquad \qquad \qquad [x^n] = x_1^n +x_2^n +x_3^n +x_4^n.
\end{equation}
Then we give the expressions for the denominators of bosonic and fermionic propagators,
\begin{eqnarray}
\Delta_B(k) &=& 4+\mu_R^2-\cos(k_1 )-\cos(k_2 )-\cos(k_3 )-\cos(k_4); \\ \nonumber
\Delta_F(k) &=& 10-4\;\sum_{\mu=1}^4 \cos(k_\mu) + \sum_{1\leq \mu < \nu \leq 4} \cos(k_\mu) \cos(k_\nu) + \mu_R^2 \nonumber
\end{eqnarray}
where $\mu_R$ is the fictitious mass for infrared regularization.
We also use $D_F=2\;\Delta_F$ and $D_B=2\;\Delta_B $ normalized in the standard way ($D_{B(F)(k)}\simeq 1/k^2$ as $k\to \infty$).
These propagators in the coordinate representation are defined as follows:
\begin{equation}
G_{B(F)}(x) = \int_{BZ} {dk \over (2\pi^4)} {e^{-ikx} \over D_{B(F)}(k) },
\end{equation}
where $BZ$ is the Brillouin zone, $\displaystyle BZ = \Big\{ p:\ -\;{\pi\over a}\leq p_\mu \leq {\pi\over a} \Big\}$;
\begin{equation}
\hat p_\mu = {2\over a} \sin \left({ p_\mu a \over q }\right), \qquad \qquad \hat p^2 = \sum_{mu=1}^4 \hat p_\mu^2;
\end{equation}
$a$ is the lattice size. In this work we set $a=1$ for the sake of simplicity.
\section{The Burgio--Caracciolo--Pelissetto method}
\subsection{Bosonic Intefgrals}
\noindent The integrals under study are defined as follows: $F(q;\tilde n) = \lim_{\delta \to 0} F_\delta(q;\tilde n)$, where
\begin{equation}\label{InitialBosonInt}
F_\delta(q;\tilde n)
= \int_{BZ} dk\;{\cos(k_1 )^{n_1} \cos(k_2 )^{n_2} \cos(k_3 )^{n_3} \cos(k_4 )^{n_4} \over \Delta_B^{q + \delta}}.
\end{equation}
Here $\delta$ is an infinitesimal parameter for an intermediate regularization \cite{Burgio}.
This parameter makes it possible to derive\footnote{Using integration by parts} the recursion relations of the form
\begin{eqnarray}\label{RRforFbosonic}
& & F(q;..., n_\mu,...) = F(q;..., n_\mu-2,...) \;-\; \\ \nonumber
&& - { (n_\mu-1) F(q-1;...,n_\mu-1,...) \over q-1+\delta} + { (n_\mu -2) F(q-1;...,n_\mu-3,...) \over q-1+\delta} \qquad (n_\mu \geq 2). \nonumber
\end{eqnarray}
With these relations and similar relations for $n_\mu \leq 1$, one can
express the integrals (\ref{InitialBosonInt}) in terms of the quantities
\begin{equation}
G_\delta (q,\mu_R^2)=\int_{BZ} {dk \over (2\pi)^4 } {1\over (\Delta_B)^{q+\delta}}.
\end{equation}
Up to terms of the order ${\cal O}(\mu_R^2)$ and ${\cal O}(\delta)$, this expression has the form
\begin{equation}\label{BosFtoGgen}
F_\delta (q,\tilde n) =
\sum_{r=q-n_1-n_2-n_3-n_4}^{0} A^{(-)}_{qr}(\delta,\tilde n) G_\delta (r,0) +
\sum_{r=1}^{q} A^{(+)}_{qr}(\mu_R^2,\tilde n) G_\delta(r,\mu_R^2),
\end{equation}
where $A^{(-)}_{qr}(\delta,\tilde n)$ have a pole singularity in $\delta$, and
$A^{(+)}_{qr}(\mu_R^2,\tilde n)$ are polynomials in $\mu_R^2$.
As for the function $G_\delta(r,\mu_R^2)$, the domains $r>0$ and $r \leq 0$
should be considered separately.
At $r>0$, $\ \delta$ can be safely set to zero
and the function $G_\delta(r,\mu_R^2)$ should be expanded in powers of $\mu_R^{-2}$:
\begin{equation} \label{G_FMRdef1}
G_\delta (r,\mu_R^2) = {1\over (2\pi)^2 \Gamma(r)} \left[ \; - \; b_{r-2} l_C + \sum_{k=1}^{r-2} { b_{r-k-2} \Gamma (k) \over (\mu_R^2)^k} \right] \; + \; J(r)\; + \; {\cal O}(\mu_R^2) + {\cal O}(\delta),
\end{equation}
where $b_n$ are the coefficients of the asymptotic expansion at $z\to \infty$ of the function\footnote{$I_0(z)$ is the Infeld function.}
\begin{equation}\label{InfeldAsExp0}
\exp(-4z) I_0^4(z) \simeq {1\over (2\pi z)^2}
\left(1+ {b_1\over z} + {b_2\over z^2} + ... \right),
\end{equation}
$l_C = \ln (\mu_R^2) + C$, and $C$ is the Euler-Mascheroni constant.
At $r<0$, $\mu_R$ can be safely set to zero and the function $G_\delta(r,0)$ should be expanded in $\delta$ as follows: $G_\delta(r,0)= B(r) + J(r)\delta + {\cal O}(\delta^2)$.
The functions $J(q)$, in their turn, obey recursion relations of the type
\begin{equation}\label{RRboson}
c_0(q) J(q) + c_1(q) J(q+1) + c_2(q) J(q+2) + c_3(q) J(q+3) + c_4(q) J(q+4) = 0
\end{equation}
derived in \cite{Burgio}; the explicit expressions for the coefficients $c_i(q)$
can be found in \cite{Rogalyov}. Thus we express $J(q)$ at $q\geq 4$ and at $q\leq 0$
in terms of $J(0), J(1), J(2)$ and $J(3)$.
It should be noted that $J(0)$ does not appear in ultimate expressions for the integrals (\ref{InitialBosonInt}).
Then one can introduce the values
\begin{equation}
Z_0 = {J(1)\over 2}, \qquad F_0 = 4\pi^2 J(2), \qquad Z_1 = 32 J(3) - 8 J(2) + {13 \over 6 \pi^2} + {1\over 4},
\end{equation}
which are equal to \cite{CapitaniPR} $ Z_0\approx0.15493339023, Z_1\approx0.10778131354, F_0\approx4.369225233874758$.
\subsection{Fermion Integrals}
In the fermionic case, we consider the quantities
$F(p,q;\tilde n) = \lim_{\delta \to 0} F_\delta(p,q;\tilde n)$, where
\begin{equation}
F_\delta(p,q;\tilde n) = \lim_{\delta \to 0} \int {d^4k\over (2\pi)^4}
{\cos^{n_1}(k_1) \cos^{n_2}(k_2) \cos^{n_3}(k_3) \cos^{n_4}(k_4) \over \Delta_B^q
\Delta_F^{p+\delta}}.
\end{equation}
With the recursion relations similar to (\ref{RRforFbosonic}),
these integrals are expressed in terms of the functions $\displaystyle G_\delta(p,q) = \int {d^4k\over (2\pi)^4} {1 \over \Delta_B^q \Delta_F^{p+\delta}},$
which can be represented in the form
\begin{eqnarray}\label{GdeltaExpansion}
G_\delta(p,q)&=& D(p,q;\mu_R^2) + B(p,q) + \delta \;(L(p,q;\mu_R^2)+J(p,q)) + O(\delta^2) , \qquad p\leq 0; \\ \nonumber
G_\delta(p,q)&=& D(p,q;\mu_R^2) + J(p,q) + O(\delta), \qquad p > 0. \nonumber
\end{eqnarray}
The divergent parts $D(p,q;\mu_R^2)$ and $L(p,q;\mu_R^2)$ in the domain of interest can be determined by
a straightforward procedure \cite{Burgio}, whereas the functions
$B(p,q)$ and $J(p,q)$ obey recursion relations of several types.
These relations and the procedure of their derivation were described in \cite{Burgio};
their explicit form (very cumbersome) is given in \cite{Rogalyov}.
With the use of these relations, the functions $F(p,q;\tilde n)$
can be represented (see \cite{CapitaniPR}, \cite{Burgio})
as linear combinations of the constants $F_0$, $Z_0$, $Z_1$ and
\begin{eqnarray}\label{Ydef2}
Y_0 &=& {J(2,0)\over 4}\; - \; {F_0 \over 16\pi^2}, \qquad Y_1 = {1 \over 48} - {1 \over 4}\; Z_0 - {1 \over 24}\; J(-1,2) +
{1 \over 12}\; J(0,1) + {1 \over 12}\; J(1,0), \\ \nonumber
Y_2 &=& { 1 \over 6 } - {1\over \pi^2} - Z_0 - { 1 \over 6 }\; J(-1,2) +
{1 \over 3}\; J(0,1) -{ 1 \over 24} \; J(1,-2) - {1 \over 12} \; J(1,-1) - \\ \nonumber
&& -
{17 \over 8}\; J(1,0) + 4\; J(1,1) - {1 \over 48}\; J(2,-2) + {25 \over 6}\; J(2,-1)
- 4\; J(2,0), \\ \nonumber
Y_3 &=& - {1\over 384\pi^2} - F_0\; {1\over 128\pi^2} + {1 \over 96}\; Z_0 -
{1 \over 48}\; J(-1,3) + {1 \over 192} \; J(0,1) + {1 \over 48}\; J(0,2) + {1 \over 48}\; J(1,1);
\end{eqnarray}
\begin{eqnarray}\label{Ydef1}
&& Y_4={J(1,0)\over 2}, \qquad Y_5 = J(1,-1), \qquad Y_6 = 2 J(1,-2), \qquad Y_7={J(2,-1)\over 2}, \\ \nonumber
&& Y_8 = J(2,-2), \qquad Y_9 = {J(3,-2)\over 2}, \qquad Y_{10} = J(3,-3), \qquad Y_{11}= 2 J(3,-4). \nonumber
\end{eqnarray}
The respective codes can be found on the web page of the ITEP Lattice group \\
{ \tt http://www.lattice.itep.ru/$\sim$pbaivid/lattpt/}. \
The results stored there are as follows:
(i) the program for a computation of $F(p,q;\tilde n)$ at $0\leq p,q \leq 9$ and $n_1+n_2+n_3+n_4
|
\leq 25$;
(ii) the values of the functions $J(p,q)$ and $B(p,q)$ at $-26 \leq p \leq 0,\ \ -56 - 2p \leq q \leq 34 $
and the values of $J(p,q)$ at $1\leq p \leq 9, \ \ -28 \leq q \leq 33 - p$; and
(iii) The explicit expressions for $F(p,q;\tilde n)$
at some particular values of $p$ and $q$ and all $n_1\leq 6$.
\section{The L{\" u}scher--Weisz method}
\noindent To outline the LW method \cite{Luscher}
of computation of two-loop diagrams in the coordinate representation,
we consider the diagram in~Fig.1, given by the expression
\hspace*{-2mm}\begin{minipage}{0.4\hsize}
\begin{picture}(170,130)(0,0)
\Text(-6,96)[tr]{\large $x$}
\Text(172,96)[tr]{\large $0$}
\ArrowArcn(80,0)(113,135,45)
\ArrowArc(80,160)(113,225,315)
\ArrowLine(0,80)(160,80)
\end{picture}
\vspace*{-9mm}
\begin{center}
Figure 1
\end{center}
\end{minipage}\hfill\begin{minipage}{0.52\hsize}
\begin{equation}\label{DiagSunSet00}
A_B(p) = \sum_{x\in\Lambda} e^{-ipx} G_B^3(x).
\end{equation}
In the bosonic case, L{\" u}scher and Weisz
calculated $A_B(0)$ and its asymptotic expansion when $p \to 0$;
they used the following representation:
\begin{eqnarray}\label{DivLambdaIn2Domains}
A_B(0) &=& G_B^3(0) + \sum_{x\in \Lambda'} G_{as}^3(x) \\ \nonumber
&& + \sum_{x\in\{{\cal F}_{N}\}} \Big( G_B^3(x) - G_{as}^3(x)\Big) \\ \nonumber
&& + \sum_{ x\in\{\Lambda' \backslash {\cal F}_{N}\}} \Big( G_B^3(x) - G_{as}^3(x)\Big), \nonumber
\end{eqnarray}
\end{minipage}
\noindent where
${\cal F}_{N} =\{ x: |x_1| + |x_2| + |x_3| + |x_4| \leq N \} $,
and $G_{as}(x)$ is an asymptotic approximation of $\displaystyle G_B(x)$ when $x\to \infty$,
\begin{equation}\label{DBas}
G_{as}(x) = {1\over [x^2] }
\; + \; \left( {2 [x^4] - [x^2]^2 \over [x^2]^4 } \right)
+ \left( 40 \; {[x^4]^2 \over [x^2]^7 }
+ 16 \; {[x^4] \over [x^2]^5 }
- 48 \; {[x^6] \over [x^2]^6 }
- 4\; {1 \over [x^2]^3 } \right) + ...
\end{equation}
In the domain ${\cal F}_{N}$, the propagator $\displaystyle G_B(x)$ can computed by the recursion formulas
\begin{equation}\label{RecRelxBosProp}
G_B(x+\hat \mu) = G_B(x - \hat \mu) + \frac{2x_\mu}{\Big(\sum_{\nu=1}^4 x_\nu\Big)}\; \sum_{\lambda=1}^4 (G_B(x)- G_B(x-\hat \lambda)),
\end{equation}
which allow to express it in terms of $G_B(0,0,0,0)=Z_0$ and $G_B(1,1,0,0) = - 1/4 + Z_1 + Z_0$.
The domain $\{\Lambda \backslash {\cal F}_{N} \}$ is chosen so that the propagator
is fitted by its asymptotic expression (\ref{DBas}) with a sufficient precision
making it possible to neglect the third sum in the formula (\ref{DivLambdaIn2Domains}).
Then the first sum can be calculated exactly using the summation formulas derived in \cite{Luscher}
and the second sum can be expressed in terms of $Z_1$ and $Z_2$ by employing
the relations (\ref{RecRelxBosProp}).
It should be emphasized that $A_B(0)$ is the coefficient of the expansion
$$
A_B(p) = {1\over a^2}\Big( A_B(0) + (pa)^2 \left[ A_1 + B_1 \ln(pa)^2 \right] + O((pa)^4) \Big),
$$
where $a$ is the length of a link; that is, $A_B(0)$ is the coefficient of the
divergent part. It does not vanish, though the ``bigmac'' diagram
depicted in Fig.1 converges in the dimensional regularization
provided that all masses and the external momentum are equal to zero.
\section{Two-loop fermionic integrals.}
In the fermionic case, calculations are performed by the same procedure, however,
{\bf we have no recursion relations similar to (\ref{RecRelxBosProp}).}
The fermionic propagator in $x$-representation
\begin{equation}
G_F(x_1,x_2,x_3,x_4)=
\int {d^4k\over (2\pi)^4}
{\cos(k_1 x_1) \cos(k_2 x_2) \cos(k_3 x_3) \cos(k_4 x_4) \over
\Delta_F}
\end{equation}
is expressed in terms of the quantities
\begin{equation}
F(p,q;n_1,n_2,n_3,n_4)=
\int {d^4k\over (2\pi)^4}
{\cos^{n_1}(k_1) \cos^{n_2}(k_2) \cos^{n_3}(k_3) \cos^{n_4}(k_4) \over \Delta_B^q
\Delta_F^{p+\delta}}
\end{equation}
by making use of the relations
\begin{equation}
\cos (nx) = 2^{n-1} \cos^n x \;+\; {n\over 2} \sum_{k=0}^{[n/2]-1}
{(-1)^{k+1} \over k+1} C_{n-k-2}^{k} (2\cos x)^{n-2k-2}.
\end{equation}
To employ the LW method outlined above, we compile a table of values of $G_F(x)$ over the domain
$ x_1\geq x_2\geq x_3\geq x_4\geq 0,\ |x| \leq 48$
and derive an asymptotic approximation of $G_F(x)$ at $|x| \to \infty$ up to the
terms of the order $\displaystyle 1/[x^2]^4$.
To treat integrals with nontrivial numerators, we should also compile the tables of the values
\begin{equation} \label{GKL-definitions}
K_{B[F]} = \int {dp \over (2\pi^4)} {(e^{-ipx}\; - 1) \over D_{B[F]}^2 (p) } \ , \quad
L_{B[F]} = \int {dp \over (2\pi^4)} {\displaystyle \left( e^{-ipx} - 1 + {x^2\over 8} \; (4\; -\; \sum_{\mu=1}^4 \cos^2 k_\mu )\;\right)\over D_{B[F]}^3 (p) } \ ,
\end{equation}
Each of these tables involves 14147 entries, each entry is a linear combination of the constants
$F_0, Z_0, Z_1$, $Y_0, Y_1, ... Y_{11}$, $\displaystyle {1\over(2\pi)^2}$, and 1 with
rational coefficients; from 5 to 20~MB per table in size.
Fortunately, they can be conveniently treated with FORM \cite{FORM}.
The precision of 20 significant digits in determination of the constants
$Y_4\div Y_{11}$ \cite{CapitaniPR}, \cite{Burgio}
is not sufficient for computation of $G_F(x)$ at $|x| > 6$.
Using the procedure proposed in \cite{Luscher} for calculation of $Z_0$ and $Z_1$,
we obtain
\begin{eqnarray}\nonumber
Y_4 &=& 0.08539036359532067913516702888533412058194147127443265(1) \\ \nonumber
Y_5 &=& 0.46936331002699614475347539705751803482046295887523184(1) \\ \nonumber
Y_6 &=& 3.39456907367713000586008689702374496453685272313733503(1) \\ \nonumber
Y_7 &=& 0.05188019503901136636490228766471579940968012757291508(1) \\ \nonumber
Y_8 &=& 0.23874773756341478520233613930386970445280194983477988(1) \\ \nonumber
Y_9 &=& 0.03447644143803223145396188144243193600121277124715784(1) \\ \nonumber
Y_{10} &=& 0.13202727122781293085314731098196596971197144795959477(1) \\ \nonumber
Y_{11} &=& 0.75167199030295682253543148590778110991011277193144803(1) \nonumber
\end{eqnarray}
At $|x|>48$, $G_F(x)$ is approximated by the function
\begin{equation} G^{(as)}_F(x) = {1\over [x^2] }
+ \left( {8 [x^4] - 4 [x^2]^2 \over [x^2]^4 } \right)
+ \left( 640 \; {[x^4]^2 \over [x^2]^7 } - 768 \; {[x^6] \over [x^2]^6 }
+ 208 \; {[x^4] \over [x^2]^5 }
- {40 \over [x^2]^3 } \right) + ...,
\end{equation}
To provide an example, let us consider the following two-loop fermionic integrals:
\begin{eqnarray}
Q_1^{BBB} &=& \int_{BZ} {d^4k \over (2\pi)^4} \; {d^4q \over (2\pi)^4} \sum_{\mu=1}^4
{\hat k_\mu^2 \hat q_\mu^2 \over D_B(k) D_B(q) D_B(r) } \\ \nonumber
Q_1^{BBF} &=& \int_{BZ} {d^4k \over (2\pi)^4} \; {d^4q \over (2\pi)^4} \sum_{\mu=1}^4
{{\hat k_\mu^2} {\hat q_\mu^2} \over D_B(k) D_B(q) D_F(r) } \\ \nonumber
\end{eqnarray}
and similar quantities with other combinations of bosonic and fermionic propagators.
We can also consider
\begin{equation}
Q_2^{BBF} = \int_{BZ} {d^4k \over (2\pi)^4} \; {d^4q \over (2\pi)^4} \sum_{\mu=1}^4
{{\hat k_\mu^2} {\hat q_\mu^2} {\hat r_\mu^2} \over D_B(k) D_B(q) D_F(r) }
\end{equation}
etc. The results of the computations are as follows:
\begin{eqnarray}
&Q_1^{BBB} = 0.042306368(1) \qquad \qquad &Q_1^{FBB} = 0.020079702(3) \\ \nonumber
&Q_1^{BBF} = 0.024555253(3) \qquad \qquad &Q_1^{FFB} = 0.00969896(1) \\ \nonumber
&Q_1^{BFF} = 0.01173224(1) \qquad \qquad &Q_1^{FFF} = 0.00576013(3) \\ \nonumber
&Q_2^{BBB} = 0.05462397818(1) \qquad \qquad &Q_2^{BBF} = 0.02659175158(3) \\ \nonumber
&Q_2^{BFF} = 0.0130373237(1) \qquad \qquad &Q_2^{FFF} = 0.0064945681(3) \nonumber
\end{eqnarray}
\section{Summary and Outlook}
The BCP algorithm has been realized on a computer.
The basic fermionic integrals $G(p,q)$ are found over a sufficiently large domain of values of $p$ and $q$.
This allows (i) to express one-loop intergals involving fermionic denominators in terms of
the constants $F_0, Z_0, Z_1$ and $Y_0\div Y_{11}$ and (ii) to compute $G_F(x)$ at $|x|\leq 96$.
The LW method is extended to the case of fermions;
asymptotic behavior of the fermionic propagator at $|x|\to \infty$ is found.
Therewith, $G_F(x)$ is expressed at $|x|\leq 48$ in terms of the constants $Y_4\div Y_{11}$,
the values of which are computed to a precision of 54 significant digits.
This is really needed for calculation of two-loop integrals.
A new feature of FORM - a possibility to work with database-like structures -
proved to be useful for summation over the domain $|x|\leq 48$.
As an illustration, several two-loop fermionic integrals are evaluated
at zero external momentum.
Operations with a table of precise values of the functions $G_{B(F)}(x)$
$K_{B(F)}(x)$ and $L_{B(F)}(x)$ allow to compute one-loop and two-loop
diagrams of the propagator type at nonvanishing external momentum.
The work is in progress!
\vspace*{3mm}
{\bf Acknowledgments:} I am grateful to V.Bornyakov, A.Kataev, H.Perlt, and A.Schiller
for stimulating discussions. This work was supported in part by
the Russian Foundation for Basic Research (grant no. 07-02-00237-a)
and by the grant for scientific schools NSh-6260.2010.2.
|
\section{Historical Overview}
Chemical evolution (CE) models follow the formation, destruction, abundance, and spatial and stellar distribution of elements created during the nucleosynthesis era of the Big Bang, and the different evolutionary stages in stars. Models are pitted against a host of observational test, such as the relative abundances of the various elements and their isotopes in meteorites, stars, and in the interstellar, intergalactic and intracluster media, the G-dwarf metallicity distribution, and the age-metallicity relation of the various systems [e.g. \citep{matteucci01,pagel97}].
CE models provide a natural framework for studying the evolution of dust since the abundance of the elements locked up in dust must be constrained by the availability of refractory elements in the interstellar medium (ISM). CE models need then to be generalized to include processes unique to the evolution of dust: the condensation efficiency of refractory elements in stellar ejecta, the destruction of grains in the ISM by expanding supernova remnants (SNRs), and the growth and coagulation of grains in clouds \cite{dwek98}.
The first dust evolution models \citep{dwek79,dwek80b} (hereafter DS) addressed the origin of the elemental depletion pattern, which was a subject of considerable debate. \cite{field74} showed that the depletion pattern correlated with condensation temperature, suggesting that it reflects the condensation efficiency of the elements in their respective sources. Such causal correlation requires that dust undergoes very little interstellar processing that can alter the depletion pattern. An equally good correlation exists between the magnitude of the elements' depletion and their first ionization potential \citep{snow75}, suggesting that the depletion pattern may instead be governed by accretion processes in molecular clouds. Pointing out the intrinsic physical correlation between the condensation temperature, the first ionization potential, and the threshold for grain destruction by sputtering, DS suggested that the depletion pattern could reflect the destruction efficiency of dust in the ISM. It is currently clear that all three processes play some role in establishing the elemental depletion pattern, since it depends on the density of the medium in which it is observed \citep{savage96}. Globally, their relative importance depends on the prevalence of and the cycling times between the different phases (hot, neutral, and molecular) of the ISM.
A more detailed review of dust evolution models is given by \cite{dwek98}. Since then, several models have been constructed to follow the evolution of dust in dwarf galaxies \citep{lisenfeld98}, Damped Ly$\alpha$ systems \citep{kasimova03}, the Milky Way galaxy \citep{zhukovska08}, and the origin of isotopic anomalies in meteorites \citep{clayton04,zinner06a,zinner06b}.
Early dust evolution models adopted the instantaneous recycling approximation, which assumes that all the elements and dust are promptly injected back into the ISM following the formation of their nascent stars. In contrast \cite{dwek98} and \cite{morgan03} constructed models that take the finite main sequence (MS) lifetimes of the stars into account. Silicate dust is primarily produced in supernovae (SN) that ``instantaneously'' recycle their products back to the ISM, whereas carbon dust is mainly produced in low-mass carbon-rich stars which have significantly longer MS lifetimes. Consequently these models predicted that the composition of the dust should evolve as a function of time.
Dust evolution models contain many still uncertain parameters such as the dust yields in the various sources and the grain destruction efficiency in the ISM. However, in spite of these uncertainties we will show that they can be very successful in predicting global evolutionary trends, namely the observed correlation of the abundance of polycyclic aromatic hydrocarbon (PAH) molecules with metallicity, and in examining the origin of dust in the high-redshift universe.
\section{Main Ingredients of Dust Evolution Models}
\subsection{The Yield of Dust in Stars}
The dust condensation efficiency, and its composition and size distribution depend very much on the environment in which it is formed. Dust can form in the quiescent outflows of AGB or Wolf-Rayet stars, or in the explosively expelled ejecta of SNe and novae. There is substantial observational evidence for the formation of dust in all these sources, however, their relative importance as sources of interstellar dust, and the composition and size distribution of the condensed dust are still uncertain, especially in SNe.
Because of the ease of formation and stability of the CO molecule, dust sources can be divided into two categories: carbon-rich sources, in which the C/O abundance ratio is larger than 1. These sources will produce carbon dust; and oxygen-rich sources with C/O $<$ 1, which will produce silicate-type dust. Such simple arguments assume that CO formation goes to completion, which may not always be the case, as suggested by the dynamical condensation models of \cite{ferrarotti06}.
SN explosions mark the death of stars more massive than $\sim8$~\msun. Figure \ref{snejecta} depicts the post-explosive composition of a 25~\msun\ star \citep{woosley88}. It depicts a typical onion-skin structure in which the composition of the different layers reflect the pre- and post-explosion nuclear burning stages of the star. Globally, the ejecta has a C/O ratio $<$~1, and should in principle be only producing silicate dust. However, in spite of the mixing between the layers caused by Rayleigh-Taylor instabilities in the ejecta, this mixing is of macroscopic nature and does not occur on an atomic level. So the layers above above $\sim~4.2$~\msun, in which C/O $>$~1, will maintain this ratio and produce carbon dust, whereas the inner layers will produce silicates. If all condensible elements precipitated out of the ejecta and formed dust, the yield of dust in a typical SN would be about 1~\msun\ \citep{kozasa89,kozasa91}.
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{fig1.eps}
\end{center}
\caption{The post-explosive composition of a 25~\msun\ star \citep{woosley88}. A typical SN can potentially produce 1~\msun\ of dust. }
\label{snejecta}
\end{figure}
The yield and composition of dust in lower mass stars depends on the C/O ratio in their atmosphere during the AGB phase of their evolution.
Figure \ref{coyield} depicts the C and O yield in stars. Supernovae yields were taken from \cite{woosley95} and AGB yields were taken from \citep{karakas03a}. Stars with masses above $\sim 8$~\msun\ produce carbon and silicate dust, irrespective of the global C/O ratio in their ejecta. The mass range of carbon rich stars depends on the initial stellar metallicity. At zero metallicity (left panel) the mass range of stars producing carbon dust is between 0.8 and 7.8~\msun. This range narrows significantly to masses between 3.0 and 3.6~\msun\ at solar metallicity (right panel). Figure \ref{co} presents a qualitative depiction of the method we use to calculate the yield of carbon and silicate dust in AGB stars. A more realistic model calculating the yield of dust in AGB stars was presented by \cite{ferrarotti06} and Hoefner (2009, this conference proceedings).
\begin{figure}
\hspace{-0.5in}
\begin{tabular}{cc}
\includegraphics[width=2.8in]{fig2a.eps}
\includegraphics[width=2.8in]{fig2b.eps}
\end{tabular}
\caption{The C and O yields in stars. Stars with masses above $\sim~8$~\msun\ become SNe and produce both silicate and carbon dust. Stars with masses below $\sim 8$~\msun\ produce either carbon or silicate dust, depending on the C/O ratio in their ejecta. The light gray area in the horizontal bar depicts the range of stellar masses in which C/O $>$ 1.
\label{coyield}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=2.5in]{fig3a.eps}
\includegraphics[width=2.5in]{fig3b.eps}
\end{tabular}
\caption{Qualitative depiction of the calculated yield of carbon and silicate dust in AGB stars. When $C/O > 1$ (left panel), the star produces only carbon dust. The dark shaded area depicts the number of carbon atoms that condense into dust. When $C/O <1$ (right panel) the star produces silicate dust, and the dark shaded area depicts the condensing elements.
\label{co}
\end{figure}
Figure \ref{dustvol} (left panel) depicts the stellar evolutionary tracks of stars with an initial solar metallicity. Also shown in the figure is their MS lifetime. Similar lifetimes are obtained for different initial metallicities \citep{portinari98}. At metallicity of $Z = 0$, the first carbon dust producing stars are about 8~\msun\ and will evolve of the MS about 50~Myr after their formation. At solar metallicities, the production of carbon by AGB stars will be delayed by about 500~Myr, when $\sim 4$~\msun\ stars evolve off the MS. Since most of the interstellar carbon dust is made in AGB stars, this delay in its formation can have important observational consequences. Carbon dust has a significantly higher visual opacity than silicates, so the opacity of galaxies will change with time, with young systems being more transparent than older ones.
Figure \ref{dustvol} (right panel) depicts the different evolutionary trends of SN- and AGB-condensed dust calculated for a CE model with by exponential star formation rate characterized by a decay time of 6~Gyr, and a Salpeter initial mass function \citep{dwek98, dwek05}.
The silicate and carbon dust yields were calculated assuming a condensation efficiency of unity in the ejecta, and grain destruction was neglected. The model therefore represents an idealized case, in which grain production is maximized, and grain destruction processes are totally ignored. Also shown in the figure are the separate contributions of AGB stars to the abundance of silicate and carbon dust. The onset of the AGB contribution to the silicate abundance starts at $t \approx 50$~Myr, when $\sim$ 8~M$_{\odot}$ stars evolve off the main sequence, whereas AGB stars start to contribute to the carbon abundance only at $t \approx 500$~Myr, when 4~M$_{\odot}$ stars reach the AGB phase. The figure also presents the dust-to-ISM metallicity ratio, which is almost constant at a value of $\sim$ 0.36.
\begin{figure}
\hspace{-0.5in}
\begin{tabular}{cc}
\includegraphics[width=2.1in]{fig4a.eps}
\includegraphics[width=3.4in]{fig4b.eps}
\end{tabular}
\caption{{\it Left panel}: The H-R diagram of stars and their main sequence lifetime. {\it Right panel}: The evolution of silicate (dashed line) and carbon (solid line) dust from SNe (bold curves) and AGB stars (light curves).
\label{dustvol}
\end{figure}
\section{The Lifetime of Interstellar Dust}
Following their injection into the ISM, the newly-formed dust particles are subjected to a variety of interstellar processes resulting in the exchange of elements between the solid and gaseous phases of the ISM, including: (a) thermal sputtering in high-velocity ($>$200 km~s$^{-1}$) shocks; (b) evaporation and shattering by grain-grain collisions in lower velocity shocks; and (3) accretion in dense molecular clouds.
Detailed description of the various grain destruction mechanisms and grain lifetimes in the ISM were presented by \citep{jones96, jones04}. In addition, SN condensates can be destroyed shortly after their formation by reverse shocks that travel through the expanding ejecta (Dwek 2005, Bianchi \& Schneider 2007, Nozawa et al. 2008).
The most important parameter governing the evolution of the dust is its lifetime, $\tau_d$, against destruction by SNRs. In an interstellar medium with a uniform dust-to-gas mass ratio, $Z_d$, this lifetime is given by \citep{dwek80b,mckee89b}:
\begin{equation}
\tau_d = {M_d(t)\over \left<m_d\right> R_{SN}} = {M_g(t)\over \misme R_{SN}}
\label{dust_life}
\end{equation}
where $M_d$ and $M_g$ are, respectively, the total mass of dust and gas in the galaxy, $\left<m_d\right>$ is the total mass of elements that are locked up in dust and returned by a single SNR back to the gas phase either by thermal sputtering or evaporative grain-grain collisions. $R_{SN}$ is the SN rate in the galaxy, so that the product $\left<m_d\right>\, R_{SN}$ is the destruction rate of dust in the ISM. The parameter $\misme \equiv \left<m_d\right>/Z_d$ is the effective ISM mass that is completely cleared of dust by a single SNR, given by \citep{dwek07b}:
\begin{equation}
\misme = \int_{v_0}^{v_f}\ \zeta_d(v_s)\ \left|{dM\over dv_s}\right|\ dv_s
\end{equation}
where $ \zeta_d(v_s)$ is the fraction of the mass of dust that is destroyed in an encounter with a shock wave with a velocity $v_s$, $(dM/dv_s)dv_s$ is the ISM mass that is swept up by shocks in the [$v_s,\ v_s+dv_s$] velocity range, and $v_0$ and $v_f$ are the initial and final velocities of the SNR.
Figure \ref{xsi} depicts the mass fraction of carbon and silicate dust that is destroyed after being swept up by a shock of velocity $v_s$ as a function of shock velocity. An updated version for the carbon and PAH destruction efficiency was presented by Jones et al. (2009, this conference proceedings). For example, in the Milky Way $M_g \approx 5\times 10^9$~\msun, $R_{SN} \approx 0.03$~yr$^{-1}$, and $\misme \approx 300$~\msun, giving a dust lifetime of $\sim 6\times 10^8$~yr.
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{fig5.eps}
\end{center}
\caption{The mass fraction of carbon and silicate dust that is destroyed as a function of shock velocity (after Jones et al. 1996). }
\label{xsi}
\end{figure}
In addition, SN condensates can be destroyed shortly after their formation by reverse shocks that travel through their expanding ejecta (Dwek 2004, Nozawa et al. 2008).
\section{PAHs and Silicate Dust as Tracers of AGB- and SN-condensed Dust}
An exciting discovery made by spectral and photometric observations of nearby galaxies with the {\it Infrared Space Observatory} ({\it ISO}) and {\it Spitzer} satellites was the striking correlation between the strength of their mid-infrared (IR) aromatic features, commonly attributed to the emission from PAHs, and their metallicity, depicted in Figure \ref{pahvol} [left panel; see \cite{galliano08a} for references]. The figure shows the rise of $F_{8/24}$, the 8~\mic-to-24\mic\ band flux ratio with galaxies' metallicity, and the existence of a metallicity threshold below which $F_{8/24}$ is equal the flux ratio of the dust continuum emission. The strength of the aromatic feature is a measure of the PAH abundance. Since PAHs are predominantly made in C-rich AGB stars, this correlation provides the first observational evidence for the delayed injection of AGB condensed dust into the ISM, provided the metallicity is a measure of the galaxies' age. The testing of this hypothesis requires first the determination of the PAH abundance in each galaxy.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=2.2in]{fig6a.eps}
\includegraphics[width=3.0in]{fig6b.eps}
\end{tabular}
\end{center}
\caption{{\it Left panel}: The observed correlation between the 8-to-24~\mic\ bands flux density and metallicity. {\it Right panel}: PAH and dust abundances derived from detailed models of galaxies' SED versus galaxies' metallicity.
\label{pahvol}
\end{figure}
PAHs are very small macromolecules, typically 50~\AA\ in diameter, that are stochastically heated by the ambient radiation field. Consequently, only a fraction of the PAHs are radiating at mid-IR wavelengths at any given time. To determine the total abundance of PAHs, including those too cold to emit in the aromatic features requires the determination of the intensity of the interstellar radiation field (ISRF) to which they are subjected.
Figure \ref{dustspec} depicts the steps used by \cite{galliano08a} in modeling the galaxies' spectral energy distribution (SED). The galaxy used for this illustrative purpose is the starburst M82. The dust model used in the calculations is the BARE-GR-S model of \cite{zubko04}, consisting of PAHs, and bare silicate and graphite grains with solar abundances constraints.
Figure 7a shows the SED of M82, and its various emission components: stellar emission at optical, and near-IR wavelengths; the PAH spectrum at mid IR wavelengths; dust emission from mid- to far-IR wavelengths
|
; and free-free and synchrotron emission at radio wavelengths.
Fig 7b depicts a fit to the dust spectrum using an ISRF characterized by a power-law distribution of radiation field intensities. PAH abundance determined by this method will {\it underestimate} the real abundance of PAHs in the galaxy compared to the method outlined below. Our models use the free-free and mid-IR emissions to constrain the gas and dust radiation from the gas and dust from H~II regions, and the far-IR and optical emission to constrain the ISRF that heats the dust in photo-dissociation regions (PDRs). The radio emission is uniquely decomposed into free-free (dashed) and synchrotron (dotted) emission components (see Figure 7d).
Massive stars are required to produce the ionizing radiation and the expanding SN blast waves that generate, respectively, the observed free-free and synchrotron emission. These stars are produced in an "instantaneous" burst of star formation, in contrast to the stars that are continuously created over the lifetime of the galaxy and mostly contribute to the optical and near-IR emission (Figure 7c).
The ionizing and a fraction of the non-ionizing radiation emitted by the starburst component that is absorbed in the H~II region is shown as a shaded area in Figure~7d. This energy is reradiated by the dust and gas, giving rise to the thermal IR and free-free emission components shown in the figure.
The non-ionizing radiation from the older stellar population and the radiation escaping the H~II regions form the diffuse ISRF that is absorbed by the dust in photodissociation regions (PDR) (Figure 7e). The shaded area depicts the fraction of the radiation from the older stellar population that is absorbed in PDRs. The absorbed radiation is reemitted by the dust, giving rise to the IR emission (figure 7f).
Figure 7g depicts the different emission components, and Figure~7h shows the fit of their sum to the SED of M82. The details of the fitting procedure are described in \cite{galliano08a}.
Using this physical fitting procedure, \cite{galliano08a} derived the abundance of PAHs, silicates, and graphite grains in 35 nearby galaxies with metallicities ranging from 1/50 to 3 times solar. The detailed physical modeling of their SEDs gives larger PAH abundances compared to models that employ a template ISRFs to heat the PAHs and the dust. In these models, such as the one depicted in Fig. 7b, PAHs are subjected to the same intense radiation field as that required to produce the mid-IR emission from hot dust. In our model, PAHs are subjected to a weaker radiation field. Since PAHs do not survive in H~II regions, all their emission originates from PDRs, which are subjected to lower intensity radiation fields than the H~II regions. So, compared to the template ISRF models, a larger amount of PAHs is required to produce the same PAH spectrum with a weaker ISRF. The resulting PAH abundances are plotted versus metallicity in Figure \ref{pahvol} (right panel). The figure shows that the observed trend of increasing 8-to-24~\mic\ band flux ratio with metallicity indeed reflects a trend of increasing PAH abundance with metallicity. The figure also shows the distinct evolutionary trends of PAHs and the far-IR emitting dust with metallicity.
\begin{figure}
\hspace{-0.5in}
\begin{tabular}{ll}
\includegraphics[width=3.5in]{fig7a.eps}
\includegraphics[width=3.5in]{fig7b.eps} \\
\includegraphics[width=3.5in]{fig7c.eps}
\includegraphics[width=3.5in]{fig7d.eps} \\
\includegraphics[width=3.5in]{fig7e.eps}
\includegraphics[width=3.5in]{fig7f.eps} \\
\end{tabular}
\end{figure}
\begin{figure}
\hspace{-0.5in}
\begin{tabular}{ll}
\includegraphics[width=3.5in]{fig7g.eps}
\includegraphics[width=3.5in]{fig7h.eps}
\end{tabular}
\caption{ Construction of the fit to the SED of the starburst galaxy M82. See text for details.
\label{dustspec}
\end{figure}
Figure \ref{chemvol} compares the evolution of the PAHs and dust components derived from the dust evolution model to the trend of PAH abundances with metallicity. For the sake of this comparison, the evolution of dust abundances as a function of time was converted to an evolution as a function of metallicity, using the age-metallicity relation derived in the model. We emphasize that the parameters used in the dust evolution model (the star formation rate, the stellar IMF) were identical to those used in the population synthesis model that was used to fit the galaxies' SED. From the dust evolution model we already derived the two distinct evolutionary trends of SN- and AGB-condensed dust (see Fig. \ref{pahvol}, right panel for the idealized example). The current figure compares these results with the derived abundance of the PAHs and of the dust that gives rise to the far-IR emission. The latter is dominated by emission from silicates, and should therefore follow the trend of the SN condensates, since most silicate grains are produced in SNe. The figure shows that the observed far-IR emitting dust falls on the evolutionary track of the calculated SN-condensed dust, and that the observed PAH abundances fall on the evolutionary track of the carbon dust that formed in AGB stars. The shaded regions in the figure represent the range of evolutionary tracks that correspond to different parameters that determine the star formation rate and grain destruction efficiencies in the models.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=5.0in]{fig8.eps}
\end{center}
\caption{ Comparison between the metallicity trends of the PAH abundance derived from the observed SED and those derived from the
chemical evolution model.
The shaded area represent the range of model prediction for different grain destruction and star formation rates. Details of the figure are described in \cite{galliano08a}.}
\label{chemvol}
\end{figure}
\section{The presence of massive amounts of dust at high redshift}
The detection of massive amounts of dust in hyperluminous IR galaxies at redshifts $z > 6$ raises challenging questions about the sources capable of producing such large amount of dust during the relatively short lifetime of these galaxies \citep{maiolino06,beelen06,morgan03}.
For example, the galaxy SDSS J1148+5251 (hereafter \jay) located at $z = 6.4$ was observed at far-IR and submillimeter wavelength \citep{bertoldi03a,robson04,beelen06}. The average IR luminosity of the source is $L_{IR} \sim 2\times 10^{13}$~\lsun, and the average dust mass is $M_d \sim 2\times 10^8$~\msun. Using the \cite{kennicutt98a} relation, one can derive a star formation rate (SFR) of $\sim 3000$~\msun~yr$^{-1}$ from the observed far-IR luminosity. For comparison, the Milky Way galaxy is about 10~Gyr old, has an average SFR of $\sim 3$~\msun~yr$^{-1}$, and contains about $5\times 10^7$~\msun\ of dust, a significant fraction of which was produced in AGB stars.
At $z=6.4$ the universe was only 890~Myr old, using standard $\Lambda$CDM parameters ($\Omega_m = 0.27$, $\Omega_{\Lambda} = 0.73$, and $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$). If \jay\ formed at $z = 10$ then the galaxy is only 400~Myr old.
If the SFR had occured at a constant rate over the lifetime of the galaxy, its initial mass should have been about 10$^{12}$~\msun, which is significantly larger than the dynamical mass $M_{dyn} \approx 5\times 10^{10}$~\msun\ of the galaxy \citep{walter04}. The high observed SFR may therefore represent a recent burst of star formation that has lasted for only about 20~Myr.
The galaxy \jay\ is therefore at most $\sim 400$~Myr old, and probably significantly younger with an age of only $\sim 20$~Myr. Adopting a current gas mass of $M_g = 3\times 10^{10}$~\msun\ for this galaxy we get that the gas mass fraction at 400~Myr is about 0.60. The dust-to-gas mass ratio is given by $Z_d \equiv M_d/M_g = 0.0067$.
A significant fraction of the dust in the Milky Way was produced in AGB stars. However, these stars are not likely to contribute significantly to the formation of dust in very young galaxies, since the low mass stars ($M \approx 3$~\msun) that produce most of the dust did not have time to evolve off the main sequence \citep{dwek98,morgan03,dwek05}.
In contrast, core collapse SNe ($M > 8$~\msun) and their post-main-sequence progenitors inject their nucleosynthetic products back into the ISM shortly ($t < 20$~Myr) after their formation, resulting in the rapid enrichment of the interstellar medium (ISM) with the dust that formed during the mass loss phase prior to the SN event, or in the explosive SN ejecta. We will hereafter attribute both contribution to the SN event, since both are return "promptly" to the ISM. But can SNe account for the large amount of dust seen in this object? The answer to this question is complicated by the fact that SNe are also the main source of grain destruction during the remnant phase of their evolution \citep{jones96,jones04}. The problem can therefore only be quantitatively addressed with CE models for the dust in these systems.
The results of the detailed dust evolution models described summarized in Figure \ref{chemvol} show that the contribution of AGB stars to metal and dust abundance can be neglected in galaxies with ages less than about 400~Myr.
The equations for the chemical evolution of the galaxy can then be considerably simplified using the instantaneous recycling approximation, which assumes that stars return their ejecta back to the ISM promply after their formation. The evolution of the dust abundance can then be written in analytical form \citep{morgan03,dwek07b}.
In particular, the yield of dust, \ydust, required to obtain a given dust-to-gas mass ratio, $Z_d$, when the galaxy reaches a given gas mass fraction \mug, is given by \citep{dwek07b}:
\begin{equation}
\yduste =Z_d\ \left[{\misme + R\ \mstare \over 1-\muge^{\nu-1}}\right]
\label{ydzd}
\end{equation}
\noindent where,
\begin{equation}
\nu \equiv {\misme + \mstare \over (1-R)\ \mstare}\qquad ,
\label{nu_eq}
\end{equation}
$R$ is the fraction of the stellar mass that is returned back to the ISM during the stellar lifetime, \mism\ is given by eq. (2), and \mstar\ is the mass of all stars born per SN event. For example, $\mstare=147$ and $50$ ~\msun, respectively, for a Salpeter and top-heavy IMF.
Figure \ref{snyield} shows how much dust an average SN {\it must} produce in order to give rise to a given dust-to-gas mass ratio, for various grain destruction efficiencies. The value of \ydust\ was calculated when \mug\ reaches a value of 0.60, the adopted gas mass fraction of \jay\ at 400~Myr.
Calculations were performed for two different functional forms of the stellar IMF: a Salpeter IMF in which $\phi(m) \sim m^{-2.35}$ and $0.1 < m$(\msun) $< 100$; and a top heavy IMF characterized by the same mass limits but a flatter slope $\phi(m) \sim m^{-1.50}$. Here, $\phi(m)$ is the number of stars per unit mass interval, normalized to unity between 0.1 and 100~\msun.
The figure shows that, for example, to produce a value of $Z_d = 0.0067$ at \mug\~=~0.60, a SN must produce about 0.4~(1.2)~\msun\ of dust for a top-heavy (Salpeter) IMF, provided the dust is not destroyed in the ISM, that is, \mism\ = 0. Even with modest amount of grain destruction, \mism\ = 100~\msun, the required SN dust yield is dramatically increased to about $1-2$~\msun, depending on the IMF.
The horizontal line in the figure corresponds to a value of \ydust\ = 0.054~\msun, the largest mass of SN-condensed dust inferred to be present in a supernova or SNR \citep{rho08,sugerman06}. Contrary to the claim by \cite{rho08}, this yield is not sufficient to account for the large amount of dust observed in high redshift galaxies, since the quoted chemical evolution models of \cite{morgan03} do not include the effect of grain destruction. The figure shows that even without grain destruction, the largest observed yield can only give rise to a dust-to-gas mass ratio of $\sim 4\times10^{-4}$. If the mass of dust in the ejecta of Cas~A represents a typical SN yield, then other processes, such as accretion onto preexisting grains in molecular clouds is needed to produce the mass of dust in J1148+5251.
\begin{figure}
\begin{center}
\includegraphics[width=5.0in]{fig9.eps}
\end{center}
\caption{The IMF-averaged yield of dust by type~II supernova, \yd, that is required to account for a given dust-to-gas mass ratio $Z_d$, is presented for different values of \mism\ given in units of \msun. Solid and dashed lines correspond to calculations done for a top-heavy and a Salpeter IMF, respectively. The horizontal dashed line near the bottom of the figure corresponds a value of $Y_d = 0.054$~\msun, the highest inferred yield of dust in supernova ejecta to date \cite{rho08}. The vertical dotted line represents the value of $Z_d$ at $\muge =0.60$. Curves are labeled by \mism\ given in units of \msun. The top two dashed (solid) horizontal lines represent IMF-averaged theoretical dust yields for a Salpeter (top-heavy) IMF, assuming 100\% condensation efficiency in the SN ejecta. }
\label{snyield}
\end{figure}
\section{Summary}
Dust evolution models have proven to be very successful in predicting global evolutionary trends in dust abundance and composition, and in analyzing the origin of dust in the early universe.
An important prediction of these models is that SN- and AGB-condensed dust should follow distinct evolutionary paths because of the different stellar evolutionary tracks of their progenitor stars. By analyzing the UV-to-radio SED of 35 nearby galaxies we have identified silicates and PAHs, respectively, as tracers of SN- and AGB-condensed dust. Our SED fitting procedure used chemical evolution, dust evolution, and population synthesis models in a consistent fashion. The models used the free-free and mid-IR emissions to constrain the gas and dust radiation from the gas and dust from H~II regions, and the far-IR and optical
emission to constrain the ISRF that heats the dust in PDRs.
The observed correlation of the intensity of the mid-IR emission from PAHs with their metallicity can then be interpreted as the result of stellar evolutionary effects which cause the delayed injection of carbon dust into the interstellar medium.
The early universe is a unique environment for studying the role of massive stars in the formation and destruction of dust.
The equations describing their chemical evolution can be greatly simplified by using the instantaneous recycling approximation, and by neglecting the delayed contribution of low mass stars to the metal and dust abundance of the ISM. Neglecting any accretion of metals onto pre-existing dust in the interstellar medium, the evolution of the dust is then primarily determined by the condensation efficiency of refractory elements in the ejecta of Type~II supernovae, and the destruction efficiency of dust by SN blast waves.
We applied our general results to \jay, a dusty, hyperluminous quasar at redshift $z = 6.4$ and found that the formation of a dust mass fraction of $Z_d = 0.0067$ in a galaxy with an ISM mass of $3\times 10^{10}$~\msun, requires an average SN to produce between 0.5 and 1~\msun\ of dust if there was no grain destruction. Such large amount of dust can be produced if if the condensation efficiency in SNe is about unity. Observationally, the required dust yield is in excess of the largest amount of dust ($\sim 0.054$~\msun) observed so far to have formed in a SN. This suggests that accretion in the ISM may play an important role in the growth of dust mass.
For this process to be effective, SNRs must significantly increase, presumably by non-evaporative grain-grain collisions during the late stages of their evolution, the number of nucleation centers onto which refractory elements can condense in molecular clouds.
\acknowledgments
This work was supported by NASA's LTSA 03-0000-065.
\newpage
|
\section{Introduction and Motivation}
\label{sec:1}
There is considerable excitement surrounding the
discovery
\cite{Jin3,Grimm,Jin4,Ketterle3,KetterleV,
Thomas2,Grimm3,ThermoScience,Salomon3,Hulet4}
of superfluidity
in the ultracold Fermi gases. What is novel about these new superfluids is
that one can tune the attractive interaction from weak (as in the BCS
limit) to strong as in the Bose Einstein condensation (BEC) regime.
These experiments will continue to impact condensed matter
physics by providing, at the least, a new class of ``materials" which elucidate a
very powerful generalization of BCS theory. A number of people
\cite{ourreview,Varenna,Strinaticuprates,randeriareview} have
also argued that this BCS-BEC crossover might be relevant to the cuprate
superconductors. Because of their anomalously short coherence length
it is claimed \cite{LeggettNature}
that these materials are ``mid-way between BCS and BEC".
That is, the attractive interaction driving the superconducting
pairing may be stronger than that in conventional superconductors.
In this way the tuneability of the interaction strength in
the Fermi gases provides an ideal model system with which to study the physics
of the short coherence length cuprates and the role of
strong attraction (generally associated with high transition
temperatures).
From a very different perspective, it
has also been argued that in future optical lattice experiments
\cite{GeorgesReview}
involving the atomic Fermi gases, one will be able to simulate repulsive
Hubbard models and thereby investigate the ``Mott physics" aspects
\cite{LeeReview} of high $T_c$ superconductivity.
While condensed matter physicists have a wealth of well-developed techniques
for characterizing electronic superconductors, the tools currently available
to the atomic physicists who study the Fermi gases are far more
limited. Moreover, it is not at all straightforward to determine something
as commonplace as the temperature in the gas, although some impressive
progress \cite{ThermoScience,ThomasUnitary,KetterleVarenna}
has been made along these lines.
This paper is devoted to
addressing one of the most powerful
techiques currently being applied to the Fermi gases: radio frequency
(RF) spectroscopy. We will show how this technique is similar to that of
photoemission in condensed matter
physics and exploit the analogy, already discussed in
the literature \cite{Jin6}, between momentum resolved
RF and angle resolved photoemission spectroscopy (ARPES). As a background
for both communities, we
review some of the experimental and theoretical
literature on RF spectroscopy (of cold gases)
and photoemission spectroscopy
(of the cuprates). We argue that there are
a number of issues which have been central to high temperature superconductivity
which would be useful to address more systematically in the ultracold Fermi gases.
Perhaps the most notable example of commonality \cite{ourreview,Varenna}
in this regard
is the ubiquitous pseudogap phase which is at the core of
current studies in the high $T_c$
superconductors and has emerged as important in the ultracold Fermi gases.
We begin by focusing on the overlap of the experimental
concepts behind photoemission experiments \cite{arpesstanford_review,arpesanl}
in the cuprates and RF spectroscopy in the atomic Fermi gases.
We will see that both experiments reflect the behavior of the
all important fermionic spectral function $A ( \bf{k}, \omega)$ which
characterizes completely the single fermion or one-particle properties of a
given many body system.
In simplistic terms, the driving force motivating the photoemission
studies in the cuprates
is to acquire an understanding of the ``mechanisms" and nature of
superconductivity. There has been a recent emphasis on high temperatures
near $T^*$, where the pseudogap turns on and on the region from above to
below the superfluid transition temperature, $T_c$.
By contrast in the ultracold gases, the RF spectra have been
used to characterize the pairing gap $\Delta$-- much like tunneling is used
in conventional superconductors.
There has been a recent emphasis on very low temperatures $T << T_c$
and in particular in quantifying the size of $\Delta$ at $T=0$.
Some of the key issues which have emerged in photoemission studies
of the
cuprates
involve (i) a characterization of the self energy contained in
the spectral function.
Different empirical models \cite{Pepin} have been deduced which, it is argued,
might ultimately hold the clue
as to the nature of the mediating boson.
(ii) Also important is the origin of the all important pseudogap.
There is a debate \cite{LeeReview,ourreview} about whether
this gap is a signature of a hidden order parameter or whether it reflects
the incipient pairing which ultimately leads to the condensed phase at
lower $T$.
(iii) It is viewed as extremely important
to arrive at an understanding of how
superconducting coherence manifests itself in this spectroscopic
experiments as one goes from the
normal to the ordered phase. This is a complicated question, given
the presence of a normal state (pseudo)gap.
Finally, other issues of interest are the nature of the order parameter
and pseudogap symmetry
(which have been shown to be consistent with $d$-wave).
In the cold gases
an underlying goal has been
to test different theories of BCS-BEC crossover, particularly
establishing the most suitable ground state and its quantitative
implications such as the pair size \cite{Ketterlepairsize}.
The parameters which quantify
the nature of
the scale-free or ``unitary" gas have also been addressed.
Of additional interest are studies on how population imbalance
\cite{Rice1,Rice2,MITPRL06,ZSSK06}
can co-exist with
superfluidity. Here new phases associated with, for example,
the exotic \cite{FFLO} Larkin- Ovchinnikov, Fulde-Ferrell (LOFF) form
of pairing have been contemplated.
Even more topical is
the behavior in the limit of extreme imbalance
\cite{MITPRL06,ZSSK06}.
One can see that, despite the similarities in
these two spectroscopic techniques, the research agenda in the two communities is
rather different. In the high temperature superconductors,
the focus has been
around the temperature regime near $T_c$.
Furthermore, quantitative issues are viewed as of considerably
less importance than arriving at a qualitative understanding, which
is still very incomplete.
By contrast in the ultracold Fermi gases the focus has been
on temperatures associated with the ground state and on arriving at
a more complete quantitative characterization.
This brings us to a major goal of the present paper
which is to suggest new directions in
the cold gas research agenda from the condensed matter
perspective. In particular we wish
to highlight differences and similarities
in the cold gases with the analogous cuprate studies.
A general theme, which takes a cue from the copper oxide
superconductors, is to focus on a characterization of (i) the
fermionic self energy, (ii) the pseudogap phase and (iii)
how superfluid coherence is established and
manifested (in these spectroscopies) at and below $T_c$.
\subsection{Comparing and Contrasting RF With
Photoemission}
\begin{figure*}
\centerline{\includegraphics[width=2.0in,clip]{ARPES2.eps} \hskip
2ex
\includegraphics[width=2.2in,clip]{RFTransitions.eps}}
\caption{
Figure on left:
Energy levels in an ARPES transition. In a paired system there are
two fermionic states which contribute to the photoemitted current. These
correspond to the red and black curves. The upper branch (red curve)
will not be occupied until the temperature is high. Here a tight
binding dispersion $\epsilon_{\bf k}$ is assume for the underlying
non-paired initial state (magenta curve). The cyan dashed line
indicates the Fermi level of the electrons and the green solid line the
dispersion of outgoing electrons.
Figure on right: Energy levels in an RF transition. $\Omega_L$ is the RF
frequency for exciting a free atom from hyperfine level 2 (maroon
line) to level 3 (green line). $\Omega_L^\prime$ is the same energy
but measured relative to the respective chemical potentials. The black
and red curves are the dispersion of the particle and hole branch of a
paired atom in level 2, with energy level given by $\mp E_{\mathbf{k}} + \mu$,
respectively.
}\label{RFtran}
\end{figure*}
Photoemission and Angle Resolved Photoemission
Spectroscopy (ARPES) have been remarkable tools for characterizing
the cuprate superconductors \cite{arpesanl,arpesstanford_review}.
Here one invokes
the ``sudden'' approximation which corresponds to
the assumption that the electron acquires the photon energy
instantaneously and emerges from the crystal surface immediately. As
a consequence, photoemission is
associated with electrons near the crystal surface. In
addition, only the momentum component in parallel with the surface is
conserved. It follows that ARPES is ideal for layered materials.
The energy levels involved in the ARPES process are shown in
Figure \ref{RFtran}a. Here, and throughout the paper, we define
the quantity $E_{\mathbf{k}}$ corresponding to the dispersion of the paired
fermions in terms of the usual BCS expression
\begin{equation}
E_{\mathbf{k}} \equiv \sqrt{ (\epsilon_{\mathbf{k}} -\mu)^2 + \Delta^2(T) }
\label{eq:13}
\end{equation}
Because of the large
photon energy $h\nu$, compared to the electron energy scale inside the
crystal, the final state of the photo-emitted electron is
essentially free so that
the energy conservation constraint is given by
$E_i = E_f -h\nu$,
where
$E_f = \mathbf{k}^2/2m_{\mbox{e}}$
is measured with an energy
analyzer. Here $m_{\mbox{e}}$ denotes electron mass. In turn, the momentum (in the known direction)
has magnitude $k=\sqrt{2m_{\mbox{e}}
E_f}$.
%
The ARPES spectrum is given by
\begin{equation}
I^{photo} (\mathbf{k},\omega) = M_0(\mathbf{k},\nu) A(\mathbf{k},\omega) f(\omega)
\label{eq:photo1}
\end{equ
|
ation}
where $M_0(\mathbf{k},\nu)$ is a matrix element which depends on the photon
energy.
Apart from the matrix element and the Fermi function $f(\omega)$,
one sees that ARPES
measures the electronic spectral function.
The energy levels involved in an RF transition are shown in
Fig.\ref{RFtran}b. Here $\Omega_L$ is the RF frequency for exciting
a free atom from hyperfine level 2 (maroon line) to level 3 (green
line). We neglect final state effects, which will be discussed
later. A significant difference between an RF and ARPES transition
is that in the RF case a dominantly large fraction ($\Omega_L$) of
the photon energy
is converted to
excite a fermion from one internal state to another.
As a
consequence, the excited atoms do not have a substantially higher
kinetic energy so that they do not leave the bulk gas immediately after
the transition until they are deliberately released.
The energy zero for an RF transition is more conveniently chosen
to be the bottom of the free atom band of state 2. In this
convention, the final state energy is $E_f = \Omega_L +
\epsilon_\mathbf{k}$, where $\epsilon_{\mathbf{k}} = k^2/2m$, and the initial state
energy is $E_i = \pm E_{\mathbf{k}} +\mu$ for the two branches shown in Fig.
1b. Therefore, the same energy conservation constraint emerges
$h\nu = E_f - E_i$. Finally the RF current (which will be derived
in Section \ref{LinearResponse}) is
\begin{equation}
I_0^{RF}(\mathbf{k}, \delta\nu) =\left. \frac{|T_k|^2}{2\pi}
A(\mathbf{k}, \omega) f(\omega)\right|_{\omega=\epsilon_{\mathbf{k}} -\delta\nu}
\label{RFc0m}
\end{equation}
where
$|T_k|^2$ is a tunneling matrix element
and, the momentum integrated current, which is the more widely studied form, is
\begin{multline}
I_0^{RF}( \delta\nu)
=\sum_{\bf k}
I_0^{RF}(\mathbf{k}, \delta\nu) \\
= \sum_{\bf k} \left. \frac{|T_k|^2}{2\pi} A(\mathbf{k}, \omega)
f(\omega)\right|_{\omega=\epsilon_{\mathbf{k}} -\delta\nu} \label{RFc0}
\end{multline}
We note an important contrast with
Eq.~(\ref{eq:photo1})
because here there is the restriction
$\omega=\epsilon_{\mathbf{k}} -\delta\nu$ which (apart from the matrix element effects)
serves to differentiate the photoemission and RF responses.
\subsection{Overview of the Literature on RF Experiments}
Experiments and theory have worked well hand in hand in developing
an understanding of the so-called ``RF pairing gap spectroscopy"
in the atomic Fermi gases. This class of experiments was
originally suggested by Torma and Zoller \cite{Torma} as a method
for establishing the presence of superfluidity. In this context an
equation equivalent to Eq (\ref{RFc0}) was derived. Later work
\cite{JS2,Torma1}, made the observation that these RF experiments,
which reflect the spectral function $A({\bf k}, \omega)$, would
observe a pairing gap $\Delta(T)$ which may be unrelated to
superconducting order (except in the strict BCS regime). This was
the beginning of a recognition that a pseudogap would be present,
which is associated with stronger-than-BCS attractive
interactions. Moreover, this pseudogap appears in the ``fermionic
regime", that is, when the fermionic chemical potential is
positive \cite{ourreview}.
An experimental ground breaking paper \cite{Grimm4}
reported the first experimental implementation of this pairing gap spectroscopy
in $^6$Li over a range of fields corresponding to the BCS, BEC and unitary
regimes. Accompanying this paper was a theoretical study \cite{Torma2}
by Torma and co-workers based on the BCS-BEC crossover
approach introduced earlier \cite{JS2}, but, importantly,
generalized to include trap effects.
This theoretical scheme is the one that will be the focus of the present paper.
The calculations showed reasonable agreement
with experiment, and subsequent work \cite{heyan} presented more
quantitative comparisons of the spectra along with theoretically-inferred
estimates of the temperature, based on an adiabatic sweep
thermometry \cite{ChenThermo}. Some of the first evidence that one was, indeed,
observing a pairing gap (or pseudogap) in the normal phase was
presented in Reference \cite{Varenna}, based on this same
thermometric approach and the data of the Innsbruck group \cite{Grimm4}.
In an important contribution Yu and Baym pointed out \cite{YuBaym}
that the theoretical framework described above and summarized in
Eq(\ref{RFc0}) missed what have now become known as ``final state
effects". Moreover, this could be seen most clearly in sum rule
constraints on the RF spectra. These final state effects can be
understood as follows. Assume as the right panel of Figure~\ref{RFtran}
that the condensed
phase involves pairing among hyperfine channels 1 and 2 and that
the excited atomic state is associated with hyperfine level 3.
While the attractive interaction $g_{12}$ drives the pairing, the
excited atoms in 3 will also experience a residual interaction
$g_{13}$, which may modify the RF spectra. In this way, these
final state effects yield corrections to the lowest order current,
shown in
Eq (\ref{RFc0}).
Interestingly, the sum rule, now known as the
``clock shift" sum rule \cite{Baym2} shows that the first moment
of the current sums to an internally consistent value, rather than
a pre-determined constant. This will be discussed in Section \ref{SumRule}
A new set of groundbreaking experiments from MIT have introduced a
powerful way of exploiting and enhancing RF spectroscopy first via
tomographic techniques \cite{MITtomo}. With the tomographic scans,
the complication of studying the spectra in a trapped
configuration can now be removed, so that the system is
effectively homogeneous. Also important was the demonstration that
the entire collection of $^6$Li superfluids with hyperfine levels
1 and 2 paired, as well as 1 and 3 as well as 2 and 3,
are
stable and can be probed in RF spectroscopy with variable RF
transitions, $\Omega_L$ (defined in the right panel of Figure~\ref{RFtran}). In this way one
has, in conjunction with a larger complex of superfluids, a way of
tuning final state effects. Moreover, it was hoped that a proper
choice of the superfluid and the RF transition
can reduce the importance of these final state corrections and
allow one to consider the simpler theory of Eq(\ref{RFc0}).
The theoretical challenge of incorporating final state
contributions has become very topical, in large part because of
the existence of data in effectively ``homogeneous" systems
through these tomographic techniques. It is only in the absence of
a trap that one can readily handle the higher order terms
introduced by Yu and Baym \cite{YuBaym}. With these corrections to
Eq(\ref{RFc0}) one may have a better opportunity to quantitatively
fit the RF spectra. Very nice calculations \cite{Strinati7,Basu}
of $I(\nu)$ in the homogeneous case consider the $T \approx 0 $
superfluid and good agreement with experiment has been
demonstrated \cite{Strinati7}. Subsequent work \cite{ourRF3} has
addressed the entire range of temperatures where one can probe the
RF contributions associated with pre-existing thermally excited
quasi-particles. These are shown as a second branch of RF
transitions in the right panel of Figure~\ref{RFtran}. The body of work
\cite{heyan,heyan2}
at general
temperatures $T$ makes the important point that the presence or
absence of superfluid order (as long as $T < T^*$)
will not lead to fundamentally
different physics. This observation is in contrast to
alternative calculations \cite{Strinati7,Stoof3,Basu} which
consider only the $T \approx 0 $ superfluid and/or separately the
normal phase.
Along with these new developments has been an experimental
and theoretical focus on
population imbalanced gases
\cite{SR06,SM06,Chien06,Rice2,ChienPRL}.
The observation \cite{KetterleRF} that extreme imbalance
may drive the system to an exotic
normal phase has captured the attention of the community.
This exotic phase appears to be associated \cite{Lobo,Chevy2} with the binding
of a small number of reverse spins to the majority states and this signature
is consistent with
RF experiments, as shown
theoretically \cite{Stoof3,Punk}. It should be stressed that
this binding is not the same as pairing which is a macroscopic
many body phenomenon. But it may, nevertheless, smoothly evolve into
pairing as one varies the concentration of reverse spins
\cite{MITtomoimb}, and in this way diminishes the population
imbalance.
With the growing appreciation for final state effects, an
interesting controversy has recently emerged concerning slightly
different data obtained on the 12 superfluid at unitarity. This
involves the original Innsbruck experiment \cite{Grimm4} and more
recent data from the MIT group \cite{Ketterlepairsize}. The latter
series of studies have led the authors to inquire as to whether
the pairing gap observations reported in Reference
\cite{Grimm4} might instead be associated with final state
effects. We comment on this possibility in Section
\ref{FinalStateCMP} of the paper, where we argue on behalf of the
original interpretation in Reference \cite{Grimm4}.
Finally, recent experiments on $^{40}$K from the JILA group
\cite{Jin6} have now demonstrated that it is possible to
measure the spectral functions directly using momentum resolved RF
pairing gap spectroscopy over a range of magnetic fields
throughout the BCS-BEC crossover. In these recent experiments
\cite{Jin6} the momentum of state 3 atoms is obtained using
time-of-flight imaging, in conjunction with 3D distribution
reconstruction
|
structure of the equations defining $k-th$ order equivalences of structures. Whereas in the theory of finite dimensional Lie groups the three fundamental theorems of Lie shine as neat and as beautiful as Botticelli's \textit{La Nascita di Venere} and Ingre's \textit{Le Printemps}, in the case of groupoids and Lie pseudo-groups they look more like Picasso's \textit{Guernica} and are a painful headache. In our case however, we only have to cope with the second theorem that fortunately is, in the present situation, very condescending and indulgent (sect.4,5,6, \cite{Cartan1937}, \cite{Cartan1938}).
\vspace{2 mm}
\noindent
Last but not least, here are a few words for "peer reviewers". Curiously enough, Sophus Lie and Élie Cartan did always row off the "main stream" for the simple reason that, at their time, essentially nobody was able to understand their writings. It took Cartan to understand what Lie did mean and Charles Ehresmann to understand Cartan. Significantly enough, Lie was Cartan's thesis adviser and Cartan was Ehresmann's adviser. As for Sophus Lie, he in fact never needed any adviser at all since he began writing in Norwegian so nobody would understand him anyway. Unfortunately, the author is unable to pinpoint anybody who did or who does (or who ever will) \textit{really} understand the full extent of Ehresmann's thoughts in all their galactic magnitude. Most probably we shall have to await for the next millennium\footnote{Il est à remarquer cependant que Ehresmann, bien qu'il construisait de très beaux arcs-en-ciel, ne s'est jamais soucié d'aller chercher le trésor se trouvant au bout. Par contre, Lie ainsi que Cartan allaient chercher désormais ce trésor sans se soucier à peindre au préalable de beaux arcs-en-ciel.}.
\section{Ehresmann's prolongation spaces}
Let \textit{P} be a differentiable manifold where, for convenience, we assume all the data of class $C^\infty$ though it would suffice to assume differentiability just up to a certain order. A \textit{finite} prolongation space of \textit{P} is a quadruple $(E,\pi,P,p)$ where \textit{E} is a differentiable manifold called the total space of the prolongation, $\pi:E\longrightarrow P$ a fibration (surmersion) and \textit{p} a prolongation operator that associates to each local diffeomorphism $\varphi$ of \textit{P} a local diffeomorphism $p\varphi$ of \textit{E} whose source and target are $\pi$-saturated open sub-sets inverse images of the source and target, respectively, of $\varphi$ and that furthermore obey the following requirements:
\vspace{4 mm}
\hspace{6 mm}\textit{i}) $p\varphi$ commutes with $\varphi$ and the projection $\pi$,
\vspace{2 mm}
\hspace{5 mm}\textit{ii}) \textit{p} is local and preserves pastings (\textit{recollements}),
\vspace{2 mm}
\hspace{4 mm}\textit{iii}) \textit{p} is a groupoid functor with respect to local diffeomorphisms ($\varphi$ being composable with $\psi$ whenever $\alpha(\psi)\cap\beta(\varphi)$ is non void, the unities being the identities on the open sets),
\vspace{2 mm}
\hspace{5 mm}\textit{iv}) Every differentiable one-parameter family $(\varphi_t)$ of local diffeomorphisms of \textit{P} prolongs, by \textit{p}, onto a one-parameter family of local diffeomorphisms of \textit{E} for which the vector field $d/dt(p\varphi_t)_{t=0}$ depends only upon $d/dt(\varphi_t)_{t=0}$ and projects onto it by $T\pi$.
\vspace{4 mm}
\noindent
We shall say that $p\varphi$ is the prolongation of $\varphi$ and, in order to simplify notations, the prolongation space will just be denoted by \textit{E}.
\noindent
Much in the same way, an \textit{infinitesimal} prolongation space of \textit{P} is a quadruple $(E,\pi,P,p)$ where the prolongation operator \textit{p} associates to each local vector field (infinitesimal transformation) $\xi$ given on \textit{P}, a \textit{prolonged} vector field $p\xi$ defined on the inverse image of the source $\alpha(\xi)$, this operation satisfying the corresponding (infinitesimal) properties:
\vspace{4 mm}
\hspace{6 mm}\textit{i}) $p\xi$ is $\pi$-projectable onto $\xi$,
\vspace{2 mm}
\hspace{5 mm}\textit{ii}) \textit{p} is local and preserves pastings,
\vspace{2 mm}
\hspace{4 mm}\textit{iii}) \textit{p} is a pre-sheaf morphism of Lie algebras.
\vspace{4 mm}
\noindent
Any finite prolongation space determines uniquely an infinitesimal prolongation space by which it is generated but what really matters is the converse that is not always true as we shall see in the sequel. Most fibre bundles considered in geometry (\textit{e.g.}, tensor bundles, Cartesian frames and co-frames, Stiefel truncated frames and co-frames, Grassmannian contact elements and their corresponding higher order analogues) are of course finite or infinitesimal prolongation spaces or both though our main interest is directed towards jet spaces. Let us also observe that prolongation spaces "compose" since the prolongation algorithm itself can be composed.
\vspace{4 mm}
\noindent
Let $\pi_0:P \longrightarrow M$ be a fibration (surmersion), denote by $J_k P$ the $k-th$ order jet bundle of local sections of $\pi_0$ and $\alpha$, $\beta$, $\rho_{hk}$ the well known projections. Following Ehresmann, we also denote by $\Pi_k P$ the groupoid of all invertible $k-$jets of the manifold \textit{P} ($k-$jets of local diffeomorphisms), by $J_k TP$ the vector bundle of all $k-$jets of local sections of $~TP\longrightarrow P~$ \textit{i.e.}, the jets of local vector fields on \textit{P} and finally by $\tilde{J}_k TP$ the fibration of all $k-$jets of local sections of the composite fibration $TP\longrightarrow P\longrightarrow M$. In the sequel, this \textit{tilde} notation will always be used for jets of sections of composite fibrations.
\newtheorem{prol}[LemmaCounter]{Lemma}
\begin{prol}
Given any fibration $\pi_0:P\longrightarrow M$, the jet space $J_k P$ has a natural infinitesimal prolongation space structure $(J_k P,\beta,P,p_k)$ where $\beta$ is the target map and where $p_k$ is the $k-th$ order standard prolongation operator for vector fields (by the target).
\end{prol}
\vspace{4 mm}
\noindent
Though the prolongation morphism $p_k$ goes back to Sophus Lie (in coordinates), we would like to add a few words so as to avoid any misunderstanding. It is \textit{not} possible to prolong, to $J_k P$, any local diffeomorphism $\varphi$ of \textit{P} since such a map can upset transversality (generic position) of a local section with respect to the $\pi_0$-fibres. However, when this condition is fulfilled, we can transform (at least locally) any $\pi_0$-section, whose image is contained in the domain of $\varphi$, into a new $\pi_0$-section and thereafter take its \textit{k}-jet. In particular, we shall then be able to prolong any $\pi_0$-projectable local diffeomorphism $\varphi$ of \textit{P} and the prolongation functor thus obtained, on projectable maps only, will of course fulfill the above stated properties of a finite prolongation space. Since we can always define a local vector field by its (local) one-parameter group $(\varphi_t)_t$ and since $\varphi_0=Id$, there is no restriction whatsoever in the prolongation procedure, to $k-th$ order, of \textit{any} local vector field defined on \textit{P} whereupon $J_k P$ becomes an authentic \textit{infinitesimal} prolongation space of \textit{P}.
\vspace{2 mm}
\noindent
Inasmuch, we can say that $TP,~J_k TP,~TM~and~J_k TM$ are prolongation spaces (finite and infinitesimal) of \textit{P} and \textit{M} respectively and, moreover, that $\tilde{J}_k TP$ is an infinitesimal prolongation space not only of \textit{TP} but also of \textit{P} for we can first prolong the local vector field $\xi$, defined on \textit{P}, to \textit{TP} and thereafter proceed with the above described "jet space prolongation". It should also be noted that $\tilde{J}_k TP$ is a (locally trivial) vector bundle with base space \textit{P} since the \textit{k-th} order tangency renders possible the vector space operations on the fibres. Finally, we would like to "stress" the condition (\textit{iii}) above by writing explicitly the equality $p(f\xi)=fp\xi$ where \textit{f} is any function.
\vspace{2 mm}
\noindent
As is well known, $(h+k)-$jets can become $h-$jets of $k-$jets and, inasmuch, $(h+k)-$jets can operate on "split" jets this motivating the following definitions:
\newtheorem{finite}[DefinitionCounter]{Definition}
\begin{finite}
a) The finite prolongation space E is said to be of order $\ell$ when, for any $k\geq 0~$, the k-jet of $p\varphi$ at the point $z\in E$ only depends upon the $(\ell+k)-$jet of $\varphi$ at the point $y=\pi (z)\in P$.
\vspace{2 mm}
\hspace{24 mm}b) The infinitesimal prolongation space E is said to be of order $\ell$ when, for any $k\geq 0$, the $k-$jet of $p\xi$, at the point $z\in E$, only depends upon the $(\ell+k)-$jet of $\xi$ at the point $y=\pi(z)\in P$.
\end{finite}
\vspace{2 mm}
\noindent
Recalling that the jet bundle $J_k TM$ identifies with the Lie algebroid of the Lie groupoid $\Pi_k M$, we can define for the above prolongation spaces of finite order and for any fixed positive integer \textit{k}:
\vspace{2 mm}
a) A left action
\begin{equation}
\Lambda_k:\Pi_{\ell+k} P~\times_P~\Pi_kE \longrightarrow \Pi_kE
\end{equation}
\vspace{2 mm}
\noindent
of the Lie groupoid $\Pi_{\ell+k}P$ on the groupoid $\Pi_{k}E$ (one can actually replace the last groupoid by the space of all $k-$jets $J_k(E,E)$) by setting
\begin{equation}
(j_{\ell+k}\varphi(\beta X),X) \longmapsto j_k(p\varphi)(\beta X)\cdotp X~,
\end{equation}
\vspace{2 mm}
\noindent
the fibre product being taken with respect to $\alpha$ , for the first factor, and with respect to $\pi\circ\beta$ , for the second factor,
\vspace{2 mm}
b) its infinitesimal generator namely, the morphism
\begin{equation}
\lambda_k:J_{\ell+k}TP~\times_P~E \longrightarrow J_k TE
\end{equation}
\vspace{2 mm}
\noindent
defined by
\begin{equation}
(j_{\ell+k}\xi (y),X)\longmapsto j_k(p\xi)(X)~,~y=\pi X~,
\end{equation}
\vspace{2 mm}
c) and, finally, its extension to an infinitesimal action on the jets of the tangent bundles
\begin{equation}
T\Lambda_k:J_{\ell+k+1}TP~\times_P~J_{k+1}TE \longrightarrow J_k TE
\end{equation}
\vspace{2 mm}
\noindent
defined by the left Lie bracket action
\begin{equation}
(j_{\ell+k+1}\xi(y),X)\longmapsto [j_{k+1}(p\xi)(\alpha X),X]~,~y=\pi\alpha X~,
\end{equation}
\vspace{2 mm}
\noindent
where we are forced to augment the order by 1 since brackets absorb one order of differentiation. The reason for putting in evidence the Lie bracket becomes apparent if we operate, as is usually done, by taking local one parameter groups and thereafter differentiating.
\vspace{2 mm}
\noindent
We now claim that the action $\Lambda_k$ is differentiable. In fact, using standard methods involving the Lie algebroid $J_kTP$ of the groupoid $\Pi_k P$ (or, if one prefers, the sheaf $\underline{J_kTP}$), we can define a local exponential map that, with the help of the property (\textit{iv}) will provide the required differentiability. It then also follows that the infinitesimal generator $\lambda_k$ as well as the infinitesimal action $T\Lambda_k$ are differentiable though this property can be proved directly by observing that the vector bundles involved are locally trivial and generated by local holonomic sections. It should also be observed that $T\Lambda_k$ is bilinear over \textbf{R} when the source and target spaces are fibered over \textit{E}.
\vspace{2 mm}
\noindent
We next observe that given a finite number of "composable" prolongation spaces, each of finite order, the composed prolongation space is also of finite order equal to the sum of the individually prescribed orders.
\vspace{2 mm}
\noindent
Other prolongation spaces that will be of our interest are those described in the following
\newtheorem{higher}[LemmaCounter]{Lemma}
\begin{higher}
To each finite or infinitesimal prolongation space $(E,\pi,P,p)$ and to each positive integer h corresponds, in a canonical way, a prolongation space $(J_h E,\alpha,P,p_h)$ verifying the following properties:
\vspace{2 mm}
a) When E is of finite order $\ell$ then $J_h E$ is of order $\ell+h$.
\vspace{2 mm}
b) The target projection $\beta:J_h E\longmapsto E$ is a surjective morphism of prolongation spaces i.e., respects the fibrations over P and commutes with the respective prolongation operations.
\vspace{2 mm}
c) More generally, the projection $\rho_{k,h}:J_h E\longrightarrow J_k E$ is a prolongation spaces morphism.
\end{higher}
\vspace{2 mm}
\noindent
Here, $J_h E$ is the set of all \textit{h}-jets of local sections of $\pi$ and is called the standard \textit{h-th order} prolongation of \textit{E}, the prolongation operation being the composite of the operation provided by \textit{E} followed by the standard jet prolongation. Needless to say, everything that was stated concerning the prolongation space $J_k P$ in the Lemma 1 can be paraphrased \textit{ipsis litteris} for the above data. Inasmuch, we can also repeat everything that was said previously for the prolongation space $\tilde{J}_k TP$ of \textit{infinitesimal variations} relative to the composite
\begin{equation*}
TP\longrightarrow P\longrightarrow M~,
\end{equation*}
\vspace{2 mm}
\noindent
the second arrow being equal to $\pi_0$, as well as for the $k-th$ order variations space $\tilde{J}_k E$ composed of all \textit{k}-jets of local sections of the composite fibration $E\longrightarrow P\longrightarrow M$, where $E\longrightarrow P$ is a prolongation space and $P\longrightarrow M$ simply a fibration giving rise to the finite or infinitesimal variations (\textit{cf.}, \cite{Kumpera1975} for the definitions). As for the bracket operation considered previously when defining an infinitesimal action, it is well defined in the present context due to the contact order conditions imposed. Finally, all the above considerations also extend to pre-sheaves $\Gamma(~)$ of local sections and enable us to operate with locally defined objects. In the sequel we shall also need the following extension of (3) namely,
\begin{equation}
\overline{\lambda}_k:J_{\ell+k}TP~\times_P~J_k E \longrightarrow TJ_k E~,
\end{equation}
\vspace{2 mm}
\noindent
defined by $(j_{\ell+k}\xi,j_k\sigma)= p_k\circ p(\xi)(j_k\sigma)$ and where $TP\longrightarrow P$ is the tangent prolongation space. In much the same way and using the prolongation operator, we can define the morphism
\begin{equation}
\tilde{\lambda}_k:J_{\ell+k}TP~\times_P~J_k E \longrightarrow J_k TE~,
\end{equation}
\vspace{2 mm}
\noindent
as well as the \textit{semi-holonomic} extension
\begin{equation}
\overline{\lambda}_{k+h}:J_{\ell+k+h}TP~\times_P~J_k E \longrightarrow J_h(TJ_k E)~.
\end{equation}
\vspace{2 mm}
\noindent
We thus see that the choices are many and, in fact, we could go on much further with the Ehresmannian game of the \textit{jeu de la théorie des jets} by considering for instance semi-holonomic, sesqui-holonomic and (definitely) non-holonomic jets but fortunately these will be of no purpose to us so we might as well forget about them right away. Furthermore, and this will be very useful, we can play the Ehresmannian game with \textit{differential forms} and \textit{co-tangent bundles} that, in this case, will act \textit{co-variantly}. As for the prolongation operation, there is of course essentially just one such operation that can however be vested under two or three garbs. Instead of prolonging by the target, as is done in the present paper, we can also prolong by the source or even, combining the two, we can prolong via the \textit{anchor}, the same one that holds Kirill anchored and not on the sail. In later sections we shall also introduce \textit{merihedric} prolongations (\textit{prolongements mériédriques de Élie Cartan}) as well as such transformations on jet spaces and higher order Grassmannians. Contrary to what is standard, there are uncountably many possibilities for the merihedric functor each one having its own merits and outstanding performance. In fact, Medusa as well as Mona Lisa claim that the merihedric setup was the magical trump and joker hidden in Cartan's sleave. Fortunately we shall need not talk about merihedric jet spaces, the standard ones being still of good use. One last remark: When differentiability is replaced by analyticity in the initial requirements for the prolongation spaces then, of course, all the other data also become analytic. We hope as well that the reader already noticed our small notational changes. Instead of the standard $j^k_x\sigma~$, we prefer to write $j_k\sigma(x)$ and, instead of $J^k$, we write $J_k$, such notations rendering more pleasant and "co-variant" their composites.
\section{Symbols}
In this section we shall often refer to former publications (\cite{Kumpera1971}, \cite{Kumpera1972}) for the notations and the results which, however, are completely standard and well known. In this sense, we denote by \textit{D} the Spencer operator, by $S^k$ the bundle of symmetric tensors and by $\delta$ the Spencer operator restricted to the principal parts.
\noindent
Our first task is to examine what happens with the \textit{kernels} and consequently write:
\begin{equation*}
0\longrightarrow S^{\ell+k}T^*P\otimes TP\times_P E\longrightarrow J_{\ell+k}TP\times_P E\longrightarrow J_{\ell+k-1}TP\times_P E\longrightarrow 0 \end{equation*}
\begin{equation*}
\hspace{20 mm}\downarrow\ell_k\hspace{32 mm}\downarrow\lambda_k\hspace{27 mm}\downarrow\lambda_{k-1}\hspace{4 mm}
\end{equation*}
\begin{equation*}
0\longrightarrow\hspace{4 mm}S^k T^*E\otimes TE\hspace{8 mm}\longrightarrow\hspace{7 mm} J_k TE\hspace{7 mm}\longrightarrow\hspace{8 mm} J_{k-1}TE\hspace{4 mm}\longrightarrow 0
\end{equation*}
\vspace{4 mm}
\noindent
as well as
\begin{equation*}
0\!\rightarrow\!S^{\ell+k}T^*P\otimes TP\times_P J_k E\!\rightarrow\!J_{\ell+k}TP\times_P J_k E\!\rightarrow\!J_{\ell+k-1}TP\times_P J_{k-1} E\!\rightarrow\! 0
\end{equation*}
\begin{equation*}
\hspace{12 mm}\downarrow\overline{\ell}_k\hspace{31 mm}\downarrow\overline{\lambda}_k\hspace{31 mm}\downarrow\overline{\lambda}_{k-1}
\end{equation*}
\begin{equation*}
0\rightarrow S^k T^*E\otimes VE\times_E J_kE\hspace{2 mm}\longrightarrow\hspace{2 mm} TJ_kE\hspace{3 mm}\longrightarrow\hspace{2 mm}TJ_{k-1} E\times_{J_{k-1}E} J_kE\!\rightarrow\! 0
\end{equation*}
\vspace{4 mm}
\noindent
and observe that the above $\overline{\lambda}_k$ is equal to $\lambda_0$ relative to the prolongation space $J_k E$ of \textit{P}, the later being of finite order $\ell+k$. More generally and with the obvious notations, $\overline{\lambda}_{k+h}(E)=\lambda_h(J_k E)$. Next, we claim that each family $(\ell_k)$ and $(\overline{\ell}_k)$ is a \textit{natural transformation} of the corresponding $\delta-cohomology~complexes$ this becoming apparent by examining the commutativity of the two diagrams below and observing that the vertical map $\overline{\ell}_k$ in the second diagram is in fact equal to
\begin{equation*}
[S^{\ell+k}T^*P\otimes TP\times_P J_1 E]\times_{J_1 E} J_k E\xrightarrow{\overline{\ell}_k\times Id}[S^k T^*E\otimes VE\times_E J_1 E]\times_{J_1 E} J_k E
\end{equation*}
\vspace{2 mm}
\noindent
the term $\overline{\ell}_k$ depending only upon its projection in $J_1 E$. In order to show the naturality we are forced, of course, to confront the above maps with the Spencer operator hence the necessity in extending the actions to the sheaf level.
\begin{sidewaysfigure}[htp!]
\vspace{125 mm}
\begin{equation*}
0 \longrightarrow S^{\ell+k}T^*P~\otimes~TP~\times_P~E\xrightarrow{\delta\times Id}T^*P~\otimes~S^{\ell+k-1}T^*P~\otimes~TP~\times_P~E\xrightarrow{\delta\times Id}\wedge^2T^*P~\otimes~S^{\ell+k-2}T^*P~\otimes~TP~\times_P~E\hspace{1 mm}\xrightarrow{\delta\times Id}\hspace{1 mm}\bf{\cdots}
\end{equation*}
\begin{equation}
\hspace{9 mm}\downarrow\ell_k \hspace{51 mm}\downarrow\pi^*~\otimes~\ell_{k-1}\hspace{50 mm}\downarrow\pi^*~\otimes~\ell_{k-2}
\end{equation}
\begin{equation*}
0\longrightarrow \hspace{4 mm}S^k T^*E~\otimes~TE\hspace{10 mm}\xrightarrow{\delta}\hspace{8 mm}T^* E~\otimes~S^{k-1} T^*E~\otimes~TE\hspace{6 mm}\xrightarrow{\delta}\hspace{8 mm}\wedge^2T^*E~\otimes~S^{k-2} T^*E~\otimes~TE\hspace{7 mm}\xrightarrow{\delta}\hspace{4 mm}\bf{\cdots}
\end{equation*}
\vspace{30 mm}
\begin{equation*}
0\rightarrow S^{\ell+k}T^*P\otimes TP\times_P J_1 E\xrightarrow{\delta\times Id}\hspace{1 mm}T^*P\otimes S^{\ell+k-1}T^*P\otimes TP\times_P J_1 E\xrightarrow{\delta\times Id}\hspace{1 mm}\wedge^2T^*P\otimes S^{\ell+k-2}T^*P\otimes TP\times_P J_1 E\xrightarrow{\delta\times Id}\bf{\cdots}
\end{equation*}
\begin{equation}
\hspace{1 mm}\downarrow\overline{\ell}_k\hspace{43 mm}\downarrow Id_{T^*P}~\otimes~\overline{\ell}_{k-1}\hspace{34 mm}\downarrow Id_{\wedge^2T^*P}~\otimes~\overline{\ell}_{k-2}
\end{equation}
\begin{equation*}
0\rightarrow\hspace{1 mm}S^kT^*E\otimes VE\times_E J_1 E\xrightarrow{\delta\times Id}T^*P\otimes S^{k-1}T^*E\otimes VE\times_E J_1 E\xrightarrow{\delta\times Id}\hspace{1 mm}\wedge^2T^*P\otimes S^{k-2}T^*E\otimes VE\times_E J_1 E\xrightarrow{\delta\times Id}\bf{\cdots}
\end{equation*}
\end{sidewaysfigure}
\noindent
We first consider the diagram (10), extend the infinitesimal generator (3) to the pre-sheaf of local sections and derive thereafter the sheaf morphism
\begin{equation*}
\underline{\lambda}_k:\underline{J_{\ell+k}TP}~\times_P~E~\longrightarrow~\underline{J_kTE},
\end{equation*}
\vspace{2 mm}
\noindent
where $\underline{\lambda}_k(\underline{\sigma}_y,q)=(\underline{\Gamma(\lambda_k)\sigma})_q~,~y=\pi(q)~,~\Gamma(~)$ denoting the set of all local sections. It now suffices to show that $\underline{\lambda}_k$ commutes with \textit{D} or, in other words, that the diagram (12) below commutes:
\vspace{8 mm}
\begin{equation*}
\underline{J_{\ell+k}TP}~\times_P~E~\xrightarrow{D~\times~Id}~(\underline{T^*P}~\times_P~E)~\otimes~
(\underline{J_{\ell+k-1}TP}~\times_P~E)
\end{equation*}
\begin{equation}
\downarrow\underline{\lambda}_k\hspace{30 mm}\downarrow~\underline{\pi}^*\otimes\underline{\lambda}_{k-1}
\end{equation}
\begin{equation*}
\underline{J_kTE}\hspace{6 mm}\xrightarrow{D}\hspace{10 mm}\underline{T^*E}~\otimes~\underline{J_{k-1}TE}\hspace{6 mm}
\end{equation*}
\vspace{4 mm}
\noindent
The commutativity is obvious on holonomic sections since \textit{D} vanishes on them and thereafter it suffices to argue as in \cite{Kumpera1972}, pg.75, using the formula (2) shown on that page. The restriction to the symbols then proves our claim for the first square of (10). To complete the proof, it suffices to adapt the diagram (12) to the other squares of (10) by iterating successively the Spencer operator. However, the previously mentioned formula (2) implies immediately the desired result as soon as the first square commutes. The commutativity of (11) is proved essentially in the same manner by observing that $\overline{\lambda}_k$ is the pullback of $\lambda_k$ via the composite $p_k\circ\sharp~$, where $~\sharp:J_kTE~\times_E~J_kE\longrightarrow \tilde{J}_kTE~$ is the fibre bundle morphism considered in \cite{Kumpera1971}, Propos.11.1 (\textit{M} standing in the place of \textit{P}). The proof can then be achieved with the help of the \textit{reduced form of the holonomic prolongation operator} acting on infinitesimal variations and the Theorem 12.1.
\vspace{4 mm}
\noindent
Since the $(\ell_k)$ and $(\overline{\ell}_k)$ are natural transformations of the $\delta$ complexes and since the $(\lambda_k)$ and $(\overline{\lambda}_k)$ commute with \textit{D} it also follows that these natural transformations are compatible (commute) with the algebraic prolongations operating on the principal parts (i.e., on the symbols).
\vspace{4 mm}
\noindent
For the sake of not omitting any useful information, let us finally observe that the total spaces of the finite prolongation spaces are always locally trivial bundles over their base spaces, a fact that is not true anymore for infinitesimal prolongation spaces (\textit{e.g.}, withdraw a point from the total space). However, when the fibres over the base space are compact and connected then these total spaces also become locally trivial, this last claim being just a special case of very general results due to Ehresmann.
\section{Lie and Cartan's notion of structure}
Let $(E,\pi,P,p)$ be a finite or infinitesimal prolongation space.
\newtheorem{struct}[DefinitionCounter]{Definition}
\begin{struct}
A finite or infinitesimal (almost-)structure of species E on the manifold \text{P} is the data provided by a global section (continuous, differentiable, analytic) of the prolongation space \textit{E}.
\end{struct}
\vspace{2 mm}
\noindent
We can define as well a structure of species \textit{E} above an open set \textit{U} of \textit{P} by simply taking a local section\footnote{Il serait malheureux de l'appeler une \textit{structure locale} car cette terminologie, due à Ehresmann, a un sens très précis autre que ci-dessus. Toute structure d'espèce \textit{E} est une espèce de structure locale.} and thereafter derive the notions of \textit{germ of structure, $k-$jet of structure, $k-th$ order contact element of a structure} inasmuch as an \textit{infinite jet of a structure} (formal structure at a point). We should also mention that the above definitions are not any of Spencer's \textit{abstract nonsense} since many of the well known structures (\textit{e.g.}, Riemannian, conformal, almost-complex, almost-symplectic, etc.) are defined by global sections of locally trivial tensor bundles with fibres homogeneous spaces of linear groups. Inasmuch, almost-product, contact, Stiefel and many other structures are defined by sections of well known bundles in homogeneous spaces. Moreover, as we shall see further, the \textit{integrability} of an almost-structures can be detected by a Pfaffian system (also a structure!) defined on \textit{E} or on one of its prolongation spaces.
\vspace{2 mm}
\noindent
Given a structure \textit{S} of species \textit{E} (a finite prolongation space), associated to it are the pseudo-group $\Gamma(S)$ of all its local automorphisms as well as the infinitesimal pseudo-algebra $\mathcal{L}(S)$ of all its local infinitesimal automorphisms (vector fields). By definition, $\varphi\in\Gamma(S)$ when $p\varphi$ leaves invariant the sub-manifold \textit{im S} or, in other terms, when $\varphi(S)\stackrel{def}{=}p\varphi\circ S\circ\varphi^{-1}=S$ in the appropriate domains. Much in the same way, $\xi\in\mathcal{L}(S)$ when $\xi$ generates a local one-parameter group $(\varphi_t)$ with all its elements in $\Gamma(S)$ which amounts to say that $p\xi$ is tangent to \textit{im S} or equivalently that the Lie derivative $\theta(p\xi)S=\frac{d}{dt}~\varphi_t^{-1}(S)|_{t=0}$ vanishes (for each fixed $Y\in im~S$, take the tangent vector to the curve issued from \textit{Y}). Quite often, it is convenient to consider (differentiable) one-parameter families of local transformations that are not necessarily local one-parameter groups. With that in mind, let us next remark that such a local one-parameter family $(\varphi_t)$ defined in \textit{P} is entirely composed of elements belonging to $\Gamma(S)$ if and only if the following two properties are fulfilled:
\vspace{2 mm}
a) $\varphi_{t_0}\in\Gamma(S)$ for some value $t_0$ (\textit{t} varies in an open interval) and
\vspace{2 mm}
b) for each \textit{u}, $T\varphi_u^{-1}\circ\frac{d}{dt}~(\varphi_t)|_{t=u}\circ\varphi_u\in\mathcal{L}(S)$.
\vspace{2 mm}
\noindent
The set of all local transformations arising from such one-parameter families is obviously equal to $\Gamma(S)$. When \textit{E} is just an infinitesimal prolongation space, we can still define $\mathcal{L}(S)$ by using the tangency with respect to \textit{im S} and afterwards defining $\Gamma_{inf}(S)$ as being the pseudo-group generated (through composition and gluing) by the set of all the elements belonging to the local one-parameter families verifying $\varphi_{t_0}=Id~$ together with the above condition (b). We can further simplify (re-parametrize) by taking $t_0=0$ and clearly $\Gamma_{inf}(S)\subset \Gamma(S)$ when \textit{E} is also a finite prolongation space, $\mathcal{L}(S)$ being its infinitesimal generator. The condition $\varphi_{t_0}=Id$ is an acceptable simplification (by composing, if necessary, with $\varphi_{t_0}^{-1}$) and enables us to initiate the one-parameter family with the automorphism \textit{Id} that can always be prolonged.
\vspace{4 mm}
\noindent
Let us now examine the \textit{defining equations} for $\Gamma(S)$ and $\mathcal{L}(S)$ hence examine the nature of the infinitesimal jets that are automorphisms of \textit{S} up to a certain order and this brings us to examine \textit{contact} properties. In general, being given a differentiable manifold \text{M} and two sub-manifolds $N,~N'$ of the same dimension \textit{p} and meeting at a point \textit{x}, we shall say, according to the general definition of higher order Grassmannians, that these two sub-manifold have a $k-th$ order contact at the point \textit{x} when there exists an invertible $k-$jet $\mu$ of \textit{N} onto $N'$ with source and target \textit{x} such that $j_k\iota (x)=j_k\iota '(x)\cdot\mu$, where $\iota$ and $\iota '$ are the inclusions of the sub-manifolds. The equivalence classes under this relation are called the $k-th$ order contact elements of dimension \textit{p} on the manifold \textit{M} and at the point \textit{x}. Let us now take a vector field $\xi$ defined in a neighborhood of \textit{x}. The composite $\xi\circ\iota$ is a local section of the vector bundle $TM|_N\longrightarrow N$ , hence its $k-$jet at the point \textit{x} is an element of $J_k(TM|_N)$ and, of course, $J_kTN\subset J_k(TM|_N)$. We shall say that $\xi$ is tangent of order \textit{k} to \textit{N} at \text{x} when $j_k(\xi\circ\iota)(x)\in J_kTN~$. Furthermore, if $N$ and $N'$ have a contact of order $k+1$ at the point \textit{x} then $TN$ and $TN'$ have, along any point of $T_x N=T_x N'$ a contact of order \textit{k} when considered as sub-manifolds of $TM$: The contact relation is obtained by extending to tangent vectors the initial contact relation. In particular, we infer that the vector fields tangent of order \textit{k} to the sub-manifold $N$ at \textit{x} are the same as those tangent to $N'$. Conversely, the last condition (on the tangency of $TN$ with $TN'$) implies the $(k+1)-st$ order tangency of $N$ with $N'$ at the point \textit{x} (\cite{Molino1977}, Propos.3, pg.20). Observing that $j_k(\xi\circ\iota)(x)=j_k\xi(x)\cdot j_k\iota(x)$ , we conclude that the $k-th$ order tangency concept for vector fields is more properly a notion of tangency of the elements of $J_kTM$ with a sub-manifold $N$ or, in a more general form, with the elements belonging to the fibre bundle of $(k+1)-st$ order $p-$dimensional contact elements of $M$ ($p=dim~N$). We finally remark that the groupoid $\Pi_k M$ operates to the left on the fibre bundle of $k-th$ order contact elements. There is an alternative definition, due to Ehresmann, stating that a $k-th$ order contact element is a linear subspace in the space of $k-th$ order tangent vectors. Though apparently simpler, this definitions seems to lack visibility. It is easy to \textit{see} tangency but not so easy to \textit{see} partial derivatives.
\vspace{4 mm}
\noindent
We now translate the above definitions of contact order into a form more suitable to our context, return to the local automorphisms and consider, to start with, a finite prolongation space $~E\longrightarrow P~$ of order $\ell$ as well as a structure \textit{S} of species \textit{E}. A local diffeomorphism $\varphi$ of \textit{P} is called a $k-th$ order automorphism of \textit{S} at the point \textit{y} whenever the image sub-manifold $p\varphi(im~S)$ has a $k-th$ order contact with \textit{im S} at the point $S\circ\varphi(y)$. Since the contact only depends on $j_k p\varphi(S(y))$ and the prolongation space is of finite order $\ell$, we infer that this notion boils down to the following: An element $Z\in\Pi_{\ell+k} P$ with source \textit{y} and target \textit{y'} is a $k-th$ order automorphism of \textit{S} when $pZ$, $k-$jet of source $S(y)$ corresponding to \textit{Z} ($(\ell+k)-$jets prolong to $k-$jets), transforms the $k-th$ order contact element of \textit{S} at $S(y)$ onto the $k-th$ order contact element of \textit{S} at $S(y')$. However, the contact element $pZ(S)$ is represented by the image of the local section $p\varphi\circ S\circ\varphi^{-1}$ where $\varphi$ is a representative of \textit{Z} i.e., $Z=j_{\ell+k}\varphi~$. Moreover, the images of two local sections $\sigma$ and $\sigma'$ of \textit{E} define the same $k-th$ order contact element at a common point $z\in E$ if and only if $j_k\sigma(\pi z)=j_k\sigma'(\pi z)$ and therefore $pZ(S)$ and \textit{S} define the same $k-th$ order contact element at $S(y')\in E$ if and only if $j_k(p\varphi\circ S\circ\varphi^{-1})(y')=j_k S(y')$ that is to say, when
\begin{equation*}
(pZ)\cdot j_kS(y)\cdot Z_k^{-1}=j_kS(y')~,\hspace{5 mm} Z_k=\rho_{k,\ell+k}Z .
\end{equation*}
\vspace{4 mm}
\newtheorem{highest}[LemmaCounter]{Lemma}
\begin{highest}
A jet $Z\in\Pi_{\ell+k}P$ is a $k-th$ order automorphism of S if and only if $Z(j_kS(\alpha Z))=j_kS(\beta Z)$ by means of the left action of $\Pi_{\ell+k}P$ on $J_kE$ (notations: $Z(~)=p_k\varphi(~)$, $\varphi$ represents Z and $p_k=p_k\circ p$). The $(\ell+k)-th$ order equation defining $\Gamma(S)$ is provided by the closed sub-groupoid (P being replaced by $\alpha(S)$ whenever S is not global)
\begin{equation}
\mathcal{R}_{\ell+k}(S)=\{Z\in\Pi_{\ell+k}P~|~Z(j_kS(\alpha Z))=j_kS(\beta Z)\}.
\end{equation}
\end{highest}
\vspace{4 mm}
\noindent
This equation is not \textit{complete} in general \textit{i.e.}, $\Gamma(S)$ might be strictly contained in $Sol~\mathcal{R}_{\ell+k}
(S)$ and, furthermore, the inclusion $\mathcal{R}_{\ell+k}(S)\supset J_{\ell+k}\Gamma(S)$ might also be strict.
\vspace{4 mm}
\noindent
We next consider the non-linear differential operator
\begin{equation*}
\mathcal{D}(S):~\varphi\in\underline{Di\!f\!f~P}~\longmapsto ~\varphi^{-1}(S)=(p\varphi^{-1})\circ S\circ\varphi\in\underline{E}
\end{equation*}
\vspace{2 mm}
\noindent
that is of order $\ell$ since it is obtained by the composition of $j_\ell$ and the fibered morphism
\begin{equation*}
\Phi(S):~Y\in\Pi_\ell P~\longmapsto~Y^{-1}(S(\beta Y))\in E
\end{equation*}
\vspace{2 mm}
\noindent
where $Y(~)$ is the action of the groupoid $\Pi_\ell P$ on \textit{E}. The $k-th$ order prolongation of $\mathcal{D}(S)$ is the operator of order $\ell+k$
\begin{equation*}
p_k\mathcal{D}(S):~\varphi\in\underline{Di\!f\!f~P}~\longmapsto ~j_k(p\varphi^{-1}\circ S\circ\varphi)=p_k\varphi^{-1}\circ j_kS\circ\varphi=
\end{equation*}
\begin{equation*}
\hspace{30 mm}=\varphi^{-1}(j_kS)\in\underline{J_kE}
\end{equation*}
\vspace{2 mm}
\noindent
and is obtained, with the aid of $j_{\ell+k}$, by means of the fibered morphism
\begin{equation*}
p_k\Phi(S):~Z\in\Pi_{\ell+k}P~\longmapsto~Z^{-1}(j_kS(\beta Z)\in J_kE
\end{equation*}
\vspace{2 mm}
\noindent
where $Z(~)$ is the action of $\Pi_{\ell+k}P$ on $J_kE$. The following short non-linear sequences
\begin{equation*}
1\longrightarrow\mathcal{R}_{\ell+k}(S)\longrightarrow\Pi_{\ell+k}P\xrightarrow{p_k\Phi(S)}(J_kE,j_kS)
\end{equation*}
\vspace{2 mm}
\noindent
and
\begin{equation*}
1\longrightarrow\underline{\Gamma(S)}\longrightarrow\underline{Di\!f\!f P}\xrightarrow{p_k\mathcal{D}(S)}(\underline{J_kE},j_kS)
\end{equation*}
\vspace{2 mm}
\noindent
are therefore exact in the set theoretical sense. When $im~p_k\Phi(S)$ is a sub-manifold of $J_kE$ and
\begin{equation*}
p_k\Phi(S):~\Pi_{\ell+k}P~\longrightarrow~im~p_k\Phi(S)
\end{equation*}
\vspace{2 mm}
\noindent
is a submersion then $\mathcal{R}_{\ell+k}(S)$ is a regularly embedded sub-manifold of $\Pi_{\ell+k}P$ and, for all \textit{h}, the equation $\mathcal{R}_{\ell+k+h}(S)$ is the prolongation in the usual sense i.e., the sub-set $~\Pi_{\ell+k+h}P\cap J_h\mathcal{R}_{\ell+k}(S)$ of $\mathcal{R}_{\ell+k}(S)~$. Taking into account the Proposition 2.1 in \cite{Gold1969}, we are tempted to replace the above two hypotheses by the unique assumption:
\begin{equation*}
p_k\Phi(S):~\Pi_{\ell+k}P~\longrightarrow~J_kE
\end{equation*}
\vspace{2 mm}
\noindent
is locally of constant rank in a neighborhood of each point belonging to $\mathcal{R}_{\ell+k}(S)$. Unfortunately (or perhaps fortunately) the above Proposition is inexact. If, with the notations of the above citation, X is reduced to a point, this Proposition would imply the remarkable statement: \textit{Théorème. Toute immersion est un plongement}. We shall return later to this matter and show that, in the specific case of the equations $\mathcal{R}_{\ell+k}(S)$, the first hypothesis can in fact be replaced by weaker conditions.
\vspace{4 mm}
\noindent
Let us now examine the infinitesimal aspects. A local vector field $\xi$ defined on \textit{P} is said to be a $k-th$ order infinitesimal automorphism of the structure \textit{S} at the point \textit{y} when the prolonged vector field $p\xi$ is tangent up to order \textit{k} to the sub-manifold $im~S$ at the point $z=S(y)$. Since $\pi\circ S=Id$, this condition can be replaced by
\begin{equation*}
j_k(p\xi\circ S)(y)=j_k(TS\circ\xi)(y),
\end{equation*}
\vspace{2 mm}
\noindent
this last condition measuring the "distancing" of $p\xi$ from $S_*\xi$ along $im~S$ in the vicinity of the point $z=S(y)$. The prolongation space \textit{E} being of finite order $\ell$, this last condition translates by the following:
An element $Y\in J_{\ell+k}TP$ of source \textit{y} is a $k-th$ order infinitesimal automorphism of \textit{S} whenever
\textit{pY}, $k-$jet of source $S(y)$ corresponding to \textit{Y}, is tangent of order \textit{k} to \text{im S} at the point
$S(y)$. We next observe that both $p\xi\circ S$ and $TS\circ\xi$ are local sections of the \textit{tangent Lie fibration} $TE\longrightarrow P$ composite of the natural projection $TE\longrightarrow E$ with $\pi$ and, consequently, the \textit{
reduced form} of the holonomic prolongation (\cite{Kumpera1971}, Théorème 12.1) shows that the vector $(p_k\xi)_{j_kS(y)}$ ($p_k=p_k\circ p$) only depends on $j_k(p\xi\circ S)(y)$ and the second line of the diagram (22.29) in the above citation, $\S$ 22, restricted to the fixed section $j_kS$ (the section being fixed, we are not forced any more to ascend to $J_{k+1}E$), shows that the jets $j_k(TS\circ\xi)(y)$ are precisely those, among the elements of $\tilde{J}_kTE$, for which the vector associated by prolongation is tangent to $im~j_kS$ at the point $j_kS(y)$. Summing up, we obtain the following
\newtheorem{lowest}[LemmaCounter]{Lemma}
\begin{lowest}
The jet $Y\in J_{\ell+k}TP$ is a $k-th$ order infinitesimal automorphism of S if and only if the vector $p_kY$ corresponding to it, at the point $j_kS(\alpha Y)$, is tangent to $im~j_kS$ ($p_kY=(p_k\xi)_{j_kS(\alpha Y)}$ where $\xi$ is a representative of \textit{Y}). Moreover, the equation of order $\ell+k$ of the Lie algebroid $\mathcal{L}(S)$ is given by
\begin{equation}
\textbf{R}_{\ell+k}(S)=\{Y\in J_{\ell+k}TP~|~p_kY\in (j_kS)_*T_{\alpha Y}P\}.
\end{equation}
\end{lowest}
\vspace{4 mm}
\noindent
Since the map $\lambda_k$ is a differentiable vector bundle morphism and the associated infinitesimal action $\Gamma(\lambda_k)$ is also a morphism of Lie algebra pre-sheafs, it follows that the linear equation \textbf{R}$_{\ell+k}(S)$ is a Lie equation (\cite{Kumpera1972}, \cite{Malgrange1972}). Moreover, $\mathcal{L}(S)=Sol~$\textbf{R}$_{\ell+k}(S)$ though, in general, this equation is not \textit{complete}. We next consider the linear differential operator
\begin{equation*}
\textbf{D}(S):~\xi\in\underline{TP}~\longmapsto~\theta(\xi)S=\frac{d}{dt} \varphi_t^{-1}(S)|_{t=0}\in\underline{VE~|~im~S}
\end{equation*}
\vspace{2 mm}
\noindent
where $VE$ is the vector sub-bundle of $TE$ composed by the $\pi-$vertical vectors and $\varphi_t$ is a local one-parameter family \textit{e.g.}, the local group, such that $\frac{d}{dt}\varphi_t|_{t=0}=\xi$. The restriction $VE~|~im S$ can be considered a bundle with base space \textit{P} since it can be identified to $S^{-1}VE$ and, according to the general definitions, $\bf{D}(S)$ is the linearization of $\mathcal{D}(S)$ along the section \textit{Id} of $\underline{Di\!f\!f~\!P}$. It is a linear operator of order $\ell$ whose associated linear morphism
\begin{equation*}
\Psi(S):~J_\ell TP\longrightarrow VE~|~im~S
\end{equation*}
\vspace{2 mm}
\noindent
is defined as follows: Let $Y=j_\ell\xi(y),~(\varphi_t)$ a local one-parameter group associated to $\xi$ and $Y_t=j_\ell\varphi_t(y)$. Then
\begin{equation*}
\Psi(S)(Y)=\frac{d}{dt}[Y_t^{-1}(Sy_t)]_{t=0},
\end{equation*}
\vspace{2 mm}
\noindent
where $y_t=\beta Y_t$ and where, for simplicity, we write $Sy_t$ though meaning $S(y_t)$. However, $Y_t^{-1}(Sy_t)=p\varphi_t^{-1}(Sy_t)=p\varphi_{-t}(Sy_t)$ and, consequently, $\Psi(S)(X)=-(p\xi)_{S(y)}+S_*(\xi_y)$. We infer that
\begin{equation}
\Psi(S)(Y)=-\textbf{v}(p\xi_{S(y)}),
\end{equation}
\vspace{2 mm}
\noindent
where \textbf{v}( ) is the vertical component of a vector following the direct sum decomposition $T_{S(y)}E=V_{S(y)}E~\oplus~S_*(T_yP)$ along \textit{im S}.
\vspace{2 mm}
\noindent
Let us now return to the general definitions of \cite{Kumpera1975}, I, $\S$17 and, in particular, to the last sequence, on pg.341, exhibiting the linear morphism $\tau_k(T\mathcal{D})$ that defines the differential operator $T\mathcal{D}$. In the present case (and adapting slightly the notations), it concerns the vertical linearization along the section $Id$ for the specific fibration $P=M~\times~M~\longrightarrow~M~$, $(x,x')~\mapsto~x$ , where $J_\ell P$ is replaced by $J_\ell M=J_\ell(M,M)$ (each section of $P$ identifies with a map $M~\longrightarrow~M$), $\tilde{J}_\ell VP~|~J_\ell M=J_\ell M~\times_M~J_\ell TM$ (fibre product with respect to the projection $\beta$) and where $p_\ell:~J_\ell M~\times_M~J_\ell TM~\longrightarrow~VJ_\ell M$ simply becomes the canonical identification. The morphism
\begin{equation*}
T\tau_\ell(\mathcal{D})=T\Phi(S):~VJ_\ell M~|~im~Id~\longrightarrow~VE~|~im~S
\end{equation*}
\vspace{2 mm}
\noindent
becomes, by means of the canonical identification, equal to $\Psi(S)$. The $k-th$ order prolongation of $\textbf{D}(S)$ is then the linear operator of order $\ell+k$
\begin{equation*}
p_k\textbf{D}(S):~\xi\in\underline{TM}~\longmapsto~j_k[\theta(\xi)S]\in\underline{J_k(VE~|~im~S)},
\end{equation*}
\vspace{2 mm}
\noindent
that we can, by means of the isomorphism $p_k:~\tilde{J}_k VE~\longrightarrow~VJ_kE$ (\cite{Kumpera1975}, Propos.12.2), replace by
\begin{equation*}
\underline{p_k}\circ p_k\textbf{D}(S):~\underline{TM}~\longrightarrow~\underline{VJ_k E~|~im~j_kS}
\end{equation*}
\vspace{2 mm}
\noindent
that, in turn, is nothing else but the linearized, along the identity section, of $p_k\mathcal{D}(S)$ (\cite{Kumpera1975}, pg.342). Furthermore, the linear morphism associated to $\underline{p_k}\circ p_k\textbf{D}(S)$ is on the one hand equal to $p_k\circ p_k\Psi(S)$ and on the other, for being a linearization and due to the canonical identification, equal to
\begin{equation}
T\tau_{\ell+k}(p_k\mathcal{D})=Tp_k\Phi(S):~VJ_{\ell+k}M~|~im~Id~\longrightarrow~VJ_k E~|~im~S.
\end{equation}
\vspace{2 mm}
\noindent
An entirely similar calculation, where we shall replace
\begin{equation*}
\varphi_t^{-1}(S)=\mathcal{D}(S)\varphi_t\hspace{4 mm}by\hspace{4 mm}\varphi_t^{-1}(j_kS)=p_k\mathcal{D}(S)\varphi_t,
\end{equation*}
\vspace{2 mm}
\noindent
will show that
\begin{equation}
p_k\circ p_k\Psi(S)(X)=-\textbf{v}(p_k\xi_{j_kS(x,x')})
\end{equation}
\vspace{2 mm}
\noindent
where $X\in J_{\ell+k}TM$ and $v$ is the vertical component of a vector in the direct sum decomposition $T_ZJ_kE=V_ZJ_kE~
\oplus~(j_kS)_*(T_xM),~Z=j_kS(x,x')$ and this implies the exactness of the sequence
\begin{equation}
0~\longrightarrow~\textbf{R}_{\ell+k}(S)~\longrightarrow~J_{\ell+k}TM~\xrightarrow{p_k\Psi(S)}~J_k(VE~|~im~S)
\end{equation}
\vspace{2 mm}
\noindent
for all $k\geq 0~$, hence $\textbf{R}_{\ell+k+h}(S)$ is the prolongation of order \textit{h} of the equation $\textbf{R}_{\ell+k}(S)$ whenever $p_k\Psi(S)$ is locally of constant rank as a linear mapping on each fibre (\textit{i.e.}, when $\textbf{R}_{\ell+k}(S)$ is a locally trivial vector bundle). Moreover, the sequence
\begin{equation}
0~\longrightarrow~\underline{\mathcal{L}(S)}~\longrightarrow~\underline{TM}~\xrightarrow{p_k\textbf{D}(S)}~\underline{J_k(VE~|~im~S)}
\end{equation}
\vspace{2 mm}
\noindent
is also exact for all $k\geq 0$.
\section{Local and infinitesimal equivalences}
In this section we study the consequences of the previous hypotheses and conditions in view of showing that the \textit{micro-differentiable structures} introduced by Pradines (\cite{Pradines1966}) and hereafter considered, do generate the desired global structures. By the canonical identification $\Pi_{\ell+k}P~\times_P~J_{\ell+k}TP~\longrightarrow~V\Pi_{\ell+k}P$ resulting from the prolongation, by the target, of local vector fields defined on \textit{P}, each sub-space $\textbf{R}_{\ell+k}(S)_y$ determines, at every point $X\in\Pi_{\ell+k}P$ with $\beta(X)=y$, a sub-space $(\Delta_{\ell+k})_X\subset V_X\Pi_{\ell+k}P$ of the same dimension as that of $\textbf{R}_{\ell+k}(S)$ and thus defines an $\alpha-$vertical distribution on $\Pi_{\ell+k}P$ (field of contact elements) whose point-wise dimension is locally constant if and only if the fibre bundle $\textbf{R}_{\ell+k}(S)$ is locally trivial. This last condition being satisfied, the distribution $\Delta_{\ell+k}$ is integrable since $\textbf{R}_{\ell+k}(S)$ is a Lie equation.
\vspace{2 mm}
\newtheorem{integer}[PropositionCounter]{Proposition}
\begin{integer}
Let S be a structure of species E and finite order $\ell$. Then $ker~Tp_k\Phi(S)=\Delta_{\ell+k}(S)$ and, furthermore, the following properties are equivalent:
\vspace{2 mm}
\hspace{2 mm}i) $p_k\Psi(S)$ is locally of constant rank,
\vspace{2 mm}
\hspace{1 mm}ii) $p_k\Phi(S)$ is locally of constant rank along the units of $\Pi_{\ell+k}P$,
\vspace{2 mm}
iii) $p_k\Phi(S)$ is locally of constant rank,
\vspace{2 mm}
\hspace{1 mm}iv) $\Delta_{\ell+k}$ is locally of constant dimension.
\vspace{2 mm}
\noindent
These equivalent conditions being verified, $\textbf{R}_{\ell+k}(S)$ is a union of integral leaves of the distribution $\Delta_{\ell+k}$.
\end{integer}
\noindent
$\bf{Proof.}$ We first observe that $\Delta_{\ell+k}$ is invariant by all the right translations of the groupoid $\Pi_{\ell+k}P$, such a translation mapping $\alpha-$fibres onto $\alpha-$fibres and preserving the targets. On the other hand, the morphism $p_k\Phi(S)$ is a \textit{differential co-variant} relative to the right action of the groupoid $\Pi_{\ell+k}P$ on itself \textit{i.e.}, the following formula holds:
\begin{equation}
p_k\Phi(S)(X\cdot Y)=Y^{-1}p_k\Phi(S)(X).
\end{equation}
\vspace{4 mm}
\noindent
We next observe that $ker~Tp_k\Phi(S)~|~im~Id=\Delta_{\ell+k}~|~im~Id$ since $p_k\circ p_k\Psi(S)$ identifies, by means of the canonical identification, to $Tp_k\Phi(S)~|~im~Id=\Delta_{\ell+k}~|~im~Id$ (\text{cf.}(14)) and that the kernel of $Tp_k\Phi(S):~T\Pi_{\ell+k}P~\longrightarrow~TJ_kE$ is equal to that of the restriction $Tp_k\Phi(S):~V\Pi_{\ell+k}P~\longrightarrow~VJ_kE~$. The invariance of $\Delta_{\ell+k}$ and the co-variance of $p_k\Phi(S)$ entails the equality everywhere and we infer the equivalence of the four stated properties. If $\mathcal{F}$ is a leaf of $\Delta_{\ell+k}$, then $Tp_k\Phi(S)~|~T\mathcal{F}=0$ and consequently $p_k\Phi(S)(\mathcal{F})$ reduces to a point. In particular, when $X\in\mathcal{F}$ and $p_k\Phi(X)=j_kS(x),~x=\alpha(X)$, then $p_k\Phi(S)(\mathcal{F})=j_kS(x)$ where after $\mathcal{F}\subset\mathcal{R}_{\ell+k}(S)$ and this achieves the proof.
\vspace{4 mm}
\noindent
Let us now observe that, for any fibration morphism $\lambda$,
\begin{equation*}
P~\xrightarrow{~\lambda~}~P
\end{equation*}
\begin{equation*}
\downarrow\hspace{12 mm}\downarrow
\end{equation*}
\begin{equation*}
M\xrightarrow{~Id~}M
\end{equation*}
\vspace{2 mm}
\noindent
the following two conditions are equivalent:
\vspace{2 mm}
\textit{a) $\lambda$ is locally of constant rank,}
\vspace{2 mm}
\textit{b) $\lambda$ is locally of vertical constant rank i.e., the rank of the restrictions of $\lambda$ to the fibres is locally constant with respect to the topology of P (and not only of the fibres),}
\vspace{2 mm}
\noindent
since we always have the relation $rank_y\lambda=dim~M+(vertical~rank)_y\lambda~$. This amounts to say that the rank of $T\lambda:~TP~\longrightarrow~TP'$ differs, at each point, from the rank of $T\lambda:~VP~\longrightarrow~VP'$ by the integer \textit{dim M} and, in particular, that $ker~T\lambda=ker~T\lambda~|~VP$. We also observe that the above properties still hold when we replace \textit{Id} by any diffeomorphism $\varphi$ of \textit{M} (and even by a diffeomorphism $M\longrightarrow M'$). In particular, when \textit{P} and \textit{P'} are fibre bundles and $\lambda$ is a morphism of such bundles, then the vertical rank (rank of $T\lambda$ restricted to the tangent space of a fibre) is equal to the rank of the restriction of $\lambda$ to the fibres.
\vspace{2 mm}
\noindent
On the other hand, and without any regularity hypotheses pending upon $p_k\Phi(S)$ or $p_k\Psi(S)$, we remark that the isotropy
\begin{equation}
(\mathcal{R}^0_{\ell+k}S)_y=\{X\in\mathcal{R}_{\ell+k}(S)~|~\alpha(X)=\beta(X)=y\}
\end{equation}
\vspace{2 mm}
\noindent
of $\mathcal{R}_{\ell+k}(S)$ at the point \textit{y} is always a closed Lie sub-group of the total isotropy $(\Pi^0_{\ell+k}M)_y$ since it is given by the "closed" conditions $X(j_kS(y))=j_kS(y)$. Its Lie algebra identifies canonically with the linear isotropy
\begin{equation}
(\textbf{R}^0_{\ell+k}S)_y=\{X\in\textbf{R}_{\ell+k}(S)~|~\alpha(X)=y~,~\beta(X)=0\}
\end{equation}
\vspace{2 mm}
\noindent
of $\textbf{R}_{\ell+k}(S)$ that on its turn is determined by the condition $p_kX=0~$. Furthermore, these two remarks show that the finite and infinitesimal $k-th$ order isotropies of \textit{S} at the point \textit{y} are entirely determined by the jet $j_kS(y)$. In particular, the jet of order $\ell$ only depends on the point $S(y)\in E$. Let us denote by $\mathcal{R}_{\ell+k}(S)_y$ the fibre, with respect to $\alpha$ and above the point \textit{y}, of $\mathcal{R}_{\ell+k}(S)$ and by $\mathcal{B}_{\ell+k}(S)_y$ its projection $\beta(\mathcal{R}_{\ell+k}(S)_y)$.
\vspace{2 mm}
\newtheorem{isotropy}[CorollaryCounter]{Corollary}
\begin{isotropy}
If \textbf{R}$_{\ell+k}(S)$ is a locally trivial vector bundle then, at any point y, the fibre $\mathcal{R}_{\ell+k}(S)_y$ is a closed and regularly embedded sub-manifold of $(\Pi_{\ell+k}P)_y$ whose connected components are integral leaves of $\Delta_{\ell+k}$. Furthermore, $\mathcal{B}_{\ell+k}(S)_y$ is a closed and regularly embedded sub-manifold of P and the triple $(\mathcal{R}_{\ell+k}(S)_y~,~\beta~,~\mathcal{B}_{\ell+k}(S)_y)$ is a locally trivial sub-fibre bundle of $(\Pi_{\ell+k}P)_y~|~\mathcal{B}_{\ell+k}(S)_y$ with structure group equal to $\mathcal{R}^0_{\ell+k}(S)_y$ (it being understood that $\alpha(S)=P$).
\end{isotropy}
\noindent
$\bf{Proof.}$ It is clear that $\mathcal{R}_{\ell+k}(S)_y$ is closed being the inverse image of the point $j_kS(y)$ relative to the map $p_k\Phi(S)_y:~(\Pi_{\ell+k}P)_y~\longrightarrow~(J_kE)_y$. Inasmuch, it is also clear that $\mathcal{R}_{\ell+k}(S)_y$ is a principal space of the isotropy group $\mathcal{R}^0_{\ell+k}(S)_y$, the orbits being the inverse images, by $\beta$, of the points of $\mathcal{B}_{\ell+k}(S)_y~$. The hypothesis on $\textbf{R}_{\ell+k}(S)$ entails, in virtue of the previous proposition, that $p_k\Phi(S)_y:~(\Pi_{\ell+k}P)_y~\longrightarrow~(J_kE)_y$ has a locally constant rank and consequently $\mathcal{R}_{\ell+k}(S)_y$ is a regularly embedded sub-manifold. Since $ker~Tp_k\Phi(S)=\Delta_{\ell+k}(S)$, we see at once that $T\mathcal{R}_{\ell+k}(S)_y=\Delta_{\ell+k}(S)~|~\mathcal{R}_{\ell+k}(S)_y$ and thus infer that the leaves of $\Delta_{\ell+k}(S)$ contained in $\mathcal{R}_{\ell+k}(S)_y$ are open sets hence, due to the connexity of the leaves, are necessarily the connected components. We denote by $\Xi$ the distribution defined on \textit{P} by $\Xi_y=\beta(\textbf{R}_{\ell+k}(S)_y)$. The right invariance of the distribution $\Delta_{\ell+k}$ or, still better, the definition itself of this distribution shows immediately that $\beta_*(\Delta_{\ell+k})_X=\Xi_{\beta (X)}$. This distribution $\Xi$ is not, in general, of locally constant dimension but is generated by a family of vector fields that is involutive and locally of finite type. In fact, every section of $\textbf{R}_{\ell+k}(S)$ determines, by projection, a vector field that is a section of $\Xi$ and, since $\textbf{R}_{\ell+k}(S)$ is a Lie equation, the bracket of two sections projects onto the bracket of their images. Moreover, since $\textbf{R}_{\ell+k}(S)$ is locally trivial, the pre-sheaf of local sections of this fibre bundle is locally of finite type (in fact, locally free) and the local finiteness property of $\Xi$ follows. It is also easy to verify, by using again the right translations of $\Pi_{\ell+k}P$, that the integral leaves of $\Xi$ are precisely the projections of the integral leaves of $\Delta_{\ell+k}$ (\textit{cf.} \cite{Hermann1962} and \cite{Turiel1976}, Chap.I, $\S~2$).
Hence, we thus conclude that $\beta(\mathcal{R}_{\ell+k}(S)_y)=\mathcal{B}_{\ell+k}(S)_y$ is a union of integral leaves of $\Xi$ and this union is discrete: For every $z\in\mathcal{B}_{\ell+k}(S)_y$, there exists an open neighborhood \textit{U} of \textit{z} in \textit{P} such that $\mathcal{B}_{\ell+k}(S)_y\cap U$ reduces to the intersection of \textit{U} with a unique integral leaf of $\Xi$. To see this, it suffices to note firstly that the leaves of $\Delta_{\ell+k}$ passing by the points of the same $\beta-$fibre of $\mathcal{R}_{\ell+k}(S)_y$ project all on the same leave of $\Xi$ and secondly, recalling that $\mathcal{R}_{\ell+k}(S)_y$ is a regularly embedded sub-manifold whose connected components are integral leaves of $\Delta_{\ell+k}~$, we shall take an open neighborhood $\mathcal{U}$, in $\Pi_{\ell+k}(S)_y~$, of a point \textit{X} contained in a $\beta-$fibre of $\mathcal{R}_{\ell+k}(S)_y$ in such a way that $\mathcal{U}\cap\mathcal{R}_{\ell+k}(S)_y$ just contains the points of a single leaf of $\Delta_{\ell+k}$. By right translation with the elements of the isotropy $\mathcal{R}^0_{\ell+k}(S)_y$, the same situation reproduces itself, with the translated open set, at every other point of the $\beta-$fibre and consequently $U=\beta(\mathcal{U})$ responds to the required property. Shrinking, if necessary, the open set $\mathcal{U}$ and recalling the regularity of the embedding of $\mathcal{R}_{\ell+k}(S)_y$, we can show further that $\mathcal{B}_{\ell+k}(S)_y\cap U$ is a \textit{slice} (in a coordinate system) and, consequently, that $\mathcal{B}_{\ell+k}(S)_y$ is a regularly embedded sub-manifold of \textit{P}. Finally, a saturation argument of $\mathcal{R}_{\ell+k}(S)_y$ by the action of the total isotropy $(\Pi^0_{\ell+k}P)_y$ and the fact that $\mathcal{R}_{\ell+k}(S)_y$ is closed, shows that $\mathcal{B}_{\ell+k}(S)_y$ is closed in \textit{P}. We thus see that $\beta:\mathcal{R}_{\ell+k}(S)_y~\longrightarrow~\mathcal{B}_{\ell+k}(S)_y$ is a \textit{surmersion} (surjective submersion) and, consequently, a (sub-)principal fibre bundle of $(\Pi_{\ell+k}P)_y~|~\mathcal{B}_{\ell+k}(S)_y$ that is locally trivial since it admits local sections and the proof is therefore complete. We shall nevertheless continue by showing, further, that the dimension of the eventually non-connected sub-manifold $\mathcal{B}_{\ell+k}(S)_y$ is constant on each connected component of $\alpha(S)$. In fact, since the dimension of $\mathcal{R}_{\ell+k}(S)$ is constant on each connected component $\mathcal{O}$ of $\alpha(S)$, we infer that the dimension of $\Delta_{\ell+k}(S)$ is constant on $\beta^{-1}\mathcal{O}$ and, consequently, that of $\mathcal{R}_{\ell+k}(S)_y$ is also constant above $\mathcal{O}$. On the other hand, the tangent space to each $\beta-$fibre of $\mathcal{R}_{\ell+k}(S)_y$ is isomorphic to the Lie algebra $\textbf{R}^0_{\ell+k}(S)_y$ of the structural group $\mathcal{R}^0_{\ell+k}(S)_y$ hence the dimension of this tangent space is constant. Lastly, since $\Xi=T_y\mathcal{B}_{\ell+k}(S)_y$ is isomorphic to the quotient of $(\Delta_{\ell+k})_Y~,~Y\in\mathcal{R}_{\ell+k}(S)_y~,~\beta(Y)=y~$, modulo the tangent space to the $\beta-$fibre at the point \textit{Y}, the result follows. To terminate, we provide an alternative proof of the above statements in view of the geometrical mechanisms that it will turn apparent and that will be of relevance in the sequel. For this, we go back to the definition of the elements of $\textbf{R}_{\ell+h}(S)$ as being the jets $j_{\ell+h}\xi(y)$ of local vector fields $\xi$ whose prolongations $p\xi$ are $h-th$ order tangent to the section \textit{S} at the point $S(y)$. However, we can see that any $Y\in\mathcal{R}_{\ell+k}(S)$ transforms $\mathcal{R}_{\ell+h}(S)_{\alpha Y}$ into $\mathcal{R}_{\ell+h}(S)_{\beta Y}$, $h<k$ , since the $k-$jet $pY$ with source $S(\alpha Y)$ and associated, by prolongation, to \textit{Y} transforms the $k-th$ order contact element of $im~S$ at the point $S(\alpha Y)$ into the corresponding contact element at the point $S(\beta Y)$. This property, however, fails when $h=k$ since $pY$ only operates on $(k-1)-$jets of vector fields on \textit{E} and the invariance does not subsist any longer, not even for the projected sub-spaces $\beta(\textbf{R}_{\ell+h}(S)_{\alpha Y})$ and $\beta(\textbf{R}_{\ell+h}(S)_{\beta Y})$. However, if we observe in general, \textit{N} being an arbitrary manifold, that $\Pi_rN$ operates on $J^0_rTN~$, then it will become apparent (\textit{cf.} Lemmas 3 and 4) that \textit{Y} transforms $\textbf{R}^0_{\ell+k}(S)_{\alpha Y}$ onto $\textbf{R}^0_{\ell+k}(S)_{\beta Y}$ since the jet $j_k(p\xi)[S(\alpha Y)]$ associated to $j_{\ell+k}\xi(\alpha Y)\in\textbf{R}^0_{\ell+k}(S)_{\alpha Y}$ belongs to $(J^0_kTE)_{S(\alpha Y)}$ and, consequently, $pY$ transforms the jet $j_k(p\xi)[S(\alpha Y)]$, $k-th$ order tangent to $im~S$ at the point $S(\alpha Y)$, into a $k-$jet of vector field tangent, up to order \textit{k}, to $im~S$ at the point $S(\beta Y)$, this later $k-$jet being precisely the one that corresponds, via prolongation, to the transformed jet $Y(j_{\ell+k}\xi(\alpha Y))\in\textbf{R}^0_{\ell+k}(S)_{\beta Y}~$. We thus infer that the fibres of $\textbf{R}^0_{\ell+k}(S)$ at the points $\alpha Y$ and $\beta Y$ are isomorphic where after the isomorphy of all the fibres along any orbit $\mathcal{B}_{\ell+k}(S)_y$ of $\textbf{R}_{\ell+k}(S)$ in \textit{P}. Since $\Xi_y=\textbf{R}_{\ell+k}(S)_y~/~\textbf{R}^0_{\ell+k}(S)_y$, we infer that $\Xi_y$ has constant dimension along $\mathcal{B}_{\ell+k}(S)_y$ if and only if it is inasmuch for $\textbf{R}_{\ell+k}(S)_y~$. In particular, this implies the constancy of dimensions for $\Xi_y$ on the intersection of $\mathcal{B}_{\ell+k}(S)_y$ with a connected component of $\alpha S$. The sub-manifold $\mathcal{B}_{\ell+k}(S)_y$ has therefore a constant dimension in each connected component of $\alpha(S)$.
\vspace{4 mm}
\noindent
The method of proof suggests a weakening of the regularity hypotheses imposed on $\textbf{R}_{\ell+k}(S)$. Accordingly, it would suffice to assume that $\textbf{R}_{\ell+k}(S)$ is locally of finite type \textit{i.e.}, that it be locally generated by a finite number of sections (differentiable sections of $J_{\ell+k}TP$ taking values in $\textbf{R}_{\ell+k}(S)$). The bracket properties of $\underline{J_{\ell+k}TP}$ would then imply that $\textbf{R}_{\ell+k}(S)$ is a Lie equation, the set of its local sections being closed under the bracket. However, in the specific case of the equation $\textbf{R}_{\ell+k}(S)$, this generalization is only illusory. In fact, the local finiteness of generators would imply lower semi-continuity for $dim~\textbf{R}_{\ell+k}(S)_y$ whereas the definition of this equation as the kernel of $p_k\Psi(S)$ would imply the upper semi-continuity. In definite, the local finiteness assumption is entirely equivalent to regularity.
\section{Micro-differentiable structures and globalization}
In this section we look for the hypotheses enabling us to endow the groupoid $\mathcal{R}_{\ell+k}(S)$ or eventually its $\alpha-$connected component with a differentiable structure compatible with its algebraic structure. To begin, we assume that the vector bundle $\textbf{R}_{\ell+k}(S)$ is locally trivial and already possesses all the regularity properties indicated in the last Proposition as well as in its Corollary. We denote by $\mathcal{R}_{\ell+k}(S)_0$ the union of all the integral leaves of $\Delta_{\ell+k}$ issued from the units of $\Pi_{\ell+k}P~$. A standard connectivity argument shows that $\mathcal{R}_{\ell+k}(S)_0$ is a sub-groupoid of $\mathcal{R}_{\ell+k}(S)$ that we shall call its $\alpha-$connected component of the units space (assumed to be connected otherwise we argue on each connected component). Every $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)_0$ is in fact the connected component of a unit in the corresponding $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)$.
\vspace{4 mm}
\noindent
According to the general definitions (\cite{Kumpera1971}, \cite{Pradines1967}), $\mathcal{R}_{\ell+k}(S)_0$ is the sub-groupoid of $\Pi_{\ell+k}P$ generated by the Lie algebroid $\underline{\textbf{R}_{\ell+k}(S)}\subset\underline{J_{\ell+k}TP}~$. Contrary to what has been said and written in the last century (\cite{Rodrigues1962}, main theorem), it is well known that Lie's Second Theorem is inexact even for transitive Pseudo-groups \textit{i.e.}, for transitive sub-groupoids of the general groupoid $\Pi_{\ell+k}P~$. In other terms, given a Lie sub-algebroid $\mathcal{A}$ of $\underline{J_kTP}$ and denoting by $\mathcal{G}$ the (algebraic) sub-groupoid of $\Pi_{\ell+k}P$ it generates (\textit{e.g.}, by integrating the $\alpha-$fibres distribution), it is not always possible to endow this sub-groupoid with a differentiable structure (of sub-manifold) in such a way that its Lie algebroid can be identified with the given one. The main obstruction rests in the non-vanishing \textit{holonomy} for the integral leaves of $\Delta_{\ell+k}~$, the distribution generated by the right translations applied to $\mathcal{A}~$, these leaves being precisely the $\alpha-$fibres of the desired sub-groupoid. Nonetheless, in our present situation, this holonomy fortunately vanishes since $\mathcal{R}_{\ell+k}(S)$ is the kernel of a differential operator (or, more precisely, the kernel of its defining morphism). We shall of course proceed in the most standard way by first endowing an open neighborhood of the units with the differentiable structure practically "imposed" by the algebroid $\underline{\textbf{R}_{\ell+k}(S)}$ and, thereafter, propagate this local differentiable structure to the $\alpha-$connected component $\mathcal{R}_{\ell+k}(S)_0~$. In order to further propagate this differentiable structure to the whole of $\mathcal{R}_{\ell+k}(S)$, we shall be forced to add an additional hypothesis. Let us also observe that we are undertaking this painstaking homework since, to our knowledge, this construction has never been fully elucidated before.
\vspace{4 mm}
\noindent
Since $\Delta_{\ell+k}$ is regular and integrable, we take, for each unit \textbf{e} in $\mathcal{R}_{\ell+k}(S)$, a \textit{foliating chart} ($\mathcal{U}_\alpha,\phi_\alpha)$ (\cite{Chevalley1946}, p.69) in such a way that $\textbf{e}\in\mathcal{U}_\alpha$ and that the leaves contained in $\mathcal{U}_\alpha$ are \textit{slices} with respect to the coordinate system $\phi_\alpha~$, where a slice means a sub-manifold diffeomorphic to an open $\textbf{p}-$cube in $\bf{R}^p$, $\textbf{p}=dim~\Delta_{\ell+k}$ (\textit{cf.} the aforementioned reference). Let us denote by $\mathcal{V}_\alpha$ the union of all the slices contained in $\mathcal{U}_\alpha$ and that contain units of $\mathcal{R}_{\ell+k}(S)$. Then $\mathcal{V}_\alpha$ is a closed and regularly embedded sub-manifold of $\mathcal{U}_\alpha~$. We set $\mathcal{U}=\bigcup~\mathcal{U}_\alpha$ and $\mathcal{V}=\bigcup~\mathcal{V}_\alpha~$. Then $\mathcal{V}_\alpha\cap\mathcal{V}_\beta$ is an open sub-set of both $\mathcal{V}_\alpha$ and $\mathcal{V}_\beta~$. Consequently, $\mathcal{V}$ is a closed and regularly embedded sub-manifold of $\mathcal{U}$ and, of course, $\mathcal{V}\subset\mathcal{R}_{\ell+k}(S)_0~$. We next observe that the projection $\alpha:\mathcal{V}\longrightarrow\alpha(S)$ is a surmersion and that the $\alpha-$fibres contained in $\mathcal{V}$ are sub-manifolds of locally constant dimension with respect to the variation of \textit{y} in $\alpha(S)$. Moreover, the germ of this sub-manifold, along the units, is uniquely determined \textit{i.e.}, does not depend on the initial choice of the foliating charts and, by construction, $\textbf{R}_{\ell+k}(S)$ can be identified with the $\alpha-$vertical tangent bundle of $\mathcal{V}$ along the units.
\vspace{4 mm}
\noindent
Due to its geometrical interest, we shall construct the same germ of sub-manifold by a procedure relying on the local constancy of the rank of $p_k\Phi(S)$. A unit $\textbf{e}\in\mathcal{R}_{\ell+k}(S)$ being fixed, there exists, due to the constancy of the rank, a neighborhood $\mathcal{U}$ of \textbf{e} in $\Pi_{\ell+k}P$ such that $\mathcal{W}=p_k\Phi(S)(\mathcal{U})$ is a regularly embedded sub-manifold of $J_kE$ and that $p_k\Phi(S):\mathcal{U}~\longrightarrow~\mathcal{W}$ is a surmersion. Furthermore, we can assume that $\overline{\mathcal{U}}=\alpha(\mathcal{U})=\alpha(\mathcal{U}\cap\mathcal{I})$, where $\mathcal{I}$ represents the sub-manifold of units of $\Pi_{\ell+k}P$ and, thereafter, the intersection $\mathcal{W}\cap im~j_kS=j_kS(\overline{\mathcal{U}})$ is a regularly embedded sub-manifold of $\mathcal{W}~$. The inverse image of $j_kS(\overline{\mathcal{U}})$ by the map $p_k\Phi(S)~|_{\mathcal{U}}$ is equal to $\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}$ and consequently, this inverse image is a closed and regularly embedded sub-manifold $\mathcal{V}$ of $\mathcal{U}$. Since $ker~Tp_k\Phi(S)=\Delta_{\ell+k}~$, we infer that the $\alpha-$fibres of $\mathcal{V}$ are integral manifolds of maximum dimension of $\Delta_{\ell+k}~$, these $\alpha-$fibres being precisely the fibres of $p_k\Phi(S):\mathcal{U}~\longrightarrow~\mathcal{W}$ above the points of $j_kS(\overline{\mathcal{U}})$. By shrinking, if necessary, the open set $\mathcal{U}$, we can be brought to the case where these fibres are slices and thus infer that any leave of $\Delta_{\ell+k}~$, issued from a point in $\mathcal{U}\cap\mathcal{I}~$, intercepts the open set $\mathcal{U}$ along a unique slice. Furthermore, the open set $\mathcal{U}$ is a foliating chart defined in a neighborhood of the unit \textbf{e} and verifies the conditions stated previously. By taking the union of all these open sets $\mathcal{U}_\alpha$ corresponding to the various units, we obtain an open neighborhood $\tilde{\mathcal{U}}$ of the units of $\mathcal{R}_{\ell+k}(S)$ in $\Pi_{\ell+k}P$ such that $\mathcal{V}=\tilde{\mathcal{U}}\cap\mathcal{R}_{\ell+k}(S)=\tilde{\mathcal{U}}\cap\mathcal{R}_{\ell+k}(S)_0$ is the union of the slices contained in the open sets $\mathcal{U}_\alpha$ that contain the units. Besides, $\mathcal{V}$ is a neighborhood of the units in $\mathcal{R}_{\ell+k}(S)$ as well as in $\mathcal{R}_{\ell+k}(S)_0~$. The germ, along these units, of the regularly embedded sub-manifold $\mathcal{V}$ is unique and will be called, together with Pradines (\cite{Pradines1966}), the \text{micro-differentiable structure} of $\mathcal{R}_{\ell+k}(S)$.
\vspace{4 mm}
\noindent
We now show that this micro-differentiable structure can be extended to a global structure defined on $\mathcal{R}_{\ell+k}(S)_0~$. Indeed, if $Y\in\mathcal{R}_{\ell+k}(S)_0$ and if $\textbf{e}=\alpha(Y)$, the continuation Theorem (\cite{Palais1957}, pg.10) enables us to define a differentiable mapping $\mu:\overline{\mathcal{U}}~\longrightarrow~\Pi_{\ell+k}P$ where $\overline{\mathcal{U}}$ is an open neighborhood of the unit \textbf{e} in the sub-manifold $\mathcal{I}$ composed by the units, $\mu(\textbf{e'}),~\textbf{e'}\in\overline{\mathcal{U}}~$, belongs to the leaf of $\Delta_{\ell+k}$ that contains \textbf{e'} ($\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)_0$) and $\mu(\textbf{e})=Y$. However, if we identify \textit{P} with the units manifold $\mathcal{I}~$, $\mu$ becomes a differentiable section of the bundle $\alpha:\Pi_{\ell+k}P~\longrightarrow~P$ defined on $\overline{\mathcal{U}}$ with values in $\mathcal{R}_{\ell+k}(S)_0$ that assumes the value \textit{Y} at the point \textbf{e}. Due to the local rank constancy of $p_k\Phi(S)$, there exists a neighborhood $\mathcal{U}$ of \textit{Y} in $\Pi_{\ell+k}P$ such that $\mathcal{W}=p_k\Phi(S)(\mathcal{U})$ is a regularly embedded sub-manifold of $J_kE$ and that $p_k\Phi(S):\mathcal{U}~\longrightarrow~\mathcal{W}$ is a surmersion. We can further suppose that $\overline{\mathcal{U}}=\alpha(\mathcal{U})=\alpha(\mathcal{U}\cap im~\mu)$ which implies that $\mathcal{W}\cap im~j_kS=j_kS(\overline{\mathcal{U}})$ is a regularly embedded sub-manifold of $\mathcal{W}$. It then follow as previously and shrinking, if necessary, the open set $\mathcal{U}$, that $\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}=\mathcal{R}_{\ell+k}(S)_0\cap\mathcal{U}$ is a closed and regularly embedded sub-manifold of $\mathcal{U}$, inverse image by the map $p_k\Phi(S)|_{\mathcal{U}}$ of $j_kS(\overline{\mathcal{U}})$, and the fibres of the fibration $p_k\Phi(S)|_{\mathcal{U}}$ coincide, above $j_kS(\overline{\mathcal{U}})$, with the $\alpha-$fibres of $\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}~\longrightarrow~\overline{\mathcal{U}}~$. It follows therefore that $\mathcal{R}_{\ell+k}(S)_0$ is endowed with the differentiable structure of a regularly embedded sub-manifold of $\Pi_{\ell+k}P$ compatible with its groupoid structure. Moreover, there exists an open set $\mathcal{U}$ in $\Pi_{\ell+k}P$ such that $\mathcal{R}_{\ell+k}(S)_0$= $\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}$ or, in other terms, that $\mathcal{R}_{\ell+k}(S)_0$ is locally closed. The open set $\mathcal{U}$ can be chosen saturated with respect to the leaves of $\Delta_{\ell+k}$ since $\mathcal{R}_{\ell+k}(S)$ as well as $\mathcal{R}_{\ell+k}(S)_0$ are already saturated. The constructions show clearly that $\textbf{R}_{\ell+k}(S)$ identifies with the $\alpha-$vertical tangent bundle of $\mathcal{R}_{\ell+k}(S)_0$ along the units and that, consequently, the sheaf $\underline{\textbf{R}_{\ell+k}(S)}$ is the Lie algebroid associated to the differentiable sub-groupoid $\mathcal{R}_{\ell+k}(S)_0$ (\cite{Kumpera1971},\cite{Pradines1967}).
\newtheorem{different}[PropositionCounter]{Proposition}
\begin{different}
Let S be a structure of species E and finite order $\ell$ verifying the equivalent conditions of the Proposition $1.$ The equation $\mathcal{R}_{\ell+k}(S)_0$ is then an $\alpha-$connected and regularly embedded Lie sub-groupoid of $\Pi_{\ell+k}P$ whose associated Lie algebroid is equal to $\underline{\textbf{R}_{\ell+k}(S)}$. There exists an open set $\mathcal{U}$ of $\Pi_{\ell+k}P~$, saturated by the integral foliation of $\Delta_{\ell+k}~$, such that $\mathcal{R}_{\ell+k}(S)_0$ is closed in $\mathcal{U}$ and, furthermore, the sequence
\begin{equation}
1~\longrightarrow~\mathcal{R}_{\ell+k}(S)_0~\longrightarrow~\mathcal{U}~\xrightarrow{p_k\Phi(S)}~(J_kE,j_kS)
\end{equation}
\vspace{2 mm}
\noindent
is exact. Conversely, when $\mathcal{R}_{\ell+k}(S)_0$ admits a structure of Lie sub-groupoid of $\Pi_{\ell+k}P$ with associated Lie algebroid equal to $\underline{\textbf{R}_{\ell+k}(S)}$, then the equivalent conditions of the Proposition $1$ are satisfied and the differentiable structure of $\mathcal{R}_{\ell+k}(S)_0$ is regularly embedded.
\end{different}
\vspace{4 mm}
\noindent
This proposition states precisely that $\mathcal{R}_{\ell+k}(S)_0$ is the non-linear Lie equation generated by the linear Lie equation $\textbf{R}_{\ell+k}(S)$ (\textit{cf.} \cite{Malgrange1970},\cite{Malgrange1972}) when $p_k\Psi$ is locally of constant rank. The converse statement of the above proposition is quite simple to prove and will be omitted. Nevertheless, it veils behind the curtains the following rather important fact: We first observe that any local section of $\textbf{R}_{\ell+k}(S)$ determines, by right translations, a right invariant vector field on $\Pi_{\ell+k}P$ that is contained in $\Delta_{\ell+k}$ and conversely. Let $\mathcal{P}$ denote the pseudo-group of local transformations operating on $\Pi_{\ell+k}P$ and generated by the flows of the above mentioned right invariant vector field. Then, each element of $\mathcal{P}$ leaves invariant the sub-groupoid $\mathcal{R}_{\ell+k}(S)_0$ though it needs not preserve the larger sub-groupoid $\mathcal{R}_{\ell+k}(S)$, this being the main obstacle towards the possibility of simply extending or prolonging the differentiable structure of the former to the later, as was done in the micro-differentiable situation.
So, let us now take care of $\mathcal{R}_{\ell+k}(S)$.
\vspace{4 mm}
\noindent
In the previous attempt to provide $\mathcal{R}_{\ell+k}(S)_0$ with a differentiable structure, there is a key point that still remains unexplored: The open sets $\mathcal{U}~$, neighborhood of the unit \textbf{e} , can be chosen in such a way that any leaf of $\Delta_{\ell+k}$ issued from a point $\textbf{e'}\in\mathcal{U}\cap\mathcal{I}$ cuts the open set $\mathcal{U}$ along a unique slice namely, the one containing \textbf{e'}. We show in fact that $\mathcal{U}$ can be chosen in such a way that any leaf of $\Delta_{\ell+k}$ meets $\mathcal{U}$ at most along a single slice (no holonomy). Since the operations of the groupoid $\Pi_{\ell+k}M$ are continuous, we can take open neighborhoods $\mathcal{V}$ and $\mathcal{W}$ of \textbf{e} such that $\mathcal{V}\subset\mathcal{U}~$, $\mathcal{W}\subset\mathcal{U}~$, $\mathcal{V}\cdot\mathcal{V}^{-1}\subset\mathcal{U}$ and $\mathcal{W}\cdot\mathcal{V}\subset\mathcal{U}~$, the operations being performed on all composable pairs. We can further assume that both neighborhoods are the domains of foliating charts, each fibre being a slice. Let us now take a leaf $\mathcal{F}$ and assume that it meets $\mathcal{V}$ along two slices $\mathcal{S}_1$ and $\mathcal{S}_2~$.
If we take a point $X\in\mathcal{S}_1$ then $\mathcal{F}_{\beta(X)}=\mathcal{F}\cdot X^{-1}$ is the leaf of $\Delta_{\ell+k}$ passing by the unit $\beta(X)\in\mathcal{U}\cap\mathcal{I}$ and consequently $\mathcal{s}_1\cdot X^{-1}$ as well as $\mathcal{S}_2\cdot X^{-1}$ are included in the slice of $\mathcal{U}$ hence also in that of $\mathcal{W}$ and containing the unit $\beta(X)$. Applying the inverse operation $\cdot X~$, we infer that $\mathcal{S}_1$ and $\mathcal{S}_2$ are both contained in a same slice of $\mathcal{U}$ and consequently in a same slice of $\mathcal{V}$ since the later is just a foliated chart restriction of $\mathcal{U}~$. More generally, we show that the same property continues to hold at each point of $\Pi_{\ell+k}P$, it being understood that \textit{P} is to be replaced by $\alpha(S)$ when \textit{S} is just a local section.
In fact, let $X=j_{\ell+k}\varphi(y)$ be an arbitrary point and let us consider the \textit{flow} $j_{\ell+k}\varphi~$. By right translations, provided by the flow elements, we establish a diffeomorphism $\tau:\beta^{-1}(\beta\varphi)~\longrightarrow~\beta^{-1}(\alpha\varphi)~$ that is compatible with $\Delta_{\ell+k}$ and consequently the leaves are transformed in leaves, the foliating charts in foliating charts and the slices in slices. Furthermore, the desired property is verified for the foliating open set $\tau(\mathcal{U})$ as soon as it is verified for $\mathcal{U}~$. In sum, for every $X\in\Pi_{\ell+k}P~$, there exists a foliating chart $\mathcal{U}$ of $\Delta_{\ell+k}$ that is a neighborhood of \textit{X} and for which any leaf of $\Delta_{\ell+k}$ meets at most along a single slice. However, these properties translate by saying that the integral foliation of $\Delta_{\ell+k}$ is \textit{simple} or, in other terms, (\cite{Palais1957}, pg.19) that there exists a differentiable structure, necessarily unique, on the quotient $\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ of the general groupoid modulo the integral leaves of $\Delta_{\ell+k}$ and in such a way that the quotient map $\zeta:\Pi_{\ell+k}M~\longrightarrow~\Pi_{\ell+k}M~/~\Delta_{\ell+k}~$ is a surmersion. Moreover, since each leaf of $\Delta_{\ell+k}$ is contained in an $\alpha-$fibre, there is a canonical projection $\overline{\alpha}$ of the quotient space onto \textit{P} that commutes with $\alpha$. As previously, we shall also denote by $\mathcal{I}$ the identity section $y\in P~\longmapsto~j_{\ell+k}Id(y)\in\Pi_{\ell+k}P$ and set $\overline{\mathcal{I}}=\zeta\circ\mathcal{I}~$. Then, of course, $\overline{\mathcal{I}}$ is a differentiable section of $\overline{\alpha}~$, the restriction $\zeta:im~\mathcal{I}~\longrightarrow~im~\overline{\mathcal{I}}$ is a diffeomorphism, $im~\overline{\mathcal{I}}$ is a regularly embedded sub-manifold of the quotient $\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ and $\mathcal{R}_{\ell+k}(S)_0=\zeta^{-1}(im~\overline{\mathcal{I}})$. Moreover, since $\mathcal{R}_{\ell+k}(S)_0$ is locally closed in a saturated open set $\mathcal{U}~$, the sub-manifold $im~\overline{\mathcal{I}}$ is locally closed in the open set $\zeta(\mathcal{U})$ (it should however be observed that $\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ needs not be separated). Finally, since $\mathcal{R}_{\ell+k}(S)$ and $\mathcal{R}_{\ell+k}(S)_y$ are closed and saturated by the leaves and since $\mathcal{R}_{\ell+k}(S)_y$ is a regularly embedded sub-manifold of $(\Pi_{\ell+k}P)_y$ whose connected components are precisely the maximal integral leaves of $\Delta_{\ell+k}~$, we conclude that $\zeta(\mathcal{R}_{\ell+k}(S))$ is closed and that $\zeta(\mathcal{R}_{\ell+k}(S)_y)$ is closed and discrete (each point is isolated) in the fibre $(\Pi_{\ell+k}P~/~\Delta_{\ell+k})_y~$.
\vspace{4 mm}
\noindent
Let us next assume that $\mathcal{R}_{\ell+k}(S)$ is endowed with a differentiable structure that makes it become a Lie sub-groupoid of $\Pi_{\ell+k}P$ and whose associated Lie algebroid is equal to $\underline{\textbf{R}_{\ell+k}(S)}$. We show that, under these conditions, the differentiable structure of $\mathcal{R}_{\ell+k}(S)$ is regularly embedded in $\Pi_{\ell+k}P~$, $\zeta(\mathcal{R}_{\ell+k}(S))$ becomes a regularly embedded sub-manifold of $\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ admitting the open subset $im~\overline{\mathcal{I}}$ and the restriction $\overline{\alpha}:\zeta(\mathcal{R}_
{\ell+k}(S))~\longrightarrow~\alpha(S)$ is \textit{étale}.
\vspace{4 mm}
\noindent
In fact, since $\mathcal{R}_{\ell+k}(S)$ is a Lie groupoid, the projection
\begin{equation*}
\alpha:\mathcal{R}_{\ell+k}(S)~\longrightarrow~\alpha(S)
\end{equation*}
\vspace{2 mm}
\noindent
is a surmersion and consequently, for any $X\in\mathcal{R}_{\ell+k}(S)$, there exists a differentiable local section $\mu$ of $\alpha$ taking its values in $\mathcal{R}_{\ell+k}(S)$ and passing through \textit{X}. Furthermore, there exists an open neighborhood $\mathcal{V}$ of \textit{X} in $\mathcal{R}_{\ell+k}(S)$ such that the fibres of $\alpha:\mathcal{V}~\longrightarrow~\alpha(\mathcal{V})$ are slices. By the hypothesis, $\underline{\textbf{R}_{\ell+k}(S)}$ is the Lie algebroid of $\mathcal{R}_{\ell+k}(S)$ hence the $\alpha-$vertical tangent bundle $V\mathcal{R}_{\ell+k}(S)$ is equal to $\Delta_{\ell+k}|\mathcal{R}_{\ell+k}(S)$. We infer that the slices of $\mathcal{V}$ are integral sub-manifolds of maximum dimension of $\Delta_{\ell+k}$ and, more generally that the $\alpha-$fibres of $\mathcal{R}_{\ell+k}(S)$ have, for connected components, the integral leaves of $\Delta_{\ell+k}~$. Let $\mathcal{F}$ be the leaf that meets $X=\mu(y)$, let us take a second $\alpha-$section $\nu$ of $\mathcal{R}_{\ell+k}(S)$ such that $\nu(y)\in\mathcal{F}$ and let us denote by $\tilde{\mathcal{V}}$ the saturated set of $\mathcal{V}$, in $\mathcal{R}_{\ell+k}(S)$, by the integral leaves of $\Delta_{\ell+k}~$. Furthermore, the continuation Theorem implies that $\tilde{\mathcal{V}}$ is an open subset of $\mathcal{R}_{\ell+k}(S)$ and, since $\nu(y)\in\tilde{\mathcal{V}}$, we infer that $\nu(z)$ and $\mu(z)$ belong to the same leaf of $\Delta_{\ell+k}$ as soon as \textit{z} is sufficiently close to \textit{y}. However, this implies that the two sections $\zeta\circ\mu$ and $\zeta\circ\nu$ of $\overline{\alpha}$ coincide in a neighborhood of \textit{y} hence enables to define a structure of differentiable sub-manifold on the image $\zeta(\mathcal{R}_{\ell+k}(S))$ in such a way that $\overline{\alpha}:\zeta(\mathcal{R}_{\ell+k}(S))~\longrightarrow~\alpha(S)$ becomes \textit{étale}. Let us finally show that this sub-manifold is regularly embedded. To do so, we recall that $\mathcal{R}_{\ell+k}(S)$ is defined as being the kernel of $p_k\Phi(S)$ and that this mapping is locally of constant rank. Therefore, we can find an open neighborhood $\mathcal{U}$ of \textit{X} in $(\Pi_{\ell+k}P)$ such that $\mathcal{W}=p_k\Phi(S)(\mathcal{U})$ is a regularly embedded sub-manifold of $J_kE$ and $p_k\Phi(S):\mathcal{U}~\longrightarrow~\mathcal{W}$ is a surmersion. We can further assume that $\overline{\mathcal
{U}}=\alpha(\mu)=\alpha(\mathcal{U})=\alpha(\mathcal{U}\cap im~\mu)$ and this implies that $\mathcal{W}\cap im~j_kS=j_kS(\overline{\mathcal{U}})$ is a regularly embedded sub-manifold of $\mathcal{W}$. We infer that $\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}$ is a closed and regularly embedded sub-manifold of $\mathcal{U}$ , inverse image of $j_kS(\overline{\mathcal{U}})$ by the map $p_k\Phi(S)|_{\mathcal{U}}$, and whose $\alpha-$fibres coincide with the fibers of $p_k\Phi(S)|_{\mathcal{U}}$ above the points of $j_kS(\overline{\mathcal{U}})$. Since, $ker~p_k\Phi(S)=\Delta_{\ell+k}$ , the $\alpha-$fibres of $\alpha:\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}~\longrightarrow~\overline{\mathcal{U}}$ are integral sub-manifolds of maximal dimension of $\Delta_{\ell+k}$ and consequently, it is possible to choose the above open set $\mathcal{U}$ as well as the open neighborhood $\mathcal{V}$ of \textit{X} in $\mathcal{R}_{\ell+k}(S)$, considered in the beginning, in such a way that $\mathcal{V}=\mathcal{R}_{\ell+k}(S)\cap\mathcal{U}$ and, moreover, that the two differentiable structures, one induced by the given structure of $\mathcal{R}_{\ell+k}(S)$ and the other induced by that of $\mathcal{U}$ , be the same and shows consequently that the given structure on $\mathcal{R}_{\ell+k}(S)$ is regularly embedded. Saturating these open sets by the leaves of $\Delta_{\ell+k}$ , we obtain much in the same way the open and saturated sub-sets
\begin{equation*}
\tilde{\mathcal{V}}=\mathcal{R}_{\ell+k}(S)\cap\tilde{\mathcal{U}}
\end{equation*}
\noindent
and
\begin{equation*}
\zeta(\mathcal{R}_{\ell+k}(S)\cap\mathcal{U})=\zeta(\mathcal{R}_{\ell+k}(S)\cap\tilde{\mathcal{U}})=\zeta(im~\mu\cap\tilde{\mathcal{U}})=im~(\zeta\circ\mu)\cap\zeta(\tilde{\mathcal{U}})
\end{equation*}
\vspace{2 mm}
\noindent
which implies that $\zeta(\mathcal{R}_{\ell+k}(S))$ is a regularly embedded sub-manifold of $\Pi_{\ell+k}P~/~\Delta_{\ell+k}~$.
\vspace{2 mm}
\noindent
Conversely, when $\zeta(\mathcal{R}_{\ell+k}(S))$ admits the structure of a regularly embedded sub-manifold, then $\mathcal{R}_{\ell+k}(S)$ admits the structure of a regularly embedded sub-manifold of $\Pi_{\ell+k}P$ that induces forcefully the regularly embedded structure of $\mathcal{R}_{\ell+k}(S)_ y~$. Consequently, the Lie algebroid associated to the Lie sub-groupoid $\mathcal{R}_{\ell+k}(S)$ is necessarily equal to $\underline{\textbf{R}_{\ell+k}(S)}$ and this implies again that the structure of $\zeta(\mathcal{R}_{\ell+k}(S))$ is \textit{étale} over $\alpha(S)$. It is further clear that $im~\overline{\mathcal{I}}$ is an open sub-manifold of $\zeta(\mathcal{R}_{\ell+k}(S))$ that however needs not be closed by virtue of the eventual non-separability of the quotient $\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ as well as that of $\zeta(\mathcal{R}_{\ell+k}(S))$.
\vspace{2 mm}
\noindent
We show as well, using the same arguments as above, that if every $X\in\mathcal{R}_{\ell+k}(S)$, the latter without any previously assigned structure, belongs to the image of a differentiable $\alpha-$section of $\Pi_{\ell+k}P$ taking its values in $\mathcal{R}_{\ell+k}(S)$, then $\mathcal{R}_{\ell+k}(S)$ is a regularly embedded sub-manifold of $\Pi_{\ell+k}P~$. Such sections are, by the way, usually obtained by composing sections of $\overline{\alpha}:\zeta(\mathcal{R}_{\ell+k}(S))~\longrightarrow~\alpha(S)$ with sections of $\zeta~$. In sum, we proved the following:
\vspace{2 mm}
\newtheorem{equivalent}[PropositionCounter]{Proposition}
\begin{equivalent}
Let S be a structure of species E and finite order $\ell$ satisfying the equivalent requirements of the Proposition $1$. Under these conditions, the following properties are also equivalent:
\vspace{2 mm}
\hspace{2 mm}i) Every element of $\mathcal{R}_{\ell+k}(S)$ belongs to the image of some differentiable section of $\alpha:\Pi_{\ell+k}P~\longrightarrow~P$ taking its values in $\mathcal{R}_{\ell+k}(S)$.
\vspace{2 mm}
\hspace{1 mm}ii) The closed subset $\zeta(\mathcal{R}_{\ell+k}(S))\subset\Pi_{\ell+k}P~/~\Delta_{\ell+k}$ is a regularly embedded sub-manifold \textit{étale} over $\alpha(S)$.
\vspace{2 mm}
iii) $\mathcal{R}_{\ell+k}(S)$ is a Lie sub-groupoid of $\Pi_{\ell+k}P$ with associated Lie algebroid equal to $\underline{\textbf{R}_{\ell+k}(S)}$.
\vspace{2 mm}
\hspace{1 mm}iv) $\mathcal{R}_{\ell+k}(S)$ is a regularly embedded Lie sub-groupoid of $\Pi_{\ell+k}P~$.
\end{equivalent}
\vspace{4 mm}
It is desirable, at this point, to examine more carefully the embedding of $\mathcal{R}_{\ell+k}(S)$. With this in mind, let us assume that the structure \textit{S} satisfies the equivalent conditions of the last Proposition whereby $\mathcal{R}_{\ell+k}(S)$ becomes a regularly embedded
Lie sub-groupoid of $\Pi_{\ell+k}P$ and let us denote by $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$ the set of right sided classes of $\Pi_{\ell+k}P$ modulo $\mathcal{R}_{\ell+k}(S)$, that is to say, the quotient modulo the relation: $X\sim Y$ if and only if $X\cdot Y^{-1}\in\mathcal{R}_{\ell+k}(S)$. These classes are no other than the orbits, by the left action (via the target), of $\mathcal{R}_{\ell+k}(S)$ on $\Pi_{\ell+k}P$ and consequently we see promptly that the right action (via the source) of $\Pi_{\ell+k}P$ on itself factors to an action (to the right) of $\Pi_{\ell+k}P$ on the above quotient $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$. We have already seen that each $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)$ is a closed and regularly embedded sub-manifold of an $\alpha-$fibre of $\Pi_{\ell+k}P$ and its connected components are the integral leaves of $\Delta_{\ell+k}~$. Let us now take a unit \textbf{e}. Since there exists a saturated open set $\tilde{\mathcal{U}}$ such that $\tilde{\mathcal{U}}\cap\mathcal{R}_{\ell+k}(S)=\tilde{\mathcal{U}}\cap\mathcal{R}_{\ell+k}(S)_0$ , we infer, in view of the previous results, that there exists a foliating chart $\mathcal{U}$ for $\Delta_{\ell+k}$ , neighborhood of \textbf{e}, such that each $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)$ issued from a point $\textbf{e'}\in\mathcal{U}\cap\mathcal{I}$ meets the open set $\mathcal{U}$ along a unique slice namely, the slice containing \textbf{e'}. Let us next observe that the right action of $\Pi_{\ell+k}P$ permutes the trajectories (orbits) of $\mathcal{R}_{\ell+k}(S)$. Arguments entirely analogous to those used previously for the integral foliation of $\Delta_{\ell+k}$ will show that, for any $X\in\Pi_{\ell+k}P$, there exists a foliating chart $\mathcal{V}$ of $\Delta_{\ell+k}$ , neighborhood of \textit{X} , such that an arbitrary trajectory of $\mathcal{R}_{\ell+k}(S)$ will meet the open set $\mathcal{V}$ in at most one slice. We find ourselves within conditions entirely analogous to those found in the Theorem 8, pg.19 of \cite{Palais1957}. In this theorem, the differentiable structure of the quotient (\textit{i.e.}, the appropriate changes of charts) is guaranteed by the transport Theorem (\cite{Palais1957}, pg.10) which however does not apply in the present case in view of the (eventual) non-connectivity of the $\alpha-$fibres of $\mathcal{R}_{\ell+k}(S)$. Nevertheless, the transport can be replaced by the following argument: Let $X\in\mathcal{R}_{\ell+k}(S)$, $\textbf{e}=\alpha(X)$ the corresponding source, $\mu$ a differentiable section of $\alpha:\mathcal{R}_{\ell+k}(S)~\longrightarrow~\alpha(S)$ assuming the value $\mu(\textbf{e})=X$ (it is essentially here that intervenes the property (\textit{i}) of the last Proposition) and $\mathcal{U}$ , respectively $\mathcal{U}'$, the domain of a foliating chart of $\Delta_{\ell+k}~$, neighborhood of \textbf{e}, respectively \textit{X}, whose intersection with any orbit of $\mathcal{R}_{\ell+k}(S)$ reduces at most to a single slice and that verifies moreover the condition $\mu(\mathcal{U}\cap\mathcal{I})\subset\mathcal{U}'$. Let $\mathcal{W}$ be a sub-manifold transverse to the slices of $\mathcal{U}$ such that $\mathcal{W}\supset\mathcal{U}\cap\mathcal{I}$ and let us set
\begin{equation*}
\mathcal{V}=\{Y\in\mathcal{W}~|~\beta(Y)\in\mathcal{U}\cap\mathcal{I}\}.
\end{equation*}
\vspace{4 mm}
\noindent
Clearly, $\mathcal{V}$ remains a transversal sub-manifold and we define the mapping $\Sigma:\mathcal{V}~\longrightarrow~\Pi_{\ell+k}P~,~\Sigma(Y)=\mu(\beta Y)\cdot Y$, and we see readily that the following properties hold:
\vspace{4 mm}
1) $\Sigma(Y)$ is contained in the orbit of \textit{Y},
\vspace{3 mm}
2) $\Sigma~|~\mathcal{U}\cap\mathcal{I}=\mu~$,
\vspace{3 mm}
3) $\Sigma(\mathcal{V})\subset\mathcal{U}'$, in shrinking if necessary the sub-manifold $\mathcal{V}$.
\vspace{4 mm}
\noindent
The condition (1) implies that $\Sigma$ is injective. Furthermore, since the right action of $\Pi_{\ell+k}P$ on itself is effective and the map
\begin{equation*}
\mu:~\mathcal{U}\cap\mathcal{I}~\longrightarrow~\mu(\mathcal{U}\cap\mathcal{I})
\end{equation*}
\vspace{2 mm}
\noindent
is a diffeomorphism of regularly embedded sub-manifolds, we infer that $\Sigma$ has injective rank (on the tangent level) and consequently $\Sigma:\mathcal{V}~\longrightarrow~\Sigma(\mathcal{V})$ is a diffeomorphism of sub-manifolds respecting the orbits (property (1)) and $\Sigma(\mathcal{V}$ is transverse to the slices of $\mathcal{U}'$. We can therefore extend the conclusions of Palais' Theorem to the space of the orbits of $\mathcal{R}_{\ell+k}(S)$ since the two charts in the quotient space originating from $\mathcal{U}$ and $\mathcal{U}'$ do as well originate from the transverse sub-manifolds $\mathcal{V}$ and $\Sigma(\mathcal{V})$ and therefore are compatible. The quotient set $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$ admits therefore a (necessarily unique) differentiable manifold structure for which the quotient map is a surmersion. We next remark that $\Pi_{\ell+k}P~/~\Delta_{\ell+k}=\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)_0$ and the diagram below is commutative,
\begin{equation*}
\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)_0~\xleftarrow{\hspace{2 mm}\zeta\hspace{2 mm}}~\Pi_{\ell+k}P~\xrightarrow{~\zeta'~}~\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)
\end{equation*}
\begin{equation*}
\downarrow\hspace{30 mm}\downarrow\hspace{30 mm}\downarrow
\end{equation*}
\begin{equation*}
P\hspace{9 mm}\xleftarrow{~Id~}\hspace{9 mm}P\hspace{9 mm}\xrightarrow{~Id~}\hspace{9 mm}P
\end{equation*}
\vspace{4 mm}
\noindent
the arrow $\Upsilon=\zeta'\circ\zeta^{-1}$ being surjective and \textit{étale}. Furthermore, $\mathcal{R}_{\ell+k}(S)=(\zeta')^{-1}(\zeta'(\mathcal{I}))$ (inverse image) and $\zeta(\mathcal{R}_{\ell+k}(S))=\Upsilon^{-1}(\zeta'(\mathcal{I}))$. The arrows $\zeta,~\zeta'$ and $\Upsilon$ are differential co-variants with respect to the right action (by the source) of $\Pi_{\ell+k}P$ on the three spaces. The quotient manifold $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$ can be obtained from $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)_0$ by identifying the points on each orbit that are deducible one from the other by the discrete action of $\mathcal{R}_{\ell+k}(S)$ and this identification is globally compatible when the equivalent properties of the above Proposition are verified. We next observe that the co-variance of $p_k\Phi(S)$ enables the factorisation of this morphism, modulo the action of $\mathcal{R}_{\ell+k}(S)$, and consequently the diagram that follows is commutative, the factored differential co-variant $p'_k\Phi(S)$ becoming an injective immersion (\textit{inmersion? Kkkk}).
\begin{equation*}
\Pi_{\ell+k}P~\xrightarrow{p_k\Phi(S)}~J_kE
\end{equation*}
\begin{equation}
\hspace{7 mm}\zeta'\downarrow\hspace{16 mm}\nearrow~p'_k\Phi(S)
\end{equation}
\begin{equation*}
\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)\hspace{5 mm}
\end{equation*}
\vspace{4 mm}
\noindent
We infer that $im~p_k\Phi(S)$ is a sub-manifold (not always regularly embedded) of $J_kE$ canonically isomorphic to the quotient $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$, the map $\alpha:im~p_k\Phi(S)~\longrightarrow~\alpha(S)$ is a fibration and
\begin{equation*}
p_k\Phi(S):~\Pi_{\ell+k}P~\longrightarrow~im~p_k\Phi(S)
\end{equation*}
\vspace{2 mm}
\noindent
is a surmersive morphism of fibrations having for basis $\alpha(S)$ (\textit{P} being replaced by $\alpha(S)$ when \textit{S} is not global). The covariance of $p_k\Phi(S)$ finally shows that $(im~p_k\Phi(S),\alpha,\alpha(S),p_k)$ is a prolongation space of order $\ell+k$ and that the map
\begin{equation*}
p'_k\Phi(S):(\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S),\overline{\alpha},\alpha(S),p_k^S)\longrightarrow(im~p_k\Phi(S),\alpha,\alpha(S),p_k)
\end{equation*}
\vspace{2 mm}
\noindent
is an isomorphism of prolongation spaces where the first term is given the quotient prolongation space structure, modulo the right action of $\mathcal{R}_{\ell+k}(S)$, of the canonical structure of $\Pi_{\ell+k}P$ resulting from the standard prolongation operation by the source (\cite{Kumpera1975}, $\S$ 16, part (b)).
\vspace{2 mm}
\newtheorem{quotient}[TheoremCounter]{Theorem}
\begin{quotient}
Let S be a structure of species E and of finite order $\ell$ such that $p_k\Psi(S)$ is locally of constant rank. Then the following conditions are equivalent:
\vspace{2 mm}
\hspace{2 mm}i) $\mathcal{R}_{\ell+k}(S)$ is a Lie sub-groupoid of $\Pi_{\ell+k}P$ whose associated Lie algebroid is equal to $\underline{\textbf{R}_{\ell+k}(S)}$.
\vspace{2 mm}
\hspace{1 mm}ii) $\mathcal{R}_{\ell+k}(S)$ is a regularly embedded (and closed) Lie sub-groupoid of $\Pi_{\ell+k}P~$.
\vspace{2 mm}
iii) There exists a differentiable structure on $\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S)$ such that the quotient map is a submersion.
\vspace{2 mm}
\hspace{1 mm}iv) The image of $p_k\Phi(S)$ admits a sub-manifold structure of $J_kE$ such that the map $p_k\Phi(S):\Pi_{\ell+k}P~\longrightarrow~im~p_k\Phi(S)$ is a submersion.
\vspace{2 mm}
\hspace{2 mm}v) Every element of $\mathcal{R}_{\ell+k}(S)$ belongs to the image of a local differentiable section of $\alpha:\Pi_{\ell+k}P~\longrightarrow~P$ taking values in $\mathcal{R}_{\ell+k}(S)$.
\vspace{2 mm}
\noindent
These equivalent conditions being verified, the quotient differential covariant
\begin{equation*}
p'_k\Phi(S):(\Pi_{\ell+k}P~/~\mathcal{R}_{\ell+k}(S),\overline{\alpha},\alpha(S),p_k^S)\longrightarrow(im~p_k\Phi(S),\alpha,\alpha(S),p_k)
\end{equation*}
\vspace{2 mm}
\noindent
is an isomorphism of prolongation spaces.
\end{quotient}
\vspace{4 mm}
\newtheorem{sub-quotient}[CorollaryCounter]{Corollary}
\begin{sub-quotient}
Let S be a structure of species E and of finite order $\ell$ such that $p_k\Psi(S)$ is locally of constant rank and let us assume further that the equation $\textbf{R}_{\ell+k}(S)$ is transitive $(\beta(\textbf{R}_{\ell+k}(S))=TP)$. Under these conditions, the equivalent properties of the previous Theorem are always satisfied namely, $\mathcal{R}_{\ell+k}(S)$ is a closed and regularly embedded Lie sub-groupoid of $\Pi_{\ell+k}P$. Moreover, $\mathcal{R}_{\ell+k}(S)$ as well as $\mathcal{R}_{\ell+k}(S)_0$ are locally trivial Lie sub-groupoids and $\mathcal{R}_{\ell+k}(S)_0$
is closed in $\Pi_{\ell+k}P$.
\end{sub-quotient}
\vspace{2 mm}
\noindent
$\bf{Proof.}$ The transitivity of $\textbf{R}_{\ell+k}(S)$ implies that the restriction of $\beta$ to each $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)$ is a submersion (that will be surjective whenever $\mathcal{R}_{\ell+k}(S)$ becomes transitive). Let $X\in\mathcal{R}_{\ell+k}(S)$, $y=\alpha(X)$, and let us take a section $\sigma$ of $\beta:\mathcal{R}_{\ell+k}(S)_x~\longrightarrow~P$ defined on an open neighborhood $\mathcal{U}$ of \textit{y} such that $\sigma(y)=\textbf{e}$ (the unit associated to \textit{y}). The map $\tau:\mathcal{U}~\longrightarrow~\Pi_{\ell+k}P~$,
$\tau(y)=X\cdot\sigma(y)^{-1}$, is an $\alpha-$section taking its values in $\mathcal{R}_{\ell+k}(S)$ and such that $\tau(y)=X~$, thus retrieving the property (\textit{v}) of the last Theorem. The submersivity of $\beta$ on each $\alpha-$fibre of $\mathcal{R}_{\ell+k}(S)$ implies the submersivity of $\alpha\times\beta:\mathcal{R}_{\ell+k}(S)~\longrightarrow~\alpha(S)\times\alpha(S)$ whereupon the possibility (\cite{Kumpera1971}) in defining local trivialisations of $\mathcal{R}_{\ell+k}(S)$ and $\mathcal{R}_{\ell+k}(S)_0$ with the help of local sections of $\alpha\times\beta$. We also remark, \textit{en passant}, that each domain of a local trivialisation admits a regularly embedded differentiable structure inherited from the isotropy, at a given point, of $\mathcal{R}_{\ell+k}(S)$ (resp. $\mathcal{R}_{\ell+k}(S)_0$) , this isotropy being a closed Lie sub-group hence regularly embedded (resp. regularly embedded hence also closed). The family of all such trivialisations is compatible and defines thereafter the regularly embedded structure of $\mathcal{R}_{\ell+k}(S)$ inasmuch as that of $\mathcal{R}_{\ell+k}(S)_0~$. We finally note that, in the transitive case envisaged, the topological nature of $\mathcal{R}_{\ell+k}(S)_0$ is entirely determined by the topological nature of its isotropy group at a point. Since this group is closed in $(\Pi^0_{\ell+k}P)_y~$, we derive that $\mathcal{R}_{\ell+k}(S)_0$ is closed in $\Pi_{\ell+k}P~$. Observe, however, that the isotropy of $\mathcal{R}_{\ell+k}(S)_0$ is not necessarily the connected component, of the unit, in the isotropy of $\mathcal
{R}_{\ell+k}(S)$.
\vspace{4 mm}
\newtheorem{inf-quotient}[CorollaryCounter]{Corollary}
\begin{inf-quotient}
Let S be a structure of species E and of order $\ell$ such that $p_k\Psi(S)$ is locally of constant rank. Under these conditions,
\vspace{2 mm}
a) $\mathcal{R}_{\ell+k+h}(S)_0$ is the standard $h-th$ prolongation of $\mathcal{R}_{\ell+k}(S)_0$ i.e.,
\begin{equation*}
\mathcal{R}_{\ell+k+h}(S)_0=\Pi_{\ell+k+h}P\cap J_h\mathcal{R}_{\ell+k}(S)_0~.
\end{equation*}
\vspace{2 mm}
b) If, moreover, S verifies the equivalent conditions of the Theorem, then $\mathcal{R}_{\ell+k+h}(S)$ is the standard $h-th$ prolongation of
$\mathcal{R}_{\ell+k}(S)$.
\end{inf-quotient}
\vspace{4 mm}
\noindent
The proof of this corollary relies on \cite{Kumpera1972} and on the following Lemma:
\vspace{2 mm}
\newtheorem{not-quotient}[LemmaCounter]{Lemma}
\begin{not-quotient}
Let $E'\longrightarrow E\longrightarrow(E'',\sigma)$ be an exact sequence of fibrations over the base space P (exact in the set theoretical sense and also in the vertical tangential sense). Then, for any integer k, the sequence of prolonged fibrations $J_kE'\longrightarrow J_kE
\longrightarrow(J_kE'',j_k\sigma)$ is also exact.
\end{not-quotient}
\vspace{4 mm}
\noindent
It is then achieved by an inductive argument on the integer \textit{k} using, at each stage, the affine structure of the kernels as well as the exactness of the sequence of linear symbols that is a consequence of the tangential exactness of the initially given sequence.
\vspace{4 mm}
\noindent
In order to prove the part (\textit{b}) of the corollary, we simply use the exactness of the sequence
\begin{equation*}
\mathcal{R}_{\ell+k}(S)~\longrightarrow~\Pi_{\ell+k}P~\xrightarrow{p_k\Phi(S)}~(J_kE,j_kS)
\end{equation*}
\vspace{2 mm}
\noindent
whose tangential exactness follows from the local constancy of the rank of $p_k\Phi(S)$ and thereafter observe that the diagram below is commutative and exact:
\begin{equation*}
\textbf{1}~\longrightarrow~J_h\mathcal{R}_{\ell+k}(S)~\longrightarrow~J_h\Pi_{\ell+k}P~\xrightarrow{J_hp_k\Phi(S)}~(J_hJ_kE,j_hj_kS)
\end{equation*}
\begin{equation*}
\hspace{50 mm}\nearrow
\end{equation*}
\begin{equation*}
\hspace{7 mm}\iota\uparrow\hspace{24 mm}\iota\uparrow\hspace{7 mm}p_hp_k\Phi(S)\hspace{12 mm}\iota\uparrow
\end{equation*}
\begin{equation*}
\hspace{23 mm}\diagup
\end{equation*}
\begin{equation*}
\textbf{1}~\longrightarrow~\mathcal{R}_{\ell+h+k}(S)~\longrightarrow~\Pi_{\ell+h+k}P~\xrightarrow{p_{h+k}\Phi(S)}~(J_{h+k}E,j_{h+k}S)
\end{equation*}
\begin{equation*}
\end{equation*}
\begin{equation*}
\uparrow\hspace{28 mm}\uparrow\hspace{32 mm}
\end{equation*}
\begin{equation*}
\end{equation*}
\begin{equation*}
\textbf{1}\hspace{28 mm}\textbf{1}\hspace{32 mm}
\end{equation*}
\vspace{4 mm}
\noindent
As for the part (\textit{a}), we simply replace, according to the Proposition 2, the previous exact sequence by
\begin{equation*}
\mathcal{R}_{\ell+k}(S)_0~\longrightarrow~\mathcal{U}~\xrightarrow{p_k\Phi(S)}~(J_kE,j_kS)
\end{equation*}
\vspace{2 mm}
\noindent
We now observe that it is essential to use hypotheses guaranteeing the appropriate differentiable structures for $\mathcal{R}_{\ell+k}(S)$ and $\mathcal{R}_{\ell+k}(S)_0$ in the lack of which the above Corollary becomes inexact, not subsisting but the inclusion
\begin{equation*}
\Pi_{\ell+h+k}P\cap J_h\mathcal{R}_{\ell+k}(S)\subset\mathcal{R}_{\ell+h+k}(S).
\end{equation*}
\vspace{2 mm}
\noindent
The above proof being rather esotheric\footnote{qualification donnée, dans les écoles des anciens philosophes, à leure doctrine secrète, réservée aux seuls initiés}, we shall transcribe it in local coordinates for the usage of the non-initiated. However, this naive transcription will be useful later. We recall that both Lie and Cartan frequently indulged into incredible calculations since they believed, presumably, that this was the first step in understanding Heaven right here from earth. Nowadays, calculations are for many a boring activity though, for others, become indispensable. Just imagine a cosmologist trying to figure out whether Einstein's constant \textit{c} is the same here in our vicinity as in, for example, \textit{Andromeda}, 60 million light years away, or in the \textit{Whirlpool Galaxy}? The same doubts arise inasmuch for $\pi$ and on how do Pythagorean circles behave in the \textit{Magellania Cloud} or on how does \textit{e} behave, does it also change + for $\times$, in \textit{Antennae}? Even at a much closer range, just beyond the neutral zone, we might ask whether the Klingon's uncertainty (undecidability) $\hbar$ is as high as for the humans or whether they are more self-confident? (\cite{Mayot1945})
\vspace{4 mm}
\begin{figure}[ht!]
\centering
\includegraphics[scale=1.7]{universe.jpg}
\caption{The Universe}
\label{fig:univerise}
\end{figure}
\vspace{4 mm}
\noindent
In order to do so, let us return to the notations in the proof of the last Proposition and take a point $X\in\mathcal{R}_{\ell+k}(S)$ as well as a differentiable $\alpha-$section $\mu:\overline{\mathcal{U}}~\longrightarrow~\Pi_{\ell+k}P$ taking its values in $\mathcal{R}_{\ell+k}(S)$ and such that $\mu(y)=X~$. By the local constancy of the rank of $p_k\Phi(S)$, there exists an open neighborhood $\mathcal{U}$ of \textit{X} in $\Pi_{\ell+k+h}P$ such that $\mathcal{W}=p_k\Phi(S)(\mathcal{U})$ is a regularly embedded sub-manifold of $J_kE$ and that $p_k\Phi(S):\mathcal{U}~\longrightarrow~\mathcal{W}$ is a surmersion. We can assume, as previously, that $\overline{\mathcal{U}}=\alpha(\mathcal{U})=\alpha(\mathcal{U}\cap im~\mu)$ which implies that $\mathcal{W}\cap im~j_kS=j_kS(\overline{\mathcal{U}})$ is a regularly embedded sub-manifold of $\mathcal{W}$ . Shrinking, if necessary, the open set $\mathcal{U}$ , let us take a finite family $(f_i)$ of independent functions whose zeros define the sub-manifold $j_kS(\overline{\mathcal{U}})$ in $\mathcal{W}$ .
The local constancy of the rank of $p_k\Phi(S)$ combined with the key property $\mathcal{W}\cap im~j_kS=j_kS(\overline{\mathcal{U}})$ forces the composite functions $F_i=f_i\circ p_k\Phi(S)$ to be also independent, their zeros defining $\mathcal{R}_{\ell+k}(S)\cap \mathcal{U}~$. We shall now complete the functions $(F_i)$ to a local coordinates system by adding some functions $(G_j)$. The condition $Y=j_{\ell+h+k}\varphi(y)\in\mathcal{R}_{\ell+h+k}(S)$ means that $p_{h+k}\Phi(S)(Y)=j_{h+k}S(y)$ or, equivalently, since $p_{h+k}=(p_k)_h~$, that
\begin{equation*}
j_h(p_k\Phi(S)(j_{\ell+k}\varphi))(y)=j_h(j_kS)(y).
\end{equation*}
\vspace{4 mm}
\noindent
Translated into coordinates, this condition reads $j_h(F_i\circ j_{\ell+k}\varphi)(y)=0~$. However, still in coordinates, if we replace $j_{\ell+k}\varphi$ by the local section $\sigma$ of $\alpha:\mathcal{R}_{\ell+k}(S)~\longrightarrow~P$ whose components along the coordinates $F_i$ are null and, along the $G_j$, are equal to those of $j_{\ell+k}\varphi~$, we shall obtain the equality $j_h(j_{\ell+k}\varphi)(y)=j_h\sigma(y)$ and consequently $Y\in\Pi_{\ell+h+k}P\cap J_h\mathcal{R}_{\ell+k}(S)$. The above argument is in fact a transversality argument of $p_k\Phi(S)$ with $im~j_kS$ in a slightly more general context since transversality of the two, in the usual context, does not hold. What holds in fact is the following: There exists, in an open neighborhood $\mathcal{W}$ of each point \textit{y} belonging to $im~j_kS~$, a local foliation for which one of its leaves is an open subset of $im~j_kS$ (for example, the foliation $f_i=c_i$ of the former open set $\mathcal{W}$) and such that, if we denote by $\mathcal{W}'$ the quotient of $\mathcal{W}$ modulo the leaves and by $\rho:\mathcal{W}~\longrightarrow~\mathcal{W}'$ the projection, the composed map $\rho\circ p_k\Phi(S)$ will be of constant rank on a sufficiently small open neighborhood $\mathcal{V}$ of \textit{y}. This argument proves the claim in part (\textit{b}) of the Corollary. As for the part (\textit{a}) it will suffice to repeat the argument placing us above the open set $\mathcal{U}$.
\vspace{4 mm}
\noindent
Let us now reassume the general case where no regularity hypothesis, on $p_k\Psi(S)$, is assumed. We already remarked that the isotropies $(\textbf{R}^0_{\ell+k}(S))_y$ and $(\mathcal{R}^0_{\ell+k}S)_y$ only depend upon the jet $j_kS(y)$ and that, in particular, the isotropy of order $\ell$ only depends on the point $S(y)$. It then follows that the \textit{symbol} $(\mathfrak{g}_{\ell}S)_y$ of $\textbf{R}_{\ell}(S)$ at the point \textit{y} also depends only on $S(y)$. Observing that $\textbf{R}_{\ell+k}(S)=ker~p_k\Psi(S)$, a simple calculation (\cite{Kumpera1975}, Lemma 23.2, \cite{Kuranishi1967}, Proposition 4.3, \cite{Ruiz1977}, Proposition 9.3) will show that the symbol $(\mathfrak{g}_{\ell+k}S)_y$ of $\textbf{R}_{\ell+k}(S)$ is the $k-th$ algebraic prolongation (\textit{espace déduit}) of the symbol $(\mathfrak{g}_{\ell}S)_y$ and consequently,
\vspace{4 mm}
\newtheorem{symbolic}[PropositionCounter]{Proposition}
\begin{symbolic}
The symbol of $\textbf{R}_{\ell+k}(S)$ at the point y only depends upon $S(y)\in E$ and this result is independent of any regularity condition requirement on the morphism $p_k\Psi(S)$. Moreover, the symbol $(\mathfrak{g}_{\ell+k}S)_y$ is the $k-th$ algebraic prolongation of $(\mathfrak{g}_{\ell}S)_y~$.
\end{symbolic}
\vspace{4 mm}
\noindent
We now examine the non-linear situation. Let $X\in\mathcal{R}_{\ell+k-1}(S)$, consider the canonical projection $\rho:\Pi_{\ell+k}P~\longrightarrow~\Pi_{\ell+k-1}P$ and define the non-linear symbol
\begin{equation*}
(g_{\ell+k}S)_X=\{Y\in\mathcal{R}_{\ell+k}(S)~|~\rho(Y)=X\}
\end{equation*}
\vspace{2 mm}
\noindent
of $\mathcal{R}_{\ell+k}(S)$ above \textit{X}. Let us show and this without any regularity hypotheses on $p_k\Phi(S)$ that $g_{\ell+k}S)_X$ is an affine sub-space of the total symbol above \textit{X} namely, the space $\{Y\in\Pi_{\ell+k}P~|~\rho(Y)=X\}$. We first observe that, for each pair $(y,z)\in P\times P$, the set
\begin{equation*}
\mathcal{R}_{\ell+k}(S)_{(y,z)}=\{X\in\mathcal{R}_{\ell+k}(S)~|~\alpha(X)=y,~\beta(X)=z\}
\end{equation*}
\vspace{2 mm}
\noindent
is a simply transitive homogeneous space of the group $\mathcal{R}^0_{\ell+k}(S)_y$ by the right action and also of the group $\mathcal{R}^0_{\ell+k}(S)_z$ by the left action. It then follows that $\mathcal{R}_{\ell+k}(S)_{(y,z)}$ is a closed and regularly embedded sub-manifold of $\Pi_{\ell+k}P_{(y,z)}$ canonically isomorphic to the left or to the right isotropy. For any $X\in\mathcal{R}_{\ell+k-1}(S)$, with $y=\alpha(X)$ and $z=\beta(X)$, the same argument shows that the symbol $(g_{\ell+k}S)_X$ is a closed and regularly embedded sub-manifold of the total symbol above \textit{X} since it is a simply transitive homogeneous space of the closed Lie group
\begin{equation*}
(g_{\ell+k}S)_y=ker~[\mathcal{R}^0_{\ell+k}(S)_y~\longrightarrow~\mathcal{R}^0_{\ell+k-1}(S)_y]
\end{equation*}
\vspace{2 mm}
\noindent
by the right action and also of the corresponding group $(g_{\ell+k}S)_z$ by the left action. Note that $(g_{\ell+k}S)_y$ is simply the symbol of $\mathcal{R}_{\ell+k}(S)$ above the unit of $\mathcal{R}_{\ell+k-1}(S)$ that identifies with \textit{y}. Let us finally show that $(g_{\ell+k}S)_X$ is a connected sub-manifold, in fact an affine sub-space of the total symbol. For this, let \textit{Z} be the projection of \textit{X} in $\mathcal{R}_{\ell}(S)$ and observe that every $Y\in(g_{\ell+k}
S)_X$ projects onto \textit{Z} by the projection $\rho_{\ell}$. We now take (\textit{at last}) local coordinate systems $(y^i)$ in an open neighborhood $\mathcal{U}$ of \textit{y} and $(z^\lambda)$ in an open neighborhood $\mathcal{V}$ of \textit{z} and consider the corresponding "jet" coordinate system $(y^i,z^{\lambda},z^{\lambda}_{\alpha})_{|\alpha|\leq r}$ on the open set $(\alpha\times\beta)^{-1}_r(\mathcal{U}\times\mathcal{V})$ of $\Pi_rP~$. Inasmuch, we also take an adapted local coordinate system $(y^i,w^{\mu})$ in an open neighborhood $\mathcal{W}$ of the point $\Phi(S)(Z)=S(y)$ in \textit{E} $(y^i=y^i\circ\pi_0)$ and there exists of course an open neighborhood $\mathcal{U}_{\ell}$ of \textit{Z} in $(\alpha\times\beta)^{-1}_{\ell}(\mathcal{U}\times\mathcal{V})$ such that $\Phi(S)(\mathcal{U}_{\ell})\subset\mathcal{W}~$. We write $\{\Phi^i,\Phi^{\mu}\}$ the components, along the coordinates $(y^i,w^{\mu})$, of the restriction $\Phi(S):\mathcal{U}_\ell~\longrightarrow~\mathcal{W}~$. Since $\Phi(S)$ is a morphism over the Identity, we infer that $\Phi^i(y^i,z^{\lambda},z^{\lambda}_{\alpha})=y^i$. We denote by $\mathcal{U}_{\ell+k}$ the inverse image, by $\rho:~\Pi_{\ell+k}P~\longrightarrow~Pi_{\ell}P~$, of the open set $\mathcal{U}_{\ell}$ and observe that the former contains
$(g_{\ell+k}S)_X$ and is endowed with the restrictions of the coordinates $(y^i,z^{\lambda},z^{\lambda}_{\alpha})_{|\alpha|\leq\ell+k}~$. Finally, denote by $\rho^{-1}(\mathcal{W})=\mathcal{W}_k$ the open set, of $J_kE$, inverse image of $\mathcal{W}$ and equipped with the natural coordinates $(y^i,w^{\mu},w^{\mu}_{\alpha})_{|\alpha|\leq k}$ derived from $(y^i,w^{\mu})$. Then $p_k\Phi(S)(\mathcal{U}_{\ell+k})\subset\mathcal{W}_k$ and the components of the restriction $p_k\Phi(S):\mathcal{U}_{\ell+k}~\longrightarrow~\mathcal{W}_k~$, with respect to the coordinates $(y^i,w^{\mu},w^{\mu}_{\alpha})_{|\alpha|\leq k}$, are precisely the functions $\{\Phi^i,\Phi^{\mu},\partial_{\alpha}\Phi^{\mu}\}_{|\alpha|\leq k}$ where $\partial_{\alpha}$ is the \textit{total derivative} of order $|\alpha|$ with respect to the variables $(y^i)$ (iterated total derivatives in jet spaces). In particular, the symbol $(g_{\ell+k}S)_X$ is defined, in the affine space of the total symbol over \textit{X}, by the equations:
\begin{equation*}
\partial_{\alpha}\Phi^{\mu}(Y)=w^{\mu}_{\alpha}(j_kS(x))=(\partial^{\alpha} S^{\mu}/\partial y^{\alpha})(y),\hspace{5 mm}|\alpha|=k~.
\end{equation*}
\vspace{2 mm}
\noindent
Since
\begin{equation*}
\partial_{\alpha}\Phi^{\mu}(Y)=\sum_{|\beta|=\ell}~(\partial\Phi^{\mu}/\partial z^{\lambda}_{\beta})(Z)\cdot z^{\lambda}_{\alpha+\beta}~+~F_{\alpha}(X)~,
\end{equation*}
\vspace{2 mm}
\noindent
where, on the right hand side, we only detail the highest order terms, the symbol $(g_{\ell+k}S)_X$ being thereafter determined by the following linear equations with constant coefficients in the variables $z^{\lambda}_{\gamma},~|\gamma|=\ell+k~$,
\begin{equation*}
\sum_{|\beta|=\ell}~(\partial\Phi^{\mu}/\partial z^{\lambda}_{\beta})(Z)\cdot z^{\lambda}_{\alpha+\beta}=(\partial^{\alpha}S^{\mu}/\partial y^{\alpha})(y)~-~F_{\alpha}(X),
\end{equation*}
\vspace{4 mm}
\noindent
defining, as pretended, a linear affine sub-space in the space of the total symbol.
\vspace{4 mm}
\noindent
Let us now glimpse at the intrinsical aspects. With the help of the canonical identification, we can see that the abelian Lie algebra $(\mathfrak{g}_{\ell+k}S)_z\subset\textbf{R}^0_{\ell+k}(S)_z$ is not only the Lie algebra of the group
$(g_{\ell+k}S)_z$ but has much more impact. Recalling the results of \cite{Kumpera1975}, $\S$ 19, each element $v\in\mathfrak{g}_{\ell+k}S)_z$ determines a vector field on the total symbol space that generates a global 1-parameter group $(\varphi_t)_t$ with the property that $\varphi_1(Y)=Y+v$ is precisely the affine operation by the vector \textit{v}. The orbits of this action are all the linear affine sub-spaces of the total symbol whose direction is given by $(\mathfrak{g}_{\ell+k}S)_z$ and finally, since the sub-spaces generated, at each point, by the above vector fields are necessarily contained in $\Delta_{\ell+k}$ , the Proposition 1 will imply that an orbit, by the above (infinitesimal) affine action of $(\mathfrak{g}_{\ell+k}S)_z~$, that contains an element of $(g_{\ell+k}S)_X$ is entirely contained in $(g_{\ell+k}S)_X~$. A dimensional
argument will also show that $dim~(g_{\ell+k}S)_X$ is equal to the dimension of
the orbits
and the previous tinkering (\textit{bricolage}) with local coordinates only serves
to prove that $(g_{\ell+k}S)_X$ is also connected hence equal to an entire orbit. Furthermore, if we consider as symbol of a non-linear equation the family of all tangent spaces to the non-linear symbol $(g_{\ell+k}S)_X~$, \textit{the tangent symbol}, we perceive that this family of tangent symbols is nothing else, by the canonical identification, than $(\mathfrak{g}_{\ell+k}S)_z~$. It then follows that the tangent symbol of $\mathcal{R}_{\ell+k}(S)$ above the point \textit{X} is the $k-th$ algebraic prolongation of the tangent symbol of $\mathcal{R}_{\ell}(S)$ at
the point \textit{Z}, hence only depends on $S(z)$. Otherwise, this last result can equally be obtained with the help of the Lemma 23.2 in \cite{Kumpera1975} or the Proposition 4.3 in \cite{Kuranishi1967} or still the Proposition 9.3 in \cite{Ruiz1977}. Let us finally observe that the groupoid structure of $\mathcal{R}_{\ell+k}(S)$ enables us to further explicit the affine structure of
the non-linear symbol. In fact, the argument in coordinates shows that the group $(g_{\ell+k}S)_z$ is connected, non-compact and actually homeomorphic to a numerical space. Being abelian, it canonically identifies with its Lie algebra $(\mathfrak{g}_{\ell+k}S)_z$ and the affine action of the latter on the total symbol above \textit{X} is nothing else but the left action by the abelian group $(g_{\ell+k}S)_z$. The symbol $(g_{\ell+k}S)_X$ is just one of the orbits of this action, the restricted action becoming simply transitive (without fixed points). Similarly, the right action of $(g_{\ell+k}S)_y$ on $(g_{\ell+k}S)_X$ is an affine space structure that coincides with the previous one as soon as we identify $(g_{\ell+k}S)_y$ with $(g_{\ell+k}S)_z$ by means of a conjugation via an element of $(g_{\ell+k}S)_X$.
\vspace{2 mm}
\newtheorem{affine}[PropositionCounter]{Proposition}
\begin{affine}
Without any regularity hypotheses on $p_k\Phi(S)$, the symbol of $\mathcal{R}_{\ell+k}(S)$ above any point $X\in\mathcal{R}_{\ell+k-1}(S)$ is an affine sub-space of the total symbol and the corresponding affine structure can be identified with the left action by the symbol $(g_{\ell+k}S)_z$ of the isotropy at the point $z=\beta(X)$. The tangent symbol above X is isomorphic to $(\mathfrak{g}_{\ell+k}S)_z$ and consequently only depends upon the point $S(z)\in E~$. In particular, the non-linear symbol and the tangent symbol above a unit $\textbf{e}\in\mathcal{R}_{\ell+k-1}(S)$ i.e., the symbol of the isotropy group of order $\ell+k$ at the point $y=\alpha(\textbf{e})=\beta(\textbf{e})$ as well as its Lie algebra $(\mathfrak{g}_{\ell+k}S)_y$ only depend upon $S(y)$.
\end{affine}
\section{The general problem}
Let $(E,\pi,P,p)$ be a finite prolongation space of order $\ell$ and let us now assume that \textit{E} hence consequently \textit{P} are paracompact spaces. Let us denote by $\Gamma=\Gamma(P)$ the general pseudo-group of all the local diffeomorphisms of \textit{P} and by $\mathcal{L}=\mathcal{L}(P)$ the pseudo-algebra of all the local vector fields (infinitesimal automorphisms). By prolongation of $\Gamma$ (resp. $\mathcal{L}$) to $J_kE$, we obtain the pseudo-group $\Gamma_{\ell+k}$ resp., the pseudo-algebra (pre-sheaf of Lie algebras) $\mathcal{L}_{\ell+k}$ . These are obtained by localisation of $p_k\Gamma$ resp., $p_k\mathcal{L}$ which means that we consider the set of all local finite or infinitesimal transformations of $J_kE$ that coincide locally with the prolonged transformations (and where $p_k=p_k\circ\pi$). Although $\Gamma$ and $\mathcal{L}$ are "Lie" at any order, this might fail to be true with the prolonged objects, the regularity of these being closely related to the geometry of the prolongation space \textit{E}. Nevertheless, we can still obtain much information concerning the formal equivalence problem as well as on other matters involving structures of species \textit{E} by examining closely the trajectories (orbits) of these prolonged pseudo-groups and pseudo-algebras. In fact, the "Lemma" is as follows:
\vspace{4 mm}
\textit{Two k-jets of structures of species E are equivalent (or two germs of structures of species E are equivalent up to order k) when the two jets find themselves on the same trajectory of $\Gamma_{\ell+k}~$}.
\vspace{4 mm}
\noindent
The prolongation space $p_k:J_kE~\longrightarrow~P$ being of order $\ell+k~$, we know that any local diffeomorphism $\varphi$ of \textit{P} prolongs to a local diffeomorphism $p_k\varphi$ defined by $p_k\varphi(X)= j_k(p\varphi)\cdot X$ . We thus obtain a left or right action of the groupoid $\Pi_{\ell+k}P$ on $J_kE$ though, for the time being, we only consider the left action. If $Z\cdot X=Z'\cdot X~$, then of course $Z^{-1}Z'\in \Pi^0_{\ell+k}P_X~$, the isotropy group of $\Pi_{\ell+k}P$ at the point \textit{X}, that we shall denote by $H_{\ell+k}(X)$ or simply $H(X)$. The action being differentiable, each isotropy group is a closed Lie subgroup of $\Pi^0_{\ell+k}P_y~,~y=\alpha(X)$ and the isotropies at two distinct points of the same orbit $\Omega(X)=\Pi_{\ell+k}P\cdot X$ are conjugate subgroups. Furthermore, the quotient space $(\Pi_{\ell+k}P)_y~/~H_{\ell+k}(X)$ of the classes, to the left, of $(\Pi_{\ell+k}P)_y$ that are orbits under the right action (by the source) of $H_{\ell+k}(X)$, is a differentiable fibre bundle in homogeneous spaces via the left action (by the target) of $\Pi^0_{\ell+k}P)_z~,~z=\beta(X)$, on the fibre above \textit{z}. The isotropy of this left action at the point $Z\in H(X)$ is equal to $H(Z\cdot X)$, it is obtained by the conjugation $H(Z\cdot X)=ZH(X)Z^{-1}$ and the diagram below is commutative:
\vspace{4 mm}
\begin{equation*}
(\Pi_{\ell+k}P)_y\hspace{7 mm}\xrightarrow{~\overline{q}~}\hspace{7 mm}\Omega(X)
\end{equation*}
\begin{equation*}
\downarrow\hspace{20 mm}\nearrow\mu
\end{equation*}
\begin{equation*}
(\Pi_{\ell+k}P)_y~/~H_{\ell+k}(X)\hspace{25 mm}
\end{equation*}
\vspace{4 mm}
\noindent
the arrow $\mu$ being bijective onto $\Omega(X)$ and differentiable as a mapping into $J_kE$. Let us now transport on $\Omega(X)$ and by means of $\mu$ the differentiable structure coming from the quotient and show that $\Omega(X)$ becomes a sub-manifold of $J_kE$ though not necessarily regularly embedded. To do so, it will suffice to show that $\mu$ , as a mapping into $J_kE$ , is an immersion (maximum injective rank) and this leads us to examine the infinitesimal prolongation.
\vspace{4 mm}
\noindent
The prolonged infinitesimal pseudo-algebra $\mathcal{L}_{\ell+k}$ induces, at each point, a subspace of the tangent space to $J_kE$ and consequently a distribution (field of contact elements) $\Delta_k$ on the manifold $J_kE$ that is generated by a family of vector fields stable under the bracket. Though involutive, this distribution can admit singularities. Since the infinitesimal prolongation operator $p_k$ is of order $\ell+k$, each subspace $(\Delta_k)_X$ is entirely determined by $(J_{\ell+k}TP)_y~,~y=\alpha(X)$ and, more precisely, the following sequence is exact:
\begin{equation*}
J_{\ell+k}TP~\times_P~J_kE~\xrightarrow{\lambda_k}~\Delta_k~\longrightarrow~0
\end{equation*}
\vspace{2 mm}
\noindent
Since the sheaf of germs of local sections of $J_{\ell+k}TP$ is free and of finite rank, the image sheaf, that is closed for the bracket and generates $\Delta_k~$, is also of finite type hence (\cite{Hermann1962},\cite{Turiel1976}) every $X\in J_kE$ is contained in a maximal integral sub-manifold $\omega(X)$ and verifies $T_Y\omega(X)=(\Delta_k)_Y$ for all $Y\in\omega(X)$, though the ensemble of these integral sub-manifolds does not form necessarily a regular foliation since their dimensions can vary. The space $J_kE$ admits therefore a partition by integral leaves, with eventual singularities, of $\Delta_k~$. Moreover, the leaf $\omega(X)$ is the set of points of $J_kE$ that can be joined from \textit{X} by piece-wise differentiable integral curves of $\Delta_k~$. Since $\Delta_k$ is generated by $\mathcal{L}_{\ell+k}~$, the leaf $\omega(X)$ is also the trajectory of $\mathcal{L}_{\ell+k}$ passing by \textit{X} hence, the set of all points of $J_kE$ that we can join to \textit{X} (or, for that matter, from \textit{X}) by piece-wise differentiable curves where each differentiable arc is the trajectory of a vector field belonging to $\mathcal{L}_{\ell+k}~$. We infer that $\omega(X)\subset\Omega(X)$ and, more generally, that $\Omega(X)$ is a union of integral leaves of $\Delta_k~$. Furthermore, since $\underline{J_{\ell+k}TP}$ is the Lie algebroid of the Lie groupoid $\Pi_{\ell+k}P$ and since
\vspace{4 mm}
a)the prolongation, by the target, of $\mathcal{L}$ to $\Pi_{\ell+k}P$ is infinitesimally transitive on each $\alpha-$fibre, the trajectories of the prolonged algebroid $\mathcal{L}_{\ell+k}$ being the connected components of the $\alpha-$fibres - maximal integral sub-manifolds of the trivial distribution $V\Pi_{\ell+k}P$ - as well as
\vspace{2 mm}
b) the infinitesimal action $p_k$ being derived from the finite action $p_k$ ,
\vspace{4 mm}
\noindent
we infer (always under the canonical identification) that
\vspace{2 mm}
c) the tangent map to $(\Pi_{\ell+k}P)_y~\longrightarrow~\Omega(X)\subset J_kE$ at the point \textit{Z} is equal to $\lambda_k:(J_{\ell+k}TP)_{\beta(Z)}~\longrightarrow~T_{Z\cdot X}J_kE$ and consequently its rank is equal to $dim(\Delta_k)_{Z\cdot X}~$,
\vspace{2 mm}
d) the Lie algebra $h(X)$ of $H(X)$ is equal to the kernel of the map
\begin{equation*}
\lambda_k:(J_{\ell+k}TP)_{\alpha(X)}~\longrightarrow~T_XJ_kE~,
\end{equation*}
\vspace{2 mm}
\noindent
and
\vspace{2 mm}
e) since $ZH(X)=H(Z\cdot X)Z~$, the kernel of the tangent map to $(\Pi_{\ell+k}P)_y~\longrightarrow~(\Pi_{\ell+k}P)_y~/~H_{\ell+k}(X)$ at the point \textit{Z} is equal to $h(Z\cdot X)=ker~(\lambda_k)_{\alpha(Z\cdot X)=\beta(Z)}$,
\vspace{2 mm}
\noindent
we conclude that
\begin{equation*}
(\mu_*)_{ZH(X)}:(J_{\ell+k}TP)_{\beta(Z)}~/~h(Z\cdot X)~\longrightarrow~(\Delta_k)_{Z\cdot X}
\end{equation*}
\vspace{2 mm}
\noindent
is an isomorphism hence $\Omega(X)$ is a sub-manifold of $J_kE$ for which the integral leaves of $\Delta_k$ are open sets. Since these leaves are the trajectories of $\mathcal{L}_{\ell+k}~$, they are disjoint and constitute the connected components of $\Omega(X)$. Finally, since the connected components of $\Pi_{\ell+k}P_y$ are the trajectories of the standard prolongation of $\mathcal{L}$ by the target, we see that the image of each connected component of $\Pi_{\ell+k}P_y$ by the map $\overline{q}$ is an integral leaf of $\Delta_k$ contained in $\Omega(X)$ and therefore the inverse image of a leaf is a union of connected components. In particular, the image of the connected component of the unit at the point \textit{y} is equal to $\omega(X)$.
\vspace{4 mm}
\newtheorem{orbital}[TheoremCounter]{Theorem}
\begin{orbital}
Each orbit $\Omega(X)$ of $\Gamma_{\ell+k}$ is a differentiable sub-manifold of $J_kE$ canonically isomorphic to $\Pi_{\ell+k}P_y~/~H_{\ell+k}(X)$ and invariant under $\mathcal{L}_{\ell+k}~$, the infinitesimal action being transitive. The quadruple $(\Omega(X),\alpha,P,p_k)$ is a finite prolongation space of order $\ell+k$ and the groupoid $\Pi_{\ell+k}P$ as well as the sheaf (pseudo-algebra) $\underline{J_{\ell+k}TP}$ operate on it. The restrictions of $~\Gamma_{\ell+k}$ and $\mathcal{L}_{\ell+k}$ to $\Omega(X)$ are finite and infinitesimal pseudo-groups and pseudo-algebras of arbitrary order. For all $k\geq h$, the canonical projection $\rho_{h,k}$ transforms every $k-th$ order orbit onto an $h-th$ order orbit and thus defines a prolongation spaces morphism. The distribution $\Delta_k$ on $J_kE$ induced by $\mathcal{L}_{\ell+k}$ is involutive and locally of finite type, its maximal integral manifolds are the orbits of $\mathcal{L}_{\ell+k}$ and each orbit of the finite action has for its connected components the orbits of the infinitesimal action. The quadruple $(\omega(X),\alpha,P,p_k)$ is an infinitesimal prolongation space of order $\ell+k$ whenever \textit{P} is connected.
\end{orbital}
\vspace{4 mm}
\noindent
The standard prolongation by the target (\cite{Kumpera1975}, $\S$ 16, item (a)) determines a canonical finite prolongation structure $(\Pi_kP,\beta,P,p^b_k)$ of finite order \textit{k} for which the $\alpha-$fibres are the trajectories\footnote{$p^s_k$ - \textit{prolongement par la source}, $p^b_k$ - \textit{prolongement par le but}.}. For each $y\in P$, the prolongation sub-space
$((\Pi_kP)_y,\beta,P,p^b_k)$ is transitive and the equivalence relation defined by any closed Lie sub-group $H\subset (\Pi^0_kP)_y$ is compatible with the prolongation operations. Consequently, the quotient quadruple $((\Pi_kP)_y~/~H,\beta,P,p^b_k)$ is a finite prolongation space of order \textit{k}.
\vspace{4 mm}
\newtheorem{sub-orbital}[CorollaryCounter]{Corollary}
\begin{sub-orbital}
The canonical isomorphism of the preceding theorem is an isomorphism of prolongation spaces
\begin{equation*}
((\Pi_{\ell+k}P)_y~/~H(X),\beta,P,p^b_{\ell+k})~\xrightarrow{~\mu~}~(\Omega(X),\alpha,P,p_k).
\end{equation*}
\end{sub-orbital}
\vspace{4 mm}
\noindent
Since $\Delta_k$ admits in general singularities, the space of orbits by the finite or infinitesimal actions is most often rather complicated. It can reduce to a finite number or to a discrete family of orbits (for the quotient topology), it can present itself as a regular foliation (continuous family of orbits) and, most often, the two options can appear simultaneously. The discrete orbits correspond geometrically to the existence of models (in coordinates) for the germs of structures or for their $k-$jets. Quite to the contrary, the continuous families of orbits apparently turn nonexistent the presence of models these being replaced by local deformations of non-equivalent structures, since the nature in itself of a model highlights and emphasizes the notion of rigidity (not to be confounded with the deformation of structures locally equivalent to a given model). When the orbits are discrete, the formal and local equivalence problems will have to be examined by methods specific to each case and using all the available techniques as well as the invariants. This is the case especially for the "modeled structures" (for example, modeled on a Lie pseudo-group) where we shall first try to establish the formal equivalence with the model and thereafter the local equivalence leading most often to an integrability problem. When the orbits are distributed along continuous families, it seems advantageous to appeal to the differential invariants of the action of $\Gamma$ on the space \textit{E} . These however are only speculations and the sole positive statement is the following:
\vspace{2 mm}
\textit{Two
|
infinite jets of structures of species E are formally equivalent if and only if their $k-$jets, for any k, belong to the same $k-th$ order orbit.}
\section{The restricted problem}
Often, mainly in physics and other domains, it is important to know the equivalence not only with respect to an arbitrary transformation but also one respecting certain additional properties (\textit{e.g.}, conservation laws) and this conveys us to what we call the restricted equivalence problem with respect to the transformations of a given pseudo-group or pseudo-algebra.
\vspace{4 mm}
\noindent
Let $\Gamma$ be a pseudo-group of local transformations operating on the manifold \textit{P} and $\mathcal{L}$ the corresponding infinitesimal pseudo-algebra (sometimes called infinitesimal pseudo-group) \textit{i.e.}, the sub-presheaf of $\Gamma(TP)$ (sections) obtained by localizing as well as pasting together all the local vector fields of \textit{P} of the form $\xi=\frac{d}{dt}~\varphi_t|_{t=0}$, where $(\varphi_t)_t$ is a local one parameter family of element of $\Gamma$. We shall say that $\Gamma$ is a Lie pseudo-group of order $k_0$ if, for any $k\geq k_0$, the following properties hold:
\vspace{4 mm}
a) $J_k\Gamma$ is a closed Lie sub-groupoid of $\Pi_kP$ and the projection
\begin{equation*}
\rho:~J_{k+h}\Gamma~\longrightarrow~J_k\Gamma
\end{equation*}
\vspace{2 mm}
is a surmersion.
\vspace{2 mm}
b) $J_{k+1}\Gamma$, considered as a differential equation on the fibration
\begin{equation*}
\alpha:~\Pi_{k+1}P~\longrightarrow~P~,
\end{equation*}
\vspace{2 mm}
\noindent
is the standard prolongation of $J_k\Gamma$.
\vspace{2 mm}
c) $J_k\Gamma$ is infinitesimally complete \textit{i.e.}, the associated linear Lie equation $\textbf{R}_k=VJ_k\Gamma|P$ (\textit{P} being identified with the units of $J_k\Gamma$ and $V=\alpha-vertical$) is equal to $J_k\mathcal{L}$ .
\vspace{2 mm}
d) $\Gamma$ is complete of order $k_0$ which means that $\varphi\in\Gamma$ if and only if $j_{k_0}\varphi$ is a section (solution) of $J_{k_0}\Gamma$.
\vspace{4 mm}
\noindent
$\textbf{Remark:}$ When (a) is verified, we can easily prove that $J_{k+1}\Gamma$ is contained in the prolongation (as a differential equation) of $J_k\Gamma$ and consequently the property (b) will follow eventually at a higher order $k_0+h$ namely, when the $\delta-$cohomology of the symbols of the linear equations $\textbf{R}_k$ become $1-$acyclic. This results locally, in an open neighborhood $\mathcal{U}$ of a point in $J_{m+1}\Gamma, m=k_0+h~$, in virtue of the prolongation theorem of \textit{Cartan-Kuranishi} (\cite{Kuranishi1967}, Theorem 10.1). The invariance of the prolongation $\textit{p}(J_m\Gamma)$ by the left action of $J_{m+1}\Gamma$ shows that the open set $\mathcal{U}$ can be chosen saturated with respect to the orbits of this action. Finally, an argument based on the constancy of the characters of an exterior differential system, similar to that employed in the proof of the finiteness theorem below, shows that equality holds, at the level $m+\mu~$, in the open set $\rho^{-1}_m(\mathcal{U})$. In the next section, we provide a global proof.
\vspace{4 mm}
\noindent
The property (b) implies the corresponding property for the linear equations $\textbf{R}_k~$. Moreover, the property (d) together with (c) imply that $\mathcal{L}$ is complete of order $k_0$ and, consequently, $\mathcal{L}$ is a Lie pseudo-algebra (infinitesimal pseudo-group) of order $k_0$ since $J_k\mathcal{L}~(=\textbf{R}_k)$ is a locally trivial vector sub-bundle of $J_kTP$ and $J_{k+1}\mathcal{L}$ is the prolongation of $J_k\mathcal{L}$ in the sense of linear equations. Finally the property (c), that will be a consequence of (a), (b) and the \textit{Cartan-Kähler} theorem when the initial data is real analytic (it will also be a consequence in other situations, especially in the transitive elliptic case), serves to assure later that any orbit of the infinitesimal action is open in the corresponding orbit of the finite action and, more precisely, is a connected component.
\vspace{4 mm}
\noindent
We shall say that $\Gamma$ is \textit{transitive} when there exists $\varphi\in\Gamma$, with $\varphi(x)=y~$, whatever the points $x,y\in P$ and that it is \textit{locally trivial} when the projection $\alpha\times\beta:J_k\Gamma~\longrightarrow~P\times P$ is a submersion (we do not assume transitivity (\cite{Kumpera1971}). On account of the property (a), it will suffice to have local triviality at order $k_0~$. We shall say that $\mathcal{L}$ is \textit{transitive} or that $\Gamma$ is infinitesimally transitive when the vector sub-space induced by $\mathcal{L}$ at every point \textit{y} in \textit{P} is equal to $T_yP$. Finally, the formal transitivity is the one linked to the linear and non-linear equations $J_k\mathcal{L}$ and $J_k\Gamma$ and coincides entirely with the transitivity above.
\vspace{4 mm}
\noindent
Let $(E,\pi,P,p)$ be a finite or infinitesimal prolongation space and $\Gamma$, resp. $\mathcal{L}$, a finite or infinitesimal Lie pseudo-group (pseudo-algebra) of order $k_0$ operating on \textit{P}. The Definition 1 can be re-written by replacing the general pseudo-group and pseudo-algebra of all local finite or infinitesimal transformations by the more specific data $\Gamma$ and $\mathcal{L}~$. In this context, we can re-write essentially all of the section 2 by replacing $\Pi_{\ell+k}P$ and $J_{\ell+k}TP$ by $J_{\ell+k}\Gamma$ and $J_{\ell+k}\mathcal{L}$ as soon as $\ell+k\geq k_0~$. We can also transcribe the considerations of the previous section where we shall replace $\Gamma_{\ell+k}$, resp. $\mathcal{L}_{\ell+k}$, by the prolongations of $\Gamma$, resp. $\mathcal{L}~$, the general equivalence problem by the restricted one and, in the Theorem 2, the groupoid $\Pi_{\ell+k}P$ by $J_{\ell+k}\Gamma$. Inasmuch, we can rewrite the sections 3 and 4 in the restricted context though certain parts and especially those concerning the morphisms $\Phi$ and $\Psi$ require some additional considerations. We shall return to this in a later section.
\vspace{4 mm}
\noindent
Still in a wider context, we can define finite and infinitesimal prolongation spaces \textit{relative} to given finite or infinitesimal pseudo-groups of transformations. In other terms, the prolongation operations are only defined for the elements of the pseudo-group or pseudo-algebra envisaged. A most relevant example is provided by the Cartan \textit{normal} prolongation spaces associated to given pseudo-groups and their quotient spaces. In replacing the general pseudo-group by a given one we can still argue as in the previous sections though, of course, we shall not forget the inequality $\ell+k\geq k_0~$.
\vspace{4 mm}
\noindent
The Lie pseudo-group $\Gamma$ is said to be analytic when the manifolds $J_{\ell+k}\Gamma,~k\geq k_0~$, are analytic sub-groupoids of $\Pi_{\ell+k}P$ (supposing of course that $\pi:E~\longrightarrow~P$ is an analytic fibration). Inasmuch, the Lie pseudo-algebra $\mathcal{L}$ is said to be analytic when the linear equations $J_k\mathcal{L}$ are analytic vector sub-bundles of $J_kTP$. Clearly, the analiticity of $\Gamma$ implies that of $\mathcal{L}$ and the converse is also true since the differentiable structure of $J_k\Gamma$ is entirely determined, in a neighborhood of the units hence everywhere, by the structure of $\textbf{R}_k~$.
\section{The role of the differential invariants - finiteness theorems}
The interesting situation from the point of view of the differential invariants is that of continuous families of orbits. We therefore assume, for the time being, that there exists an integer $k_1$ such that, for $k\geq k_1$, the orbits of the action of $\Gamma_{\ell+k}~$ on $J_kE$ or rather those of the infinitesimal action of $\mathcal{L}_{\ell+k}~$ are distributed along a regular foliation \textit{i.e.}, the integrable distribution $\Delta_k$ is locally of constant dimension. A first integral of $\Delta_k$ (a function that is locally constant on each integral leaf of $\Delta_k$ or, equivalently, a function whose differential $df$ vanishes on $\Delta_k$) will be called a \textit{differential invariant of order k} of the Lie pseudo-group $\Gamma$, resp. of the pseudo-algebra $\mathcal{L}~$, and relative to the prolongation space \textit{E}. Since $\Delta_k$ is assumed to be regular there exists, in a neighborhood of each point in $J_kE~$, a fundamental system of independent differential invariants their number (rank) being equal to the
co-dimension of $\Delta_k~$. On the other hand, with the aid of the formal derivatives (total derivatives) in the jet manifolds, it is possible to ascend (lift), in a non-trivial way, differential invariants defined on $J_kE$ to new differential invariants defined on $J_{k+1}E~$. Lie's finiteness theorem for the differential invariants states essentially that the invariants of any order are generated by those of a certain finite order together with all their successive formal derivatives. The mechanism involving the differential invariants presumes of course certain regularity hypotheses as well as specific technicalities that eventually will lead us to the Fundamental Theorem of Sophus Lie (\cite{Lie1884}) and we shall try to describe these in the most succinct manner by referring as much as possible to \cite{Kumpera1975}. Since the specific case of prolongation spaces and of the formal equivalence of structures, our main concern, isn't but a special case of the general problem discussed in the previous reference, it is possible to simplify two of the hypotheses and the form under which we state the Lie Theorem for its applications in the equivalence problem. We shall in fact provide a much more precise statement than the one claimed in the Theorem 23.6 (\textit{loc.cit.}).
\vspace{4 mm}
\noindent
Let us first remark that the problem in \cite{Kumpera1975} consists in taking an arbitrary fibration (surmersion) $~\pi:P~\longrightarrow~M~$ together with a sheaf $\mathcal{L}$ (Lie sheaf) of vector fields on \textit{P} and thereafter study the differential invariants of $\mathcal{L}$ in the realm of the standard prolongation spaces $J_kP$ above \textit{P}. Here, quite to the contrary, we are given an infinitesimal Lie pseudo-algebra $\mathcal{L}$ of order $k_0$
on the manifold \textit{P}, an infinitesimal prolongation space $(E,\pi,P,p)$ of finite order $\ell$ and study the differential invariants of the infinitesimal action of $\mathcal{L}$ on the prolongation spaces $J_kE~$. We can re-conduce our considerations to the above mentioned context by simply considering the prolonged sheaf $\mathcal{L}_{\ell}=p\mathcal{L}$ defined on the space \textit{E} and study the differential invariants of $\mathcal{L}_\ell$ by the techniques and methods found in \cite{Kumpera1975}. However, the present methods are far more reaching and accurate.
\vspace{4 mm}
\noindent
We examine initially the hypothesis $H_1$ (\textit{loc.cit.}, pg.363) or, by preference, the weaker hypothesis on pg.378.
\vspace{2 mm}
$H'_{1,Y}$: There exists an integer $k_1$ such that, for any $k\geq k_1~$, the fibre space $(\tilde{L}_V)_k$ with base space $J_{k+1}E$ is of constant rank in the neighborhood of each point $Y_{k+1}~$.
\vspace{2 mm}
\noindent
Let us recall (see \cite{Kumpera1975} for the notations) that $(\tilde{L}_V)_k\subset J_{k+1}P~\times_P~\tilde{J}_kVE$ and that this fibre space is the image of $\tilde{L}_k=\tilde{J}_k\mathcal{L}$, $\mathcal{L}$ being a Lie sheaf over \textit{P}, by the verticalisation operation described in terms of the exact sequence (22.29) in \cite{Kumpera1975}. However, in the present case\footnote{The "tilde" notation refers to the composite fibration $TE~\longrightarrow~E~\longrightarrow~P$ and where $TE$ is also replaced by $VE$}, we start with an infinitesimal Lie pseudo-algebra $\mathcal{L}$ of order $k_0$ defined on \textit{P}, consider its prolongation $\mathcal{L}_\ell=p\mathcal{L}$ to \textit{E} and $\tilde{L}_k$ becomes $\tilde{J}_k(\mathcal{L}_\ell)$. Under these conditions, $(\tilde{L}_V)_k$ is the image of $J_{k+1}E~\times_P~J_{\ell+k}\mathcal{L}$ by the mapping
\begin{equation*}
J_{k+1}E~\times_P~J_{\ell+k}TP~\longrightarrow~J_{k+1}E~\times_P~\tilde{J}_kVE
\end{equation*}
\vspace{2 mm}
\noindent
defined by
\begin{equation*}
(j_{k+1}\sigma(y),j_{\ell+k}\xi(y))~\longmapsto~j_k[(p\xi)\circ\sigma-(T\sigma\circ\xi)](y).
\end{equation*}
\vspace{2 mm}
\noindent
Let us next consider the exact sequence
\begin{equation*}
0~\longrightarrow~\mathcal{N}_{\ell+k}~\longrightarrow~J_{k+1}E~\times_P~J_{\ell+k}\mathcal{L}~\longrightarrow~(\tilde{L}_V)_k~\longrightarrow~0
\end{equation*}
\vspace{2 mm}
\noindent
where $\mathcal{N}_{\ell+k}$ denotes the kernel. We thus see that the regularity of $(\tilde{L}_V)_k$ can be replaced, when $\ell+k\geq k_0$, by that of $\mathcal{N}_{\ell+k}$ for which the defining equation is given by
\begin{equation*}
j_k[(p\xi)\circ\sigma-(T\sigma\circ\xi)](x)=0.
\end{equation*}
\vspace{2 mm}
\noindent
This equation can be envisaged as a linear differential equation of order $\ell+k$ in $J_{\ell+k}\mathcal{L}$ (or as well in $J_{\ell+k}TP$) whose coefficients depend on the parameters in $J_{k+1}E~$. Consequently, the regularity of this equation in the neighborhood of a jet $Z_{k+1}\in J_{k+1}E$, is closely related to the geometry of the prolongation space \textit{E} in the neighborhood of $\beta(Z_{k+1})$.
\vspace{2 mm}
\noindent
As for the other two hypotheses on the pg.363, we can partly weaken $H_{2,X}$ by taking into account that $J_{\ell+k}\mathcal{L}$ is a Lie equation hence the distribution $\Delta_k$ automatically satisfies the involutivity condition. However, we shall be forced to strengthen the part concerning regularity. Inasmuch, we shall strengthen the point-wise hypothesis $H_{3,X}$ by a local condition, its most efficacious verification criterion being provided by the Proposition 25.4 in \cite{Kumpera1975} on account of its Corollary. We therefore consider the following hypotheses:
\vspace{4 mm}
$H_1:$ For any $k\geq k_1~(\geq k_0-\ell),$ the vector bundle $\mathcal{N}_{\ell+k}$ has constant rank in a neighborhood of $Z_{k+1}$ and we denote by $k_1(Z)$ the integer where-after $(\Delta_{k-1,k})_{Z_k}$ (the kernel) becomes involutive.
\vspace{4 mm}
$H_2:$ There exists a family $(\mathcal{U}_k)_{k\geq k_2(Z}$ of open neighborhoods of the jets $Z_k$ such that $~\rho_{k,k+h}:\mathcal{U}_{k+h}~\longrightarrow~\mathcal{U}_k~$ is a fibration and $\Delta_k$ has constant dimension on $\mathcal{U}_k$ .
\vspace{4 mm}
$H_3:$ $\beta(\textbf{R}_{k_3(Z)}(\mathcal{L}))_{Z'})=T_{\beta(Z')}P$ for all $Z'\in\mathcal{U}_{k_3}(Z)$ and for some integer $k_3\geq k_2$ .
\vspace{4 mm}
\noindent
The last hypothesis assures the existence of a local basis of \textit{admissible} formal derivations of order $k_3$ centered around \textit{Z}, admissible meaning that such derivations transform differential invariants into differential invariants. Before stating the desired theorem, let us examine a little closer the above hypotheses in order to better discern their meaning.
\vspace{5 mm}
\noindent
1. Let $(\Delta_{k-1,k})_{Z'}$ be the kernel of $~T\rho_{k-1,k}:
\Delta_k~\longrightarrow~\Delta_{k-1}~$ at the point $Z'\in J_{k+1}E$ (this mapping being always surjective). The first hypothesis serves to prove that there is an order $k'$ such that the kernel $(\Delta_{k,k+1})_{Z_{k+1}}~$, $k\geq k'$, is contained, by means of the canonical identification, in the algebraic prolongation of $(\Delta_{k-1,k})_{Z_k}$ and consequently becomes equal to it from an order $k''$ onwards or, in other terms, the Spencer $\delta-$complex constructed with the kernels $(\Delta_{k-1,k})_{Z_k}$ becomes $1-$acyclic for $k\geq k''$. Likewise, it will become involutive beginning with an integer that we shall denote by $k_1(Z)$. This hypothesis alone enables us to prove the \textit{asymptotic stability} result (\cite{Kumpera1975}, Theorems 22.1 and 23.1) that in turn and with the aid of the hypotheses $H_{2,Z}$ and $H_{3,Z}$, leads to the Lie Theorem (\textit{loc.cit.}, Theorem 23.6).
\vspace{4 mm}
\noindent
2. The hypothesis $H_2$ assures a sufficient number \textit{i.e.}, a complete set of $k-th$ order differential invariants defined on the open set $\mathcal{U}_k~$.
\vspace{4 mm}
\noindent
3. The hypothesis $H_3$ enables us to obtain a sufficient number of $(k+1)-st$ order differential invariants by taking admissible formal derivations of the $k-th$ order differential invariants (and, of course, lifting also the latter up to order $k+1$).
\vspace{4 mm}
\noindent
4. One shows that the finiteness property of the differential invariants takes place at the stage $k~\rightsquigarrow~k+1$ (\textit{i.e.}, for the germs of invariants at the points $Z_k$ and $Z_{k+1}$ respectively) if and only if $(\Delta_{k,k+1})_{Z_{k+1}}$ is the algebraic prolongation of $(\Delta_{k-1,k})_{Z_k}$ (\cite{Kumpera1975}, Lemmas 23.3 and 23.5).
\vspace{4 mm}
\noindent
We next remark that the three hypotheses, \textit{per se} independent, are not strictly necessary to prove the desired results. In fact, the hypothesis $H_{2,X}$ underlying the theorem 23.8 in \cite{Kumpera1975} is considerably weaker than $H_2$ . However, the asymptotic stability, consequence of $H'_{1,Z}~$, joint to $H_{3,Z}$ imply the local regularity and the integrability of the distribution $\Delta_k~$, for $k>k_2$ , in view of the Corollary 5 (\textit{loc.cit.}, pg.377, conditions (I) and (II)). Viewed from another angle, we note that solely the hypotheses $H_2$ and $H_3$ will, in virtue of the lemma 23.3 in \cite{Kumpera1975}, imply that $(\Delta_{k,k+1})_{Z_{k+1}}\subset p(\Delta_{k-1,k})_{Z_k}$ and we thus obtain the asymptotic stability of the kernels starting from a certain integer $k''$. These remarks simply show that the usage of the above three hypotheses admits a certain flexibility, the appropriate choices being conditioned to the results looked for.
\vspace{4 mm}
\noindent
At present we choose $H_2$ and $H_3$ as underlying hypotheses and fix the order $\mu=k_1(Z)$ where after the symbols $(\Delta_{k-1,k})_{Z_k}$ become involutive (the hypothesis $H_1$ only reappearing later when the regularity of the $\Delta_k$ becomes apparent). We can further assume that $k_2(Z)~<~k_1(Z)~$. The hypothesis $H_2$ implies that the kernels $\Delta_{\mu,\mu+1}$ and $\Delta_{\mu-1,\mu}$ are of constant dimension in $\mathcal{U}_{\mu+1}$ and $\mathcal{U}_\mu$ respectively, and further $(\Delta_{\mu-1,\mu})_{Z_\mu}$ is involutive, $(\Delta_{\mu,\mu+1})_{Z_{\mu+1}}$ being its algebraic prolongation. According to the Theorem 23.6 (\textit{loc.cit.}), there exists an open neighborhood $\mathcal{U}_{\mu+1}$ of $Z_{\mu+1}$ such that $~(\Delta_{\mu,\mu+1})_{Z'_{\mu+1}}=p(\Delta_{\mu-1,\mu})_{Z'_{\mu}}~$ for all $Z'_{\mu+1}\in\mathcal{U}_{\mu+1}$ and consequently the finiteness property of the differential invariants is verified at the step $~\mathcal{U}_{\mu}~\rightsquigarrow~\mathcal{U}_{\mu+1}~$, $\mathcal{U}_{\mu}=\rho(\mathcal{U}_{\mu+1})$. Let us next observe that the \textit{characters} $\tau_i$ of $~(\Delta_{\mu-1,\mu})_{Z'_{\mu}}~,~Z'_{\mu}\in\mathcal{U}_{\mu}~$, are lower semi-continuous. The dimensions of $\Delta_{\mu-1,\mu}$ and $\Delta_{\mu,\mu+1}$ being constant, the characterization of the involutivity (\cite{Kumpera1975}, $\S$ 24, property 8, \cite{Kuranishi1967}, proposition 6.1) implies the existence of an open neighborhood $\mathcal{W}_{\mu}$ of $Z_{\mu}$ in which the kernels $(\Delta_{\mu-1,\mu})_{Z'_{\mu}}~$, $Z'_{\mu}\in\mathcal{W}_{\mu}~$ are all involutive, the characters $\tau_i$ remaining constant. Let us denote by $\mathcal{W}_{\mu+1}$ the inverse image of $\mathcal{W}_{\mu}$ with respect to the projection $~\rho:\mathcal{U}_{\mu+1}~\longrightarrow~\mathcal{U}_{\mu}~$ and, similarly, define $\mathcal{W}_{\mu+h}$ considering $~\rho:\mathcal{U}_{\mu+h}~\longrightarrow~\mathcal{U}_{\mu}~$. Furthermore, denote by $(\Delta'_{\mu+2})_{Z'}$ the sub-space of $T_{Z'}J_{\mu+2}E~$, ${Z'}\in\mathcal{W}_{\mu+2}~$, defined by
\begin{equation*}
(\Delta'_{\mu+2})_{Z'}=ker_{Z'} \{\rho^*_{\mu+1,\mu+2}df, \partial_\varphi df~|~Z'_{\mu}\in\mathcal{W}_{\mu},~ f\in(\mathfrak{I}_{\mu+1})_{Z''},
\end{equation*}
\begin{equation*}
\varphi\in\mathcal{R}_{\mu+1}(\mathcal{L})_{Z''},~Z''=\rho_{\mu+1,\mu+2}Z'\},
\end{equation*}
where $\mathfrak{I}_{\mu+1}$ denotes the algebra of all differential invariants of order $\mu+1~$. Since the elements of $\mathcal{R}_{\mu+1}(\mathcal{L})_{Z''}$ are admissible, the inclusion $~\Delta'_{\mu+2}\supset\Delta_{\mu+2}~$ holds and the lemma 23.3 in \cite{Kumpera1975} shows furthermore that
\begin{equation*}
dim~(\Delta'_{\mu+2})_{Z'}=dim~(\Delta_{\mu+1})_{Z''}+dim~p(\Delta_{\mu,\mu+1})_{Z''}~.
\end{equation*}
\vspace{2 mm}
\noindent
However, $(\Delta_{\mu,\mu+1})_{Z''}$, $Z''\in\mathcal{W}_{\mu+1}$, is involutive it being the prolongation of an involutive space and the property 9 in $\S$ 24 of \cite{Kumpera1975} or else, the Proposition 9.4 in \cite{Kuranishi1967} shows, in view of the constancy of the characters $\tau_i~$, that $~dim~p(\Delta_{\mu,\mu+1})_{Z''}~$ is constant in $~\mathcal{W}_{\mu+1}~$, the characters of these prolonged spaces being also constant. Returning to the point $Z_{\mu+2}~$, we perceive that this dimension is equal to $~dim~(\Delta_{\mu+1,\mu+2})_{Z_{\mu+2}}$ and consequently that $~dim~(\Delta'_{\mu+2})_{Z'}=dim~(\Delta_{\mu+2})_{Z'_{\mu+2}}~$. Furthermore, this entails, in virtue of the constancy of the dimensions of $\Delta_{\mu+2}~$, that $~(\Delta'_{\mu+2})_{Z'}=(\Delta_{\mu+2})_{Z'}$ for all $Z'\in\mathcal{W}_{\mu+2}~$. We thus infer that the finiteness property for the differential invariants is verified at the step $~\mathcal{W}_{\mu+1}~\rightsquigarrow~\mathcal{W}_{\mu+2}~$. An inductive argument will finally prove, based on the constancy of the characters, that the finiteness property is verified at the stage $~\mathcal{W}_{\mu+h}~\rightsquigarrow~\mathcal{W}_{\mu+h+1}$ since the involutivity as well as the constancy of the characters is preserved by prolongation.
\vspace{2 mm}
\noindent
Let us observe that the involutivity property of the kernels $(\Delta_{\mu-1,~\mu})_{Z'_\mu}~$ together with the regularity of the $\Delta_k~$, $k\geq\mu-1~$, on the open sets $\mathcal{U}_k~$, that "fibrate" one upon the other, serve uniquely to ensure the existence of a family of open neighborhoods $\mathcal{W}_k$ of $Z_k~$, fibering one above the other in such a way that, along every element $Z'\in~proj~lim~\mathcal{W}_k~$, the consecutive kernels of the $\Delta_k$ constitute a $1-$acyclic Spencer $\delta-$com-plex. In the applications, this local $1-$acyclicity property, sole to assure the finiteness mechanism of the differential invariants, might be verified long before the involutivity. The argument as well as the aims of the above discussion are somewhat quite the opposite of what has been looked for in the $\S$ 24 of \cite{Kumpera1975} where the problem posed was the regularity of the trajectories.
\vspace{4 mm}
\newtheorem{finiteness}[TheoremCounter]{Theorem (of finiteness)}
\begin{finiteness}
Let $(E,\pi,P,p)$ be an infinitesimal prolongation space, $Z\in J_{\infty}E$ an infinite jet of a structure of species E and $\mathcal{L}$ an infinitesimal pseudo-algebra (Lie pseudo-algebra) operating on P. Assuming that the hypotheses $H_2$ and $H_3$ are satisfied at the point Z, we write $\mu=k_1$ and take a family of n $(=dim~P)$ local sections $\varphi_i$ of $\mathcal{R}_{\mu+1}(\mathcal{L})$, defined in a neighborhood of $Z_{\mu+1}~$, such that $\{\beta\varphi_i(Z_{\mu+1})\}$ generates the tangent space $T_yP~,~y=\alpha(Z)$ \textit{i.e.}, the family $\{\varphi_i\}$ is a local basis of admissible formal derivations in the neighborhood of $Z_{\mu+1}~$. Under these conditions, there exists a family $(\mathcal{W}_k)_{k\geq\mu}~$, each $\mathcal{W}_k$ being an open neighborhood of $Z_k~$, such that:
\vspace{4 mm}
\hspace{5 mm}$i)\hspace{2 mm}\rho_{k,k+h}:~\mathcal{W}_{k+h}~\longrightarrow~\mathcal{W}_k$ is a fibration.
\vspace{3 mm}
\hspace{4 mm}$ii)\hspace{2 mm}(\Delta_{k,k+1})_{Z'_{k+1}}=p(\Delta_{k-1,k})_{Z'_k}~,\hspace{2 mm}Z'_{k+1}\in\mathcal{W}_{k+1}$.
\vspace{3 mm}
\hspace{3 mm}$iii)\hspace{2 mm}(\Delta_k)_{Z'_k}=ker_{Z'_k}\{df~|~f\in\mathfrak{I}\},\hspace{2 mm}Z'_k\in\mathcal{W}_k$.
\vspace{3 mm}
\hspace{4 mm}$iv)\hspace{2 mm}(\Delta_{k+1})_{Z'_{k+1}}\!=ker_{Z'_{k+1}}\{\rho^*_{k,k+1}df,~\partial{\varphi_i}df~|~f\in\mathfrak{I}_k,~1\leq i\leq n\}$,
\vspace{3 mm}
\hspace{11 mm}$Z'_{k+1}\in\mathcal{W}_{k+1}$.
\vspace{3 mm}
\hspace{5 mm}$v)\hspace{2 mm}(\Delta_{k+1})_{Z'_{k+1}}\!=ker_{Z'_{k+1}}\{\rho^*_{k,k+1}df,~\partial{\varphi_i}df~|~f\in\mathfrak{I}_k$,
\vspace{3 mm}
\hspace{11 mm}$\varphi_i\in\mathcal{R}_{\mu+1}(\mathcal{L})_{Z'_{\mu+1}}\},\hspace{2 mm}Z'_{k+1}\in\mathcal{W}_{k+1}$.
\vspace{3 mm}
\hspace{4 mm}$vi)\hspace{2 mm}\{\rho^*_{k,k+1}df,~\partial{\varphi_i}df~|~f\in\mathfrak{I}_k~,~1\leq i\leq n\}_{Z'_{k+1}}$
\vspace{3 mm}
\hspace{11 mm}\textit{generates} $(d\mathfrak{I}_{k+1})_{Z'_{k+1}}),\hspace{2 mm}Z'_{k+1}\in\mathcal{W}_{k+1}$.
\end{finiteness}
\vspace{4 mm}
\noindent
The interest of the finiteness theorem for the equivalence of structures is due to the fact that it enables us to translate the equivalence by \textit{only} a finite number of conditions (equality of the values taken by a finite number of differential invariants). We terminate here this awfully long "preamble" and will retake the effective study of the equivalence problem in part II of this paper where diverse \textit{mises en scène} shall be examined. As a last word, we should say that all the previous discussion can be carried out in the context of prolongation spaces \textit{relative} to given finite Lie pseudo-groups or infinitesimal Lie pseudo-algebras and also it is worthwhile to recall that Sophus Lie provided some of the most remarkable contributions. Surprisingly, the formula 25.5, concerning the bracket of formal and holonomic derivations (\cite{Kumpera1975}) is already written in his work \cite{Lie1884} (see also \cite{Kumpera1967}, \cite{Molino1972}).
\bibliographystyle{plain}
|
\section{Introduction}
The great majority of main-sequence stars are in the mass range of 0.07\,M$_{\odot}$ to 10\,M$_{\odot}$, which is also the mass range of white dwarf (WD) progenitors \citep{Doherty2015}. According to the literature \citep{Fontaine2001,Heger2003}, up to 97\% of all stars in our Galaxy will eventually evolve to WDs. Based on spectroscopic features, WDs mainly consist of DAs and DBs. They are the most explored types of single WDs, one reason is that they account for 90\% of WDs. The specific number of this fraction recently was found vary with the effective temperature \citep{Rolland2018,Cunningham2020}, based on $Gaia$ spectroscopic sample, after significant fraction of cool WDs were discovered \citep{Gentile2019}. Another reason is that they are objects of great importance, providing crucial information in various fields. For instance, the simple cooling mechanism of WDs makes it easier to obtain relatively accurate ages. Therefore, for research on the age of Galactic stellar halo \citep{Guo2016,Kilic2019,Guo2019}, WDs are important age estimation tools after their physics are well-understood. By studying mass distribution \citep{Kepler2007,Holberg2016,Hollands2018} and luminosity functions \citep{Harris2006,Munn2017,Lam2019} of WDs, multiple astrophysical processes of scientific importance can be learnt, in particular, the initial mass function and binary interactions. Meanwhile, WDs can serve as accurate records of star formation and reveal the evolution history of the Milky Way \citep{Krzesinski2009,Rowell2013}. Additionally, for stars that will eventually evolve to WDs, studies of their initial-final mass relation depend on both single WDs in clusters \citep{Catalan2008,Kalirai2008,Kalirai2009} and WDs in binaries \citep{Catalan2008b,Zhao2012}, These reseaches will help to provide important information on the evolution of our Galaxy \citep{Kilic2017}.
The goal to establish a large WD database has been pursued for more than three decades. \cite{McCook1987} spectroscopically identified 1\,279 WDs that has been updated to 2\,249 entries by \cite{McCook1999}. The total number of WDs has increased greatly, with the development of large surveys, e.g. Palomar-Green \citep{Green1986} and Sloan Digital Sky Survey (\citealt{York2000}, SDSS) particularly \citep{Kleinman2004,Eisenstein2006}. By using the data release (DR) 7 \citep{Abazajian2009}, \cite{Kleinman2013} spectroscopically identified about 20\,000 WDs. Recently, \cite{Kepler2015} made $\sim$ 9\,000 new identifications of WDs from SDSS DR10 \citep{Ahn2014}. More recently, around 6\,000 and 20\,000 WDs were identified from SDSS DR12 and DR14, respectively \citep{Kepler2016,Kepler2019}.
The Large Area Multi-Object fibre Spectroscopic Telescope (LAMOST) pilot survey started in 2012, and the first phase was completed five years later. After a transition from 2017 September to 2018 June, the survey began its second phase. There has been medium resolution (resolving power $R\sim$ 7500) spectroscopy carried out during bright nights in the second phase, together with lower resolution observations during dark nights ($R\sim$ 1800). In the five years of the first phase, several studies on LAMOST WDs have been conducted. \cite{Zhao2013} identified 70 DAs from the LAMOST pilot survey. \cite{Zhang2013} presented a catalog of 230 DAs by fitting Sersic profiles to Balmer lines of spectra. Combining the spectral type results from LAMOST pipeline, the Balmer line equivalent width measurements, and the colour-colour cut method, \cite{Guo2015b} identified 1056 DAs, 34 DBs, 276 white dwarf main sequence (WDMS) binaries and other spectral types of WDs in LAMOST DR2. \cite{Gentile2015} also reported the discovery of 253 new WDs in LAMOST DR3. By exploiting a well characterised magnitude limited DA star sample selected from LAMOST Galactic anticeter, space density, formation rate, luminosity and mass functions of DA WDs were studied \citep{Rebassa2015}. Recently, a catalog of 876 WDMS binaries identified in LAMOST DR5 are presented by \citet{Ren2018}.
Apart from spectroscopic surveys, the studies of WDs has opened a new window since the $Gaia$ data released \citep{Gaia2016}. Based on $Gaia$ data, local complete WD samples have been built separately for 20\,pc \citep{Hollands2018} and 40\,pc \citep{Tremblay2020}. There are other much larger but not volume complete WD sample discovered in $Gaia$ DR2 \citep{Gaia2018}. \cite{Jimenez2018} presented a catalogue of 73\,221 WD candidates extracted from DR2. More recently, more than 260\,000 high-confidence WD candidates were discovered in DR2, as well \citep{Gentile2019}. Evidence for merged WDs has also been revealed \citep{Kilic2018}. By adopting accurate positions, proper motions and parallaxes, detailed studies on WD kinematics have been conducted by several research groups \citep{Bovy2017,Gaia2018b,Rowell2019,Torres2019}.
In this work, we present a catalogue of WDs discovered in LAMOST DR5\footnote{Spectra available at \url{http://dr5.lamost.org/}}, as well as their physical parameter estimation. In Section 2, we present the selection methods of DAs and DBs. The estimation of the parameters for DAs and DBs (effective temperature, surface gravity, mass, and cooling age) is described in Section 3. We discuss our white dwarf target selection for the second phase of LAMOST in Section 4. Finally, we present our summary in Section 5.\\
\section{Candidates selection}
\subsection{LAMOST observation}
LAMOST is a 4-m reflecting Schmidt telescope, equipped with a multi-object spectrograph that has a 20\,deg$^{2}$ field of view and 4\,000 fibres. The LAMOST project began its one-year pilot study in 2012, followed by a five-year first phase survey. There is only low resolution spectroscopy in the first phase. The spectral resolving power is $R \sim 1800$, covering a wavelength range $3800-9000$ \,\AA. With typical exposure times of 1.5\,h, the limiting magnitude can reach 20.5\,mag \citep{Cui2012}. The first phase survey is composed of two major parts \citep{Zhao2012}. The first part is the LAMOST Experiment for Galactic Understanding and Exploration (LEGUE) survey, focused on understanding the structure and evolution of the Milky Way \citep{Deng2012}. The second part is the LAMOST Extra-Galactic Survey of galaxies (LEGAS), whose purpose is to study the large-scale structure of the Universe. The LEGUE survey consists of three smaller surveys, where each is selected for distinct purposes \citep{Chen2012,Carlin2012,Yuan2015}. The targets are mainly chosen from the Galactic disc, spheroid, and anti-center. Although LAMOST is a spectroscopic survey only, there are also photometric data provided by different astronomers from various photometric catalogs, e.g. XSTPS-GAC \citep[Xuyi Schmidt Telescope Photometric Survey of the Galactic Anti-center]{LiuXW2014,Guo2018}, 2MASS, SDSS, Kepler, NVSS etc. More than 50\% of the input catalog entries at least have $g$, $r$, $i$ magnitudes, mainly from XSTPS-GAC and SDSS. In LAMOST DR5, there are almost 10 million spectra (details in Table \ref{tab1}). While the second phase of LAMOST is ongoing, this work is based on the completed first phase of LAMOST, in particular DR 3, 4, and 5.
\begin{table}
{\tiny
\begin{center}
\caption{Updated spectral statistic of each data release for first phase.\newline\newline
Note: STAR, GALAXY, QSO and UNKNOWN are class assigned by LAMOST pipeline. Numbers may change as spectral reduction and classification pipelines update.}
\label{tab1}
\begin{tabular}{p{0.4cm} p{1.5cm} c c c c c}\hline
Survey & DATE &STAR & GALAXY & QSO & UNKNOWN & TOTAL \\
\hline
Pilot & 2011-10-24 2012-06-17 & 837,056 & 8,045 & 1,227 & 118,060 & 964,388 \\
DR1 & 2012-09-28 2013-06-03 & 1,536,045 & 12,734 & 6,035 & 127,198 & 1,682,012\\
DR2 & 2013-09-10 2014-06-03 &1,504,329 & 30,432 & 6,382 & 91,399 & 1,632,542\\
DR3 & 2014-09-10 2015-05-30 & 1,516,147 & 26,288 & 8,753 & 88,956 & 1,640,144 \\
DR4 & 2015-09-12 2016-06-02 & 1,551,394 & 39,498 & 13,954 & 96,680 & 1,701,526 \\
DR5 & 2016-09-09 2017-06-16 & 1,226,472 & 36,093 & 14,782 & 119,885 & 1,397,232\\\hline
First phase & 2011-10-24 2017-06-16 & 8,171,443 & 153,090 & 51,133 & 642,178 & 9,017,844\\\hline
\end{tabular}
\end{center}
}
\end{table}
\begin{table}
{\small
\begin{center}
\caption{Classification for the 3522 spectra of 3069 sources.\newline
}
\label{tab_stat}
\begin{tabular}{c c c}\hline
Type & Spectra & Sources\\
\hline
DA/DA: & 2625 & 2281 \\
DB/DB: & 182 & 166 \\
DC/DC: & 62 & 58 \\
DCA/DCQ & 2 & 2 \\
DAH/DBH/DAP & 34 & 31\\
DZ/DZ: & 36 & 33 \\
DZA/DZB & 6 & 4\\
DAZ/DAZe & 6 & 5\\
DBAZ/DBAZ: & 3 & 2\\
DBZ/DBZA & 7 & 4\\
DAB/DBA/DBA: & 76 & 64 \\
DO/DAO/DOA/DBOA$^{a}$ & 23 & 20 \\
DQ/DQ: & 19 & 19 \\
CV/CV: & 130 & 106\\
(DA,DB,DC,DAH,DBA)+M/DA+(K,DQ)$^{b}$ & 311 & 274 \\\hline
\end{tabular}
\begin{tablenotes}
\item
Notes: \newline
Spectral types listed in this paper followed the definitions in Section 2.2 from \cite{Kepler2019}. \newline
Especially, notation ":" means uncertainty mainly due to low S/N.\newline
Notation "e" means emission line present in the spectrum.\newline
$^{a}$DBOA type means its a Helium dominated white dwarf with He II 4\,686 \AA\,line and mild Balmer absorption lines.\newline
$^{b}$White dwarf binary. White dwarf with M, K type star or DQ white dwarf.\newline
\end{tablenotes}
\end{center}
}
\end{table}
\subsection{DA selection}
The raw spectral data are processed with the LAMOST 2D pipeline \citep{Luo2015}. The standard procedures of dark current subtraction, bias subtraction, cosmic ray removal, 1D spectral extraction, sub-exposures combination are performed in the process of 2D pipeline. Next, the 1D pipeline is used to perform spectral classification, then calculate the radial velocity for each spectrum. Spectral classification adopted by 1D pipeline is done by template matching. The most important part is to construct a spectral classification template library. The early version of this library for LAMOST is built firstly by excluding outliers using local outlier probabilities, then principal component analysis was used to reconstruct spectra. More description can be found in \cite{Wei2014}. The updated version of this library is constructed through clustering algorithm \citep{Kong2019}. The pipeline classification determines object class (STAR, QSO, GALAXY and UNKNOWN), and subclass, e.g.\ (B2, G6, WD). With better pipeline development and more information, there are more correctly classified spectra in DR5. We successfully identify numerous false positives from our previous catalog, mostly spectra with low signal to noise ratio, which we marked ":" for uncertain about the classification (presented in Table\,\ref{tab5}).
Machine learning has drawn increasing attention in astronomy, where spectral classification is an important potential application \citep{Jiang2013,Liu2014,Li2018}. Thus we applied machine learning algorithm to select DA candidates in this work, as another independent method besides adopting pipeline subclass. There are various types of machine learning algorithms, e.g.\ support vector machine, Bayesian networks, decision tree learning. Based on our previous experience performing spectral classification \citep{Bai2018,Bai2019}, we choose an algorithm called random forest \citep[RF]{Breiman2001}. It performs better than other algorithms, in terms of time cost and accuracy.
The RF algorithm operates by constructing a multitude of decision trees at training and outputting the class that is the mode of the classification of the individual trees \citep{Breiman2001}. Random decision forests correct for decision trees's habit of overfitting to their training set. The RF algorithm implementation used in this work is a supervised algorithm, and adopts 100 trees (estimators). All flux values of a spectrum are used as parameters. The largest weight is given to parameters in the Balmer line wings. For each spectrum, a probability value will be produced by the RF algorithm. Then a simple binary classification is used. If the probability of being a DA star is greater than 50\%, this object will be classified as a DA candidate. Otherwise, it will be classified as non-DA. This algorithm implementation is from scikit-learn python package \citep{Pedregosa2011}, and information of the specific model can be found in their website \footnote{\url{https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html}}. Considering the fact that observed spectra, even for the same source, vary from different surveys (i.e. for the same faint source, SDSS spectra are likely to have higher signal-to-noise (S/N) than LAMOST spectra. And SDSS spectra are better flux calibrated, compared to LAMOST spectra etc.), we first built a DA training sample from LAMOST spectra. First of all, there are 379 DA spectra of S/N greater than 10 in SDSS $g$ band selected from our previous catalogue \citep{Guo2015b}. Secondly, we randomly selected 100 spectra with the same S/N limit for each spectral type classified by LAMOST pipeline, which is around 6k of non-DA spectra in total. Together, these 379 DA spectra and $\sim$ 6k non-DA spectra are used to train the model, then the model is applied to a randomly selected 50k spectra sample for initial test. Based on our test result, most of the contaminants come from A stars. It is understandable, because our RF algorithm mostly uses the broad and deep Balmer line profile of DA stars to select candidates. Some A stars have similar profiles in Balmer line regions. Thus, we added an additional training model specific for separating DA and A stars. The main idea is to distinguish A stars by using their multiple absorption line features in the near-infrared region. Therefore, the largest weight is given to that region. This time, same 379 known DA stars and another 1\,000 A type star spectra (classified by LAMOST pipeline) with S/N above 10 are selected to train the model. Then this additional model is applied to DA candidates identified by first model. It produces a probability value as well. But this time, if the probability is greater than 50\%, this object is classified as A type star. Consequently, its label is changed from DA to non-DA. At last, this code is applied to the whole data set in LAMOST DR 3, 4, \& 5. In total, there are 6\,662 unique spectra candidates selected by our RF algorithm. Each spectrum is visually inspected by eye, 1\,710 DAs are identified.
In addition, we used a conventional colour-colour cut (Formula 1-4 in \citealt{Eisenstein2006}, and Table 1 in \citealt{Girven2011}) as supplementary methods to select DAs. Even though LAMOST is basically a spectroscopic survey, there are imaging surveys like XSTPS-GAC and SDSS, providing $g$, $r$, $i$ magnitudes for more than 50\% of the input catalogue entries. Around 1.46 million entries in LAMOST DR 3, 4 \& 5 have $u$, $g$, $r$, $i$, and $z$ magnitudes, mostly from SDSS. Using method from \cite{Eisenstein2006}, 43\,461 white dwarf candidate spectra are selected, while 847 DA star candidate spectra are selected adopting method from \cite{Girven2011}. For pipeline selection, we select spectra that are classified by the pipeline as WD, WDMagnetic, DoubleStar, or CarbonWD as DA candidates. Spectral types of WDMagnetic, DoubleStar, and CarbonWD are also included, because small number of DA can be misclassified by the pipeline and identification of those types of WD is a part of our goals as well. Those methods result in 6\,662 unique candidate spectra from RF algorithm, 43\,461 candidate spectra from \cite{Eisenstein2006}, 847 DA candidate spectra from \cite{Girven2011}, and 7\,024 DA candidates from pipeline selection. In total 2\,620 bona fide DA spectra and many various types of WDs are identified after visual inspection (See Table. \ref{tab_stat}). After cross-matching with the literature, 393 DAs are new identifications \citep{Kleinman2004,Kleinman2013,Zhao2013,Zhang2013,Guo2015b,Kepler2015,Kepler2016,Kepler2019}.
One should note that our strategy in selecting DA stars is to include as many DAs as possible with manageable candidate size. This may result in considerable contamination rate, but rely on visual inspection, contaminants are able to be removed and a relatively complete DA sample can be obtained. Thus, we emphasis the importance of visual inspection of candidate spectra in this work. According to our identification, the efficiency (defined as the ratio of correctly identified DAs to the total number of objects identified as DA candidates) of LAMOST pipeline is $2\,234/7\,024=31.8\%$. But more accurately, there are 3869 spectra classified by LAMOST pipeline as DA (i.e. subcalss WD). Spectra classified as subclass WDMagnetic and DoubleStar are also included in pipeline method to ensure Magnetic WDs and WDMS binaries are included in our main catalogue. Therefore, the actual efficiency of LAMOST pipeline should be $2\,234/3\,869=57.7\%$. The main contaminants are low S/N spectra. Of the 3\,869 spectra classified as subclass WD, only 928 spectra have S/N greater than 5 in $u$, $g$, $r$, $i$, and $z$ bands. Substantial type A and F stars with very noisy spectra have been wrongly matched with DA templates. The efficiency of RF algorithm is $1\,710/6\,662=25.7\%$. Similarly, the main contaminants are still A stars (3\,645 of 6\,662), together with some F stars and hot subdwarfs. Even though the efficiency of Eisenstein colour cut is only $1\,172/43\,461=2.7\%$, majority of other WD types (except DA and DB) are identified through this method. Contaminants are various, more than half of the candidates selected by Eisenstein colour cut method are QSOs, along with B, A, F stars and other types of WDs. The efficiency of Girven colour cuts is $373/847=43.9\%$. The main contaminants are type G, A, B stars and QSOs. There are 216 DAs missed by LAMOST pipeline and RF algorithm. After a closer look at those DAs, about 100 DAs are found to be misclassified by pipeline as A stars mostly, and B, F stars. Those DAs display relatively narrow absorption lines, compared to the majority of DAs. But they are indeed located in the WD region in the colour-colour plot and $Gaia$ colour-magnitude diagram. This indicates the DA templates used by LAMOST pipeline lack low mass DAs. Same problem exists in the RF method, since its training sample is based on our previous catalogue, and identification of low mass DA is generally difficult before $Gaia$. Apart from those 100 missed DAs, another 100 missed DAs have very low S/N spectra (S/N<5 in all SDSS bands). It appears that broad absorption line features in those very noisy spectra have not been picked up by template matching and RF algorithm.
Regarding the completeness ratio of our identified DA sample to all DAs observed by LAMOST, we applied simple and rough ways to estimate, aiming to evaluate the robustness of our DA identification methods. From four methods, LAMOST pipeline identified 2\,234 DA spectra, two colour cut methods identified 753 spectra, and RF algorithm identified 1\,710 spectra. Most DA spectra are discovered by two or more methods. Because there is no independent LAMOST imaging survey, substantial LAMOST sources have no required colour to perform the selection, DA spectra identified via colour cut method is highly incomplete. Therefore, only pipeline and RF algorithm identified DAs are used to estimate the completeness. There are 1\,819, 1\,523, and 2\,106 DA spectra with S/N no less than 5 in $g$ band identified by LAMOST pipeline, RF method and four methods combined, respectively. The number of common spectra identified by both pipeline and RF is 1\,379. Thus, the percentage of pipeline missed DA spectra is about $(2\,106-1\,819)/1\,819=18\%$. Since pipeline and RF together identified 1\,963 unique DA spectra, the percentage of pipeline and RF missed spectra is about $(2\,106-1\,963)/1\,963=7\%$. Therefore, the total missed DA spectra could be $2\,106\times18\%+2\,106\times7\%=527$, which means a lower limit of completeness should be around $2\,106/(2\,106+527)=80\%$. With the help of pipeline and RF method, we managed to identify 1\,963 DA spectra with S/N no less than 5 in $g$ band, which means an upper limit of completeness of pipeline and RF combine is 1\,963 /\,2\,106=93\%. In summary, the completeness ratio of identified DA sample to all DAs observed by LAMOST is in the range of 80\% to 93\%. It shows the estimated completeness is consistent, both indicate the number of DA spectra that are not included in our catalogue is not significant.
Another separate test has been carried out by cross matching LAMOST DR 3, 4, \& 5 spectra with most recently published SDSS DR 14 DA catalogue first \citep{Kepler2019}, there are 272 DA common sources with S/N ratio no less than 5 in $u$, $g$, and $r$ band. Then those sources were cross matched with our DA catalogue, five spectra were found to be missing in our DA catalogue. Therefore, the completeness estimated this way is $5/272=98\%$, consistent with our previous estimation. Note that those five missed sources were added to our catalogue, making the total number of identified DA spectra 2\,625.
To further illustrate the ability of different methods in identifying DA spectra as a function of S/N, Fig. \ref{fig01} is here to show the statistics of DA spectra identified by LAMOST pipeline and RF versus S/N of LAMOST spectra in SDSS $g$ band. DA spectra identified by colour cut methods are not considered here as their sample are biased by lacking photometric data. All panels have shown that pipeline method are better than our RF method in identifying more DA spectra and yielding less contaminants, regardless of S/N. Bottom panel indicates that pipeline method yields much more contaminants at low S/N than at high S/N, while the contamination rate of RF method seems not much affected by S/N of LAMOST spectra. Nonetheless, more than 60\% of RF method selected candidates are contaminants.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig1.eps}
\caption{The colour-colour cut selection, based on SDSS $u$-$g$ and $g$-$r$, de-reddened. The black dots are LAMOST sources with $u$, $g$, $r$ magnitudes, while red dots are WD candidates selected following \protect\cite{Eisenstein2006}, blue dots are DA stars selected following \protect\cite{Girven2011}. Magenta lines have separated different regions defined by \protect\cite{Eisenstein2006}. Bottom left is WD candidate region, while right region is main-sequence and blue horizontal branch region. Black dots above the horizontal magenta line are mostly quasars.}
\label{fig00}
\end{figure}
\begin{figure*}
\center
\includegraphics[angle=0,width=1.0\textwidth]{fig2.eps}
\caption{Examples of various WD spectral types identified in LAMOST. In order to focus on the relevant spectral features, this figure only shows the wavelength range 3500--8000\,\AA. Classified types and their unique LAMOST spectral IDs are also shown.}
\label{fig0}
\end{figure*}
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig3.eps}
\caption{The statistics of DA spectra number identified by different methods as a function of S/N in SDSS $g$ band. Top panel: Accumulated number histograms of LAMOST pipeline, RF algorithm, and all methods identified DA spectra in blue, red and cyan colour, respectively. Middle panel: Number percentage of LAMOST pipeline (blue line) and RF algorithm (red line) identified DA spectra in all DA spectra identified as a function of S/N. Bottom panel: Blue line represents number percentage of LAMOST pipeline identified DA spectra in candidate spectra selected by pipeline, while red line represents number percentage of RF algorithm identified DA spectra in candidate spectra selected by RF method. }
\label{fig01}
\end{figure}
\subsection{DB selection}
Because there are no DB templates in the model database of LAMOST classification pipeline, we only applied RF machine learning method to identify DB stars.
There are 1\,842 objects in SIMBAD that are listed as DB stars. Those known DBs in SIMBAD were cross-matched with all spectra from LAMOST DR5, resulting in 178 matches, where 150 of these are classified as UNKNOWN by LAMOST pipeline. A large portion of those 150 spectra have very low S/N ratio that are not able to be identified by eyes. Thus, 58 of 178 known DBs are not recovered by our DB selection. Because we can not identify them based on their unrecognisable noisy LAMOST spectra alone, and these noisy spectra have no use for any analysis, those are not included as DBs in this paper. In order to select a high quality sample to best represent DB features, we only chose those LAMOST spectra with S/N $>10$ in the SDSS $g$ band. There are 45 spectra that meet this criterion and form the training sample. Next, after removal of the 178 known SIMBAD DB spectra, roughly 1\,000 spectra with S/N>10 were randomly selected from LAMOST DR5. Most of those 1\,000 spectra are different types of star, together with galaxies and QSOs. DB spectra are ensured to be excluded by careful spectral inspections. Similar to RF method applied to identify DAs, we used all unsmoothed 45 DB and $\sim$1\,000 non-DB spectra to construct the supervised DB classifier for the RF algorithm. All flux values
|
of a spectrum are adopted as parameters. The largest weight is given to parameters in the region of Helium absorption line wings. A simple DB or non-DB classification is used. An object is classified as a DB candidate, when its probability of being DB star is greater than 50\%. Or this object will be classified as non-DB. In the final step, all spectra in DR\,3, 4 \& 5 are fed to the classifier, DB candidates are obtained.
There are 12\,572 DB candidates selected by the RF algorithm. After visual inspection, 182 spectra of 166 sources were found to be bona fide DBs. After cross-matching with previous published catalogs, 46 DBs are new identifications \citep{Kleinman2004,Kleinman2013,Kepler2015,Kepler2016,Guo2015b}. We believe that higher contamination rate in DB selection is caused by smaller training sample size relative to DA training sample. In addition, a second training is applied to distinguish DA and A star in DA machine learning selection.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig4a.eps}
\includegraphics[angle=0,width=0.5\textwidth]{fig4b.eps}
\caption{Balmer line fitting examples for J041010.37+180222.8 (upper panel, S/N=107) and J071004.93+292403.2 (lower panel, S/N=28). For both cases, the black solid lines are observed and normalised DA spectra, while red dashed lines are the best fit model. From bottom to top of each panel, fitted absorption lines are H$\beta$, H$\gamma$, H$\delta$, and H$\epsilon$, respectively.}
\label{fig1}
\end{figure}
\subsection{Other WD spectral types}
Besides DA and DB stars, other spectral types of WD are identified as well, as a result of our effort in searching for DA stars. They are from the colour-colour cut selection method from \cite{Eisenstein2006}, \cite{Girven2011} and visual inspection of spectra that are classified as WDMagnetic, DoubleStar and CarbonWD. Colour-colour cut from \cite{Eisenstein2006} yielded more types of WD than any other methods. But not all LAMOST sources have SDSS photometry, so WD types except DA and DB are highly incomplete.
There are 5 DAZs, 4 DBZs and 2 DBAZs identified in this work. Two of five DAZs are classified as WD by LAMOST pipeline, Three of four DBZs and one of two DBAZs are selected as DB candidates by RF algorithm. The rest $3+1+1=5$ are discovered by Eisenstein's colour cut method. Based on a rough estimation, the fraction of those special metal polluted WDs in Eisenstein's colour cuts selected WD candidates is about $5/43\,461=0.011\%$, and the fraction of this colour cuts selected candidates in LAMOST DR3, 4 \& 5 that have SDSS photometry is $43\,461/1\,455\,566=2.98\%$. Therefore, the number of those special WDs that have been missed in LAMOST because they do not have SDSS photometry is $(4\,740\,458-1\,455\,566)\times(0.011\%\times2.98\%)\approx11$. Besides those three types of WDs, magnetic WDs (DAH, DBH, and DAP) mainly come from pipeline classified WDMagnetic, while pipeline classified DoubleStar identified most WDMS binaries and CVs. And all the spectral types identified in this work with their corresponding numbers are listed in Table.\ \ref{tab_stat}. A small part of basic parameters of all WDs identified in LAMOST DR 3, 4 \& 5, including designation, ra, dec, observation date, unique spectral ID, and WD subtype, has shown in Table.\ \ref{tab2}.
\section{Parameter determination}
\subsection{$T_{\rm eff}$ and log $g$ for DA stars}
In order to ensure the accuracy of the derived parameters for DAs, only spectra with S/N $>15$ in SDSS $g$ band are fitted. For sources with multiple observations, those with highest $g$ band S/N are used. To estimate the effective temperature and surface gravity of DA stars, absorption line profiles from H$\beta$ to H$\epsilon$ were fitted to theoretical spectral models. Before performing the spectral fitting, both the observed and model absorption line profiles were trimmed and normalised following standard procedures \citep{Liebert2005}. The next step is to use a minimisation technique to fit the line profile. The technique adopted here is the well known non-linear least-squares method of Levenberg-Marquardt, that is based on a steepest descent method. To fit the best model template, the open source {\sc idl} package {\sc mpfit} was used \citep{Markwardt2009}. The DA atmosphere models used here are provided by D.\ Koester (2015, private communication) and are based on those described in \cite{Koester2010}. The errors for $T_{\rm eff}$ and log $g$ were determined by stepping the parameters about the minimum $\upchi^{2}$. The difference, which is calculated between each current minimum $\upchi^{2}$ and the previous true minimum $\upchi^{2}$, corresponds to 1$\sigma$ for a given number of free model parameters is regarded as the error.
Notably, we introduced {\em Gaia} $G\rm_{BP}$-$G\rm_{RP}$ colour into the spectral fitting process. First, a sample of SDSS identified DAs with relatively high precision $T_{\rm eff}$ is constructed. Next, an empirical relation is established between $T_{\rm eff}$ and their corresponding {\em Gaia} colour. Then for a new DA spectrum to be fitted, an initial effective temperature value was given by the empirical relation, based on its {\em Gaia} colour. The initial value was then used as one input parameter in {\sc mpfit} code to start model fitting iterations (More details in Zhang et al. in prep). After this preparation, it's the same pure conventional spectral fitting to theoretical model spectra to obtain the final $T_{\rm eff}$ and log $g$. By using {\em Gaia} data in this way, the final effective temperature will be determined more robustly and accurately than our previous study \citep{Guo2015b}. Two examples of Balmer line fitting are shown in Fig.\ref{fig1}. For comparison purposes, we chose one spectrum with high S/N (J041010.37+180222.8), and another spectrum with low S/N (J071004.93+292403.2). Figure \ref{fig2} shows a $\upchi^{2}$ contour plot of $T_{\rm eff}$ and log $g$ for the source J041010.37+180222.8. The best-fitted values for $T_{\rm eff}$ and log $g$, together with their uncertainties are estimated from the minimum converged residual $\upchi^{2}$. The median error of $T_{\rm eff}$ and log $g$ in our DA sample is roughly 8\% and 4\%, respectively.
\begin{figure}
\center
\includegraphics[angle=0,width=0.53\textwidth]{fig5.eps}
\caption{$\upchi^{2}$ contour plot of $T_{\rm eff}$ and log\,$g$ determination for DA J041010.37+180222.8. Red cross represents final result with its uncertainties.}
\label{fig2}
\end{figure}
\subsection{$T_{\rm eff}$ and log $g$ for DB stars}
To ensure the number of DB stars with parameters that can be compared with the literature, we adopt a S/N limit of 10. The DB spectrum with highest S/N was chosen for model fitting in the case that a source has multiple spectra. Also, because there can be a large number of helium absorption lines, the fitting process is complicated by their normalisation. Therefore, the parameter determination for DB spectrum is usually done by fitting the entire spectrum to the theoretical model. But in this case, flux calibration errors may affect the accuracy of parameter estimation. Flux calibration is difficult for LAMOST spectra, owing to e.g.\ flat-fielding over large fields of view, use of pseudo-standard stars, and possible unknown reddening \citep{Du2016}. To evaluate which is more suitable for LAMOST DB spectral fitting, by fitting only Helium absorption lines after continuum normalisation (similar to DA spectral fitting) or fitting the entire flux calibrated spectrum directly, a sample of a few tens of known DB stars from \cite{Koester2015} is selected to perform a less formal test. Those DB stars are common sources of our LAMOST DB sample and DB catalogue from \cite{Koester2015} with relatively small errors of derived $T_{\rm eff}$ and log $g$. Two sets of parameters are derived from these two different methods, who performed on LAMOST spectra. Next, both results were compared with SDSS catalogue values, and we found that fitting the entire flux calibrated spectrum yielded more consistent results. Therefore, we chose this method despite flux calibration concerns. The DB models used in the fits were pure He models, kindly provided by \cite{Koester2010}. Fig. \ref{fig3} shows two examples of DB stars where the entire spectra were fitted by models. One is DB spectrum with high S/N (J164718.38+322832.8, S/N=57), while the other has lower S/N (J012148.23-001053.0, S/N=29). The best-fitted $T_{\rm eff}$ and log $g$, as well as their uncertainties are also obtained from the minimum converged residual $\upchi^{2}$. It is noted that the effective temperature and surface gravity for DB training sample we adopted ranges from 11\,722\,K to 33\,894\,K and 7.6 to 8.7 dex, respectively, according to SIMBAD records. Therefore, even though these ranges should cover most of DB stars, it is still possible that a small number of DBs with $T_{\rm eff}$ and log $g$ outside these ranges are missing in our DB catalogue.
\begin{figure*}
\center
\includegraphics[angle=0,width=0.7\textwidth]{fig6a.eps}
\includegraphics[angle=0,width=0.7\textwidth]{fig6b.eps}
\caption{DB Helium absorption lines model fit comparison between J164718.38+322832.8 (upper panel, S/N=57) and J012148.23-001053.0 (lower panel, S/N=29). The black solid lines are observed DB spectra, while red dashed lines are best fit model spectra.}
\label{fig3}
\end{figure*}
\subsection{Mass and cooling age}
Once the $T_{\rm eff}$ and log $g$ are obtained from the model fitting, their mass and cooling age can be estimated based on evolutionary models \citep{Fontaine2001} \footnote{The cooling model can be downloaded from the Web site: \url{http://www.astro.umountreal.ca/\textasciitilde bergeron/CoolingModels/}.}. For DAs, the cooling models in \cite{Wood1995} was used for the carbon-core with thick ($q_{\rm H}= M_{\rm H}/M_{\star}=10^{-4}$) hydrogen layers with $T_{\rm eff}$ greater than 30,000 K. For models with effective temperature less than 30,000 K, the cooling models for carbon-oxygen cores in \cite{Fontaine2001} with thick ($q_{\rm H}=10^{-4}$) hydrogen layers are used. For DBs, the cooling models in \cite{Wood1990} was adopted for the carbon-core with thick ($q_{\rm He}= M_{\rm He}/M_{\star}=10^{-4}$) helium layers \citep{Bergeron2001}.
An example of mass and cooling age determination is demonstrated in Figure \ref{fig34}. The resulting distributions of mass and cooling age are shown in Figure \ref{fig4} and Figure \ref{fig6}, respectively. There are 1316 DAs and 70 DBs with good quality spectra that can be used to construct these distributions.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig7.eps}
\caption{One example of mass and cooling age determination, based on effective temperature and surface gravity. Black cross represents the location of this example with uncertainties in the effective temperature vs. surface gravity space. Different lines represent theoretical models with correspondent mass. Determined mass and cooling age are shown in the upper left with their errors.}
\label{fig34}
\end{figure}
Looking at Fig. \ref{fig4}, the DA mass range is from 0.3\,M$_{\odot}$ to 1.2\,M$_{\odot}$, with a peak near 0.62\,M$_{\odot}$. This is in agreement with the mean mass of 0.613\,M$_{\odot}$ from \cite{Tremblay2011} and our previous research \citep{Guo2015b}. But our peak mass is slightly less massive than the mean mass of 0.649\,M$_{\odot}$ from \cite{Kepler2015}. As is well established in the literature \citep[and references therein]{Rebassa2015} as well as in \cite{Guo2015b}, two obvious, non-single Gaussian distribution features present near 0.4\,M${_{\odot}}$ and 1.0\,M$_{\odot}$. However, these features are not evident in our combined DA mass distribution of DR 1-5.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig8.eps}
\caption{Distributions of derived DA parameters, including DAs with updated parameters from LAMOST DR2. From top to bottom: mass, effective temperature, surface gravity, cooling age and inverted parallax from Gaia DR2.}
\label{fig4}
\end{figure}
Of all the fitted LAMOST DAs, 506 stars are found to have fitted parameters in the SDSS DR14 white dwarf catalogue \citep{Kepler2019}. In Fig. \ref{fig5}, we compare our results with \cite{Kepler2019}. In general, both sets of results are in good agreement. Sources common to both studies with higher S/N LAMOST spectra exhibit higher consistency between the studies. Also, the derived surface gravities and thus corresponding masses are more discrepant at the high and low ends. Our study systematically underestimates masses at the high end, and overestimates mass at the low end, relative to the SDSS WD study. In terms of S/N, the difference is displayed clearly in the bottom panel of Fig. \ref{fig5}. Regarding DAs with LAMOST spectra S/N of less than $\sim$50, the SDSS spectra quality are obviously better than LAMOST spectra. However, as for DAs with LAMOST spectra S/N of greater than 50, the LAMOST spectra quality are clearly better. We suggest our derived parameters are reliable for S/N above 30. Similarly, there are 1059 stars found to have fitted parameters in the {\em Gaia} DR2 WD catalogue \citep{Gentile2019}. In Fig. \ref{fig51}, our parameters and results from \cite{Gentile2019} are compared. Results from both catalogues are generally agree with each other. However, the derived surface gravity and mass data are clearly more disperse than the data from comparisons between LAMOST and SDSS, despite S/N of LAMOST spectra. It is understandable that parameters like surface gravity and mass derived from spectral fitting are generally more accurate and reliable. Moreover, derived $T_{\rm eff}$ from $Gaia$ are much more consistent with LAMOST $T_{\rm eff}$ when they are low. This implies derived $T_{\rm eff}$ from $Gaia$ for relatively cool WDs are generally reliable, while for hotter WDs, derived $T_{\rm eff}$ should be used cautiously.
\begin{figure}
\center
\includegraphics[angle=0,width=0.55\textwidth]{fig9.eps}
\caption{Comparisons between LAMOST DA parameters (DAs with updated parameters from LAMOST DR2 are also included) and SDSS parameters from \protect\cite{Kepler2019}. From top to bottom: mass, effective temperature, surface gravity and S/N. Black circles represent DAs with S/N of LAMOST spectra between 10 and 30, while blue crosses are DAs with S/N of LAMOST spectra greater than 30. The red solid lines are unit slope relation.}
\label{fig5}
\end{figure}
\begin{figure}
\center
\includegraphics[angle=0,width=0.55\textwidth]{fig10.eps}
\caption{Comparisons between LAMOST DA parameters (DAs with updated parameters from LAMOST DR2 are also included) and {\em Gaia} DR2 WD parameters from \protect\cite{Gentile2019}. From top to bottom: mass, effective temperature and surface gravity. Black circles represent DAs with S/N of LAMOST spectra between 10 and 30, while blue crosses are DAs with S/N of LAMOST spectra greater than 30. The red solid lines are unit slope relation.}
\label{fig51}
\end{figure}
With respect to the parameter distributions of the 70 relatively high S/N DBs, the peak in mass is located near 0.65\,M$_{\odot}$, and the peak in log\,$g$ distribution is around 8.0. These distributions are consistent with the literature \citep{Eisenstein2006,Kleinman2013}. It is difficult to draw any further conclusions from these distributions, because there are so few sources.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig11.eps}
\caption{Distributions of derived DB parameters. From top to bottom: mass, effective temperature, surface gravity, cooling age and inverted parallax from {\em Gaia} DR2.}
\label{fig6}
\end{figure}
Among the 70 DBs with derived parameters, 42 have parameters in \cite{Kleinman2013} and \cite{Koester2015} , but only 11 sources have derived masses. In Fig. \ref{fig7}, from top to bottom panel, these are comparisons of mass, effective temperature, surface gravity and S/N between our results and results from \cite{Kleinman2013} and \cite{Koester2015}, respectively. Based on these direct comparisons, our results are generally consistent with results from SDSS. It is also true that DBs with higher S/N tend to have more consistent parameters than DBs with lower S/N. And SDSS DB spectra have higher S/N than SDSS spectra, when the S/N of LAMOST DBs are less than $\sim$ 50. Even though there are only three sources, it's likely the opposite when the S/N of LAMOST DBs are greater than $\sim$ 50.
\begin{figure}
\center
\includegraphics[angle=0,width=0.5\textwidth]{fig12.eps}
\caption{Comparisons between LAMOST DB parameters and SDSS parameters from \protect\cite{Kleinman2013} and \protect\cite{Koester2015}. From top to bottom: mass, effective temperature, surface gravity and S/N. Black circles represent DBs with S/N of LAMOST spectra between 10 and 20, while blue crosses are DAs with S/N of LAMOST spectra greater than 20. The red solid lines are unit slope relation.}
\label{fig7}
\end{figure}
\section{On going and future WD candidate selection for LAMOST}
We have been selecting potential WD targets for future, low-resolution, LAMOST spectral fibre selection. In this section, we describe our pre-{\em Gaia} DR2 criteria for the second phase survey. Even though with the release of {\em Gaia} 2$^{\rm nd}$ data release, a white dwarf candidates sample as large as 486\,641 has been revealed \citep{Gentile2018}, and their $T_{\rm eff}$, log\,$g$, radii and mass can be inferred from accurate photometry and trigonometric parallax. However, the importance of obtaining the optical white dwarf spectra can not be neglected. One important application is to discover more rare WDs with emission lines caused by gaseous debris disk or irradiation from unseen substellar object \citep{Gansicke2006,Melis2012,Guo2015a,Manser2019}. Another example is to identify more metal polluted WDs (DZs), which can be a unique tool to analyse the chemical composition of their tidally disrupted planetesimal \citep{Gansicke2012,Farihi2013}. As a matter of fact, in order to obtain WD optical spectra, future large multi-object spectroscopic surveys are planning to observe these white dwarf candidates from {\em Gaia}. For instance, SDSS-V \citep{Kollmeier2017}, 4most \citep{de Jong2012}, WEAVE \citep{Dalton2012}, and DESI \citep{DESI2016}.
At the end of the LAMOST first phase survey in 2017 June, we began a process of selecting WD candidates, based on reduced proper motion. The photometric data we used is taken from the Xuyi Schmidt Telescope Photometric Survey of the Galactic Anti-center \citep[XSTPS-GAC]{LiuXW2014,Guo2018} and Pan-STARRS \citep{Kaiser2010}, while the proper motion data are from the Gaia-PS1-SDSS \citep[GPS1]{Tian2017} proper motion catalog. XSTPS-GAC is a photometric sky survey in the SDSS $gri$ bands, and because LAMOST sources are selected from XSTPS-GAC, it is highly suitable for WD candidate identification. The total number of sources in XSTPS-GAC is 110\,168\,720, and after cross-matching with GPS1 within 3\,arcsec, there are 56\,059\,242 sources in common. It is worth mentioning that this is a simple direct cross-matching without considering the epochs of observations and proper motion of stars. We use equation \ref{equation1} to calculate reduced proper motion, and plot this against $g-i$ in order to identify WD candidates, where Fig.\ \ref{fig8} shows our exact parameter cuts.
Common sources are plotted in the $g$-$i$ colour versus RPM space, over-plotted with the density contours. Red triangles represent 2\,030 known DAs among these common sources, while blue triangles represent 127 known DBs in the same sample. Based on their location, WDs should locate in the same region. Therefore, we defined a triangle region to separate the WDs and the other types of star. The triangle region is defined by cyan solid line, X axis and Y axis. Black dots are common sources locate in the defined region, which represent selected white dwarf candidates.
Our reduced proper motion cuts yielded 30\,441 WD candidates. Next, we cross-matched these objects with the literature, finding 20\,628 sources are previously unrecognised WD candidates (at that time). These candidates were entered into the target selection software designed for LAMOST. Some of those white dwarf candidates have already been observed and included in DR6, while the rest will appear in the 2nd phase.
\begin{equation}
RPM=g+5\times log(PM)+5
\label{equation1}
\end{equation}
\begin{figure}
\center
\includegraphics[angle=90,width=0.5\textwidth]{fig13.eps}
\caption{Colour g-i versus RPM. The coloured density map and contours demonstrate the majority of common stars locate at the central region. Red triangles represent known DAs in the common source sample, while blue triangles represent known DBs. The cyan solid line is defined to separate WDs from majority of stars. The black dots are selected white dwarf candidates, based on the RPM approach.}
\label{fig8}
\end{figure}
After the second release of {\em Gaia} data in 2018 April, \cite{Gentile2018} has revealed a white dwarf candidate sample as large as 486\,k. Unfortunately, our selection of WDs for LAMOST second phase has completed in 2017. Nonetheless, plan has been made to utilise this high fidelity catalog to form a proper sample for LAMOST to observe in the future.
\section{Summary}
In this work, we study the DAs and DBs identified in LAMOST DR5. The LAMOST pipeline classification, colour-colour cut and random forest method were utilised to select DA candidates. Visual inspection produced 2\,620 authentic DA spectra. For DBs, a random forest, machine learning algorithm was used to select the candidates. From 12\,572 DB candidates, 182 were visually confirmed. Regarding the efficiency, larger training sample in the future can definitely be helpful. After cross-matching with SIMBAD and the literature, 393 DAs and 46 DBs are found to be new identifications.
Those identified WDs were analysed. For all DAs where their spectral S/N $>10$, and DBs with same S/N $>10$, their spectra were fitted with white dwarf atmosphere models to derive effective temperature and surface gravity \citep{Koester2010}. From these two parameters the mass and cooling age were calculated from evolutionary models.
Proper motions, magnitude and parallax of {\em Gaia} DR2 are also provided in our catalog (Table. \ref{tab3}). In addition, distributions of mass, effective temperature, surface gravity, cooling age and distance were built to study the property of our sample.
For DAs, the peak of the mass distribution is in general agreement with previous results. Both low- and high-mass residuals are present in the mass distributions. For DBs, the peak of the mass distribution is located around 0.65 M$_{\odot}$, and also consistent with other studies. However, there are only 70 DBs with sufficiently high S/N to derive reliable parameters, and the resulting distributions are too sparsely populated to draw any conclusions. The parameters of LAMOST DAs and DBs have been compared for sources in common with previous work, and found to be in good agreement for spectra with relatively high S/N. We also found that for spectra with LAMOST S/N less than $\sim$ 50, SDSS spectra tend to have larger S/N, while for spectra with S/N greater than $\sim$ 50, LAMOST spectra have larger S/N.
Even after the release of {\em Gaia} DR2 \citep{Gaia2018} and the white dwarf candidates catalog in {\em Gaia} \citep{Gentile2018}, the observation of optical white dwarf spectra is still of great importance.
Optical spectral identification is of increasing importance in the {\em Gaia} era. Therefore, we selected 20\,628 white dwarf candidates from XSTPS-GAC and GPS1, then put them into the input catalog of low-resolution observation for the transition year and 2$^{\rm nd}$ phase of LAMOST, which began in 2017. Since the white dwarf candidates catalog from {\em Gaia} DR2 has been published, we will try to add those observable candidates into LAMOST inout catalog in the near future as well.
\section*{Acknowledgments}
We thank the anonymous referee for helpful suggestions and advices, which improved this paper a lot. We thank M.Barlow, J.Farihi and A.Swan for a careful reading of the manuscript and helping with the improvement of English writing. We also thank D. Koester and P. Bergeron for providing white dwarf models. Balmer/Lyman lines in the models were calculated with the modified Stark broadening profiles of \cite{Tremblay2009}, kindly made available by the authors. This work is supported by the National Natural Science Foundation of China (Grant No. 11890694, No. 11473001 and 11078006). The authors also acknowledge support by the National Key Basic Research Program of China (Grant No. 2014CB845700). We also acknowledge the support from the 2m Chinese Space Station Telescope project: CMS-CSST-2021-A10. This work was partially supported by the Scholar Program of Beijing Academy of Science and Technology (DZ:BS202002). This work is also supported by the China Postdoctoral Science Foundation (Grant No. 2017M610695) and the Astronomical Big Data Joint Research Centre, co-founded by the National Astronomical Observatories, Chinese Academy of Sciences and the Alibaba Cloud.
The LAMOST FELLOWSHIP is supported by Special Funding for Advanced Users, budgeted and administrated by the Centre for Astronomical Mega-Science, Chinese Academy of Sciences.
The Guo Shou Jing Telescope (the Large Sky Area Multi-Object Fibre Spectroscopic Telescope, LAMOST) is a National Major Scientific Project which is built by the Chinese Academy of Sciences, funded by the National Development and Reform Commission, and operated and
managed by the National Astronomical Observatories, Chinese Academy of Sciences.
\section*{DATA AVAILABILITY}
The spectra data underlying this article are publicly available from LAMOST archive (\url{http://dr5.lamost.org/}). Table \ref{tab2}, \ref{tab3}, \ref{tab5}, and \ref{tab6} are partially shown in the article. They are fully accessible through publisher in supplementary material. Table \ref{tab5} and \ref{tab6} are WDs identified in our previous work (LAMOST DR2) with updated classifications and parameters.
|
\section{Introduction}
Presently, graphene is considered as a prospective material for nanoelectronics
and nanophotonics.\cite{geim07,bona10} Among the various experimental
techniques, the Raman spectroscopy has proven to be an indispensable
tool for investigation of this material.\cite{ferr06,mala09a,dres10,zoly11}
Graphene has a single Raman-active phonon $E_{2g}$ observed as an
intense line (the G band) in the first-order Raman spectra. The second-order
spectra of graphene with low defect density has several intense bands,
which originate from scattering of electrons and holes by two phonons
of the same/different frequency and non-zero momentum and are called
overtone/combination bands. The appearance of intense second-order
bands can be explained by the double-resonant (DR) scattering mechanism.\cite{thom00,reic04}
These bands contain valuable information on the phonon dispersion\cite{sait02,grun02,mafr07,grun09}
and the electron-phonon and electron-electron matrix elements.\cite{bask09}
The Raman spectra of graphene with defects show additional bands,
which arise from DR scattering of electrons and holes by phonons and
defects.
The theoretical investigation of the two-phonon DR scattering in graphene
has been performed using various approximations: replacement of the
electron-photon and electron-phonon interactions with constants, using
a constant electronic lifetime, considering only high-symmetry directions
in the Brillouin zone, as well as exact DR conditions. The predicted
dispersive behavior of the Raman bands\cite{thom00} and the frequency
shift of the Stokes and anti-Stokes Raman bands\cite{canc02} have
been found in quantitative agreement with the experimental data. It
has also been realized that the integration over the entire Brillouin
zone of graphene is essential for predicting the Raman intensity.\cite{maul04a,naru08}
While most of the theoretical papers focus on the most intense overtone
band, in a recent study, all two-phonon Raman bands with observable intensity have been calculated using an electron-phonon matrix element derived within a nearest-neighbor $\pi$-band tight-binding model.\cite{vene11}
The dominant contribution to the two-phonon bands from different parts of the
Brillouin zone\cite{vene11,naru11} and from different scattering processes\cite{vene11} has also been discussed. The polarization dependence of the most intense
overtone band has been studied experimentally and theoretically.\cite{yoon08}
The progress, made so far in the modeling of the DR bands, has been
achieved either with simple $\pi$-band tight-binding models, or with
more sophisticated models but relying on approximations of the
electron-phonon matrix element, and
in most cases concerns only a few intense bands. Our experience
indicates that, although the Raman shift of the bands is not sensitive
to the used matrix element, their Raman intensity crucially depends
on it.
Here, we calculate the two-phonon DR Raman spectra of graphene using
a non-orthogonal tight-binding (NTB) model, which implements parameters
derived from a density functional theory (DFT) study and thus has no adjustable parameters.\cite{pore95}
In particular, the electronic\cite{popo04} and phonon\cite{popo06}
dispersion, the electron-photon and electron-phonon matrix elements,\cite{popo05}
as well as the electronic linewidth\cite{popo06a} are all obtained
within this model. The NTB model for electrons and phonons in graphene
is introduced in Sec. II. The calculated two-phonon DR Raman spectra
of graphene and its polarization dependence are discussed in Sec.
III. The paper ends up with conclusions (Sec. IV).
\section{Theoretical part}
\subsection{The NTB model}
We use a NTB model with four valence electrons per carbon atom to
calculate the electronic dispersion of graphene.\cite{popo04} This
model is based on matrix elements of the Hamiltonian and overlap matrix
elements derived from DFT\cite{pore95} and therefore
it does not rely on any adjustable parameters. It also allows one
to estimate the total energy and the forces on the atoms. This feature
is utilized for relaxation of the atomic structure. Up to a few electron
volts away from the Fermi energy, the electronic structure of graphene
has the form of conic valence and conduction bands (Dirac cones) with
a common apex (the Dirac point) at two non-equivalent special points,
K and K$^{'}$, of the Brillouin zone. This specific form of the electronic
bands plays an important role in the enhancement of the two-phonon
Raman scattering through the DR mechanism.
The dynamical model of graphene uses a dynamical matrix derived by
a perturbative approach within the NTB model.\cite{popo06} The electron-photon
and electron-phonon matrix elements are calculated explicitly.\cite{popo05}
The summation over the Brillouin zone in the first-order perturbation
term of the dynamical matrix is performed over a $40\times40$ mesh
of $\mathbf{k}$ points, for which the phonon frequencies converge
within $1$ cm$^{-1}$. The calculated in-plane phonon branches
of graphene, after scaling by a factor of $0.9$, agree fairly
well with the available experimental data\cite{popo06} (Fig. 1).
The phonons with displacement in the graphene plane (in-plane phonons)
can interact with electrons and thus can contribute to the Raman spectra.
Those with atomic displacement perpendicular to graphene (out-of-plane
phonons) are less well reproduced but they do not contribute to the
spectra.
It will be shown below that only phonons, close to the high-symmetry
directions $\Gamma$K, $\Gamma$M, and KM of the Brillouin zone, are
of major importance for the two-phonon spectra. The phonon branches
along these directions will be denoted, as usual, by two-letter acronyms describing
their vibrational pattern: the letters O and A stand for ``optical''
and ``acoustic'', respectively; the letters L, T, and Z denote in-plane
longitudinal, in-plane transverse, and out-of-plane atomic displacement,
respectively. The acronyms for the branches along the KM direction
will be primed. Alternatively, for each wavevector, the phonons with
be ascribed the index $\nu$, $\nu=1,...,6$, in order of increasing
frequency. The phonons with a certain $\nu$ can belong to branches
with a different vibrational pattern. For example, a phonon with $\nu=6$
can belong to the LO or TO branch. It will also be argued that only
phonons close to the $\Gamma$ and K points give a significant contribution
to the two-phonon Raman spectra. Such phonons will be denoted by acronyms
ending with @$\Gamma$ and @K. For example, LO phonons close to the
$\Gamma$ point will be denoted by LO@$\Gamma$ and TO phonons close
to the K point along the KM direction will be denoted by TO$^{'}$@K
(see, Fig. 1).
\subsection{The double-resonant processes}
|
The amplitude for two-phonon DR Raman scattering processes in graphene
is described by fourth-order terms in perturbation theory.\cite{mart83}
The underlying processes include virtual scattering of electrons/holes by phonons
between states of the Dirac cone at the K point or the K$^{'}$ point,
or between states of the Dirac cones at the K and K$^{'}$ points.
The momentum is conserved in each virtual process but the energy is
conserved only for the entire DR process. Below, we will consider
only Stokes processes. In this case, a two-phonon DR process includes
an absorption of a photon with a creation of an electron-hole pair,
two consecutive processes of scattering of an electron/hole with creation
of a phonon, and a recombination of the electron-hole pair with
an emission of a photon (Fig. 2). There are altogether eight such
processes.\cite{kurt02,vene11} The total two-phonon Raman intensity
is given by the expression
\begin{equation}
I\propto\sum_{f}\left|\sum_{c,b,a}\frac{M_{fc}M_{cb}M_{ba}M_{ai}}{\left(E_{i}-E_{c}-i\gamma\right)\left(E_{i}-E_{b}-i\gamma\right)\left(E_{i}-E_{a}-i\gamma\right)}\right|^{2}\delta\left(E_{i}-E_{f}\right)\label{a2}
\end{equation}
Here, the inner sum is the scattering amplitude. $E_{u}$, $u=i,a,b,c,f$,
are the energies of the initial ($i$), intermediate ($a,b,c$), and
final ($f$) states of the system of photons, electrons, holes, and
phonons. In the initial state, only an incident photon is present
and, therefore, $E_{i}=E_{L}$, where $E_{L}$ is the incident photon
energy. In the final state, there is a scattered photon and two created
phonons. $M_{uv}$ are the matrix elements for virtual processes between
initial, intermediate, and final states. In particular, $M_{ai}$
and $M_{fc}$ are the matrix elements of momentum for the processes
of creation and recombination of an electron-hole pair, respectively.
$M_{ba}$ and $M_{cb}$ are the electron/hole-phonon matrix elements.
$\gamma$ is the sum of the halfwidths of pairs of electronic and
hole states, and will be referred to as the electronic linewidth.
The electron-photon and electron-phonon matrix elements, and the electronic
linewidth are calculated explicitly.\cite{popo05,popo06a} The Dirac
delta function ensures energy conservation for the entire process.
In the calculations, it is replaced by a Lorentzian with a halfwidth
of $5$ cm$^{-1}$. The summation over the intermediate states runs
over all valence and conduction bands, and over all electron wavevectors
$\mathbf{k}$. The summation over the final states runs over all phonon
branches and phonon wavevectors $\mathbf{q}$. For both summations,
convergence is reached with a $800\times800$ mesh of $\mathbf{k}$
and $\mathbf{q}$ points in the Brillouin zone.
For the discussion of the polarization dependence of the Raman intensity
of the two-phonon bands, it is advantageous to rewrite Eq. (\ref{a2})
in the form
\begin{equation}
I\propto\sum_{f}\left|\mathbf{e}_{S}\cdot R\cdot\mathbf{e}_{L}\right|^{2}\delta\left(E_{i}-E_{f}\right)\label{a4}
\end{equation}
Here, $\mathbf{e}_{L}$ and $\mathbf{e}_{S}$ are the polarization
vectors of the incident and scattered laser light, respectively, and $R$ is the
Raman tensor. Everywhere below we consider only backscattering geometry
in accord with the usual experimental Raman setup for graphene and,
therefore, the polarization vectors lie in the graphene plane.
\section{Results and Discussion}
\subsection{Electronic linewidth}
The electronic linewidth $\gamma$ is due to a large extent to scattering
of the electrons (holes) by phonons and other electrons (holes). The
majority of the published reports assume that $\gamma$ is energy-independent.
Recently, it has been argued that in undoped graphene $\gamma$ is
dominated by electron-phonon processes and the expression $\gamma=9.44E+3.40E^{2}$
has been derived, where the energy separation between the valence
and conduction bands $E$ is in eV and $\gamma$ is in meV.\cite{vene11}
Here, the electronic linewidth was calculated by summing up the contributions
of all electron/hole-phonon scattering processes for all phonons in
the Brillouin zone as a function of $E$. The obtained energy dependence
was approximated in the range $[1.0,3.5]$ eV with the expression
\begin{equation}
\gamma=12.60E+3.45E^{2}\label{b2}
\end{equation}
For energies in this energy range, $\gamma$ changes more than four
times from $16$ to $86$ meV (Fig. 3).
\subsection{Overtone bands}
The calculated overtone Raman spectrum for $E_{L}=2.0$ eV and parallel light polarization along a zigzag line of carbon bonds is shown in
Fig. 4 (bottom). It has two intense bands ($2D$ and $2D^{'}$) and
three weaker ones ($2D^{3}$, $2D^{4}$, and $2D^{''}$) and a much
weaker one ($2D^{5}$). There are also other bands of in-plane phonons
close to the $\Gamma$ and K points but they are either very weak,
or are in the shoulders of intense bands, and in both cases are practically
unobservable.
The assignment of the overtone spectrum can be performed by analyzing
the contribution of phonons with different $\nu$ and from different
parts of the Brillouin zone of graphene. First, the contributions
of the phonons with $\nu=2,...,6$ are given in Fig. 4 (top five graphs).
The bands in these spectra originate from pairs of phonons TA@$\Gamma$
($\nu=2$), LA@$\Gamma$ and TA@K ($\nu=3$), LA@K ($\nu=4$), TO@K
($\nu=5$), and TO$^{'}$@K and LO@K ($\nu=6$) (see also Fig. 1).
The contributions for $\nu=5,6$ are by about three orders of magnitude
larger than the remaining ones.
Secondly, the assignment of the Raman bands to definite phonons at
the $\Gamma$ and K points is supported by the analysis of the contributions
to the spectra from different parts of the Brillouin zone for parallel
light polarization with averaging over all orientations in the graphene
plane. In Fig. 5, the regions with major contribution to the bands
are given by shaded areas. In particular, the $2D^{3}$ band comes
from phon
|
given by $2\pi /P$. Thus when $t$ is one quarter of the period $P$ after primary eclipse, the compact object is approaching with speed $v_x$ and is receding with speed $v_x$ a quarter of a period before primary eclipse.
I now assume that a shell of wind lights up suddenly (say in H$\alpha$) at time $t$ but became detached a time $s$ earlier, the centroid moving with speed $v_x$ tangential to the orbit. For a delay $s$ of $0.25P$, the most redshifted centroid is observed when $t=P$ rather than $0.75P$. The phasing of the wind centroid relative to the photometric ephemeris only requires a delay of $\sim$2 days. The final step is to suppose that the H$\alpha$ emission dies away over a timescale of several days. This will affect the phase and also makes an average of the line-of-sight velocity of the H$\alpha$ centroid, thereby reducing the amplitude of $v_r$ below $v_x$. This may be calculated by specifying some emission function of $s$ with a delay parameter $\tau$ and a duration parameter $T$; $f(s; \tau ,T)$. At time $t$, the centroid of the shell detached a time $s$ earlier has recessional velocity given by
\begin{equation}
v_r(t,s) = -v_x\sin\omega (t-s).
\end{equation}
The value perceived at time $t$ is obtained by averaging over all $s$, using $f(s;\tau ,T)$ as the weight. The important point is that for durations of a few days the average over $s$, $<v_r(t)>$, represents very well the data in both amplitude and phase. For the purpose of illustration I have used two different functions for the emission factor $f(s)$. In the first case I supposed a rectangular profile as a function of $s$, with a duration of $T$. This switches on at $s= \tau - T/2$ and switches off at $s= \tau + T/2$, it being supposed that $\tau$ exceeds $T/2$. The weighted average is then
\begin{equation}
<v_r(t)> = -v_x\sin\omega (t-\tau)\sin (\omega T/2)/(\omega T/2).
\end{equation}
This is not a realistic form, but it makes the point and the structure is easy to visualise. For the particular case of the parameter $\tau = T/2$ (ignition immediately on launch) and duration time $T=P/2$, the centroid of the wind is most redshifted at orbital phase 0, one quarter of a period late, and the amplitude is $v_x$sinc$(\pi /2)$, which for $v_x$ 175 km s$^{-1}$ is 111 km s$^{-1}$. These results are very close to the behaviour of the data. The parameters $T$ and $\tau$ might exhibit some fluctuation with time - winds can be gusty.
A probably more realistic form is to suppose exponential decay of the emission factor after the initial light up. In this case the duration parameter $T$ is the decay time of the exponential, and there is a delay $\tau$ between launch and ignition. The weighted average recession is now
\begin{equation}
<v_r(t)> = \frac{v_x}{1 + (\omega T)^2} [-\sin\omega (t-\tau) + \omega T\cos\omega (t-\tau)] .
\end{equation}
For the simple case of $\omega T=1$, the decay time $T$ is about two days, the amplitude of the oscillation is $v_x/\sqrt 2$ (124 km s$^{-1}$) and the recession velocity is greatest at orbital phase 0.875 for the case of instant ignition, $\tau =0$. Thus these simple models have demonstrated that both the amplitude and the phase of the centroid of the broad component of H$\alpha$, relative to the photometric ephemeris, are easily understood in terms of a wind that becomes detached from the orbiting source and decays away in H$\alpha$ over a few days. The amplitude and phase of the wind centroid is reconciled with the way in which the line-of-sight wind speed varies with the nodding of the disk (Blundell, Bowler \& Schmidtobreick 2008).
\section{Two observations from more violent times}
The period from JD +245 to + 274 was a period of calm before the storm. The persistent pattern in H$\alpha$ of two narrow components with a broad wind swinging between them was overwhelmed by an optical outburst commencing around JD +290, followed by a radio flare. Between JD +291 and +295 the speed of the wind, as measured by the width $\sigma$ of the broad component, rose from $\sim$600 km s$^{-1}$ to $\sim$1200 km s$^{-1}$ and then remained at the higher
|
figure. The history is presented in some detail in Blundell, Schmidtobreick \& Trushkin (2011) - see Fig.8 therein. In the lower panel of Fig.3, I show the motion of the centroid of the wind in H$\alpha$ over the period JD +294 to +310, as displayed in Fig.4 (c) of Blundell, Schmidtobreick \& Trushkin (2011). The amplitude is certainly consistent with the orbital speed of the compact object, and despite the paucity of data, the first minimum (blueshift) occurs at approximately JD +298, an orbital phase only marginally above 0.25. Similarly, the following maximum redshift occurs at $\sim$ JD +304, marginally above orbital phase 0.75. These data seem to have a better memory of the motion of the source of the wind than the slower winds in quieter times; in terms of the exponential decay model the data suggest a decay time of perhaps one day or less.
Over this same period many of the He I spectra are rendered unusable by the proximity of the moving H$\alpha$ lines of the jets, but after JD +294 the O I triplet commencing at 7772 \AA\ exhibits a pronounced P Cygni absorption trough. These O I spectra may also be found in Fig.2 of Schmidtobreick \& Blundell (2006), and from that figure I have extracted the motion of the deepest point of the trough. This is shown in the upper panel of Fig.3. The pattern is erratic but does show some memory of the orbital motion of the compact object, with an amplitude of $\sim$ 100 km s$^{-1}$, with minimum at orbital phase $\sim$ 0.5 and maximum at JD +306, orbital phase close to 1. These data suggest that the O I 7772 \AA\ line is formed in the slow equatorial wind from the disk, with a speed $\sim$150 km s$^{-1}$ and that (absorption) decay times $\sim$2 days are appropriate. [ The O I 7772 \AA\ line is not accessible during the period JD +245 to +274 because of the redshifted moving H$\alpha$ line.]
The notion of a detached wind affecting the magnitude and phase of Doppler shifts may also be relevant to certain absorption spectra in the blue. These have features similar to mid-A supergiants and are probably formed in the wind (Barnes et al 2006).
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=9cm,trim=0 0 0 80]{wigglateO.eps}
\includegraphics[width=9cm,trim=0 0 0 50]{wigglateH.eps}
\caption{ Lower panel: the motion of the centroid of the fast H$\alpha$ wind during the outburst episode. Upper panel: the motion of the blueshifted absorption trough in O I 7772 \AA\ .}
\label{fig:timesequence}
\end{center}
\end{figure}
\section{Discussion}
The broad component of the stationary H$\alpha$ line was identified as formed in a wind from the accretion disk of SS 433 on the basis of the line of sight speed varying with the nodding of the disk (Blundell, Bowler \& Schmidtobreick 2008). The amplitude and phase of the motion of the centroid of this component have now been shown to be entirely consistent with this origin, provided the emission decay time within the wind is a few days. The broad component of the H$\alpha$ is indeed formed in the wind on a scale comparable to or greater than that of the binary system (Gies et al 2002). In contrast, the two narrow components show no sign of an origin within the wind; indeed, the narrow components of both H$\alpha$ and He I are not consistent with such an origin but are well described in terms of emission from a fairly close circumbinary disk, stimulated by the intense radiation from the vicinity of the compact object (Bowler 2010, 2011b). It is interesting that in that circumbinary disk model the decay time for H$\alpha$ emission must again be of the order of a few days. Thus there are two phenomena, not directly related, that are described very well by the phenomenological assumption that H$\alpha$ emission from dense plasma has, following ignition, a decay time of the order of days. There remains the question of by what mechanisms H$\alpha$ radiation is sustained for several days after ignition. That at least one mechanism exists enabling clumps of plasma to exhibit H$\alpha$ decay times of a few days is certain: the individual bolides ejected at about one quarter the speed of light in the jets for which SS 433 is famous are visible in H$\alpha$ for several days ( see for example Vermeulen et al 1993, Gies et al 2002, Blundell, Bowler \& Schmidtobreick 2007 ).
|
\section{Introduction}
Diffusive shock acceleration (DSA) is an established mechanism to
convert bulk kinetic energy into a non-thermal distribution of
relativistic particles with a maximum energy much larger than the
average energy of particles in the plasma. This theory explains well
the spectrum of Galactic cosmic rays (CR) with energies up to
$\sim$3~PeV, accelerated in supernova remnant shocks
\citep[see][for a review]{Bell_rev_14}. The most energetic CR,
i.e. particles with energies up to $50$~EeV, are accelerated outside
the Galaxy but the origin of these particles is still
unknown. Relativistic shocks in extragalactic sources like Gamma Ray
Bursts and Active Galactic Nuclei have been proposed as candidates
\cite[e.g.][]{Gallant,Murase}. However, theoretical models
\citep{Pelletier, lemoine-pelletier-10, Sironi_13, Brian_14} show
magnetic field amplification at ultra-relativistic shocks on
scales much smaller than the Larmor radius $r_{\rm g}$ of particles
being accelerated which precludes CR acceleration to EeV energies unless
other processes can be found to amplify the magnetic field on larger scales.
Hotspots are usually detected at the jet termination region in
type II Fanaroff-Riley (FR) radiogalaxies \citep{FR}. The location
of the hotspot is coincident with the downstream region of the jet
termination shock, where particles accelerated by the shock emit
synchrotron radiation. Therefore, hotspots are suitable places to
study DSA in high velocity shocks.
We model the emission from radio to X-rays in the southern hotspot of
the FR~II source 4C74.26 using data provided in
\cite{Erlund_TwoShocks}. We determine that the compact radio
emission traces out the location of the shock where the magnetic
field $B$ is amplified by plasma instabilities up to
$\sim$100~$\mu$G, and it damps rapidly
downstream of the shock. The turnover in the synchrotron spectrum
between infrared (IR) and optical wavelengths implies that the CR
electron scattering length is much longer than the Larmor radius
and consistent with the amplified magnetic field being structured on
very small scales and comparable with the ion skin-depth
$c/\omega_{\rm pi}$.
Cosmic ray acceleration is consequently very slow and electrons are
accelerated to only $E_{e,\rm max} \sim 0.3$~TeV. A similarly low acceleration
rate for ions would limit their energy to $\sim 100$~TeV
and preclude proton acceleration to EeV energies.
\section{The giant FR II galaxy 4C74.26}
The FR~II galaxy 4C74.26 is located at redshift $z = 0.104$
($\sim$0.5~Gpc from Earth)\footnote{Throughout this paper we use the
cosmology $H_0 = 71$~km~s$^{-1}$~Mpc$^{-1}$, $\Omega_0 = 1$ and
$\Lambda_0 = 0.73$. One arcsecond represents$1.887$~kpc on the plane of
the sky at $z = 0.104$.}. Two X-ray sources separated by
$\sim$10$^{\prime\prime}$ were detected with Chandra
\cite[by][]{Erlund_TwoShocks} at the termination region of the
southern jet, as shown in Figure~\ref{sketch} (upper). In the present
work we study the southern X-ray source, with a luminosity
$L_{\rm x} \sim 10^{41}$~erg~s$^{-1}$ at 2~keV and called ``the southern arc''.
The shape of this emission is arc-like with a characteristic size
$l_{\rm x} \sim 10^{\prime\prime}$, and encloses a compact radio source.
\begin{figure}
\vspace{-5cm}
\includegraphics[width=0.4\textwidth]{fig1_a.eps}\\
\includegraphics[width=0.4\textwidth]{sketch4.eps}
\caption{\emph{Upper:} the hotspot(s) at the southern jet of
the FR~II galaxy 4C74.26 \cite[adapted and rotated by
$\sim$60$^{\circ}$ from][]{Erlund_TwoShocks}.
White and yellow contours are X-rays and radio data, respectively.
Red and green correspond to IR and optical, respectively.
\emph{Lower:} sketch of our model of the southern arc (not to scale).
The synchrotron radio-to-optical radiation is located within
the compact MERLIN emitter, whereas IC X-ray emission is produced
in a more extended region.
\label{sketch}}
\end{figure}
\subsection{The southern arc}
Compact radio emission from the southern arc was detected
with the MERLIN high resolution interferometer ($\nu_{\rm r} = 1.66$~GHz)
with a flux $f_{\rm r} \sim$0.04~Jy, and luminosity
$L_{\rm r} \sim$ 1.9$\times$10$^{40}$~erg~s$^{-1}$ per unit logarithmic
bandwidth $\delta \nu = \nu$. This emission is located in a
region of width $l_{\rm r} < 1^{\prime\prime}$ on the plane of the sky.
In addition, faint and diffuse radiation was
detected at IR ($\nu_{\rm ir} =$ 1.36$\times$10$^{14}$~Hz) and optical
($\nu_{\rm opt} =$ 6.3$\times$10$^{14}$~Hz) bands, with fluxes
$\sim$8.4$\times10^{-6}$ and 2.82$\times$10$^{-7}$~Jy,
respectively, and located in a region of width $\gtrsim l_{\rm r}$.
However, there is a linear structure (in both bands) that traces the
brightest edge
of the MERLIN radio emission, and seems to be cupped within it.
Two factors indicate that the southern arc of X-ray emission is not
synchrotron.
First, $l_{\rm x} > l_{\rm r}$ is inconsistent with the X-ray emitting electrons
being more energetic and therefore cooling more rapidly
as they advect away from the shock.
Second, the steep spectrum between IR and optical
\citep[see Fig.~13 in][]{Erlund_TwoShocks} indicates the maximum
energy of (synchrotron) emitting electrons. We note that
similar characteristics are observed in other sources
\cite[e.g.][]{sources}.
\cite{Erlund_TwoShocks}
suggested that the multi-wavelength emission from the southern arc
is produced by non-thermal electrons, emitting synchrotron
radiation from radio to optical, and up-scattering the cosmic microwave
background (CMB) photons (with energy $\sim$7$\times10^{-4}$~eV
and energy density $U_{\rm cmb} = $6$\times$10$^{-13}$~erg~cm$^{-3}$)
to the X-ray domain.
In this scenario, the radio-to-IR spectral index is $\alpha = 0.75$
(the synchrotron flux density at frequency $\nu$ is
$f_{\nu} \propto \nu^{-\alpha}$), which corresponds to $p = 2.5$
in the non-thermal electron energy distribution when
the emission is produced in the same volume.
\section{The hotspot as a magnetic field damping region}
In this work we consider the same emission mechanisms as in
\cite{Erlund_TwoShocks}, but allow the synchrotron and Inverse
Compton (IC) emission to be
produced in regions with different spatial extents.
In particular; electrons accelerated at the shock emit
synchrotron radiation from radio to optical in a compact region
behind the shock, whereas the
IC X-ray emission is located in an extended region.
In Fig.~\ref{sketch} (lower) we sketch our model.
The synchrotron (s) and IC cooling length of electrons with Lorentz factor
$\gamma$ is $l_{\rm s,ic}(\gamma) = t_{\rm s,ic}(\gamma) v_{\rm sh}/r$, where
$t_{\rm s,ic}(\gamma)$ is the cooling timescale.
The shock velocity is approximately the same as the jet velocity which we
take characteristically to be $v_{\rm sh} = 10^{10}$~cm~s$^{-1}$
($\sim c/3$ and Lorentz factor $\Gamma_{\rm sh} \sim 1.06$)
in line with observations
of similar objects \cite[see][and references therein]{Steenbrugge_08}.
We use $r = 7$ as the shock compression ratio
for a non-relativistic shock whose downstream thermal pressure is
dominated by relativistic electrons, although $r \sim 4$ may
still apply if non-relativistic ions dominate the pressure downstream of the
shock \citep{Kirk_00}. Our conclusions
are not sensitive to the exact value of $r$.
\subsection{Inverse Compton X-ray emission}
The IC X-ray emission is produced by electrons with
$\gamma_{\rm x} \sim 10^3$ and
$l_{\rm ic}(\gamma_{\rm x})\sim10^{4} (v_{\rm sh}/10^{10}\,{\rm cm\,s^{-1}})$~arcsec,
which is much larger than $l_{\rm x}$.
The synchrotron cooling length $l_{\rm s}(\gamma_{\rm x})$
is also greater than $l_{\rm x}$, unless the magnetic
field in the X-ray emitting region is $\sim 360$~$\mu$G.
However, such a large magnetic field would produce synchrotron radio
radio emission much brighter than $L_{\rm r}$ (see next section).
Furthermore, we show below that the amplified magnetic field, of the order
of $\sim$100~$\mu$G, is confined to a small volume close to the shock.
Therefore, adiabatic expansion is probably the dominant cooling mechanism
as the particles flow out of the hotspot.
Unless $\Gamma_{\rm sh} \gtrsim 10$, X-ray emitting electrons are
non-thermal and follow a power law energy distribution
$\propto \gamma^{-p}$ \cite[e.g.][]{Mixed_Ne}.
Assuming that
the X-ray emitting volume is $V_{\rm x} \sim 300$~arcsec$^{3}$,
the energy density of these non-thermal electrons is
$\sim$10$^{-9} (\gamma_{\rm min}/50)^{-0.5}$~erg~cm$^{-3}$
where the power law terminates at a minimum Lorentz factor $\gamma_{\rm min}$.
The magnetic field with the same energy density is
$\sim 100(\gamma_{\rm min}/50)^{-0.25}$~$\mu$G.
These results correspond to the case where $p = 2.5$
(see however Sect.~\ref{ir}). In the following sections we take
$100$~$\mu$G as a fiducial magnetic field since it represents equipartition
between magnetic and relativistic electron energy densities and is typical
of other hotspots \cite[e.g.][]{B_equip}.
\subsection{Synchrotron emission}
Considering that $\gamma(\nu) \sim 4.5\times10^{-4}(\nu/B)^{0.5}$
is the Lorentz factor of electrons emitting synchrotron radiation at
$\nu$ in a magnetic field $B$, $l_{\rm s}$ can be written as
\begin{equation}
\frac{l_{\rm s}(\nu)}{[\prime\prime]} \sim 12
\left(\frac{\nu}{\rm GHz}\right)^{-0.5}
\left(\frac{B}{100\,{\rm \mu G}}\right)^{-1.5}
\left(\frac{v_{\rm sh}}{10^{10}\,{\rm cm\,s^{-1}}}\right).
\label{l_cool}
\end{equation}
\subsubsection{Radio}
\label{radio}
MERLIN emitting electrons have
$\gamma_{\rm r} \equiv \gamma(\nu_{\rm r})\sim
2\gamma_{\rm x}(B/100\,\mu{\rm G})^{-0.5}$.
If both radio and X-ray emission are produced by non-thermal
electrons that follow the same power-law energy distribution,
$L_{\rm x}/L_{\rm r} \sim (\gamma_{\rm x}/\gamma_{\rm r})^{3-p}(U_{\rm cmb}/U_{\rm mag})
V_{\rm x}/V_{\rm r}$ and
$B \sim 100$~$\mu$G corresponds to $V_{\rm x}/V_{\rm r} \sim 5\times10^3$,
where $U_{\rm mag} = B^2/8\pi$ and $V_{\rm r}$ is the volume of the
synch
|
rotron emitter\footnote{Note
that if $V_{\rm x} = V_{\rm r}$, an unrealistically small magnetic field of
$0.3$~$\mu$G would be needed to explain the observed fluxes.}.
Such a large ratio between emitting volumes is not implausible provided the
magnetic field is
inhomogeneous in the shock downstream region and the synchrotron
emitter consists of features smaller than the MERLIN point spread function
(FWHM $0.15^{\prime\prime}$) as seen in parts of the MERLIN data.
The synchrotron cooling length of MERLIN emitting electrons is
$l_{\rm s}(\nu_{\rm r}) \sim 9.3^{\prime\prime}(B/100\,{\rm \mu G})^{-3/2}
(v_{\rm sh}/10^{10}\,{\rm cm\,s^{-1}})$,
and a very large magnetic field of
$\sim 2.4 \,(v_{\rm sh}/10^{10}\,{\rm cm\,s^{-1}})^{2/3}$~mG
would be required to match
$l_{\rm s}(\nu_{\rm r}) = 0.1^{\prime\prime} \sim l_{\rm r}$.
This result suggests that the downstream extent of the compact emission
detected at $\nu_{\rm r}$ is not the result of fast synchrotron cooling,
as we can confirm when we take into account the IR emission.
\subsubsection{Infrared}
\label{ir}
The synchrotron cooling length of IR emitting electrons is
$l_{\rm s}(\nu_{\rm ir}) \sim 0.03^{\prime\prime}(B/100\,{\rm \mu G})^{-3/2}
(v_{\rm sh}/10^{10}\,{\rm cm\,s^{-1}})$, indicating that these particles
radiate most of their energy within $l_{\rm r}$.
(This angular distance is not resolved by the IR observations.)
This is consistent with a radio-to-IR electron energy spectral index of
$p \sim 2.5$ with the cooling break in the spectrum occuring
close to IR wavelengths\footnote{Note that the relationship
$p = (r + 2)/(r -1)$ breaks down for
mildly relativistic shocks \citep{Kirk_00, Bell_11} or when non-linear
feedback is important \cite[e.g.][]{Amato_05}.}.
Note that if the emitting volume were determined by synchrotron
cooling, $l_{\rm s}(\nu) \propto \nu^{-0.5}$ giving $p = 2\alpha = 1.5$
since $\alpha$ is measured to be $0.75$.
This very hard spectrum is unlikely since it diverges toward high energy
and would be remarkable in hotspots,
supporting the conclusion that the downstream radio extent $l_{\rm r}$
must be determined by factors other than synchrotron cooling.
As we discuss in Sect.~\ref{mfa} this may be the result of the damping of
the magnetic field \cite[see][for a review]{Klara_review}.
\subsubsection{Optical}
Optical emission produced by synchrotron radiation of
electrons with $\gamma(\nu_{\rm opt}) \sim \gamma(\nu_{\rm ir})$
is almost co-spatial with the IR emission, and this explains the
linear structure cupped within $l_{\rm r}$\footnote{The faint diffuse
IR and optical emission may be the result of
CMB photons up-scattered by electrons with $\gamma \sim 50$.}.
The synchrotron turnover $\nu_{\rm c}$ between $\nu_{\rm ir}$ and $\nu_{\rm opt}$
indicates that the maximum energy of non-thermal electrons is
$E_{e,\rm max} \sim \gamma(\nu_{\rm c}) m_ec^2 \sim
0.3 (\nu_{\rm c}/\nu_{\rm ir})^{0.5} (B/100\mu{\rm G})^{-0.5}$~TeV.
\section{Magnetic field amplification}
\label{mfa}
The amplification of the magnetic field at strong shocks in supernova
remnants was demonstrated by \cite{Vink-Laming} and \cite{Berezhko-03},
deriving the magnetic field from $l_{\rm s}$ downstream of the shock.
A theoretical explanation was provided by \cite{Bell_04}
showing that non-resonant hybrid instabilities are capable of enhancing
the magnetic field by orders of magnitude.
Magnetic field amplification is also responsible for
$B \sim 100$~$\mu$G in the southern arc in 4C74.26 since
it is much larger than the expected value in the jet upstream of the
termination shock \cite[e.g.][]{B_jet}.
Bohm diffusion (electron mean free path $\lambda \sim r_{\rm g}$) in a
$\sim$100~$\mu$G magnetic field would be expected to
accelerate electrons with synchrotron X-ray emitting energies as seen in
supernova remnants \citep{Stage_06}.
However, $\nu_{\rm c} \sim \nu_{\rm ir,opt}$
determined by a competition between shock acceleration and synchrotron cooling
indicates that
acceleration is slow and therefore that the electron diffusion coefficient $D$
is much larger than the Bohm value $D_{\rm Bohm}$:
\begin{equation}
\frac{D}{D_{\rm Bohm}} \sim 10^6 \left(\frac{v_{\rm sh}}{10^{10}\,{\rm cm\,s^{-1}}}
\right)^2
\left(\frac{\nu_{\rm ir}}{\nu_{\rm c}}\right),
\end{equation}
independent of $B$ \cite[see e.g.][]{Casse}.
Such a large diffusion coefficient in an amplified magnetic field
is expected if it is structured on a scale $s$ much smaller than the
Larmor radius of the electrons being accelerated.
Small angle scattering by magnetic field randomly orientated in cells
of size $s$ produces $D\sim (r_{\rm g}/s) D_{\rm Bohm}$
and then
\begin{equation}
\frac{s}{\rm cm} \sim 10^{7} \left(\frac{\nu_{\rm c}}{\nu_{\rm ir}}\right)^{1.5}
\left(\frac{B}{100\,{\rm \mu G}}\right)^{-1.5}
\left(\frac{v_{\rm sh}}{10^{10}\,{\rm cm\,s^{-1}}}\right)^{-2}.
\label{lambda}
\end{equation}
In comparison the ion skin-depth is
$c/\omega_{\rm pi} \sim 2.3 \times10^9 \,(n/10^{-4}\,{\rm cm^{-3}})^{-0.5}$~cm,
where $n$ is the particle density downstream of the shock (assumed to be
$7$ times the jet density), and
\begin{eqnarray}
\frac{s}{c/\omega_{\rm pi}} \sim & 0.01
\left(\frac{\nu_{\rm c}}{\nu_{\rm ir}}\right)^{1.5}
\left(\frac{v_{\rm sh}}{10^{10}\,{\rm cm\,s^{-1}}}\right)^{-2} \nonumber\\
{}& \left(\frac{B}{100\,{\rm \mu G}}\right)^{-1.5}
\left(\frac{n}{10^{-4}\,{\rm cm^{-3}}}\right)^{0.5}.
\end{eqnarray}
Given the uncertainties in the parameter values, the approximate nature of
the theoretical models, and the wide range of the spatial scales ($s$,
$r_{\rm g}$, $l_{\rm r}$), it is not significant or surprising that our estimate
of $s/(c/\omega_{\rm pi})$ differs from unity by a factor of $\sim 0.01$.
The order of magnitude similarity of $s$
and $c/\omega_{\rm pi}$ supports the contention that
shock-generated small-scale turbulence scatters non-thermal electrons during
diffusive shock acceleration. This is consistent with \cite{Sironi_11}
who discuss the various processes related to the Weibel instability
that excite turbulence on the characteristic scale of $c/\omega_{\rm pi}$.
Simulations show that magnetic field generated by the Weibel instability
decays downstream of the shock because of its relatively small scalelength
\citep{Sironi_11,Bret_13, Sironi_13}.
This would account for the cut-off of synchrotron emission in 4C74.26
far short of the synchrotron cooling distance of radio-emitting electrons
($l_{\rm r} \ll l_{\rm s}(\nu_{\rm r})$).
These electrons continue up-scattering CMB photons, thus producing IC
X-ray emission downstream of the shock after the MERLIN
radio emission has ceased.
\subsection{Limit on ions maximum energy}
Since the response of highly relativistic ions is similar to that
of electrons with the same energy in a tangled amplified magnetic field,
we can expect protons to have a similar ratio of $D/D_{\rm Bohm}$.
Protons can be accelerated to higher energies than electrons because their
radiative losses are minimal,
but the maximum energy to which they are accelerated is limited
because their acceleration time is increased by the ratio $D/D_{\rm Bohm}$.
The Hillas parameter $v_{\rm sh}\,B\,R$ \citep{Hillas}, where
$R \sim 2^{\prime\prime}$ ($\sim 2.8$~kpc) is the
characteristic length of the source, would suggest proton acceleration to
$\sim$100~EeV in the termination shock of 4C74.26,
but the maximum energy is reduced to only $\sim$100~TeV
if $D\sim 10^6\,D_{\rm Bohm}$ since the Hillas parameter assumes
$D\sim D_{\rm Bohm}$ and is otherwise reduced by the factor $(D/D_{\rm Bohm})^{-1}$.
Another perspective on the same effect is that the mean free path for
scattering by small-scale turbulence $\lambda \sim r_{\rm g}^2/s$ is larger
than the size of the system if $s \sim c/\omega_{\rm pi}$ and $r_{\rm g}$
is the Larmor radius of an EeV proton.
This result suggests that the mildly relativistic termination shock in
4C74.26 is a poor accelerator of UHECR.
\section{Conclusions}
We model the radio to X-ray emission in the
southern hotspot of the FR~II galaxy 4C74.26.
Our study is based on three key observational features:
1) Compact MERLIN emission region: it is too thin to be the result of
fast synchrotron cooling (Sect.~\ref{radio}).
2) The radio to IR spectrum ($\alpha = 0.75$) is too flat for the emitting
volume to be determined
by synchrotron cooling through this wavelength range (Sect.~\ref{ir}).
3) The turnover of the synchrotron spectrum at IR/optical frequencies
requires $\lambda \gg r_{\rm g}$ for any reasonable shock velocity
(Sect.~\ref{mfa}).
These three features fit well in a scenario in which the MERLIN radio
emission traces out the region where the magnetic field is amplified
by plasma instabilities with small length scale (e.g. Weibel).
The magnetic field
in equipartition with non-thermal electrons in the MERLIN emission region
is $\sim$100~$\mu$G and similar to the values obtained by other authors
\cite[e.g.][]{B_equip}. An unrealistically large magnetic field
$\sim 2.4\,(v_{\rm sh}/10^{10}\,{\rm cm\,s^{-1}})^{2/3}$~mG would be needed to
explain the compact radio emission in terms of synchrotron cooling.
If $B \sim 100$~$\mu$G in the synchrotron emission region,
the maximum energy of non-thermal electrons is $\sim$0.3~TeV.
If ions are accelerated
as well, protons with energy $\sim$0.3~TeV diffuse also with mean free path
$\lambda \gg r_{\rm g}$. If $\lambda$ is similarly larger than the Larmor
radius at higher proton energies,
then the maximum proton energy at the termination shock of 4C74.26
is only $100$~TeV instead of the $100$~EeV indicated by the Hillas parameter.
This may have important implications for the understanding of the origins
of UHECR.
\acknowledgments
We thank the referees for the constructive reports.
The research leading to this article has received funding
from the European Research Council under the European
Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement
no. 247039. We acknowledge support from the UK Science and Technology
Facilities Council under grant number ST/K00106X/1.
|
\section{Introduction}
The problem of filter sensitivities evaluation plays a key role in many areas of research; for instance, in state estimation and parameter identification realm~\cite{mehra1972,Bastin1988}, in the field of optimal input design~\cite{Mehra1974,Gupta1974}, in information theory for computing the Fisher information matrix~\cite{zadrozny1994,klein1995,klein2000} {\it etc}. In this paper we explore linear discrete-time stochastic systems where the associated Kalman filter (KF) is used for estimating the unknown dynamic states. Therefore, the standard approach for computing the filter sensitivities (with respect to unknown system parameters) is a direct differentiation of the KF equations. This conventional methodology is comprehensively studied in~\cite{Mehra1974,zadrozny1989,klein2000direct}. The shortcoming of this strategy is a numerical instability of the conventional KF (and its derivatives) with respect to roundoff errors discussed in~\cite{VerhaegenDooren1986,Verhaegen1989}. Due to this fact, special attention has been paid in the KF community for designing robust KF implementation methods. The most popular techniques belong to the class of square-root (SR) or UD factorization-based methods; see~\cite{KaminskiBryson1971,Bierman1977,Sayed1994,ParkKailath1995} and many others. These algorithms imply the Cholesky decomposition and its modification for the corresponding covariance matrix factorization~\cite{Bierman1977,KailathSayed2000,GrewalAndrews2001}. We may note that the Cholesky decomposition exists and is unique when the symmetric matrix to be decomposed is positive definite~\cite{Golub1983}. If it is a positive semi-definite, then the Cholesky decomposition still exists, however, it is not unique~\cite{Higham1990}. Further encouraging KF implementation methods might be found with the use of singular value decomposition (SVD). Some evidences of better estimation quality obtained under the SVD-based approach exist in the field of nonlinear filtering; for instance, see discussion in~\cite{Zhang2008,straka2013,Zhang2014} and others. For linear filtering problem examined in this paper, the first SVD-based KF was, to the best of our knowledge, designed in~\cite{WangSVD1992}. Our recent analysis exposes that the mentioned SVD-based filter can be further improved for enhancing its numerical robustness. This result is comprehensively studied in~\cite{KulikovaOnline}, where some new stable SVD-based KF implementations are designed.
Despite the existence of inherently more stable SR-, UD- and SVD-based KF variants, the problem of robust filter derivative computation is seldom addressed in practice because of its complicated matter. The solution to the mentioned problem heavily relies on the use of matrix differential calculus. The first SR-based {\it information-type} algorithm for the KF derivative computations belongs to Bierman {\it et al.} and was appeared in 1990; see~\cite{Bierman1990}. Alternatively, the SR-based {\it covariance-type} method was proposed in~\cite{Kulikova2009IEEE} as well as the UD-based scheme designed in~\cite{Tsyganova2013IEEE}. Later on, a general ``differentiated'' SR-based methodology was designed for both orthogonal and J-orthogonal transformations involved in the filtering equations (and their derivatives) in~\cite{Kulikova2013IEEE,Kulikova2015,Kulikova2016}. Alternatively, in this technical note we develop the SVD-based approach for the KF derivative computation. We show that the new technique is algebraically equivalent to the conventional ``differentiated'' KF, but it improves the robustness against roundoff errors as well as the existing ``differentiated'' SR- and UD-based methodologies. However, motivated by the results obtained in nonlinear filtering realm, we expect that the newly-designed SVD-based method outperforms the previously derived algorithms while solving the parameters estimation problem, especially when the error covariance is ill-conditioned.
\section{Filter sensitivity equations: conventional approach} \label{sec:state}
Consider the state-space equations
\begin{align}
x_{k} & = F(\theta) x_{k-1}\!+B(\theta)u_{k-1}\!+G(\theta) w_{k-1}, \quad k \ge 1, \label{eq:st:1} \\
z_k & = H(\theta) x_k+v_k, \; v_k \sim {\mathcal N}\left(0,R(\theta)\right), \; w_k \sim {\mathcal N}\left(0,\Omega(\theta)\right) \label{eq:st:2}
\end{align}
where $z_k \in \mathbb R^m$, $u_k \in \mathbb R^d$, $x_k \in \mathbb R^n$ and $\theta \in \mathbb R^p$ are,
respectively, the vectors of available measurements, the known deterministic control input, the unknown dynamic state and the unknown system parameters that need to be estimated from the available experimental data, $\{z_1, \ldots, z_N \}$. The process and the measurement noises are independent Gaussian zero-mean white-noise processes that are also independent from the initial state $x_0 \sim {\mathcal N}\left(\bar x_0,\Pi_0(\theta)\right)$. The covariances are assumed to be $\Omega(\theta) \ge 0$, $R(\theta) > 0$ and $\Pi_0(\theta) \ge 0$.
Equations~\eqref{eq:st:1}, \eqref{eq:st:2} represent a set of the state-space models (SSMs). Each of them corresponds to a particular system parameter value. This means that for any fixed value of $\theta$, say $\hat \theta^*$, the system matrices are known, i.e. there is no uncertainty in model~\eqref{eq:st:1}, \eqref{eq:st:2}. For simplicity, throughout the paper we write $F$ {\it etc.} instead of $F(\hat \theta^*)$ {\it etc.} when evaluating at the fixed point $\hat \theta^*$. The associated KF yields the linear minimum least-square estimate of the unknown dynamic state that can be recursively computed via the equations~\cite[Theorem~9.2.1]{KailathSayed2000}:
\begin{align}
e_k & = z_k-H\hat x_{k|k-1}, \qquad \hat x_{1|0} = \bar x_0, \quad k \ge 1, \label{kf:1} \\
K_{p,k} & = FP_{k|k-1}H^TR_{e,k}^{-1}, \quad R_{e,k} = R+HP_{k|k-1}H^T, \label{kf:2} \\
\hat x_{k+1|k} & = F \hat x_{k|k-1}+Bu_k+K_{p,k}e_k \label{kf:3}
\end{align}
where $\{ e_k \}$ are innovations of the discrete-time KF. The important property of the KF for Gaussian SSMs is $e_k \sim {\cal N}\left(0,R_{e,k}\right)$. The $P_{k|k-1}=\E{ (x_{k}-\hat x_{k|k-1})(x_{k}-\hat x_{k|k-1})^{T}}$ is the one-step ahead predicted error covariance matrix computed as follows:
\begin{equation}
P_{k+1|k} = FP_{k|k-1}F^T\!+G\Omega G^T \! - K_{p,k}R_{e,k}K_{p,k}^T, \; P_{1|0} \! = \Pi_0. \label{kf:4}
\end{equation}
The conventional approach for deriving the related sensitivity model is based on differentiation of the corresponding filtering equations. Let $A(\theta) \in \mathbb R^{m\times n}$, $B(\theta) \in \mathbb R^{n\times q}$ be matrices, which entries are differentiable functions of the parameter vector $\theta \in \mathbb R^{p}$. The $m \times n$ matrix
$\Di{A} = \partial A/ \partial \theta_i$ implies the partial derivative of the $A$ with respect to the $i$-th component of $\theta$, $i=1, \ldots p$. The $m \times n$ matrix $\D{A} = \sum_{i=1}^{p} \DDi{A} \cdot \DD{\theta_i}$ is the differential form of first-order derivatives of $A(\theta)$. Taking into account the matrix product rule of differentiation~\cite[p. 955]{neudecker1969}: $\D{\left(AB\right)} = \DD{A}B + A\DD{B}$, and the fact $\D I = 0$, we derive $\D{\left(A^{-1}\right)} = -A^{-1} \DD{A} A^{-1}$ for any square and invertible matrix $A$ (it is also known as the Jacobi's formula);
see also~\cite[p. 546]{zadrozny1989}. Using these differentiation rules, the necessary differentials of~\eqref{kf:1}-\eqref{kf:4}
can be written as follows~\cite{zadrozny1989,klein2000direct}:
\begin{align}
\D{e_k} & = -\left[ \DD{H}\hat x_{k|k-1} + H\DD{\hat x_{k|k-1}}\right], \label{diff:kf:1} \\
\D{\hat x_{k+1|k}} & = \DD{F} \hat x_{k|k-1}+ F \DD{\hat x_{k|k-1}} + \DD{B} u_{k} \nonumber \\
& + \DD{K_{p,k}}e_k + K_{p,k}\DD{e_k}, \label{diff:kf:2} \\
\D{K_{p,k}} & = \DD{F}P_{k|k-1}H^TR_{e,k}^{-1}+F\DD{P_{k|k-1}}H^TR_{e,k}^{-1} \nonumber \\
& + FP_{k|k-1}\DD{H^T}R_{e,k}^{-1} \nonumber \\
& - FP_{k|k-1}H^TR_{e,k}^{-1}\DD{R_{e,k}}R_{e,k}^{-1}, \label{diff:kf:3a} \\
\D{R_{e,k}} & = \D{R}+\DD{H}P_{k|k-1}H^T+H\DD{P_{k|k-1}}H^T \nonumber \\
& +HP_{k|k-1}\DD{H^T}, \label{diff:kf:3b} \\
\D{P_{k+1|k}} & = \DD{F}P_{k|k-1}F^T + F\DD{P_{k|k-1}}F^T \nonumber \\
& + FP_{k|k-1}\DD{F^T} +\DD{G}\Omega G^T + G\DD{\Omega}G^T \nonumber \\
& + G\Omega \DD{G^T} - \DD{K_{p,k}}R_{e,k}K_{p,k}^T \nonumber \\
& - K_{p,k}\DD{R_{e,k}}K_{p,k}^T - K_{p,k}R_{e,k}\DD{K_{p,k}^T}. \label{diff:kf:4}
\end{align}
In deriving the equations above we take into account that $\D{z_k} = 0$ and $\D{u_k} = 0$, because the observations $z_k$ and the control input $u_k$ do not depend on the parameters (i.e. their realizations are independent of variations in $\theta$) and therefore have a differential equal to zero.
We may also note that except for the scalar factor $\D{\theta_i}$, $\Di{A}$ is a special case of $\D{A}$, so that to obtain partial-derivative forms from differential forms, we only have to everywhere replace operator $\D{\left(\cdot\right)}$ with $\Di{\left(\cdot\right)}$ for
$i=1, \ldots p$~\cite[p. 546]{zadrozny1989}. Hence, from~\eqref{diff:kf:1}~-- \eqref{diff:kf:4} we obtain a set of $p$ vector equations, known as the {\it filter sensitivity equations}, for computing $\Di{\hat x_{k+1|k}}$, $i=1, \ldots p$, and a set of $p$ matrix equations, known as the {\it Riccati-type sensitivity equations}, for computing $\Di{P_{k+1|k}}$, $i=1, \ldots p$. This approach for the KF sensitivity model derivation is called the ``differentiated KF''. Its main drawback is a numerical instability of the conventional KF~\eqref{kf:1}~-- \eqref{kf:4} and inherently its derivative~\eqref{diff:kf:1}~-- \eqref{diff:kf:4} with respect to roundoff errors.
The goal of this paper is to design a robust methodology for updating the ``differentiated'' KF equations above in terms of SVD factors (and their derivatives) of the error covariance matrices $P_{k|k-1}$ instead of using the full matrices $P_{k|k-1}$ (and their derivatives).
\section{SVD factorization-based Kalman filtering \label{filters}}
To the best of our knowledge, the first SVD-based KF was by Wang {\it et al.} and appeared in 1992; see Eqs~(17), (22), (23) in~\cite[pp.~1225-1226]{WangSVD1992}. Our recent research shows that although that implementation is inherently more stable than the KF~\eqref{kf:1}~-- \eqref{kf:4}, it is still sensitive to roundoff and poorly treats ill-conditioned problems. The cited analysis exposes that the SVD-based filter can be further improved for enhancing its numerical robustness. This result is comprehensively studied in~\cite{KulikovaOnline}, where new stable SVD-based KF implementations are designed. The readers are referred to the cited paper for the detailed derivations, numerical stability discussion and proofs. Here, we briefly outline the principle steps for construction of the most advanced SVD-based KF variant. Next, we extend it to a stable filter sensitivities computation, which is the main purpose of this study.
Consider the SVD factorization~\cite[Theorem~2.8.1]{Tyrtyshnikov2012}: suppose $A \in {\mathbb C}^{m\times n}$, ${\rm rank}\: A = r$. There exist positive numbers $\sigma_1\geq\ldots\sigma_r>0$ and unitary matrices $W \in {\mathbb C}^{m\times m}$ and $V \in {\mathbb C}^{n\times n}$ such that
\[
A = W\Sigma V^*, \,
\Sigma =
\begin{bmatrix}
S & 0 \\
0 & 0
\end{bmatrix} \in {\mathbb C}^{m\times n}, \; S={\rm diag}\{ \sigma_1,\ldots,\sigma_r\}
\]
where $V^*$ is the conjugate transpose of $V$.
The diagonal entries of $\Sigma$ are known as the singular values of $A$. The non-zero $\sigma_i$ ($i=1, \ldots, r$) are the square roots of the non-zero eigenvalues of both $A^*A$ and $AA^*$.
If $A$ is a square matrix such that $A^*A=AA^*$, then the $A$ can be diagonalized using a basis of eigenvectors according to the spectral theorem, i.e. it can be factorized as follows: $A = QDQ^*$ where $Q$ is a unitary matrix and $D$ is a diagonal matrix, respectively. If $A$ is also positive semi-definite, then the spectral decomposition above, $A = QDQ^*$, is also a SVD factorization, i.e. the diagonal matrix $D$ contains the singular values of $A$. For the SSMs examined in this paper, the initial error covariance $\Pi_0 \in {\mathbb R}^n$ is a symmetric positive semi-definite matrix and, hence, the spectral decomposition implies $\Pi_0 = Q_{\Pi_0}D_{\Pi_0}Q_{\Pi_0}^T$ where $Q_{\Pi_0}$ and $D_{\Pi_0}$ are the orthogonal and diagonal matrices, respectively. It is also a SVD factorization, i.e. the factor $D_{\Pi_0}$ contains the singular values of $\Pi_0$.
Now, we are ready to present the SVD-based KF implementation developed recently in~\cite{KulikovaOnline}. Instead of conventional recursion~\eqref{kf:1}-\eqref{kf:4} for $P_{k|k-1}$, we update only their SVD factors, $\{Q_{P_{k|k-1}}, D^{1/2}_{P_{k|k-1}}\}$, at each iteration step of the filter as shown below.
\textsc{Initial Step} ($k=0$). Apply the SVD factorization for the initial error covariance matrix $\Pi_0 = Q_{\Pi_0}D_{\Pi_0}Q_{\Pi_0}^T$ and, additionally, for the process and measurement noise covariances: $\Omega = Q_{\Omega}D_{\Omega}Q_{\Omega}^T$ and $R = Q_{R}D_{R}Q_{R}^T$ , respectively. Set the initial values as follows: $Q_{P_{1|0}} = Q_{\Pi_0}$, $D^{1/2}_{P_{1|0}} = D^{1/2}_{\Pi_0}$ and $\hat x_{1|0} = \bar x_0$.
\textsc{Measurement Update} ($k=1, \ldots, N$). Build the pre-arrays from the filter quantities that are currently available and, then, apply the SVD factorizations in order to obtain the corresponding SVD factors of the updated filter quantities as follows:
\begin{align}
\underbrace{
\begin{bmatrix}
D^{1/2}_{R} Q^T_{R} \\
D^{1/2}_{P_{k|k-1}}Q^T_{P_{k|k-1}}H^T
\end{bmatrix}
}_{\rm Pre-array}
=
\underbrace{
\mathfrak{W}_{MU}^{(1)}
\begin{bmatrix}
D_{R_{e,k}}^{1/2} \label{svd:1} \\
0
\end{bmatrix}
Q_{R_{e,k}}^T
}_{\rm Post-array \: SVD \:factors}, \\
\bar K_{k} = \left(Q_{P_{k|k-1}}D_{P_{k|k-1}}Q^T_{P_{k|k-1}}\right)H^TQ_{R_{e,k}}, \label{svd:K} \\
\underbrace{
\begin{bmatrix}
D_{P_{k|k-1}}^{1/2}Q_{P_{k|k-1}}^T\left(I - K_k H\right)^T \\
D^{1/2}_{R} Q^T_{R} K_{k}^T
\end{bmatrix}
}_{\rm Pre-array}
=
\underbrace{
\mathfrak{W}_{MU}^{(2)}
\begin{bmatrix}
D_{P_{k|k}}^{1/2} \\
0
\end{bmatrix}
Q_{P_{k|k}}^T
}_{\rm Post-array \: SVD \:factors} \label{svd:2}
\end{align}
where we denote $K_k = \bar K_k D^{-1}_{R_{e,k}}Q^T_{R_{e,k}}$. The matrices $\mathfrak{W}_{MU}^{(1)} \in {\mathbb R}^{(m+n)\times (m+n)}$, $Q_{R_{e,k}} \in {\mathbb R}^{m\times m}$ and $\mathfrak{W}_{MU}^{(2)} \in {\mathbb R}^{(n+m)\times (n+m)}$, $Q_{P_{k|k}} \in {\mathbb R}^{n\times n}$ are the orthogonal matrices of the corresponding SVD factorizations in~\eqref{svd:1}, \eqref{svd:2}. Next, $D_{R_{e,k}}^{1/2} \in {\mathbb R}^{m\times m}$ and $D_{P_{k|k}}^{1/2} \in {\mathbb R}^{n\times n}$ are diagonal matrices with square roots of the singular values of $R_{e,k}$ and $P_{k|k}$, respectively.
It can be easily seen that the required SVD factors of the innovation covariance $R_{e,k}$, i.e. $\{Q_{R_{e,k}}, D_{R_{e,k}}^{1/2}\}$, and {\it a posteriori} error covariance matrix $P_{k|k}$, i.e. $\{Q_{P_{k|k}}, D_{P_{k|k}}^{1/2}\}$, are directly read-off from the post-array factors in~\eqref{svd:1} and \eqref{svd:2}, respectively. Finally, find {\it a posteriori} estimate $\hat x_{k|k}$ through equations
\begin{equation}
\hat x_{k|k} = \hat x_{k|k-1}\!+\bar K_{k}D^{-1}_{R_{e,k}}\bar e_k, \; \; \bar e_k = Q_{R_{e,k}}^T\!\!\left(z_k-H\hat x_{k|k-1}\right)\!. \label{svd:3}
\end{equation}
\textsc{Time Update} ($k=1, \ldots, N$).
Build the pre-array and apply the SVD factorization to obtain {\it a priori} error covariance SVD factors $\{Q_{P_{k+1|k}}, D_{P_{k+1|k}}^{1/2}\}$ as follows:
\begin{align}
\underbrace{
\begin{bmatrix}
D^{1/2}_{P_{k|k}}Q^T_{P_{k|k}}F^T\\
D^{1/2}_{\Omega} Q^T_{\Omega} G^T
\end{bmatrix}
}_{\rm Pre-array}
=
\underbrace{
\mathfrak{W}_{TU}
\begin{bmatrix}
D_{P_{k+1|k}}^{1/2} \\
0
\end{bmatrix}
Q_{P_{k+1|k}}^T
}_{\rm Post-array \: SVD \:factors} \label{svd:5}
\end{align}
and find {\it a priori} estimate $\hat x_{k+1|k}$ as follows:
\begin{align}
\hat x_{k+1|k} & = F\hat x_{k|k} + Bu_{k}. \label{svd:6}
\end{align}
The SVD-based KF implementation above is formulated in two-stage form. Meanwhile, following~\cite{ParkKailath1995}, the conventional KF~\eqref{kf:1}~--\eqref{kf:4} is expressed in the so-called ``condensed'' form. Nevertheless, these KF variants are algebraically equivalent. It is easy to prove if we take into account the SVD factorization $A=\mathfrak{W} \Sigma \mathfrak{V}^T$ and the properties of orthogonal matrices. Indeed, for each pre-array to be decomposed we have $A^TA = (\mathfrak{V} \Sigma \mathfrak{W}^T)(\mathfrak{W}\Sigma \mathfrak{V}^T) = \mathfrak{V} \Sigma^2 \mathfrak{V}^T$. Next, by comparing both sides of the obtained matrix equations, we come to the corresponding SVD-based KF formulas. The detailed derivation can be found in~\cite{KulikovaOnline}.
\section{Filter sensitivity equations: SVD-based approach} \label{Sec:SVD:diff}
To begin constructing the ``differentiated'' SVD-based method for computing the filter sensitivities, we pay attention to the underlying SVD-based filter and remark that it is formulated in the so-called array form. This makes the modern KF algorithms better suited to parallel implementation and to very large scale integration (VLSI) implementation as mentioned in~\cite{ParkKailath1995}. Each iteration of the SVD-based filter examined has the following pattern: given a pre-array $A \in {\mathbb R}^{(k+s)\times s}$, compute the post-array SVD factors $\mathfrak{W} \in {\mathbb R}^{(k+s)\times (k+s)}$, $\Sigma \in {\mathbb R}^{(k+s)\times s}$ and $\mathfrak{V} \in {\mathbb R}^{s\times s}$ by means of the SVD factorization
\begin{align} \label{numSVD}
A & = \mathfrak{W}\:\Sigma \:\mathfrak{V}^T, \quad
\Sigma =
\begin{bmatrix}
S \\
0
\end{bmatrix}, \quad S={\rm diag}\{ \sigma_1,\ldots,\sigma_s\}
\end{align}
where the matrix $A$ is of full column rank, i.e. ${\rm rank} \:A = s$; the $\mathfrak{W}$, $\mathfrak{V}$ are orthogonal matrices and $S$ is a diagonal matrix with singular values of the pre-array $A$.
The goal of our study is to develop the method that naturally extends formula~\eqref{numSVD} on the post-array SVD factors' derivative computation. More precisely, the computational procedure is expected to utilize the pre-array $A$ and its derivative $A'_{\theta}$ for reproducing the SVD post-arrays $\{\mathfrak{W}, \Sigma, \mathfrak{V}\}$ together with their derivatives $\{\mathfrak{W}'_{\theta}, \Sigma'_{\theta}, \mathfrak{V}'_{\theta}\}$. To achieve our goal, we prove the result presented below. We also bear in mind that the SVD post-array factor $\mathfrak{W}$ is of no interest in the presented SVD-based KF for performing the next step of the filter recursion and, hence, the quantity $\mathfrak{W}'_{\theta}$ is not required to be computed.
\begin{Lm}
\label{lemma:1} Consider the SVD factorization in~\eqref{numSVD}.
Let entries of the pre-array $A(\theta)$ be known differentiable functions of a scalar parameter $\theta$. We assume that $\sigma_i(\theta) \ne \sigma_j(\theta)$, $j\ne i$, for all $\theta$. Given the derivative of the pre-array, $A'_{\theta}$, the following formulas calculate the corresponding derivatives of the post-arrays:
\begin{align}
\Sigma'_{\theta} & =
\begin{bmatrix}
S'_{\theta} \\
0
\end{bmatrix}, & S'_{\theta} & = {\rm diag} \left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}, \label{lemma:d:1} \\
\mathfrak{V}'_{\theta} & = \mathfrak{V} \left[ \bar L_2^T-\bar L_2 \right] && \label{lemma:d:2}
\end{align}
where $ \left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}$ denotes the main $s \times s$ block of the matrix product $\mathfrak{W}^T A'_{\theta} \mathfrak{V}$, and
$\bar L_2$ is a strictly lower triangular matrix, which entries are computed as follows:
\begin{align}\label{barL2}
(\bar l_2)_{ij} & =\displaystyle\frac{{\bar u}_{ji}\sigma_j+{\bar l}_{ij}\sigma_i}{\sigma_i^2-\sigma_j^2},
& i & = 2,\ldots, s, \; j=1,\ldots, i-1.
\end{align}
In equation above, the quantities ${\bar u}_{ji}$ and ${\bar l}_{ji}$ denote the entries of matrices $\bar L$ and $\bar U$, respectively. The $\bar L$, $\bar U$ are strictly lower and upper triangular parts of the matrix product $\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}$, respectively.
\end{Lm}
\begin{proof}
By differentiating~\eqref{numSVD} with respect to $\theta$, we obtain
\begin{equation}\label{numdiffSVD}
A'_\theta=\mathfrak{W}'_\theta\Sigma \mathfrak{V}^T+\mathfrak{W} {\Sigma}'_\theta \mathfrak{V}^T + \mathfrak{W} \Sigma \: (\mathfrak{V}^T)'_{\theta}.
\end{equation}
Having applied a right-multiplier $\mathfrak{V}$ and a left-multiplier $\mathfrak{W}^T$ to equation~\eqref{numdiffSVD}, we have
\begin{equation}\label{baseSVD1}
\mathfrak{W}^T A'_{\theta} \mathfrak{V} = \left[\mathfrak{W}^T \mathfrak{W}'_{\theta}\right]\Sigma +\Sigma'_{\theta}+\Sigma \left[(\mathfrak{V}^T)'_{\theta}\mathfrak{V} \right].
\end{equation}
In deriving the equation above we take into account the properties of any orthogonal matrix $Q$, i.e. $QQ^T=Q^TQ=I$.
It is also easy to show that for any orthogonal matrix $Q$ the product $Q'_{\theta}Q^T$ is a skew symmetric matrix. Indeed, by differentiating both sides of $QQ^T=I$ with respect to $\theta$, we get $Q'_{\theta}Q^T+Q\left(Q^T\right)'_{\theta}=0$, or in the equivalent form $Q'_{\theta}Q^T=-\left(Q'_{\theta}Q^T\right)^T$. The latter implies that the matrix $Q'_{\theta}Q^T$ is skew symmetric.
For the sake of simplicity we introduce the following notations: $\Upsilon= \mathfrak{W}^T \mathfrak{W}'_{\theta}$ and $\Lambda=\mathfrak{V}^T\mathfrak{V}'_{\theta}$. As discussed above, the matrices $\Upsilon \in {\mathbb R}^{(k+s)\times (k+s)}$ and $\Lambda \in {\mathbb R}^{s\times s}$ are skew symmetric, because $\mathfrak{W}$ and $\mathfrak{V}$ are orthogonal matrices. Hence, we have $\Lambda^T = -\Lambda$. Taking into account this fact, we obtain the following partitioning of the matrix form of equation~\eqref{baseSVD1}:
\[
\begin{bmatrix}
\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}\\
\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{k\times s}
\end{bmatrix}
\! = \!
\begin{bmatrix}
[\Upsilon]_{s\times s} & [\Upsilon]_{s\times k}\\
[\Upsilon]_{k\times s} & [\Upsilon]_{k\times k}
\end{bmatrix}
\begin{bmatrix}
S\\
0
\end{bmatrix}
\!+\!
\begin{bmatrix}
S'_{\theta}\\
0
\end{bmatrix}
\!-\!
\begin{bmatrix}
S\\
0
\end{bmatrix}
\Lambda.
\]
From the equation above, we derive the formula for the main $s\times s$ block of the matrix product $\mathfrak{W}^T A'_{\theta} \mathfrak{V}$
\begin{equation}\label{baseSVD4:new}
\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s} =
[\Upsilon]_{s\times s}S + S'_{\theta} - S\Lambda.
\end{equation}
Hence, the diagonal matrix $S'_{\theta}$ obeys the equation
\begin{equation}\label{baseSVD4}
S'_{\theta} = \left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s} - [\Upsilon]_{s\times s}S + S\Lambda.
\end{equation}
Now, let us discuss formula~\eqref{baseSVD4} in details. Recall the matrices $\Upsilon$ and $\Lambda$ are skew symmetric matrices and, hence, their diagonal entries are equal to zero. The multiplication of any skew symmetric matrix by a diagonal matrix does not change the matrix structure, i.e. the diagonal entries of the matrix products $[\Upsilon]_{s\times s}S$ and $S\Lambda$ are equal to zero as well. Meanwhile, the matrix $\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}$ is a full matrix and contains a diagonal part. Hence, from equation~\eqref{baseSVD4} we conclude that diagonal matrix $S'_{\theta}$ is, in fact, a diagonal part of the main $s\times s$ block of the matrix product $\mathfrak{W}^T A'_{\theta} \mathfrak{V}$. This completes the proof of formulas in equation~\eqref{lemma:d:1}.
Finally, we need to compute $\mathfrak{V}'_{\theta}$ where $\Lambda=\mathfrak{V}^T \mathfrak{V}'_{\theta}$. Since $\mathfrak{V}$ is an orthogonal matrix, we obtain $\mathfrak{V}'_{\theta} = \mathfrak{V} \Lambda$. Next, any skew symmetric matrix can be presented as a difference of a strictly lower triangular matrix and its transpose. Hence, the skew symmetric matrices $\left[\Upsilon\right]_{s\times s}$ and $\Lambda$ can be represented as follows:
\begin{align} \label{L:decomposition}
\left[\Upsilon\right]_{s\times s} & = \bar L_1^T- \bar L_1 & \Lambda & = \bar L_2^T- \bar L_2
\end{align}
where $\bar L_1$ and $\bar L_2$ are strictly lower triangular matrices.
Next, we split the matrix product $\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}$ into strictly lower triangular, diagonal and strictly upper triangular parts, i.e. $\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s} = \bar L + D + \bar U$. It was proved above that $S'_{\theta}=D$. Taking into account this fact, the substitution of both formulas in~\eqref{L:decomposition} into~\eqref{baseSVD4} yields
\begin{equation}\label{baseSVD4:eq1}
\underbrace{D}_{S'_{\theta}}
= \underbrace{\bar L + D + \bar U}_{\left[\mathfrak{W}^T A'_{\theta} \math
|
frak{V}\right]_{s\times s}} - \underbrace{\left[\bar L_1^T- \bar L_1\right]}_{[\Upsilon]_{s\times s}}S + S \underbrace{\left[\bar L_2^T- \bar L_2\right]}_{\Lambda}.
\end{equation}
Hence, we obtain
\begin{equation}\label{baseSVD6}
\bar L+\bar U
=
[{\bar L}_1^T-\bar L_1]S
-
S[{\bar L}_2^T-\bar L_2].
\end{equation}
In~\eqref{baseSVD6}, the $\bar L$, $\bar L_1$, $\bar L_2$ are strictly lower triangular matrices, the $\bar U$ is a strictly upper triangular matrix and $S$ is a diagonal. Hence, equation~\eqref{baseSVD6} implies
\[
\left\{
\begin{array}{lcl}
\bar U & = & {\bar L}_1^T S-S{\bar L}_2^T,\\
\bar L & = & -{\bar L}_1 S+S{\bar L}_2.\\
\end{array}
\right.
\]
It can be solved with respect to entries of $\bar L_2$ as follows:
\[
(\bar l_2)_{ij}=\displaystyle\frac{{\bar u}_{ji}s_{j}+{\bar l}_{ij}s_i}{s_i^2-s_j^2}, \; i=2, \ldots, s, \; j=1,\ldots,i-1.
\]
The formula above is exactly equation~\eqref{barL2}. Having computed the entries $(\bar l_2)_{ij}$ we can form the matrix $\Lambda = \bar L_2^T- \bar L_2$ in~\eqref{L:decomposition} and, then, compute the derivative $\mathfrak{V}'_{\theta} = \mathfrak{V} \Lambda$. This completes the proof of~\eqref{lemma:d:2} and Lemma~1.
\end{proof}
\begin{remark}
The assumption of singular values of $A(\theta)$ being distinct for all values of parameter $\theta$ is necessary for avoiding the division by zero in formula~\eqref{barL2}. In future, if possible, we will intend for relaxing this restriction, which reduces the practical applicability of the proposed method.
\end{remark}
For readers' convenience, Algorithm~1 provides a pseudocode for the computational scheme derived in Lemma~1.
\begin{codebox}
\Procname{$\proc{Algorithm 1. Differentiated SVD}(A, A'_{\theta})$}
\zi {\bf Input:} $A$, $A'_{\theta}$ \qquad \Comment{\it \small Pre-array and its derivative}
\li Apply SVD from~\eqref{numSVD} to the pre-array $A$. Save $\mathfrak{W}$, $S$, $\mathfrak{V}$.
\li Compute the matrix product $\mathfrak{W}^T A'_{\theta} \mathfrak{V}$.
\li Extract the main $s\times s$ block $M=\left[\mathfrak{W}^T A'_{\theta} \mathfrak{V}\right]_{s\times s}$.
\li $M=\bar L + D + \bar U$. \Comment{\it \small Split into strictly lower triangular, diagonal}
\zi \phantom{$M=\bar L + D + \bar U$.} \Comment{\it \small and strictly upper triangular parts}
\li Given $\bar L$, $\bar U$ and $S$, compute the lower triangular $\bar L_2$ by~\eqref{barL2}.
\li Find $\mathfrak{V}'_{\theta} = \mathfrak{V} \left[ \bar L_2^T-\bar L_2 \right]$.
\li Find $S^{\prime}_{\theta} = D$. Hence, $\Sigma^{\prime}_{\theta} = \left[S^{\prime}_{\theta} \; | \; 0 \right]^T$.
\zi {\bf Output:} $\Sigma$, $\mathfrak{V}$ and $\Sigma'_{\theta}$, $\mathfrak{V}'_{\theta}$ \Comment{\it \small Post-arrays and their derivative}
\end{codebox}
The theoretical result presented in Lemma~1 can be further applied to the SVD factorization-based KF discussed in Section~\ref{filters}. The obtained computational scheme is summarized in Algorithm~2 and shown in the form of a pseudocode. The new ``differentiated'' SVD-based KF extends the underlying SVD-based filter on the derivative computation (with respect to unknown system parameters) for updating the corresponding filter sensitivities equations. The method can be used for replacing the conventional ``differentiated KF'' approach discussed in Section~\ref{sec:state} by inherently more stable approach, which is preferable for practical implementation. Finally, we would like to remark that any ``differentiated'' filtering technique consists of two parts: i) the underlying KF variant, and ii) its ``differentiated'' extension used for the filter sensitivities computation.
\begin{codebox}
\Procname{$\proc{Algorithm~2. Differentiated SVD-based KF}(\bar x_0, \Pi_0)$}
\zi {\bf Initial Step} ($k=0$).
\li \; $\Omega = Q_{\Omega}D_{\Omega}Q_{\Omega}^T$ and $R = Q_{R}D_{R}Q_{R}^T$ \Comment{\it \small SVD factorization}
\li \; $\Pi_0 = Q_{\Pi_0}D_{\Pi_0}Q_{\Pi_0}^T$ \qquad \qquad \qquad \quad \Comment{\it \small SVD factorization}
\li \; Set $Q_{P_{1|0}} = Q_{\Pi_0}$, $D^{1/2}_{P_{1|0}} = D^{1/2}_{\Pi_0}$ and $\hat x_{1|0} = \bar x_0$.
\li \; Set $\Di{Q_{P_{1|0}}}\!\! = \Di{Q_{\Pi_0}}$, $\Di{D^{1/2}_{P_{1|0}}}\!\! = \Di{D^{1/2}_{\Pi_0}}$, $\Di{\hat x_{1|0}} = 0$.
\zi {\bf Measurement Update:} ($k=1, \ldots, N$).
\li \; Build pre-array $A$ in~\eqref{svd:1} and its derivatives $\Di{A}$, $i=\overline{1,p}$.
\li \; $\left[ \Sigma, \: \mathfrak{V}, \:\Di{\Sigma}, \:\Di{\mathfrak{V}} \right] \: \leftarrow$ \verb"Differentiated SVD"($A$, $\Di{A}$).
\li \; $\left\{ D^{1/2}_{R_{e,k}}, \: \Di{D^{1/2}_{R_{e,k}}} \right\} \: \leftarrow$ read-off from $\Sigma$, $\Di{\Sigma}$ ($i=\overline{1,p}$).
\li \; $\left\{ Q_{R_{e,k}}, \: \Di{Q_{R_{e,k}}} \right\} \: \leftarrow$ read-off from $\mathfrak{V}$, $\Di{\mathfrak{V}}$ ($i=\overline{1,p}$).
\li \; Find $\bar K_{k}$ from~\eqref{svd:K} and $K_k = \bar K_k D^{-1}_{R_{e,k}}Q^T_{R_{e,k}}$.
\li \; $\Di{\bar K_{k}} = \Di{\left(Q_{P_{k|k-1}}D_{P_{k|k-1}}Q^T_{P_{k|k-1}}H^TQ_{R_{e,k}}\right)}$, $i=\overline{1,p}$.
\li \; Build pre-array $A$ in~\eqref{svd:2} and its derivatives $\Di{A}$, $i=\overline{1,p}$.
\li \; $\left[ \Sigma, \: \mathfrak{V}, \:\Di{\Sigma}, \:\Di{\mathfrak{V}} \right] \: \leftarrow$ \verb"Differentiated SVD"($A$, $\Di{A}$).
\li \; $\left\{ D^{1/2}_{P_{k|k}}, \: \Di{D^{1/2}_{P_{k|k}}} \right\} \: \leftarrow$ read-off from $\Sigma$, $\Di{\Sigma}$ ($i=\overline{1,p}$).
\li \; $\left\{ Q_{P_{k|k}}, \: \Di{Q_{P_{k|k}}} \right\} \: \leftarrow$ read-off from $\mathfrak{V}$, $\Di{\mathfrak{V}}$ ($i=\overline{1,p}$).
\li \; Find {\it a posteriori} estimate $\hat x_{k|k}$ and $\bar e_k$ from~\eqref{svd:3}.
\li \; $\Di{\bar e_k} = \DDi{Q_{R_{e,k}}^T}\left[z_k- H\hat x_{k|k-1}\right]$
\zi \; $\phantom{\Di{\bar e_k}} - Q_{R_{e,k}}^T\left[\DDi{H}\hat x_{k|k-1}+H\DDi{\hat x_{k|k-1}}\right]$, \; $i=\overline{1,p}$.
\li \; $\Di{\hat x_{k|k}} = \Di{\hat x_{k|k-1}} + \DDi{\bar K_{k}}D^{-1}_{R_{e,k}}\bar e_k + \bar K_{k}D^{-1}_{R_{e,k}}\DDi{\bar e_k}$
\zi \; $\phantom{\Di{\hat x_{k|k}}} - \bar K_{k}D^{-1}_{R_{e,k}}\DDi{D_{R_{e,k}}}D^{-1}_{R_{e,k}}\bar e_k$, \; $i=\overline{1,p}$.
\zi {\bf Time Update:} ($k=1, \ldots, N$).
\li \; Build pre-array $A$ in~\eqref{svd:5} and its derivatives $\Di{A}$, $i=\overline{1,p}$.
\li \; $\left[ \Sigma, \: \mathfrak{V}, \:\Di{\Sigma}, \:\Di{\mathfrak{V}} \right] \: \leftarrow$ \verb"Differentiated SVD"($A$, $\Di{A}$).
\li \; $\left\{ D^{1/2}_{P_{k+1|k}}, \: \Di{D^{1/2}_{P_{k+1|k}}} \right\} \: \leftarrow$ read-off from $\Sigma$, $\Di{\Sigma}$ ($i=\overline{1,p}$).
\li \; $\left\{ Q_{P_{k+1|k}}, \: \Di{Q_{P_{k+1|k}}} \right\} \: \leftarrow$ read-off from $\mathfrak{V}$, $\Di{\mathfrak{V}}$ ($i=\overline{1,p}$).
\li \; Find {\it a priori} estimate $\hat x_{k+1|k}$ from equation~\eqref{svd:6}.
\li \; $\Di{\hat x_{k+1|k}} = \DDi{F}\hat x_{k|k} + F\DDi{\hat x_{k|k}}+\DDi{B} u_{k}$, $i=\overline{1,p}$.
\zi {\bf End.}
\end{codebox}
At the same manner, one can naturally augment any existing SVD-based KF variant (see, for instance, the algorithms in~\cite{WangSVD1992,KulikovaOnline}) or potentially new SVD-based KF implementation on the corresponding filter sensitivities computation.
Finally, taking into account the properties of orthogonal matrices, it is not difficult to show that the
negative log likelihood function (LF) given as~\cite{Schweppe1965}:
\[
{\mathcal L}\left(\theta, Z_1^N\right)= c_0 +
\frac{1}{2} \sum \limits_{k=1}^N \left\{
\ln\left(\det R_{e,k}\right)+ e_k^T R_{e,k}^{-1}e_k \right\}
\]
can be rewritten in terms of the SVD filter variables $Q_{R_{e,k}}$, $D_{R_{e,k}}$ and $\bar e_k$ appeared in equations~\eqref{svd:1}~--~\eqref{svd:6} as follows:
\begin{equation}
{\mathcal L}\left(\theta, Z_1^N\right) =
c_0 + \frac{1}{2} \sum \limits_{k=1}^N \left\{
\ln\left(\det D_{R_{e,k}}\right)+ \bar e_k^T D^{-1}_{R_{e,k}}\bar e_k
\right\} \label{svd:LLF}
\end{equation}
where $Z_1^N=\{z_1,\ldots, z_N\}$ is $N$-step measurement history
and $c_0$ is a constant value where $c_0 = \frac{Nm}{2}\ln(2\pi)$.
Taking into account that the matrix $D_{R_{e,k}}$ is diagonal and using the Jacobi's formula, $\D{\left(A^{-1}\right)} = -A^{-1} \DD{A} A^{-1}$,
from~\eqref{svd:LLF} we obtain the expression for the log LF gradient evaluation in terms of the SVD filter variables and their derivatives computed in the newly-developed Algorithm~2 (for each $i=1, \ldots, p$):
\begin{align}
\Di{{\mathcal L}}\left(\theta, Z_1^N\right) & =
\frac{1}{2}\sum \limits_{k=1}^N \left\{
\tr { \DDi{D_{R_{e,k}}}D^{-1}_{R_{e,k}}} +2\DDi{ \bar e_k^T}D^{-1}_{R_{e,k}}\bar e_k
\right. \nonumber \\
& \left. -\bar e_k^TD^{-1}_{R_{e,k}} \DDi{D_{R_{e,k}}} D^{-1}_{R_{e,k}}\bar e_k
\right\}. \label{diff:svd:LLF}
\end{align}
\section{Numerical experiments } \label{experiments}
By using simple test problem, we would like to demonstrate thoroughly each step of the method summarized in Algorithm~1.
\begin{exmp}\label{ex:1}
Given pre-array $A(\theta)$ and its derivative $A'_{\theta}$
\begin{equation*}\label{A:1}
A(\theta) =
\left[\begin{array}{rr}
-2\theta & \sin(\theta) \\
2\theta & \theta^2 \\
\sin^2{(\theta)} & 1/3\:\theta^3\\
\theta & 2\theta^2-1\\
\cos^2{(\theta)} & \theta^3+\theta^2
\end{array}\right],
\end{equation*}
compute the corresponding SVD post-arrays $\Sigma$, $\mathfrak{V}$ and their derivative ${\Sigma}'_{\theta}$, $\mathfrak{V}'_{\theta}$ at the point $\hat \theta=0.5$.
\end{exmp}
Table~1 illustrates each step of the computational scheme in Algorithm~1. To assess the accuracy of computations, we compute
$l_{\infty} = \left|\left|{(A^T A)}'_{\hat\theta =0.5}-{(\mathfrak{V} \Sigma^2 \mathfrak{V}^T)}'_{\hat\theta=0.5}\right|\right|_{\infty}$. This quantity should be small. Indeed, taking into account the properties of diagonal and orthogonal matrices, from~\eqref{numSVD} we have $A^T A = \mathfrak{V} \Sigma^T \mathfrak{W}^T \mathfrak{W} \Sigma \mathfrak{V}^T = \mathfrak{V} \Sigma^2 \mathfrak{V}^T$. Hence, the derivatives of both sides of the last formula should coincide as well. In our numerical experiment we obtain $l_{\infty} = 1.99 \cdot 10^{-15}$. This justifies the correctness of computations via Algorithm~1 and confirms the theoretical derivations in Lemma~1.
\begin{table}[h]
\caption{Algorithm~1 illustrative calculations for Example~1} \label{tab:matrix}
\centering
\begin{tabular}{|l|l|}
\hline
{\bf Input} & Pre-array: $\left.A \right|_{\hat \theta = 0.5} = \left[
\begin{smallmatrix*}[r]
-1.0000 & 0.4794\\
1.0000 & 0.2500\\
0.2298 & 0.0417\\
0.5000 & -0.5000\\
0.7702 & 0.3750
\end{smallmatrix*}
\right]^{\phantom{M^M}}$ \\[10pt]
& Pre-array derivative: $\left.A'_{\theta}\right|_{\hat \theta = 0.5}=
\left[
\begin{smallmatrix*}[r]
-2.0000 & 0.8776\\
2.0000 & 1.0000\\
0.8415 & 0.2500\\
1.0000 & 2.0000\\
-0.8415 & 1.7500
\end{smallmatrix*}
\right]$ \\[10pt]
\hline
Line 1. &
$\mathfrak{W}=\left[
\begin{smallmatrix*}[r]
-0.6070 & 0.4848 & 0.1556 & 0.2057 & 0.5745\\
0.5723 & 0.4035 & 0.0539 & -0.5533 & 0.4478\\
0.1323 & 0.0735 & 0.9579 & 0.1059 & -0.2197\\
0.3159 & -0.5593 & 0.0946 & 0.4337 & 0.6247\\
0.4321 & 0.5329 & -0.2152 & 0.6724 & -0.1756
\end{smallmatrix*}
\right]^{\phantom{M^M}}$ \\[15pt]
& $\Sigma=\left[
\begin{smallmatrix*}[r]
1.7061 & 0\\
0 & 0.8185\\
0 & 0\\
0 & 0\\
0 & 0
\end{smallmatrix*}
\right]$, $\mathfrak{V}=\left[
\begin{smallmatrix*}[r]
0.9967 & 0.0811\\
-0.0811 & 0.9967
\end{smallmatrix*}
\right]$ \\[10pt]
Line 2. & Compute $M=
\left[
\begin{smallmatrix*}[r]
2.2959 & -1.6522 \\
1.1584 & 0.5691 \\
0.5177 & -0.1427 \\
-0.2470 & -2.2944 \\
-1.8181 & -0.8517
\end{smallmatrix*}
\right]$. \\[15pt]
Line 3. & Extract $\left[M\right]_{2\times2} =
\left[
\begin{smallmatrix*}[r]
2.2959 & -1.6522 \\
1.1584 & 0.5691
\end{smallmatrix*}
\right]$ \\[10pt]
Line 4. &
Split $\left[M\right]_{2\times2} \! = \! \left[
\begin{smallmatrix}
0 & 0\\
1.1584 & 0
\end{smallmatrix}
\right]\!+\!
\left[
\begin{smallmatrix}
2.2959 & 0\\
0 & 0.5691
\end{smallmatrix}
\right]
\!+\!
\left[
\begin{smallmatrix}
0 & -1.6522\\
0 & 0
\end{smallmatrix}
\right]$\!\!\!
\\[10pt]
Line 5. & Compute
$\bar L_2 =
\left[
\begin{smallmatrix}
0 & 0\\
0.8348 & 0
\end{smallmatrix}
\right]$\\[10pt]
Line 6. & $\left.\mathfrak{V}'_{\theta}\right|_{\hat \theta = 0.5} = \left[
\begin{smallmatrix*}[r]
0.0677 & -0.8321\\
0.8321 & 0.0677
\end{smallmatrix*}
\right]$ \\[10pt]
Line 7. & $\left.\Sigma'_{\theta}\right|_{\hat \theta = 0.5} = \left[
\begin{smallmatrix*}[c]
2.2959 & 0\\
0 & 0.5691\\
0 & 0\\
0 & 0\\
0 & 0
\end{smallmatrix*}
\right]$ \\[10pt]
\hline
{\bf Output} &
Post-arrays:
$\left.\Sigma\right|_{\hat \theta = 0.5} = \left[
\begin{smallmatrix*}[c]
1.7061 & 0 \\
0 & 0.8185 \\
0 & 0\\
0 & 0\\
0 & 0
\end{smallmatrix*}
\right]^{\phantom{M^M}}$ \\[10pt]
& \phantom{Post-arrays:} $\left. \mathfrak{V}\right|_{\hat \theta = 0.5} = \left[
\begin{smallmatrix*}[r]
0.9967 & -0.0811 \\
-0.0811 & -0.9967
\end{smallmatrix*}
\right]$ \\
& Post-arrays' derivative: $\left.\Sigma'_{\theta}\right|_{\hat \theta = 0.5}$ and $\left.\mathfrak{V}'_{\theta}\right|_{\hat \theta = 0.5}$ (Lines 6,7) \\
\hline
\end{tabular}
\end{table}
Next, we wish to demonstrate how the novel method for the filter sensitivities evaluation (Algorithm~2) works in practice. For that, we consider the parameter estimation problem where the gradient-based optimization method is applied for finding the optimal value of unknown system parameters. We test the conventional ``differentiated'' KF (Eqs~\eqref{kf:1}~-- \eqref{diff:kf:4} in Section~\ref{sec:state}) and the previously derived SR- and UD-based ``differentiated'' KF variants from~\cite{Kulikova2009IEEE} and~\cite{Tsyganova2013IEEE}, respectively, against the new ``differentiated'' SVD-based KF (Algorithm~2). As discussed in Section~\ref{Sec:SVD:diff}, all ``differentiated'' methods consist of two parts and, hence, they compute the Log LF and its gradient simultaneously. These values are utilized by a gradient-based optimization method for maximizing the log LF with respect to system parameters. Our library of codes is implemented in MATLAB where we use the built-in optimization method \verb"fminunc".
\begin{exmp}\label{ex:2}
Consider a linearized version of the in-track motion dynamic when a satellite travels in a circular orbit~\cite{Rauch1965}:
\begin{align*}
x_{k} & =
\begin{bmatrix}
1 & 1 & 0.5 & 0.5 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0.606
\end{bmatrix}
x_{k-1} + w_{k-1}, & \Omega & = \begin{bmatrix}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & q_1
\end{bmatrix} \\
z_k & =
\begin{bmatrix}
1 & 1 & 1 & 1\\
1 & 1 & 1 & 1+\delta
\end{bmatrix}
x_k + v_k, & R & = \begin{bmatrix}
\theta^2 \: \delta^2 & 0 \\
0 & \theta^2 \: \delta^2
\end{bmatrix}
\end{align*}
where $q_1 = 0.63 \cdot 10^{-2}$, $x_0 \sim {\mathcal N}(0, \theta^2 I_4)$ and $\theta$ is the unknown system parameter that needs to be estimated. In contrast to~\cite{Rauch1965}, we wish to test both well-conditioned and ill-conditioned situations. For that, following~\cite{GrewalAndrews2001}, we simulate the roundoff by parameter $\delta$. It is assumed to be
$\delta^2<\epsilon_{roundoff}$, but $\delta>\epsilon_{roundoff}$
where $\epsilon_{roundoff}$ denotes the unit roundoff
error\footnote{Computer roundoff for floating-point arithmetic is
characterized by a single parameter $\epsilon_{roundoff}$,
defined in different sources as the largest number such that either
$1+\epsilon_{roundoff} = 1$ or $1+\epsilon_{roundoff}/2 = 1$ in
machine precision. }. When $\delta \to \epsilon_{roundoff}$, i.e. the machine precision limit, the problem above becomes ill-conditioned. By varying the ill-conditioning parameter $\delta$, we are able to explore some numerical insights of each method assessed.
\end{exmp}
The numerical experiment is organized as follows. For each fixed value of ill-conditioning parameter $\delta$, the SSM in Example~2 is simulated for $\theta^* = 5$ to generate $N=100$ measurements. Next, the unknown system parameter $\theta$ is estimated from the available experimental data, $Z_1^N = \{z_1, \ldots, z_N \}$, by using gradient-based adaptive KF techniques examined, i.e. by four ``differentiated'' KF methods mentioned earlier in this Section. For a fair comparison, each ``differentiated'' algorithm utilizes the same data $Z_1^N$ and the same initial value for the optimization method, $\hat \theta^{(0)} = 1$. Next, the obtained optimal estimate $\hat \theta^*$ is compared with the ``true'' value of $\theta^* = 5$ for assessing the estimation quality of each method. We repeat the experiment $M=100$ times and calculate {\it a posterior} mean of the estimate, the root mean squared error (RMSE) and the mean absolute percentage error (MAPE) over $100$ Monte Carlo runs.
\begin{table*}
\renewcommand{\arraystretch}{1.3}
\caption{Effect of roundoff errors in ill-conditioned test problems in Example~2; exact $\theta^* =5$, $100$ Monte Carlo runs} \label{MC-estimators}
\centering
\begin{tabular}{c||c|c|c||c|c|c||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{``differentiated" KF} &
\multicolumn{3}{c||}{``differentiated" SR-based KF} & \multicolumn{3}{c||}{``differentiated" UD-based KF} & \multicolumn{3}{c}{``differentiated" SVD-based KF}\\
\cline{2-13}
$\delta$ & Mean & RMSE & MAPE\% & Mean & RMSE & MAPE\% & Mean & RMSE & MAPE\% & Mean & RMSE & MAPE\% \\
\hline
$ 10^{-1\phantom{0}}$ & 5.0046 & 0.2485 & 3.8829 & 5.0046 & 0.2485 & 3.8829 & 5.0046 & 0.2485 & 3.8829 & 5.0046 & 0.2485 & 3.8829\\
$ 10^{-2\phantom{0}}$ & 4.9649 & 0.2784 & 4.2892 & 4.9649 & 0.2784 & 4.2883 & 4.9649 & 0.2784 & 4.2883 & 4.9649 & 0.2784 & 4.2883\\
$ 10^{-3\phantom{0}}$ & 5.2764 & 0.7027 & 9.7757 & 5.0083 & 0.3555 & 5.7217 & 5.0083 & 0.3555 & 5.7217 & 5.0083 & 0.3555 & 5.7217 \\
$ 10^{-4\phantom{0}}$ & 8.8812 & 4.1440 & 77.623 & 4.9879 & 0.3715 & 5.8595 & 4.9879 & 0.3715 & 5.8596 & 4.9879 & 0.3715 & 5.8597\\
$ 10^{-5\phantom{0}}$ & 0.2803 & 8.0217 & $>$100\%& 4.9509 & 0.3352 & 5.6154 & 4.9508 & 0.3353 & 5.6162 & 4.9509 & 0.3352 & 5.6150\\
$ 10^{-6\phantom{0}}$ & -0.1315 & 7.2403 & $>$100\%& 4.9310 & 1.0362 & 6.8368 & 4.9323 & 1.0333 & 6.8265 & 5.0288 & 0.3138 & 4.8826\\
$ 10^{-7\phantom{0}}$ & $-$ & $-$ & $-$ & 4.9298 & 0.3658 & 5.8586 & 4.9268 & 0.3562 & 5.6883 & 4.9249 & 0.3507 & 5.5674 \\
$ 10^{-8\phantom{0}}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & 5.0437 & 0.3757 & 6.0712 & 5.0493 & 0.3790 & 6.0946 \\
$ 10^{-9\phantom{0}}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & 6.0119 & 1.2179 & 20.762 & 5.9738 & 1.1853 & 20.106 \\
$ 10^{-10\phantom{0}}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & 6.7496 & 2.6030 & 49.405 & 6.7021 & 2.5286 & 49.252 \\
\hline
\end{tabular}
\end{table*}
Having carefully analyzed the obtained numerical results summarized in Table~2, we make a few important conclusions.
First, all ``differentiated'' KF variants work equally well when $\delta$ is about $10^{-1}$ and $10^{-2}$, i.e. when the problem is not ill-conditioned. This confirms that all ``differentiated'' techniques are algebraically equivalent. Second, among all methods examined, the conventional approach (``differentiated'' KF) shows the worst performance. It degrades faster than any other algorithms when $\delta \to \epsilon_{roundoff}$. Furthermore, the line in Table~2 means that MATLAB can not even run the algorithm. Third, we analyze the outcomes obtained by other methods tested and observe that the UD- and SVD-based ``differentiated'' techniques produce a better estimation quality than the SR-based counterpart. This conclusion is reasonable if we recall that in this paper we do not explore the filtering algorithms, but their differential form for the KF sensitivities computation. Any existing ``differentiated'' SR-based scheme requires the triangular matrix inversion $R_{e,k}^{1/2}$ that is a square-root factor of the innovation covariance $R_{e,k}$; see Eq~(6) in~\cite{Kulikova2009IEEE}. In contrast, the UD- and SVD-based ``differentiated'' methods involve the inversion of only diagonal matrix $D_{R_{e,k}}$; see~\eqref{diff:svd:LLF} and Eq~(8) in~\cite{Tsyganova2013IEEE}. Finally, we observe that the new SVD-based approach slightly outperforms the UD-based counterpart when $\delta \to \epsilon_{roundoff}$.
In summary, the previously derived UD- and the new SVD-based techniques provide the best estimation quality when solving parameter estimation problem by the gradient-based adaptive filtering methodology. This creates a strong background for their practical use. In our ill-conditioned test example, the new SVD-based approach even slightly outperforms the UD-based counterpart.
|
\section*{Supplemental Material}
This supplemental material conveys additional results on the weak coupling regime, the effects of a harmonic trap, correlation functions and mutual information in the strong coupling phase separated regime, and a derivation of the spinful $PXP$ Hamiltonian at half and full filling.
\section{Weak coupling Rydberg dressed regime}
To obtain the weak-coupling Hamiltonian, see Eq. \eqref{eqn:cos} and above it, we linearize the tight-binding dispersion around $\pm k_F$. Without interactions, the bosonized Hamiltonian takes the form:
\begin{equation}
H^{nonint}_\eta = \frac{1}{2\pi} \int dx \left[ v (\pi \Pi_\eta (x))^2 + v (\nabla \phi_\eta(x))^2 \right],
\end{equation}
where $v_F= \frac{d \varepsilon}{dk}_{k=k_F}=\frac{2t}{a}\sin(k_F a)$, where $a$ is the lattice constant of the order of $\alpha$. Next we transform the lattice interaction Eq.~\eqref{eq:rydbergint} to the momentum space:
\begin{equation}
V_d(q) = \frac{1}{N}\sum_n
e^{-i q a n } V_d(a n).
\end{equation}
One can write $V_d(a n) = \int \frac{d k}{2\pi} e^{i k a n} V_C(k) = \frac{1}{L}\sum_{k=\frac{2\pi l}{L}} e^{i k a n} V_C(k)|_{L\to\infty}$, where $V_C(k) = \int dx V_d(x) e^{ikx}$ is defined in the continuum (C) limit $a\rightarrow 0$. Using the Poisson summation formula $\sum_n e^{i n g} = N \sum_m \delta_{g-2\pi m}$ one obtains:
\begin{equation}
V_d(q) = \frac{1}{L}\sum_m V_C\left(q+\frac{2\pi m}{a}\right).
\end{equation}
As $V_C(k)$ decays as $e^{-|k|r_c/2}$ for $r_c\gg a$ the $m\neq0$ terms can be neglected and $V_d(q)\approx \frac{1}{L} V_c(q)$ used in the main text is obtained.
Apart from the spin-spin correlations shown in Fig.~\ref{fig:weak}(a), we also calculate the density-density correlation
$\langle n_{L/2 \pm x} n_{L/2} \rangle$
for the two (positive and negative backscattering) phases in Fig.~\ref{fig:rhorho}.
As the charge sector is always gapless the density-density correlation decays as power-law in both cases.
The density-density and spin-spin power-law exponents in the repulsive backscattering region are both $-(K_\rho + K_\sigma)$.
We extracted the exponent for $N_p = 120$ from the spin-spin correlator (dashed line in Fig.~\ref{fig:weak}(a)) and plotted a dashed line with the same slope in Fig.~\ref{fig:rhorho}.
One can confirm that the power (slope) of the two matches well in the large $x$ limit.
\begin{figure}[b!]
\centering
\includegraphics[width=0.9\linewidth]{rhocorr4.png}
\caption{The density-density correlation function for $N_p = 70$ (blue) and $N_p = 120$ (yellow) particles in a $L = 200$ chain.
All parameters are identical to that of Fig.~\ref{fig:weak}.
The density-density correlation follows a power-law scaling on both cases.
}
\label{fig:rhorho}
\end{figure}
\section{Effect of trap potential}
The calculations in the main text focus on an optical lattice but ignores the presence of the harmonic trap potential.
In real experiments, the harmonic trap is inevitable and will affect the previously shown results. While we expect this to have a weak effect on the weak coupling phases, we need to confirm that the phase separated regime in Fig.~\ref{fig:strong}(b) survives. We include a harmonic trap by adding to the Hamiltonian a term $\sum_{r,\sigma} V_r c_{r\sigma}^{\dag}c_{r\sigma}$ with $V_r = \frac{1}{2} m\omega^2 r^2$.
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\linewidth]{trapN1.png}
\includegraphics[width=0.49\linewidth]{trapEE1.png}
\includegraphics[width=0.49\linewidth]{trapN2.png}
\includegraphics[width=0.49\linewidth]{trapEE2.png}
\caption{
Left accumulated particle $\mathcal{N}_L(x)$ as a function of position and the bipartite entanglement entropy in a $L = 400$ chain in the presence of a harmonic trap potential.
(a) $N_p = 167$ and $\frac{1}{2} m\omega^2 x_{\textrm max}^2 = t$, (b) $N_p = 163$ and $\frac{1}{2} m\omega^2 x_{\textrm max}^2 = 2t$.
The yellow dashed lines are the linear fit near the center ($x = 200$) and the deviation from this line indicates different filling (and thus different phase) away from the center.
}
\label{fig:trap}
\end{figure}
Here, we take the experimental setup in Ref.~\onlinecite{Guardado-Sanchez2021} with $t = h \times 1.7$ kHz and $a = 752$ nm.
Fig.~\ref{fig:trap} is the case of $N_p = 167, 163$ particles in a $L=400$ chain with a trap potential $\frac{1}{2} m\omega^2 x_{\textrm max}^2 = t, 2t$, respectively with $x_{\textrm max}=L/2$.
This corresponds to a trap frequency of order $10^2$Hz, which is safely larger than the typical harmonic trap.
It is clear that the slope (average particle density) at the center is different from that of the two boundaries.
The bipartite entanglement entropy also shows distinct behavior between center and boundaries, and the entanglement is low at the center regions, following the trend seen in Fig.~\ref{fig:strong}(c). Here, the center region is the magnetic domain wall and the charge like puddles are to its side.
From these results, while the overall structure of the phase separated state is modified by the trap
we conclude that the phase separated regime of the model will not be prohibited by the harmonic trap in experiments and we expect can be directly measured.
\section{Correlations in the phase separated regime}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.32\linewidth]{PS_rhorho.png}
\includegraphics[width=0.325\linewidth]{PS_sigsig.png}
\includegraphics[width=0.30\linewidth]{MI.png}
\caption{
Correlation functions for the same parameters in Fig.~\ref{fig:strong}(b,c).
($L=400$, $N_p=170$ in the strong coupling regime)
(a) The density-density correlation function where $x_0 = 87$ [blue], $212$ [yellow], $324$ [green] which are the centers of the $c=1$ puddles in Fig.~\ref{fig:strong}(c).
(b) The spin-spin correlation function where $x_0 = 154$ [blue], $263$ [yellow] which are in between the $c=1$ puddles.
(c) The mutual information (MI) between two points $x$, $y$ in the system.
The MI shows three puddles which corresponds to the three $c=1$ regions, and the information spread is blocked by the magnetized region.
}
\label{fig:MI}
\end{figure*}
In this section, we further investigate the phase separated state in the context of correlations to provide additional results that imply the presence of Luttinger liquid puddles.
In the main text Fig.~\ref{fig:strong}(c), we claimed that the three regions with $\bullet$$\bullet$$\circ$$\circ$$\circ$ patterns have a central charge of $c=1$ which results from the gapless charge degrees of freedom and is inherited from the weak coupling limit of the problem.
We check this directly calculating the density-density correlation function and observing its functional form.
In Fig.~\ref{fig:MI}(a) we plot the density-density correlation function for the same parameters in Fig.~\ref{fig:strong}(b,c), where the center $x_0$ is chosen as the center of the three $c=1$ regions.
One can observe slow power-law decay of the correlation, although the exponent is hard to extract due to the small size of each region.
In contrast, the spin-spin correlation function decays exponentially, even in the magnetized region.
In Fig.~\ref{fig:MI}(b) shows the fast decay of such correlation, where $x_0$ is the center of the magnetized region (in between the $c=1$ regions).
Finally, we plot the mutual information (MI) between two points $x$, $y$ in the system.
The MI between two subsets $A$ and $B$ of the system is defined as~\cite{Amico08}:
\begin{align}
\mathcal{I}(A, B) = \mathcal{S}(A) + \mathcal{S}(B) - \mathcal{S}(A\cup B),
\end{align}
where $\mathcal{S}(A)$ is the entanglement entropy (or von Neumann entropy) introduced in the main text.
Fig.~\ref{fig:MI}(c) plots the MI between all pair of points in the system, and shows that the MI is large only between points within the same $c=1$ region.
This shows the natural result that mutual information is large in gapless regions, and also the interesting feature that the gapped regions serve as a barrier of information spread between the gapless $c=1$ puddles.
\section{Bosonization analysis at strong coupling}
We now present details of the bosonization discussion regarding phases for arbitrary commensurate filling. For a filling equal to $\frac{p}{q}$, $p,q$ being mutually prime integers, an additional (umklapp) term appears in the low-energy theory, corresponding to simultaneous scattering of $q$ fermions from $-k_F$ to $k_F$. This can be written
|
as \cite{schulz_mott}
\begin{equation}
H_{um}=U_{um}\int dx \cos(q\sqrt{2} \varphi_\rho) \cos^q(\sqrt{2}\varphi_\sigma)
\end{equation}, where for weak coupling $U_{um}\sim U (U/t)^{q-2}$ \cite{schulz_mott,Giamarchi}. For even $q$, the most relevant part of $\cos^q(\sqrt{2}\varphi_\sigma)$ is just a constant, such that $U_{um}$ is relevant for $K_\rho<4/q^2$, leading to a charge gap in that case. Apart from half filling, this implies that the density degree of freedom can remain gapless below a threshold value of $U$. For odd $q$, $\cos(\sqrt{2}\varphi_\sigma)$ has to be kept. At weak coupling, if $V_C(2k_F)<0$ this term is fixed $\cos(\sqrt{2}\varphi_\sigma)=\pm1$ and the same condition for $K_\rho$ is valid , while for $V_C(2k_F)>0$ one can use the fixed-point value $K_\sigma=1$ \cite{schulz_mott} such that $K_\rho< 3/q^2$ is required. In the latter case, a gap will open simultaneously in the density and spin excitations. The regions of $1/5$ filling of Fig. \ref{fig:strong} (b,c) actually conform to weak coupling expectations. In that case $k_F r_c\approx 5$ such that $V_C(2k_F)/V_0 \approx - 0.16$ and a spin gap is expected, while the density gap requires strong interactions $K_\rho<0.16$. Note that the observed strong modulation of density occurs at $2k_F$, same as for a weak-coupling density wave. The
$\odot$%
$\odot$%
$\circ$%
$\circ$%
$\circ$%
$\circ$ regions, on the other hand, has filling $1/4$ with a less stringent requirement for the charge gap $K_\rho<0.25$, allowing to understand the development of a charge gap there. While in weak coupling, a spin gap would have been also expected due to $V_C(2k_F)/V_0\approx-0.026$, its absence points to inapplicability of the continuum weak coupling analysis for $1/4$ filling here, possibly due to extremely small value of $V_C(2k_F)/V_0$ and finite size effects. The above arguments also suggest that for 1/6 filling of Fig.~\ref{fig:strong}(a), there will be a spin gap $V_C(2k_F)/V_0\approx-0.2$ and no charge gap ($K_\rho<0.11$ is required), consistent with the numerical result. The charge modulation has the period $6$ (see Fig.~\ref{fig:strong2}(a)), corresponding to the usual $2k_F=2\pi/6$ Luttinger liquid correlations \cite{Giamarchi}, in contrast to the period $3$ ($4k_F$) modulation expected in the insulating state \cite{schulz_mott,Giamarchi}.
The bipartite entanglement entropy (see Fig.~\ref{fig:strong2}(b)) also shows clear $c=1$ scaling.
\begin{figure}[t!]
\centering
\includegraphics[width=0.49\linewidth]{rhocorr5.png}
\includegraphics[width=0.49\linewidth]{PS_EE3.png}
\caption{
(a) The power-law scaling density-density correlation for the system in Fig.~\ref{fig:strong}(a), $N_p = 140$. (b) The bipartite entanglement entropy as a function of the cut position. The solid line is the CFT scaling form (Eq.~\eqref{eq:ee}) with $c=1$.
}
\label{fig:strong2}
\end{figure}
\section{Optical tweezers and the $PXP$ limit}
In the limit of no tunneling (i.e. focusing on the optical tweezer set up as opposed to an optical lattice) and strong coupling we determine the effective many-body Hamiltonian, recovering the $PXP$ model in the limit of a fully filled Fermi sea and identify a magnetic $PXP$ model with a direct exchange interaction in the limit of half-filling.
The $PXP$ limit assumes a strong mixing of ground and Rydberg state $\Omega \gg \Delta$. In this case, the dominant interaction is the repulsion between adjacent Rydberg atoms, that is also larger than $\Omega$. Essentially, the interaction forbids atoms in Rydberg state to occupy neighboring sites. For spinless bosons, Rydberg Hamiltonian reduces then to the $PXP$ model, $H = \sum_i P_{i-1}X_i P_{i+1}$~\cite{sun2008,olmos2012,turner2018}.
$P_i = (1-Z_i)/2$ is a projector to the ground state.
The typical $PXP$ model studied intensively in the context of quantum many-body scars and slow dynamics focused on bosonic Rydberg atoms.
Here, we consider fermionic Rydberg atoms which introduces the spin degrees of freedom to the original problem.
First, consider a chain of Rydberg atoms with two particles (spin up and down) per site.
The $X_i$ operator connects the $|g\uparrow, g\downarrow\rangle$ state with the $(|R\uparrow,g\downarrow\rangle + |g\uparrow,R\downarrow\rangle)/\sqrt{2}$ state and thus this model reduces to the original $PXP$ model.
This is reasonable in the sense that each site has two fermions which can be considered as a bosonic degree of freedom.
Next, we consider a Rydberg atom chain with the filling of one particle per site. Neglecting the coupling of the laser to the spin degree of freedom (which is appropriate for $s$-like states), the Hamiltonian reduces to the original $PXP$ model with additional spin degeneracy, i.e.
\begin{align}
H_{PXP}^{spin} = \sum_i P_{i-1} X_i P_{i+1},
\end{align}
where $X = | R \uparrow\rangle\langle g\uparrow| + | R \downarrow\rangle\langle g\downarrow| + \textrm{h.c.}$ is the Pauli matrix in the $eg$-space, $P = | g \uparrow\rangle\langle g\uparrow| + | g \downarrow\rangle\langle g\downarrow|$ is the projection to the ground state.
The spin degeneracy, however, can be lifted by interactions due to the exchange process.
Both exchange and supexchange interactions depend on wavefunction overlap \cite{auerbach2012interacting} and can be of the same order. Therefore, one can expect that: (1) they decrease with distance fast (2) they are larger for the Rydberg state.
As the Rydberg interaction energy prohibits nearest neighbor atoms to be both in the Rydberg state, the next possibility is to either have spin-spin interactions with a neighboring ground-state atom, or a next-neighbor Rydberg atom. Combining both contributions, the spin exchange term can be written as:
\begin{equation}
\begin{gathered}
H_{exch}^{spin} = \sum_i J_{Rg} \vec{S}_i \cdot \vec{S}_{i+1} (P_i Q_{i+1} + Q_i P_{i+1})
\\
+ J_{RR} \vec{S}_i \cdot \vec{S}_{i+2} Q_i Q_{i+2},
\end{gathered}
\end{equation}
where $Q = 1 - P$.
For the direct exchange mechanism one can provide the following estimates for the couplings $J_{Rg}$ and $J_{RR}$. The Rydberg-Rydberg exchange coupling takes the form\cite{auerbach2012interacting} :
\begin{equation}
\begin{gathered}
J_{RR} = \frac{1}{2} \int d^3 {\bf r} d^3{\bf r}' V_R(|{\bf r} - {\bf r}'|) \psi^*_R({\bf r}) \psi_R({\bf r}')\times
\\
\times\psi_R^*({\bf r}'+2 a \hat{x} \bf) \psi_R({\bf r}+2 a \hat{x} \bf),
\end{gathered}
\end{equation}
where $\hat{x}$ is a unit vector along the chain direction and $V_R(|{\bf r} - {\bf r}'|) \propto 1/|{\bf r} - {\bf r}'|^6$, and $\psi_{g,R}({\bf r})$ correspond to ground and excited state atomic wavefunctions. One observes then that the integral is dominated by ${\bf r} \approx {\bf r}'$ and is determined by the overlap between Rydberg wavefunctions displaced by $2a$. As the size of the Rydberg wavefunction scales with principal quantum number as $n^2$, the integral can be estimated to scale (taking the integrand value in the middle between two atoms) as $\sim \exp\left( - 2 \frac{2a}{a_B^* n^2}\right)$, where $a_B^*$ is the Bohr radius for the atom.
The Rydberg-ground exchange coupling is determined by:
\begin{equation}
\begin{gathered}
J_{Rg} = \frac{1}{2} \int d^3 {\bf r} d^3{\bf r}' V_{Rg}(|{\bf r} - {\bf r}'|) \psi^*_R({\bf r}) \psi_R({\bf r}')\times
\\
\times\psi_g^*({\bf r}'+a \hat{x} \bf) \psi_g({\bf r}+ a \hat{x} \bf),
\end{gathered}
\end{equation}
where $V_{Rg}(|{\bf r} - {\bf r}'|)$ is the interaction between Rydberg and ground state atom, that is likely to be much weaker than the Rydberg interaction. The scaling of this integral with $n$ can be estimated analogously to be $\sim \exp\left( - \frac{a}{a_B^* n^2}- \frac{a}{a_B^*}\right)$.
Unfortunately, in typical tweezer setups \cite{bernien2017} $a\gg a_B^* n^2$, suggesting that the exchange interactions are likely quite small for the current set ups.
\end{appendix}
\end{document}
|
\section{Introduction}
Given a pointed space $X$, denote by $\mathcal E(X)$ the group of
(based) self homotopy equivalences, i.e., the group of
automorphisms of $X$ in the pointed homotopy category. From now on
we shall consider connected complexes of finite type $X$ which are
either finite or with finitely many non trivial homotopy groups.
We denote by $\dim X=N$ its topological or homotopical dimension.
Unless explicitly stated otherwise, all spaces will be of this
kind.
Although the computation of $\mathcal E(X)$ is known to be a hard task,
there are two classical and key results that impose to this group
important structural constraints:
On one hand, a theorem of Sullivan \cite[Theorem 10.3]{su} and
Wilkerson \cite[Theorem B]{wi} states that $\mathcal E(X)$ is finitely
presented. This was originally proved for simply connected spaces
and later on generalized to virtually nilpotent spaces by Dror,
Dwyer and Kan \cite[Theorem 1.1]{ddk}. The main step in the proof
is to show that $\mathcal E(X_\mathbb Q)$ is an algebraic group and that
$\mathcal E(X)$ is commensurable with an arithmetic subgroup of
$\mathcal E(X_\mathbb Q)$. As a consequence, it can be shown that there exists
a finite bound for the finite orders of elements of $\mathcal E(X)$.
On the other hand we have the following theorem due to Dror and
Zabrodsky:
\begin{theorem}\label{principal}{\em \cite[Theorem B]{d-z}} Let $G$ be a subgroup of
$\mathcal E(X)$ which acts nilpotently on $\pi_{\le N}(X)$. Then $G$ is
itself nilpotent. In particular, $\mathcal E_{ \sharp }^m(X)$ is nilpotent.
\end{theorem}
Recall that, for $0\le m\le\infty$, $\mathcal E_{ \sharp }^m(X)$ is the
distinguished subgroup of $\mathcal E(X)$ formed by those classes
inducing the identity on the homotopy groups up to $m$. In other
words, $$ \mathcal E_{ \sharp }^m(X)=\text{ker}\bigl(\mathcal E(X)\longrightarrow
\Pi_{i\le m}\text{aut}\,\pi_iX\bigr). $$ If $\dim X=N$ we shall
denote $\mathcal E_{ \sharp }^N(X)$ simply by $\mathcal E_{ \sharp }(X)$.
Here, we present a slightly different proof for this well known result
in which we use a broader study of self homotopy equivalences in
the homotopy category ${\cal L}^*$ of (based) spaces with local
coefficients. Recall (see \cite[Chap.VI]{white} for instance) that
objects in this category are pairs $(X,{\cal M})$ in which $X$ is a
(based) topological space and ${\cal M}=\{M_x\}_{x\in X}$ is a local
coefficient system in $X$. On the other hand, a morphism
$(f,\Theta)\colon (X,{\cal M})\to(Y,{\cal H})$ is a pair formed by a
based map $f\colon X\to Y$ and a morphism $\Theta\colon
f^*{\cal H}\to{\cal M}$ of local coefficient. By $f^*{\cal H}$ we denote,
as usual, the local coefficient system on $X$ induced by $f$,
i.e., $(f^*{\cal H})_x=H_{f(x)}$. For each $x\in X$ we shall denote
by $\Theta_x\colon H_{f(x)}\to M_x$ the corresponding group
morphism at $x$. After considering the appropriate homotopy
notion, one obtains the homotopy category ${\cal L}^*$. The
group of self homotopy equivalences of an object $(X,{\cal M})\in
{\cal L}^*$ shall be denoted by ${\mathcal E}(X;\calm)$. Then, we prove:
\begin{theorem}\label{local} Let $X$ be a finite Postnikov piece
and let $G\subset {\mathcal E}(X;\calm)$ be a subgroup which acts nilpotently
on both $\pi_*(X)$ and ${\cal M}$. Then, $G$ acts nilpotently on $H^*(X;{\cal M})$.
\end{theorem}
At the sight of Theorem \ref{principal}, and taking into account
the bound of finite orders of elements of $\mathcal E(X)$ plus the
existence of a ``fracture lemma" for this group (see \cite[Theorem
8.2]{Rutter}), it has been of interest to study whether $\mathcal E_{ \sharp }(X)$
satisfies the same structural restrictions when taking
$p$-localization, $p$-completion or considering $\mathcal E_{ \sharp p }(X)$. This
denotes the subgroup of $\mathcal E(X)$ formed by those classes which
induce the identity on the homotopy groups of $X$ with
coefficients on $\mathbb Z/p$, up to the dimension of $X$, i.e., $$
\mathcal E_{ \sharp p }(X)=\text{ker}\bigl(\mathcal E(X)\longrightarrow \Pi_{\le
N}\text{aut}\,\pi_i(X;\mathbb Z/p)\bigr). $$ As examples of this, we
mention two interesting results for a given nilpotent space $X$:
Maruyama proved \cite[Theorem 0.1]{ma} that
$\mathcal E_{ \sharp }(X)_{(p)}=\mathcal E_{ \sharp }^N(X_{(p)})$ while, on the other hand, M{\o}ller
showed \cite[Theorem 4.3]{mo} that
$$\mathcal E_{ \sharp }(X_{\mathbb Z_p})=Ext(\mathbb Z/p^\infty,\mathcal E_{ \sharp }(X))=
\mathcal E_{ \sharp }(X_p{\displaystyle\hat{}}). $$ Here and henceforth,
$(-)_{\mathbb Z_p}$ denotes $H_*(-;\mathbb Z/p)$-localization while
$(-)_{(p)}$ and $(-)_p{\displaystyle\hat{}}$ are the classical
localization and completion on the prime $p$.
In this paper we plan to continue this investigation extending
Theorem \ref{principal} above, considering a subgroup of $\mathcal E(X)$
which acts nilpotently in the homotopy groups of $X$ localized,
completed or with coefficients in $\mathbb Z/p$. Concerning this purpose
we prove:
\begin{theorem}\label{dos} Assume that $\pi_1(X)$ is a nilpotent group and let $G$ be a subgroup of $\mathcal E(X)$ which acts
nilpotently on $\pi_{\le N}(X)_{(p)}$, for $p$ any prime number
and $0$. If the nilpotency orders of all these actions are bounded
by a fixed integer, then G is nilpotent.
\end{theorem}
\begin{remark} {\em Observe that in the theorem above the condition of $\pi_1(X)$ being nilpotent is essential. Otherwise,
choose any finite simple group $G$ which is known to be
generically trivial, i.e., $G_{(p)}=\{1\}$ for $p$ any prime
number or zero. On the other hand observe that the map $G\to \text{aut}\,
G$ given by inner automorphisms is a monomorphism. Indeed, its
kernel is the center of $G$ which is trivial since $G$ is simple.
This inclusion renders the non nilpotent group $G$ as a subgroup
of $\mathcal E\bigl(K(G,1)\bigr)$ which acts nilpotently in the
localized homotopy group.
In a similar way, we may even produce an example of a solvable,
non nilpotent group, of homotopy equivalences acting nilpotently
in the localized homotopy groups of the space. Consider the
symmetric group $\Sigma_3$ and observe that
${\Sigma_3}_{(2)}=\mathbb Z/2$ while ${\Sigma_3}_{(p)}=1$ for $p\not=2$.
Again, $\Sigma_3\subset\mathcal E\bigl(K(\Sigma_3,1)\bigr)$ is a
solvable non nilpotent group acting nilpotently on any
${\Sigma_3}_{(p)}$.}
\end{remark}
A more subtle and slightly different situation is given when
considering nilpotent actions of subgroups of self homotopy
equivalences on the Frattini factor of the homotopy groups. Recall
that given a group $G$, the Frattini subgroup $\Phi(G)$ is the
intersection of all maximal proper subgroups of $G$. The quotient
$G/\Phi(G)$ is called the Frattini factor.
\begin{theorem}\label{uno} Assume that
$\pi_{\le N}(X)$ is a finite nilpotent group and let $G$ be a
subgroup of $\mathcal E(X)$ which acts nilpotently on $\pi_{\le
N}(X)/\Phi\bigl(\pi_{\le N}(X)\bigr)$. Then, G is nilpotent.
\end{theorem}
In particular, taking into account that for an abelian $p$-group
$G$ its Frattini factor is precisely $G\otimes\mathbb Z/p$, we obtain
the following:
\begin{corollary}\label{nuevo} Assume that
$\pi_{\le N}(X)$ is a finite abelian group and let $G$ be a
subgroup of $\mathcal E(X)$ which acts nilpotently on $\pi_{\le
N}(X)\otimes\mathbb Z/p$ for any prime $p$. Then, G is
nilpotent.\hfill$\square$
\end{corollary}
Notice that, by the Universal Coefficients Theorem for homotopy,
$\pi_*X\otimes\mathbb Z/p=\text{Ext}(\mathbb Z/p,\pi_*X)$ is a subgroup of
$\pi_*(X;\mathbb Z/p)$. Hence, as an immediate consequence of Corollary
\ref{nuevo} above we get:
\begin{corollary}\label{coeficientes} Assume that
$\pi_{\le N}(X)$ is a finite abelian group and let $G$ be a
subgroup of $\mathcal E(X)$ which acts nilpotently on $\pi_{\le
N}(X;\mathbb Z/p)$ for any prime $p$. Then, G is
nilpotent.\hfill$\square$
\end{corollary}
Having studied the nilpotency of a general subgroup of $\mathcal E(X)$,
we now focus on the group $\mathcal E_{ \sharp p }(X)$ and give necessary conditions
for it to be nilpotent.
\begin{theorem}\label{tres} Let $X$ be a space for which $\pi_{\le N}(X)$ is a finite abelian $p$-group. Then
$\mathcal E_{ \sharp p }(X)$ is nilpotent and $\mathcal E_{ \sharp p }(X)/\mathcal E_{ \sharp }(X)$ is a finite $p$-group.
\end{theorem}
\begin{theorem}\label{cuatro}
Let $X$ be a space for which $\pi_{\le N}(X)$ is a finitely
generated abelian group. Then $\cap_{p\,\,\text{prime}}\mathcal E_{ \sharp p }(X)$ is
nilpotent.
\end{theorem}
\begin{remark}\label{importante} {\em Observe that in general
$\mathcal E_{ \sharp p }(X)$ is bigger than $\mathcal E_{ \sharp }(X)$. For instance consider
$X=K(\mathbb Z/{p^r},n)$, $r,n\ge 2$. Obviously $\mathcal E_{ \sharp }(X)=\{1\}$, while
the automorphism $\rho$ of $\mathbb Z/{p^r}$ given by
$\rho(1)=p^{r-1}+1$ induces a non trivial element of $\mathcal E_{ \sharp p }(X)$.
Indeed, by the Universal Coefficients Theorem for homotopy, $$
\pi_*(X,\mathbb Z/p)=\pi_n(X,\mathbb Z/p)\oplus \pi_{n-1}(X,\mathbb Z/p), $$ in
which\hfill\break
\medskip
\noindent$\pi_n(X,\mathbb Z/p)=\hom(\mathbb Z/p,\mathbb Z/{p^r})$ {and}
$\pi_{n-1}(X,\mathbb Z/p)=\text{Ext}(\mathbb Z/p,\mathbb Z/{p^r})=\mathbb Z/p.$
\medskip
\noindent Trivially $\rho$ induces the identity on both. Note that
this example also shows that even
$\cap_{p\,\,\text{prime}}\mathcal E_{ \sharp p }(X)$ can be bigger than $\mathcal E_{ \sharp }(X)$.}
\end{remark}
The paper is organized as follows: in the next section we collect
the results we shall need from group theory and from which
Theorems \ref{dos} and \ref{uno} are immediately deduced. Theorem
\ref{local} and \ref{principal} are proved in section \S3.
Finally, in section \S4 we establish Theorems \ref{tres} and
\ref{cuatro}.
\section{From group theory}
We begin by recalling some basic facts. If $G$ is a group acting
on another group $A$ (i.e., $A$ is a $G$-group), the $n$-th $G$-commutator subgroup
$\Gamma^n_G(A) \subset A$ is the group generated by
$\{(ga^{-1})a\,|\, g\in G,\, a\in\Gamma_G^{n-1}(A)\}$, being
$\Gamma_G^0(A)=A$. The action is then nilpotent of nilpotency
order $r$, $\text{nil}\,_GA=r$, if this is the smallest integer for which
$\Gamma^r_G(A)=\{1 \}$. The group $G$ also acts in each
$\Gamma_G^n(A)$ and
$\Gamma_G^m\bigl(\Gamma_G^n(A)\bigr)=\Gamma_G^{m+n}(A)$.
Statements of next sections shall heavily rely in the following
results:
\begin{lemma}\label{lema1} Let $A$ be a $G$-group. Then:
\begin{itemize}
\item[(i)] $\Gamma_G^1(A)$ is a normal subgroup of $A$ and the $G$-action induced on $A/\Gamma_G^1(A)$ is trivial.
\item[(ii)] The quotient morphism $A{\stackrel{q}{\longrightarrow}} A/\Gamma_G^1(A)$ is equivariant and initial with
respect to trivial actions, i.e., every equivariant morphism
$A{\stackrel{f}{\longrightarrow}}H$, in which the $G$-action on
$H$ is trivial, factors uniquely through $q$.
\end{itemize}
\end{lemma}
\begin{proof} (i) is trivial. For (ii) observe that, for any $f$ as in the lemma, $\Gamma_G^1(A)\subset\ker f $.
\end{proof}
\begin{lemma}\label{lema2} Let $A$ be a $G$-group. If $A$ is nilpotent then, for any $m$, $\Gamma_G^m(A)_{(p)}=\Gamma_G^m(A_{(p)})$.
\end{lemma}
\begin{proof} Since $\Gamma_G^m(A)=\Gamma_G^{1}\bigl(\Gamma_G^{m-1}(A)\bigr)$, once we show that
$\Gamma_G^1(A)_{(p)}=\Gamma^1_G(A_{(p)})$ an easy induction proves
the lemma. As localization is an exact functor in the category of
nilpotent groups, the localization morphism $f:A\to A_{(p)}$
restricts to $f:\Gamma_G^1(A)\to \Gamma_G^1(A)_{(p)}$. Hence, we
may consider $\Gamma_G^1(A)_{(p)}$, as well as
$\Gamma_G^1(A_{(p)})$, as subgroups of $A_{(p)}$. Then, for any
$g\in G$ and $a\in A$, the trivial identity
$\bigl(gf(a)^{-1}\bigr)f(a)=f\bigl((ga^{-1})a\bigr)$ shows
equality of both subgroups.
\end{proof}
\begin{proposition}\label{uf} The group $G$ acts nilpotently on the nilpotent group $A$ if and only if $G$ acts
nilpotently on $A_{(p)}$ for $p$ any prime number or zero and all
these nilpotency orders are bounded.
\end{proposition}
\begin{proof} Assume $G$ acts nilpotently on $A$, i.e., $\Gamma_G^m(A)=\{1\}$ for some $m$. Hence, by Lemma \ref{lema2}
and for any $p$, $\Gamma_G^m(A_{(p)})=\{1\}$.
Conversely, assume $\text{nil}\,_GA_{(p)}\le m$, for all $p$ ($p$ a prime
number or $0$), and let $a$ be an element of $\Gamma_G^m(A)$. If
$a$ has finite order, say it is a $q$-element, then it obviously
survives under the $q$-localization morphism
$\Gamma_G^m\bigl(A)\to\Gamma^m_G(A)_{(q)}$. For a general group,
elements of infinite order are not guaranteed to survive under
rationalization (for instance, the rationalization of the free
product of two finite groups is trivial while it contains elements
of infinite order). However for a nilpotent group, which is our
case, one can easily show by induction on the nilpotency order of
the group, that any element of infinite order is not sent to zero
under rationalization. Taking into account, again by Lemma
\ref{lema2}, that $\Gamma^m_G(A)_{(p)}=\Gamma^m_G(A_{(p)})=\{1\}$,
it follows that $a=1$ and the proof is complete.
\end{proof}
\begin{proposition}\label{nuevolema} Let $G$ be a group acting on a finite nilpotent group $A$ in such a way that the
induced action on the Frattini factor $A/\Phi(A)$ is nilpotent.
Then, the $G$-action on $A$ is also nilpotent.
\end{proposition}
\begin{proof} Recall \cite[5.1]{Gorestein} that the Frattini subgroup of a group $A$, $\Phi(A)$, is defined to be the intersection of all its maximal proper subgroups. The Frattini factor of $A$ is $A/\Phi(A)$.
Observe in the first place that, since $\Phi(A)$ is a characteristic subgroup of $A$, i.e., it is invariant under any automorphism of $A$, $G$ in fact induces a
natural action on the Frattini factor $A/\Phi(A)$ which, by
hypothesis, is nilpotent. Hence, since $A/\Phi(A)$ is nilpotent,
the induced action on
$\bigl(A/\Phi(A)\bigr)_{(p)}=A_{(p)}/\Phi(A)_{(p)}$ is also
nilpotent by Lemma \ref{lema2}. Next, observe that for any finite
group $A$, $\Phi(A)_{(p)}=\Phi(A_{(p)})$. Indeed, this is
immediate from the definition taking into account that
localization commutes with limits, in particular, with
intersections (see for instance \cite{hmr}). Therefore, we conclude that $G$ acts nilpotently on
$A_{(p)}/\Phi(A_{(p)})$. Considering $\varphi\colon
G\to\text{aut}\,\bigl(A_{(p)}/\Phi(A_{(p)})\bigr)$ via this action, and
taking into account that $A_{(p)}/\Phi(A_{(p)})$ is a finite
$p$-group, we may apply \cite[Corollary 5.3.3]{Gorestein} to
obtain that $\varphi(G)$ is also a $p$-group. But the action of a
$p$-group on another $p$-group is always nilpotent, and therefore
$G$ acts nilpotently on $A_{(p)}$. Since this is the case for any
$p$ and $A$ is finite we may apply Proposition \ref{uf} and the
proposition follows.
\end{proof}
From these results we immediately deduce:
\bigskip
\noindent {\it Proof of Theorems \ref{dos} and \ref{uno}.} Apply
directly Propositions \ref{uf} and \ref{nuevolema} above to the
subgroup $G$ of $\mathcal E (X)$ to obtain that $G$ acts nilpotently on
$\pi_{\le N}(X)$. Then, the result follows from Theorem
\ref{principal}. \hfill$\square$
\bigskip
Closely related to Proposition \ref{nuevolema}, we have the
following:
\begin{proposition}\label{propodos} Let $G$ be a group acting on an abelian $p$-group $A$ which has an exponent $p^n$. If
$G$ acts nilpotently on $A\otimes\mathbb Z/p$, then it does so on $A$
and $$ \text{nil}\,_GA\le n\cdot\text{nil}\,_GA\otimes\mathbb Z/p. $$
\end{proposition}
\begin{proof} Call $r=\text{nil}\,_GA\otimes\mathbb Z/p$ and observe that $\Gamma_G^m(A\otimes\mathbb Z/p)=\Gamma_G^m(A)\otimes\mathbb Z/p$ for any
$m$. Therefore, since $\Gamma_G^r(A\otimes\mathbb Z/p) =0$,
$\Gamma_G^r(A)\subset pA$. Assume, as induction hypothesis, that
$\Gamma_G^{kr}(A)\subset p^kA$, for $k<n$. Hence, $$
\Gamma_G^{nr}(A)=\Gamma_G^{(n-1)r}\bigl(\Gamma_G^r(A)\bigr)\subset
\Gamma_G^{(n-1)r}(pA)= p\Gamma_G^{(n-1)r}(A)\subset p^nA. $$ Since
$A$ has $p^n$ as exponent, the proposition follows.
\end{proof}
\begin{proposition}\label{propouno} Let $A$ be a finite abelian
$p$-group and let $G\subset\text{aut}\,(A)$ be such that $\sigma\otimes
{\mathbb Z/p}=1_{A\otimes\mathbb Z/p}$ for each $\sigma\in G$. Then $G$ is a
$p$-group.
\end{proposition}
\begin{proof} As $A$ is a finite abelian $p$-group,
the Frattini factor $A/\Phi(A)$ (respec. the projection
$A\rightarrow A/\Phi(A)$) is naturally identified with
$A\otimes\mathbb Z/p$ (respec.\ the map $A\rightarrow A\otimes\mathbb Z/p$).
Now, if $G$ is not a $p$-group, there exists a non trivial
$p^\prime$-automorphism $\sigma\in G$ which, by hypothesis and
using the identification above, induces the identity on the
Frattini factor of $A$. But according to \cite[Theorem
5.1.4]{Gorestein}, the only $p^\prime$-automorphism that induces
the identity on the Frattini factor of a $p$-group is the
identity. Thus $G$ must be a $p$-group.
\end{proof}
As an immediate consequence we get:
\begin{corollary}\label{corolario} In the conditions of the proposition above, the action of $G$ on $A$ is nilpotent.
\end{corollary}
\begin{proof} Indeed, recall that the action of a
$p$-group $H$ on another $p$-group is always nilpotent.
\end{proof}
In which follows $G$ is a group acting on another group $A$. Given
$g,h\in G$ and $a\in A$, we use the following usual notation: $$
[a,g]=a^{-1}(ga),\quad [g,a]=(ga^{-1})a,\quad
[g,h]=g^{-1}h^{-1}gh. $$ Hence, the following, which can be
considered as a variation of the Witt-Hall identity \cite[Theorem
5.1]{magnus}, is obtained by direct calculation.
\begin{lemma}\label{jo}
For any $f,g\in G$ and $b\in A$, the following identity holds
|
:
$$\big[[f^{-1},g^{-1}],gb\big]b^{-1}\big[[g,b^{-1}], f\big]
b\big[[f,b],fgf^{-1}\big]=1.$$
\end{lemma}
\begin{lemma}\label{jo2} Let $H$ be a subgroup of $G$ and $K$ a normal subgroup of $H$. Then,
$$
\bigr[[H,K],A\bigr]\subset\langle\big[K,[H,A]\bigr],\bigl[H,[K,A]\bigr]\rangle.
$$
\end{lemma}
\begin{proof}
Making $f^{-1}=h$, $g^{-1}=k$ and $gb=a$ in Lemma \ref{jo}, it
follows that $$
\bigl[[h,k],a\bigr]=\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\bigr]
g^{-1}a^{-1}\bigl[h^{-1},[k^{-1},g^{-1}a]\bigr]g^{-1}a. $$ As $K$
is normal in $H$, $\bigl[h^{-1}k^{-1}h,[h^{-1},g^{-1}a]\bigr]\in
\big[K,[H,A]\bigr]$. On the other hand, as commutators are normal
subgroups,
$$g^{-1}a^{-1}\bigl[h^{-1},[k^{-1},g^{-1}a]\bigr]g^{-1}a\in
g^{-1}a^{-1} \bigl[H,[K,A]\bigr]g^{-1}a=\bigl[H,[K,A]\bigr], $$
and the lemma follows.
\end{proof}
\begin{lemma}\label{jo3} If the action of $G$ on $A$ is nilpotent, then
for each $n,m\ge 0$, $$ [\Gamma^n(G),\Gamma_G^m(A)]\subset
\Gamma_G^{n+m+1}(A). $$ In particular,
$
[\Gamma^n (G), A]\subset \Gamma_G^{n+1}(A).
$
\end{lemma}
\begin{proof} Set $\text{nil}\,_GA=r$. If $m\ge r$ the assertion is obvious. Assume the lemma holds for all $n$ and $m\le 1$ and let us prove it for $m=0$ by
induction on $n$:
Trivially,
$
[\Gamma^0(G),\Gamma_G^0(A)]=[G,A]= \Gamma_G^{1}(A)$. Finally, $$
\begin{aligned}
{[\Gamma^n(G),A]}&=\bigl[[G,\Gamma^{n-1}(G)], A\bigr]\subset
\text{(By Lemma \ref{jo2})}\\ &\subset \langle
\bigl[\Gamma^{n-1}(G),[G,A]\bigr],
\bigl[G,[\Gamma^{n-1}(G),A]\bigr]\rangle\subset \text{(By
induction)}\\ &\subset\langle [\Gamma^{n-1}(G),\Gamma^1_G(A)],
[G,\Gamma^n_G(A)]\rangle\subset \text{(Again by induction)}\\
&\subset\Gamma_G^{n+1}(A).\\
\end{aligned}
$$
\end{proof}
\begin{proposition}\label{jo4}
Let $G$ be a subgroup of $\text{aut}\,(A)$. Then, $\text{nil}\, G\le \text{nil}\,_GA-1$.
\end{proposition}
\begin{proof}
Assume $\text{nil}\,_GA=r$. By Lemma \ref{jo3},
$[\Gamma^{r-1}(G),A]\subset \Gamma_G^r(A)=\{1\}$, and therefore
$\Gamma^{r-1}(G)=\{1\}$.
\end{proof}
\section{Self homotopy equivalence of spaces with local coefficients}
As stated in the Introduction, and following the notation and
approach of the standard reference \cite[Chap. VI.2]{white}, in
this section we consider self homotopy equivalences in the
homotopy category ${\cal L}^*$ of based spaces with local
coefficients. Observe that a self homotopy equivalence of an
object $(X,{\cal M})\in {\cal L}^*$ is given by $(f,\Theta)\colon
(X,{\cal M})\to (X,{\cal M})$ in which $f\colon X\to X$ is a based
homotopy equivalence and $\Theta\colon {\cal M}\to{\cal M} $ is an
isomorphism of the coefficient system ${\cal M}$. Note that such a
self equivalence $(f,\Theta)$ acts in $\pi_*(X)$ by $\pi_*f$, in
${\cal M}$ by $\Theta$, and in $H^*(X;{\cal M})$ by $H^*(f,\Theta)$.
It is also convenient to recall how cohomology classes with local
coefficients are represented by maps into the ``twisted
Eilenberg-MacLane space'' (see \cite[Chapter 5.2]{baues},
\cite{Gitler}, \cite{moller2} or \cite{siegel} for precise
details). Let $K({\cal M},n)$ be a fixed realization of the
Eilenberg-MacLane space of type $(M_{x_0},n)$ being $M_{x_0}$ the
group of the system ${\cal M}$ at the base point. On the other hand,
denote by $L({\cal M},n)$ the space obtained by applying the Borel
construction to the universal fibration $\pi_1(X)\to\widetilde
K\stackrel{q}{\to} K(\pi_1(X),1)$ and the space $K({\cal M},n)$,
i.e.,
$$ L({\cal M},n)=\widetilde K\times_{\pi_1(X)} K({\cal M},n), $$
and it fits into the fibration $$ K({\cal M},n)\longrightarrow
L({\cal M},n)\stackrel{p}{\longrightarrow}K(\pi_1(X),1),\quad
p(a,b)=q(a). $$ Then, for a given space $X$, $H^n(X;{\cal M})$ is
in one to one correspondence with the set $[X,L({\cal
M},n)]_{K(\pi_1(X),1)}$ of homotopy classes of maps over
$K(\pi_1(X),1)$ from $X$ to $L({\cal M},n)$.
\bigskip
\noindent {\it Proof of Theorem \ref{local}.} To avoid excessive
notation we shall not distinguish between a homotopy class and a
map which represents it. First, observe that, if $(f,\Theta)$ is a
self homotopy equivalence of $(X,{\cal M})$ and $\alpha\colon X\to
L({\cal M},n)$ represents a class of $H^*(X;{\cal M})$,
$H^n(f,\Theta)(\alpha)$ is represented by the map $$
X\stackrel{f}{\longrightarrow}
X\stackrel{\alpha}{\longrightarrow}L({\cal M},n)\stackrel{\xi}{\longrightarrow}L({\cal M},n)
$$ in which $\xi$ is defined by the action of $\Theta_{x_0}$ on
$M_{x_0}$. Explicitly, for $(a,b)\in L({\cal M},n)$,
$\xi(a,b)=(a,\overline{\Theta}_{x_0}b)$ where and
$\overline{\Theta}_{x_0}$ is the realization of $\Theta_{x_0}$.
Observe that $\xi$ is well defined as $\Theta\colon {\cal M}\to{\cal M}$
is a morphism of local coefficient systems.
Moreover, if $\alpha,\beta\colon X\to L({\cal M},n)$ are in
$H^n(X;{\cal M})$, they coincides after composing with the fibration
$p\colon L({\cal M},n)\to K(\pi(X),1)$, and therefore, for each $x\in
X$, $\alpha(x),\beta(x)$ live in the same fiber $K({\cal M},n)$ of
$p$. Hence, $\alpha$ and $\beta$ can be added up on $K({\cal M},n)$
and the resulting map $\alpha+\beta$ represents precisely their
sum as cohomology classes with twisted coefficients.
We shall prove the theorem by induction on the length of the
Postnikov decomposition of $X$. Assume $X=K(\pi,m)$ and let
$(f,\Theta)\colon (X,{\cal M})\to(X,{\cal M})$ be a self equivalence.
Then, in view of the above, for any $n$-cohomology class
$\alpha\colon K(\pi,m)\to L({\cal M},n)$,
$H^n(f,\Theta)(\alpha)-\alpha$ is represented by the map
$K(\pi,m)\to L({\cal M},n)$ which, fiberwise, is
$\overline\Theta_{x_0}\alpha f-\alpha$. Writing
$\overline\Theta_{x_0}\alpha f-\alpha= \overline\Theta_{x_0}\alpha
f- \overline\Theta_{x_0}\alpha + \overline\Theta_{x_0}\alpha
-\alpha$ it is straightforward, using the nilpotency hypothesis,
to show that the $s$-th commutator of the action of $G$ on
$H^*(X;{\cal M})$ vanishes as long as
$s\le\max\{\text{nil}\,_G\pi_*(X),\text{nil}\,_G{\cal M}\}$.
Assume the theorem holds for $X=X^{r-1}$ and let $X=X^r$ be a
$r$-dimensional Postnikov piece. Consider the Serre spectral
sequence with local coefficients on ${\cal M}$ associated to the
fibration $$ K(\pi_r(X),r)\to X\to X^{r-1}. $$ whose $E_2$-term is
$$ E_2^{*,*}=H^*\bigl(X^{r-1};{\cal H}^*(K(\pi_r(X),r);{\cal
M})\bigr). $$ Note that $G$ acts naturally in the base, total
space and fiber of this fibration, and hence, it does so in all
the terms of the spectral sequence. The same argument used for
$r=1$ shows that $G$ acts nilpotently on the local coefficient
system ${\cal H}^*(K(\pi_r(X),r);{\cal M})$ and therefore, by
induction hypothesis, $G$ acts nilpotently on
$H^*\bigl(X^{r-1};{\cal H}^*(K(\pi_r(X),r);{\cal M})\bigr)$.
As the spectral sequence converges, the action of $G$ on the
associated graded module of $H^*(X;{\cal M})$ is nilpotent. Finally,
reasoning by induction on the filtration degree we deduce that the
$G$-action on $H^*(X;{\cal M})$ is also nilpotent.\hfill$\square$
In particular, for any space $X$ and any $j$ we may consider the
local coefficient system given by $\pi_jX$. In this case, any self
homotopy equivalence $f\in\mathcal E(X)$ can be seen as a self homotopy
equivalence $(f,\pi_jf^{-1})\in\mathcal E(X;\pi_jX)$. Hence, any
subgroup of $\mathcal E(X)$ may be considered as a subgroup of
$\mathcal E(X,\pi_jX)$ which then acts naturally on $H^*(X;\pi_jX)$ when
considering local coefficients. In this context, the theorem
above reads:
\begin{corollary}\label{noveas} Let $X$ be a finite Postnikov piece and let $G$ be a subgroup of
$\mathcal E(X)$ which acts nilpotently on $\pi_*(X)$. Then, for any $j$,
$G$ acts nilpotently on $H^*(X;\pi_j)$.\hfill$\square$
\end{corollary} This result is used in the proof of Theorem \ref{principal} that we now present:
\bigskip
\noindent {\it Proof of Theorem \ref{principal}.} Consider the
restriction to $G$ of the exact sequence $
{1}\to\mathcal E_{ \sharp }(X)\to\mathcal E(X)\to \Pi_{i\le N}\text{aut}\,\pi_i(X) $: $$
{1}\to\mathcal E_{ \sharp }(X)\cap G\to G\to \Pi_{i\le N}\text{aut}\,\pi_i(X). $$
The image of $G$ under this morphism, call it $\widetilde G$, is
a subgroup of automorphism of the group $\pi_{\le N}(X)$ in which
$G$ acts nilpotently by hypothesis. Then, by Proposition
\ref{jo4}, $\widetilde G$ is itself nilpotent and $\text{nil}\,\widetilde
G< \text{nil}\,_G\pi_{\le N}(X)$. Therefore, if we prove that $G$ acts
nilpotently on $\mathcal E_{ \sharp }(X)$, then (see for instance \cite[Proposition
4.1]{hmr}) $G$ would be nilpotent and
\begin{equation}\label{nose1}
\text{nil}\, G< \text{nil}\,_G\mathcal E_{ \sharp }(X)+\text{nil}\,_G\pi_{\le N}(X).
\end{equation} For that, observe in the first place that $[X,X]\cong[X^N,X^N]$, where
$X^N$ denotes the $N$-th Postnikov stage of $X$, and this
bijection restricts to an isomorphism $\mathcal E_{ \sharp }(X)\cong\mathcal E_{ \sharp }(X^N)$. On
the other hand, consider the exact sequence
\begin{equation}\label{nose2} {1}\to A_j\to \mathcal E_{ \sharp }(X^j)\to \mathcal E_{ \sharp }(X^{j-1})
\end{equation} where $\mathcal E_{ \sharp }(X^j)\to
\mathcal E_{ \sharp }(X^{j-1})$ is just the obvious restriction and $A_j$ its
kernel. Since $G$ acts on any $\mathcal E_{ \sharp }(X^j)$ and $\mathcal E_{ \sharp }(X^1)={1}$, it
will be enough to show that $G$ acts nilpotently on every $A_j$ to
conclude, by an easy induction, that it does so on
$\mathcal E_{ \sharp }(X^N)=\mathcal E_{ \sharp }(X)$.
By classical obstruction theory of liftings (see \cite[Chapter
6.6]{white}) recall that, for $j\ge 2$, there is a bijection
$\varphi\colon B_j\to H^j(X^j;\pi_j)$ where
\begin{itemize}
\item
The cohomology is taken with local coefficients.
\item $B_j$ is the set of homotopy classes of $[X^j,X^j]$ which restrict to the identity on $X^{j-1}$, i.e., homotopy
classes of liftings of $X^j\to X^{j-1}$ to $X^j$.
\item $\varphi(g)=\delta(g,1)$ is the difference cochain of degree $j$ between $g$ and the identity on $X^j$.
\end{itemize}
Recall also that, in general,
$\delta(g,f)=\delta(g,1)+\delta(1,f)$ and that
$\delta(gh,fh)=H^j(h)\bigl(\delta(g,f)\bigr)$. Moreover, if
$h\in\mathcal E_{ \sharp }(X^j)$, $\delta(hg,hf)$ is the image of $\delta(g,f)$
under the map $H^j(X^j;\pi_j)\to H^j(X^j;\pi_j)$ induced by $h$ on
$\pi_j$.
From now on, as in Corollary \ref{noveas}, any $f\in\mathcal E(X)$, and
thus in $G$, shall be considered as a self homotopy class
$(f,\pi_jf^{-1})\in\mathcal E(X,\pi_j)$. Hence, restricting $\varphi$ to
$A_j$ we obtain a map $\varphi\colon A_j\hookrightarrow
H^j(X^j;\pi_j)$ which is a $G$-map with respect to the action
$g\cdot f=g^{-1}fg$, $g\in G$, $f\in A_j$, and the usual action
on $H^j(X^j;\pi_j)$: if $\alpha\in H^j(X_j,\pi_j)$, and $g\in G$,
$g\cdot\alpha$ is the cohomology class represented by the map
$$ X^j{\stackrel{g}{\longrightarrow}}
X^j{\stackrel{\alpha}{\longrightarrow}}
L(\pi_j,j){\stackrel{\xi}{\longrightarrow}} L(\pi_j,j),\qquad
\alpha\in H^j(X^j;\pi_j), $$ with $\xi\colon L(\pi_j,j)\to
L(\pi_j,j)$ induced by $\pi_*(g^{-1})$.
Moreover, this restriction is a group morphism. Indeed, given
$f,h\in A_j$,
$\varphi(fh)=\delta(fh,1)=\delta(f^{-1}fh,f^{-1})=\delta(h,f^{-1})=\delta(h,1)+\delta(1,f^{-1})=\delta(h,1)+\delta(f,1)=
\varphi(f)+\varphi(h)$. As an immediate consequence we then obtain
that $A_j$ is an (abelian!) subgroup of $H^j(X^j;\pi_j)$.
Finally, by Corollary \ref{noveas}, $G$ acts nilpotently on
$H^j(X^j;\pi_j)$ for any $j$. Hence, it does so on $A_j$ and $$
\text{nil}\,_G\,A_j\le \text{nil}\,_G\,H^j(X^j;\pi_j). $$ Thus, by induction on
$j$ using repeatedly \cite[Proposition 4.1]{hmr}, one easily sees
in view of (\ref{nose2}) that
$\text{nil}\,_G\,\mathcal E_{ \sharp }(X)=\text{nil}\,_G\,\mathcal E_{ \sharp }(X^N)\le\sum_{j=2}^N\text{nil}\,_G\,A_j\le\sum_{j=2}^N\text{nil}\,_G\,H^j(X^j;\pi_j)$
and the theorem follows.\hfill$\square$
\section{Groups which fix the homotopy groups}
In this section we establish Theorems \ref{tres} and \ref{cuatro}.
\bigskip
\noindent {\it Proof of Theorem \ref{tres}.} Let
$\alpha\in\mathcal E_{ \sharp p }(X)$. Then, for each $i\le\dim X$, the morphism
$\pi_i(\alpha;\mathbb Z/p)\colon\pi_i(X,\mathbb Z/p)\to\pi_i(X,\mathbb Z/p)$ is just
the identity. On the other hand, the Universal Coefficients
Theorem for homotopy yields the following split short exact
sequence $$
0\longrightarrow\text{Ext}(\mathbb Z/p,\pi_{i+1}X)\longrightarrow\pi_i(X;\mathbb Z/p)\longrightarrow
\hom(\mathbb Z/p;\pi_iX)\longrightarrow 0. $$ Thus, both
$\text{Ext}(\mathbb Z/p,\pi_{i+1}\alpha)$ and $\hom(\mathbb Z/p;\pi_i\alpha)$
are the identity. But observe that
$\text{Ext}(\mathbb Z/p,\pi_{i+1}X)=\pi_{i+1}X\otimes\mathbb Z/p$ so that
$\pi_{i+1}\alpha\otimes\mathbb Z/p=1_{\pi_{i+1}X}$.
Hence, by Proposition \ref{propouno}, for each $i\le\dim X$, the
image of $\mathcal E_{ \sharp p }(X)$ in $\text{aut}\,(\pi_iX)$ is a $p$-group and then, by
Corollary \ref{corolario}, the action of $\mathcal E_{ \sharp p }(X)$ on $\pi_iX$ is
nilpotent. Thus, by Theorem \ref{principal}, $\mathcal E_{ \sharp p }(X)$ is
nilpotent. On the other hand, notice that $\mathcal E_{ \sharp }(X)$ is precisely
the kernel of the obvious map $\mathcal E_{ \sharp p }(X)\to\Pi_{i\le\dim
X}\text{aut}\pi_i(X)$. Hence, as we just proved that the image of
this map is a $p$-group, $\mathcal E_{ \sharp p }(X)/\mathcal E_{ \sharp }(X)$ is a finite $p$-group,
and the proof is complete. \hfill$\square$
\bigskip
\noindent {\it Proof of Theorem \ref{cuatro}.} Write
$\pi_iX=\mathbb Z^{n_i}\oplus\bigl(\oplus_{p\,\,\text{prime}}T_p(\pi_iX)\bigr)$
in which $T_p(\pi_iX)$ is the group of $p$-torsion elements in
$\pi_iX$. Now, if $\alpha\in\cap_{p\,\,\text{prime}}\mathcal E_{ \sharp p }(X)$, then
for each $i\le\dim X$,
$\pi_i(\alpha)|_{T_p(\pi_iX)}\in\text{aut}\,\bigl(T_p(\pi_iX)\bigr)$. Let
$z_1,\ldots,z_{n_i}$ be generators of $\mathbb Z^{n_i}\subset\pi_iX$.
Then $\pi_i(\alpha)(z_k)=\sum_{j=1}^1{n_i}a_{k,j}z_j+\omega$,
where $\omega$ is the torsion part. But this element has to
coincide with $z_k$ mod $p$, for all prime $p$. Therefore, also
for any $p$, $ a_{k,j}=0(\text{mod}\,p) $ for $k\not=j$ and
$a_{k,k}=1(\text{mod}\, p)$, for $1\le k\le n_i$. The only
possible solution is $a_{k,j}=0$, $k\not=j$, and $a_{k,k}=1$. In
other words, $\pi_i(\alpha)(z_k)=z_k+\omega$, in which $\omega$ is
a torsion element.
Adding up, for any element $\gamma\in\pi_iX$,
$\pi_i(\alpha)\gamma-\gamma$ is a torsion element in $\pi_iX$.
This is equivalent to say that the $1$-commutators of the action
of $\cap_{p\,\,\text{prime}}\mathcal E_{ \sharp p }(X)$ on $\pi_iX$ live in the
torsion part of $\pi_iX$. However, by Corollary \ref{corolario},
the action of $\cap_{p\,\,\text{prime}}\mathcal E_{ \sharp p }(X)$ on the torsion
part is nilpotent and therefore, the action on $\pi_iX$ is also
nilpotent. Apply Theorem \ref{principal} and the proof is
complete.\hfill$\square$
\begin{remark} {\em We end up by noting that the hypothesis of Theorem \ref{tres} are necessary. Indeed, consider
$X=K\bigl((\mathbb Z/2)^2,n\bigr)$ and observe that, for a prime $p$
different from $2$, $\mathcal E_{ \sharp p }(X)=\mathcal E(X)=GL_2(\mathbb Z/2)\cong \Sigma_3$
which is not nilpotent.
On the other hand, take $X=K(\mathbb Z^2,n)$ for which
$\mathcal E(X)=GL_2(\mathbb Z)$. In this case $\mathcal E_{ \sharp }(X)=\{1\}$ and, for any
prime $p$, $\mathcal E_{ \sharp p }(X)$ fits in the following short exact sequence $$
\{1\}\to\mathcal E_{ \sharp p }(X)\to GL_2(\mathbb Z)\to GL_2(\mathbb Z/p)\to\{1\} $$ where the
surjection $GL_2(\mathbb Z)\to GL_2(\mathbb Z/p)$ is just the mod-$p$
reduction. Hence $\mathcal E_{ \sharp p }(X)=\mathcal E_{ \sharp p }(X)/\mathcal E_{ \sharp }(X)$ is an infinite, non
nilpotent group.}
\end{remark}
\nocite{*}
\bibliographystyle{plain}
|
\section{Introduction}
A wide range of minimally invasive therapies have been developed for cancer treatment, additionally to open surgery \cite{Ahmed.2014,mauri2017technical,tomasian2018percutaneous}. One of these methods is the use of microwave ablation (MWA). Especially for smaller tumors, MWA shows promising results for treatment \cite{tehrani2020use}. As the minimal ablative margin (MAM) is crucial for the local tumor progression (LTP), it is of greatest importance to assess if the malignancy has been adequately and completely treated, regardless of the etiology. For each millimeter increase of the MAM, a 30\% reduction of the relative risk for LTP was found. The MAM is especially important as the only significant independent predictor of LTP (p = 0.036) \cite{laimer2020minimal}. During the intervention, magnetic resonance (MR) imaging offers several advantages like a good soft-tissue contrast without the need of contrast agent, the free orientation and positioning of single slice scans and the possibility to accurately track changes in the temperature inside the tissue \cite{Gorny.2019,Kagebein.2018c,Rieke.2008,Senneville.2007}.\\
\\
\textbf{Contribution.} In this work, we propose a novel approach for the creation of a volumetric thermometry map without the development of a fully 3D sequence. The introduced 2.5D thermometry method utilizes any common 2D gradient-echo (GRE) sequences. Therefore, possible temporal limitations are less restricting than for the 3D sequences and images with higher resolution may be acquired offering standard thermometry accuracy of around $1^\circ$C deviation while being more robust towards MR inhomogeneities \cite{Gorny.2019}. We will show that our method is well-suited to reconstruct the actual coagulation zone after thermal ablation.\\
\\
\textbf{Related Work.} Zhang et al.\cite{zhang2019variable} propose a golden-angle‐ordered 3D stack‐of‐radial multi-echo spoiled gradient‐echo sequence with a variable flip angle. The image reconstruction is performed offline offering a temporal resolution between 2s-5s. Jiang et al.\cite{Jiang.2020} use an accelerated 3D echo-shifted sequence and the Gadgetron framework for image reconstruction. Temporal resolution lies at around 3s with a temperature error of less than $0.65^\circ$C. Quah et al.\cite{quah2020simultaneous} are aiming at an increased volume coverage for thermometry without multiple receive coils. An extended k‐space hybrid reconstruction was used, yielding an error of $<1^\circ$C and an acquisition time of 3.5s for each image. Fielden et al.\cite{Fielden.2018} present a comparison study between cartesian, spiral-out and retraced spiral-in/out (RIO) trajectories. Using the 3D RIO sequence, they achieved a true temporal resolution of 5.8s with a temporal standard deviation of $1.32^\circ$C. Marx et al.\cite{Marx.2014} introduced the MASTER sequence for volumetric MR thermometry acquisition, acquiring six slices in around 5s. In a later work \cite{Marx.2017} they use optimized multiple-echo spiral thermometry sequences, which yield a better precision than the usual 2D Fourier transform thermometry. Image acquisition takes between 7s-11s. Svedin et al.\cite{svedin2018multiecho} make use of a multi-echo pseudo-golden angle stack-of-stars sequence and offline image reconstruction using MATLAB. They achieved a temporal resolution of around 2s and a spatial average of the standard deviation through a time of $0.3-1.0^\circ$C. Odéen et al.\cite{odeen2016mr} propose the use of a 3D gradient recalled echo pulse sequence with segmented EPI readout. To estimate the temperature change, they also integrate a bioheat equation. They achieved a temperature root mean square error of $1.1^\circ$C. Golkar et al.\cite{golkar2018fast} introduce a fast GPU based simulation approach for cryoablation monitoring. The reconstruction takes 110s and the final result shows a Dice coefficient of 0.82. A summarize of the related work in comparison to our method is shown in Table \ref{tab:RelatedWork}.
\begin{table}[ht]
\centering
\caption{Overview about the related work in comparison to this work. Every work has been observed according to the following: 1) The kind of image sequence used. 2) The online or offline capability of the reconstruction framework. 3) The temporal resolution of the whole image acquisition in seconds. 4) The temperature accuracy in °C. 5) The resulting Dice Score similarity measurement if available.}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& \begin{tabular}[c]{@{}c@{}}Image \\ Sequence\end{tabular} & \begin{tabular}[c]{@{}c@{}}Reconstruction \\ Framework\end{tabular} & \begin{tabular}[c]{@{}c@{}}Temporal \\ Resolution {[}s{]}\end{tabular} & \begin{tabular}[c]{@{}c@{}}Temperature \\ Accuracy {[}°C{]}\end{tabular} & Dice Score \\ \hline
Zhang et al.{[}20{]} & 3D & offline & 2 - 5 & --- & --- \\ \hline
Jiang et al.{[}6{]} & 3D & \begin{tabular}[c]{@{}c@{}}online \\ (Gadgetron)\end{tabular} & 3 & \textless 0.65 & --- \\ \hline
Quah et al.{[}13{]} & Stack of 2D & hybrid & 3.5 & \textless 1 & --- \\ \hline
Fielden et al.{[}3{]} & 3D & online & 5.8 & 1.32 & --- \\ \hline
Marx et al.{[}10{]} & Stack of 2D & --- & 5 & 1.3 & --- \\ \hline
Marx et al.{[}9{]} & Stack of 2D & online & 7 - 11 & 0.29 - 0.65 & --- \\ \hline
Svedin et al. {[}17{]} & 3D & \begin{tabular}[c]{@{}c@{}}offline \\ (MATLAB)\end{tabular} & 2 & 0.3 - 1.0 & --- \\ \hline
Odeen et al.{[}12{]} & Stack of 2D & \begin{tabular}[c]{@{}c@{}}offline \\ (MATLAB)\end{tabular} & 2.4 - 4.8 & \textless 1.1 & --- \\ \hline
Golkar et al.{[}4{]} & 3D & --- & 110 & --- & 0.82 \\ \hline
This work & Single 2D & online & variable & 1 & 0.75 \\ \hline
\end{tabular}
\label{tab:RelatedWork}
\end{table}
\section{Material and Method}
\label{sec:MaterialAndMethod}
\subsection{Image Acquisition}
\label{subsec:ImageAcquisition}
The proposed 2.5D thermometry relies on sampling the volume of interest (VOI) using a common 2D GRE sequence and [1,...,n] different orientations. The GRE sequence can directly reconstruct magnitude and phase images simultaneously. To ensure a proper sampling of the VOI in our setup the GRE sequence is rotated by $22.5^\circ$ around the applicator's main axis. This results in an evenly distributed sample of eight different orientations. To increase the spatial resolution, the angles between the acquired scans should be as high as possible, resulting in the following acquisition order: $0 ^\circ$, $90 ^\circ$, $45 ^\circ$, $135 ^\circ$, $22.5 ^\circ$, $112.5 ^\circ$, $67.5 ^\circ$, and $157.5 ^\circ$. To reduce the delay between image acquisition and visualization of the volumetric thermometry map, the SIEMENS Healthineers Access-I Framework was integrated. The framework allows for fetching the image data directly from SIEMENS MR devices without an intermediate imaging archive system.
\subsection{2.5D Thermometry Reconstruction}
\label{subsec:25DThermometryReconstruction}
Before treatment starts, reference phase images are acquired for each of the eight orientations. Each newly acquired phase image will start computing the up-to-date 2D thermometry map for the current orientation during the treatment. To do so, the proton resonance frequency shift (PRFS) method is used as described by Rieke et al. \cite{Rieke.2008}. The temperature $T$ based on the PRFS is computed using the following Equation
\begin{equation}
\label{eq:PRFS}
T = \frac{\phi(t) - \phi (t_0)}{\gamma \alpha B_0 TE} + T_0
\end{equation}
with $\phi(t) - \phi (t_0)$ defining the phase difference between the current time point $\phi(i)$ and the reference timepoint $\phi(i_0)$, $\gamma = 42,576\frac{MHz}{T}$ representing the gyromagnetic ratio of hydrogen protons, $\alpha = 0.01\frac{ppm}{\Delta T}$ representing the proton resonance frequency change coefficient, $B_0$ representing the used magnetic field strength and $TE$ representing the used echo time. The constant $T_0$ needs to be added to the temperature since Equation \ref{eq:PRFS} otherwise only computes the temperature change, neglecting the tissue's base temperature. The Access-I integration and 2D thermometry computation were implemented as modules using MeVisLab 3.4.1 \cite{ritter2011medical}.
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{MICCAI2021Workflow_v6.PNG}
\caption{Schematic overview of the proposed method.}
\label{fig:MethodOverview}
\end{figure}
The 2.5D thermometry reconstruction itself was implemented using C++. A schematic overview of the method can be seen in Figure \ref{fig:MethodOverview}. To handle the voxel values during slice rotation every cartesian coordinate was mapped to the corresponding cylindrical coordinate representation using Equation \ref{eq:CartesianToPolar},
\begin{align}
\label{eq:CartesianToPolar}
P_r(x,y,z) &= P_c(r, \theta, z)\\
r &= \sqrt{\left(x-x_c \right)^2 + \left(y-y_c\right)^2}\nonumber \\
\theta &= atan2\left(\frac{x-x_c}{y-y_c}\right)\nonumber
\end{align}
where $x,y$ represents the Cartesian coordinates of the current voxel and $x_c,y_c$ represents the Cartesian coordinates of the centerline corresponding to the applicator's axis for every slice $z$ in the reconstructed volume. Upon acquisition of the reference images, a multi-dimensional population map is created. For each voxel $(x_i,y_i,z_i)$ in the reconstructed volume, this population map holds information about the radius $r$ and angle $\theta$ of the cylindrical coordinates, the general interpolation weight $I_w$, the adjacent interpolation partner coordinates $IP_{left}(x,y)$ and $IP_{right}(x,y)$ in the 2D live data as Cartesian representation and the weights $w_1$ and $w_2$ of those interpolation partners. The weights may be acquired using Equation \ref{eq:Weights},
\begin{align}
\label{eq:Weights}
w_{1} &= \left\vert\frac{\theta_{IP_{left}} - \theta_{i}}{\theta_{IP_{left}} - \theta_{IP_{right}}}\right\vert\\
w_2 &= 1 - w_1\nonumber
\end{align}
|
with $\theta_{i}$ representing the cylindric angle of the current Voxel $i$ and $\theta_{IP_{left}}$, $\theta_{IP_{right}}$ representing the orientation angles of the left and right interpolation partners, respectively. The 2D population map can be applied to every slice of the final 3D output volume, reducing the computational power needed. During the intervention, every acquired live image triggers the reconstruction of the up-to-date 2.5D thermometry map. Here, the heat value for each voxel is reconstructed using Equation \ref{eq:Reconstruction},
\begin{equation}
\label{eq:Reconstruction}
T_i = I_w \cdot (w_{1} \cdot T_{IP_{left}} + w_{2} \cdot T_{IP_{right}})
\end{equation}
with $T_i$ representing the temperature of the current voxel $i$ and $T_{IP_{left}}$, $T_{IP_{right}}$ representing the temperature of the adjacent interpolation partners. Occurring vessels or other structures, which cause a heat sink effect are segmented during the intervention planning. Subsequently, the segmented structure is saved as an additional Look-Up Volume. Here, each voxel can be checked if it is part of a heat sink structure. Using this knowledge, the interpolation weight $I_w$, which ranges between $[0,1]$, may be adjusted. Figure \ref{fig:Reconstruction} shows a single dimension of the population map for parameter weighting, a reconstructed heat map, a coagulation estimation based on an empirically defined threshold and the corresponding ground truth segmentation. The source code is available for download at \url{https://github.com/jalpers/2.5DThermometryReconstruction}.
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{ImageReconstruction_v3.png}
\caption{A) Example population map for output weights color coded on gray scale. B) Reconstructed volumetric heat map. C) Estimated coagulation necrosis based on a threshold of $57^\circ$C. D) Manually segmented ground truth.}
\label{fig:Reconstruction}
\end{figure}
\subsection{Evaluation}
\textbf{Phantom design.} To create a first proof of concept, a pilot study was conducted to evaluate the 2.5D thermometry reconstruction using 13 bio protein phantoms as described by Bu Lin et al.\cite{BuLin.2008}. The coagulation zone's visibility in the post-treatment data sets increased by adding a contrast agent ($0,5\mu mol/L$ Dotarem) to the phantoms. For six phantoms, additional polyvinyl chloride (PVC) tubes with a diameter of $5mm$ and a wall thickness of $1mm$ were integrated into the phantoms (three single-tubes, three double-tubes) to simulate a possible heat sink (HS) effect.\\
\\
\textbf{Experimental setup.} The applicator of the permittivity feedback control MWA system (MedWaves Avecure, Medwaves, San Diego, CA, USA, 14G) was placed inside the phantom by sight and secured in position. Subsequently, the phantoms were placed inside a $1,5T$ MR scanner (Siemens Avanto, Siemens Healthineers, Germany). The coaxial cables connected to the applicator and MW generator were led through a waveguide. Chokes and electrical grounding measures were added as described by Gorny et al.\cite{Gorny.2019} to reduce radio frequency interference. In the case of the perfusion phantoms, the PVC tubes were led through the wave guide. They were connected to a diaphragm pump and a water reservoir outside the scanning room. A flow meter (SM6000, ifm electronic, Essen, Germany) was interposed between the reservoir and the pump, providing a flow rate of $800mL/min$. Observations showed a moderate HS effect using this setup with a maximum antenna power of 36W. Additionally, temperature sensors were inserted in two phantoms to experimentally verify the temperature accuracy of $1 ^\circ C$.
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{experimental_setup_v1.2.jpg}
\caption{Experimental evaluation setup. Flexible tubes (blue) lead the water (a) through a flow meter (b), a diaphragm pump (c) and the bio protein phantom (e). The coaxial cables (red) connect the applicator with the MW generator d).}
\label{fig:StudyDesignSketch}
\end{figure}
Right before treatment, ten reference phase images were acquired and averaged for each orientation to compensate for static noise. The MWA duration was set to 15 minutes with a temperature limit of $90^\circ$C. The GRE sequence offers a slice thickness of $5mm$, a field of view (FOV) of $256mm * 256mm$, a matrix of $256 * 256$, and a bandwidth of $260 Hz/Px$. Image acquisition took around $1.1s$ with a $5s$ break to simulate the temporal resolution for a breathing patient. The TE was $3.69ms$, the TR $7.5ms$, and the flip angle $7 ^\circ$. For post-treatment observation a 3D turbo spin echo (TSE) sequence (TE = $156ms$, TR = $11780ms$, flip angle = $180 ^\circ$, matrix = $256 * 256$, FOV = $256mm * 256mm$, bandwidth = $40 Hz/Px$, slice thickness = $1mm$) was used. The 3D TSE allows for proper visualization of the real coagulation zone due to a very high tissue contrast. Extraction of the coagulation ground truth was done manually by a clinical expert using MEVIS draw (Fraunhofer MEVIS, Bremen, Germany). All data sets used are available for download at \url{http://open-science.ub.ovgu.de/xmlui/handle/684882692/89}.\\
\\
\textbf{Statistical evaluation.} Final evaluation of the acquired data was performed using the dice similarity coefficient (DSC) as explained in Equation \ref{eq:DSC}
\begin{equation}
\label{eq:DSC}
DSC = \frac{2*TP}{2*TP + FP + FN}
\end{equation}
with $TP$ representing the true positives, $FP$ the false positives and $FN$ the false negatives. Additionally, the standard error of the mean (SEM) was computed at a confidence level of 95\% (p = 0.05) using Equation \ref{eq:SEM}
\begin{align}
\label{eq:SEM}
\sigma &= \sqrt{\frac{\sum(x_i-\bar{x})^2}{N-1}}\\
SEM &= \frac{\sigma}{\sqrt{N}} * 1.96\nonumber
\end{align}
with $\sigma$ representing the standard deviation, $x_i$ the current sample, $\bar{x}$ the mean value and $N$ the sample size. To compute the SEM at a confidence level of 95\% it has to be multiplied by 1.96, which is the approximated value of the 97.5 percentile of the standard normal distribution.
\section{Results}
Summarized evaluation results can be seen in Figure \ref{fig:SummarizedResults}. Empirically determined coagulation thresholds were set between $51^\circ$C and $61^\circ$C depending on each phantom's pH value.
\begin{figure}[ht]
\centering
\includegraphics[width=1\textwidth]{DiceSensitivityResults_v4.png}
\caption{Summarized evaluation results for phantoms without HS effect, phantoms with HS effect and the overall results. Note that the data range [0,0.3] was left out because no data points are present in that range.}
\label{fig:SummarizedResults}
\end{figure}
It is noticeable that the DSCs for HS phantoms show a very high SEM with $0.70\pm0.15 (\pm21.25\%)$ and $0.74\pm0.06 (\pm8.49\%)$ regarding the sensitivity. The high range results from a corrupted dataset due to heavy artifacts within the image data. Leaving the corrupted dataset out of the evaluation, the SEM shows a significantly lower deviation of $0.76\pm0.062 (\pm8.07\%)$ and $0.77\pm0.048 (\pm6.25\%)$ for the DSC and sensitivity, respectively. Observations show a slightly higher DSC and sensitivity for phantoms without any HS effect. Here, the values range from $0.79\pm0.04 (\pm 4.53\%)$ and $0.79\pm0.04 (\pm 5.55\%)$, respectively. Evaluation showed an overall SEM for the DSC of $0.75\pm0.07 (\pm9.76\%)$ and a SEM for sensitivity of $0.77\pm0.04 (\pm4.99\%)$. To evaluate the computational effort, every major step was performed 100 times. The creation of the population map and the heat sink look up volume took $25.53ms\pm 3.33ms$ and $3.91s\pm 0.59s$, respectively. These two steps need to be done just once before start of the treatment. The reconstruction of the 2.5D thermometry map was performed in $18.02ms\pm 5.91ms$ on a customary workstation (Intel( R) Core(TM) i5-6200U CPU, double-core 2.30GHz, 8GB RAM, Intel(R) HD Graphics 520). This reconstruction will be performed every time a new image is acquired during treatment.
\section{Discussion and Conclusion}
The aim of this work is to develop a volumetric thermometry map, which can be applied to a wide variety of clinical setups. Therefore, our work heavily relies on the up-to-date standard 2D GRE sequence for image acquisition. This allows for the standard accuracy of the thermometry up to $1.0^\circ$C. Nonetheless, the sampling of the 3D volume also results in some disadvantages, which need to be addressed in the future. First, the diffusion of the heat inside the tissue is not linear over time. Therefore, it would be necessary to include an adaptive temporal and spatial resolution depending on the current intervention time. A new study should be conducted to identify an optimal sequence protocol for this 2.5D thermometry approach. Second, we found that the reconstruction sometimes shows stair-case artifacts. Because only one image is acquired every few seconds, the time difference between adjacent orientations may be very high. The temperature difference for each voxel dependent on the applicator's radius may be computed and applied to the corresponding voxel on every other out-of-date data to compensate for this error. This transfer of the heat gradient may improve the reconstruction accuracy. Another approach may be the use of a model-based reconstruction to take different tissue characteristics into account. To pseudo-increase the temporal resolution, bio heat transfer simulations may also be included during reconstruction. The acquired live data may be able to adjust the simulation parameters to increase the simulation accuracy. Finally, our study only performs on bio protein phantoms. Results show a proof of concept for the proposed method, but it still has to be evaluated in real tissue and a more realistic clinical environment. Therefore, perfused ex vivo livers may be a way to go in the future. Additionally, we currently assume a breath-holding state or at least a breath-triggered image acquisition. Research shows that a wide range of interventional registration methods is available, but further investigations in this area still need to be done to create an applicable method. The last issues arise because of the MR inhomogeneity during image acquisition. The slightest disturbances may result in heavy image artifacts. Proper shielding of the MW generator is needed to reduce the SNR loss over time thus increasing the thermometry and reconstruction accuracy. \\
\\
In conclusion, we proposed a novel method for 2.5D thermometry map reconstruction based on common GRE sequences rotated around the applicator's main axis. A pilot study was conducted using bio protein phantoms to simulate cases with possible heat sink effects and without. The evaluation shows promising results regarding the DSC of the reconstructed 2.5D thermometry map and a manually defined ground truth. Future work should address the reconstruction method's improvement by integrating further apriori knowledge like the estimated shape of the heat distribution. Furthermore, a more realistic study should be conducted with bigger sample size and real tissue. In sum, the method shows a high potential to improve the clinical success rate of minimally invasive ablation procedures without necessarily hampering the standard clinical workflow of the individual clinician.
|
\section{Introduction}
The inflationary paradigm in the early Universe lays a strong foundation for successful description of the CMB and large scale structure (LSS) of the Universe so far \cite{Starobinsky:1980te,Starobinsky:1979ty,Akrami:2018odb} with the predictions of spectral index $n_s=0.9649\pm 0.0042$ for $50-60$ e-foldings of quasi-de Sitter expansion. Besides its apparent success, some CMB anomalies seem to arise and challenge any {current} model of inflation.
Understanding the observed anomalies in the latest Planck's CMB data could further elucidate some fundamental aspects of the quantum nature of the fluctuations which are believed to seed the large scale structure (LSS) of the Universe. Concretely, one of the main CMB anomalies is the presence of a hemispherical power asymmetry (HPA) at large angular scales \cite{Akrami:2019bkn} which has been seen from the NASA's Wilkinson Microwave Anisotropy Probe (WMAP) to the latest Planck data and its significance is standing now at $3.3\sigma$ \cite{Akrami:2014eta}.
In short, the HPA is an asymmetry in the two point temperature correlations on opposite hemispheres of the sky, with slightly higher power in the southern ecliptic hemisphere and slightly lower power in the northern ecliptic hemisphere \cite{Creswell:2021eqi}. This is a significant challenge to our standard model of (inflationary) cosmology, which predicts isotropy \cite{Yeung:2022smn,Aluri:2022hzs}.
This anomaly seems to appear significantly at low-multipoles $\ell =2-64$ \cite{Schwarz:2015cma}. The HPA is phenomenologically represented by the following form of the position dependent primordial power spectrum parametrized as \cite{Lyth:2013vha,Lyth:2014mga}
\begin{equation}
\mathcal{P}_\mathcal{R}\left( k,\,\hat{\boldsymbol{n}}\right) \simeq \mathcal{P}_{\mathcal{R}\:\: iso} (k)\left( 1+2A(k) \hat{\textbf{p}}\cdot\hat{\textbf{n}}\right) \,,
\label{pwa}
\end{equation}
where $\mathcal{P}_{\mathcal{R}\:\: iso} (k)$ is the statistically homogeneous and isotropic power spectrum, $A(k)$ is the amplitude of the observed dipolar asymmetry which is constrained as $\vert A \vert = 0.066\pm 0.021$ ($3.3\sigma$) at the large angular scales $\ell <64$ or at the wave numbers $k \lesssim 0.0045\textrm{ Mpc}^{-1}$. The HPA seems to be present even up to the multipoles $~600$ as seen by the Planck 2013, Hansen \emph{et al.} and Aiola \emph{et al.} \cite{Planck:2013lks,Hansen:2008ym,Aiola:2015rqa}.
Here $\hat{\boldsymbol{p}}$ is the direction of maximal symmetry and $\hat{\textbf{n}}=\frac{\boldsymbol{x}}{x_{\text{ls}}}$ is the line of sight from earth and $x_{\text{ls}} = 14,000 \textrm{Mpc}^{-1}$ is the co-moving distance to the surface of last scattering.
Following \eqref{pwa} we can derive \cite{Lyth:2013vha}
\begin{equation}
A(k) = \frac{\mathcal{P}_\mathcal{R}\left( k,\,\hat{\textbf{n}} \right) - \mathcal{P}_\mathcal{R}\left( k,\,-\hat{\textbf{n}} \right) }{4\mathcal{P}_{\mathcal{R}\:\: iso}}\,.
\label{Akg}
\end{equation}
The best explanation for the HPA, so far, invoked an additional modulated superhorizon fluctuation with a wavelength larger than the last scattering surface, whose interactions with the short wavelength curvature perturbations introduces large scale dependent primordial non-Gaussianities \cite{Schmidt:2012ky,Erickcek:2008sm,Erickcek:2008jp,Byrnes:2015asa,Byrnes:2015dub,Namjoo:2014nra}. It was first noted in \cite{Lyth:2013vha,Lyth:2014mga} that both the HPA and and the power suppression (PS) at low-multipoles \cite{Akrami:2019bkn} could be addressed in a single framework in the context of scale dependent primordial non-Gaussianities and curvaton-type scenarios. This later has been generalized and phenomenologically studied in model independent way \cite{Adhikari:2015yya,Byrnes:2016zxb}. In addition, the HPA has been addressed via a spatial dependent parametrization of the $\alpha$-vacua Bogoliubov coefficients of de~Sitter space \cite{Ashoorioon:2015pia}, in the context of anisotropic inflationary models involving gauge fields \cite{Maleknejad:2012fw} and also with spatial dependent initial conditions for the inflaton field \cite{Donoghue:2007ze}. So far all the studies ascertained that the HPA cannot be explained by the standard single-field slow-roll inflationary scenario \cite{Akrami:2014eta,Hansen:2008ym}.
Despite the above mentioned had hoc explanations for the HPA \cite{Akrami:2019bkn}, Planck data actually hints at it for both 2-point and 3-point correlations (even the equilateral shape for which three wave numbers of the modes are equal). This means that the anomaly seems to be there in all the $n$-point correlations, which begs for a fundamental explanation. Therefore, the answer must be there somewhere in the description of inflationary quantum fluctuations.
The main goal of this paper is to investigate whether the HPA can be explained through a new understanding of the quantization of the inflationary perturbations without introducing any new parameters and any new degrees of freedom. Thus we only focus on single field slow-roll inflation and propose a new vacuum for inflationary quantum fluctuations. Being more precise, we pay careful attention to the discrete symmetries such as $\mathcal{C}\mathcal{P}\mathcal{T}$ which is expected to be spontaneously broken in a dynamical spacetime. Our scheme of quantization exactly exploits this last aspect leading to a promising explanation for the HPA anomaly, with additional prediction for primordial in gravitational wave spectrum as well.
Throughout the paper, we use the metric signature $\left( -,\,+,\,+,\,+ \right)$, overdot and $^\prime$ denotes derivative with respect to cosmic time (t) and conformal time ($ \tau$) respectively, overbar denotes background values for flat Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime described by the metric $ds^2= -dt^2+a^2(t)d\textbf{x}^2$, $H=\frac{\dot{a}}{a}$ denotes the Hubble parameter.
Four-dimensional indices are labelled by small Greek letters and three dimensional quantities are denoted by $i,j =1,2,3$. We set $\hbar=c=1$ and $M_p = \frac{1}{\sqrt{8\pi G}}=1$.
\section{Brief review of inflationary quantum fluctuations in single field inflation}
\label{sec:review}
In this section, we review standard inflationary quantum fluctuations and the corresponding derivation of observables \cite{Martin:2015dha,Baumann:2018muz}. In standard single field inflation, we start with a classical action for a canonical scalar field with a suitable choice of potential.\footnote{Note that inflationary scenarios like Starobinsky and Higgs inflation can be brought to this form via a conformal transformation \cite{Kehagias:2013mya}.}
\begin{equation}\label{action-S}
S= \int d^4x\sqrt{-g} \left[ \frac{M_p^2}{2} R- \frac{1}{2} (\partial_\mu \phi)( \partial^\mu \phi )-V\left( \phi \right)\right] \,,
\end{equation}
To implement inflationary framework of expansion, we first solve the equations of motion of \eqref{action-S} in flat FLRW spacetime and obtain a quasi-de Sitter solution described by the following conditions on the slow-roll parameters:
\begin{equation}
\epsilon = -\frac{\dot{H}}{H^2}\ll 1,\quad \eta = \frac{\dot{\epsilon}}{H
\epsilon} \ll 1\,\text{,}
\label{sldef}
\end{equation}
which can be computed from the Friedmann and Raychaudhuri equations given by
\begin{equation}
H^2 = \frac{1}{3}\left( \frac{\dot{\phi}^2}{2}+V(\phi) \right),\quad \dot{H} = -\frac{\dot{\phi}^2}{2}\,.
\label{Freq}
\end{equation}
The next step is to study the scalar and tensor perturbations around the obtained inflationary background solution. When it comes to the scalar part we have perturbations arising from metric fluctuations ($g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}$) and also the scalar field fluctuations ($\phi = \bar{\phi}+\delta\phi$). The metric fluctuations can be represented by the Arnowitt-Deser-Misner (ADM) metric of the form
\begin{equation}\label{ADMmetric}
ds^2 = - a^2\left( \tau \right) \Big(N^2d\tau^2+ \gamma_{ij} \left( dx^i+N^i d\tau \right) \left( dx^j+N^j d\tau \right) \Big)\,.
\end{equation}
where $d\tau = \frac{dt}{a}$ is conformal time, $ N$ and $N_i$ are the lapse and shift functions respectively. In the unitary gauge, we fix
$\delta\phi=0$ then we linearly expand the ADM metric as
\begin{equation}\label{eq:ADMfluctuation}
N= 1+\delta N,\quad N_i = a\partial_i\chi,\quad \gamma_{ij} = \left( 1+2\zeta \right)\delta_{ij}+h_{ij}\,,
\end{equation}
where $\chi$ is a scalar function of spacetime, $\zeta $ is the curvature perturbation , $h_{ij}$ is the transverse and traceless spin-2 fluctuation. From the linear perturbed equations of motion we obtain the following constraints
\begin{equation}
\begin{aligned}
\delta N & = \frac{\dot{\zeta}}{H} \\
\partial^2\chi & = -\frac{\partial^2 \zeta}{H}+ \epsilon a^2\dot{\zeta}
\end{aligned}
\end{equation}
{Substituting \eqref{ADMmetric} into the action \eqref{action-S} and expanding to second order in the fluctuations, we find the second order action for the scalar perturbation}:
\begin{equation}\label{scalar}
\delta^{(2)}S_{s} = \frac{1}{2}\int d\tau d^3x a^2\frac{\phi^{\prime 2}}{\mathcal{H}^2} \Bigg[ \zeta^{\prime 2} -\left( \partial\zeta \right)^2 \Bigg]
\,\text{,}
\end{equation}
where $\mathcal{H} = \frac{a^\prime}{a}= aH$.
Similarly we obtain the following second order action for tensor fluctuations:
\begin{equation}
\delta^{(2)}S_h = \frac{1}{8} \int d\tau d^3x a^2\Bigg[ h_{ij}^\prime h_{ij}^\prime -\partial_k h_{ij}\partial^k h^{ij} \Bigg]
\,\text{.}
\end{equation}
What we have so far is totally classical and we haven't promoted any fields and conjugate momenta to operators. To quantize the fluctuations we first redefine the fields as
\begin{equation}
v = a\frac{\dot{\phi}}{H}\zeta,\quad u_{ij} = \frac{a}{2} h_{ij} \,,
\label{rescaleeq}
\end{equation}
The second order actions in these new variables (which are famously known as Mukhanov-Sasaki variables) become
\begin{equation}
\delta^{(2)}S_s = \frac{1}{2} \int d\tau d^3x \Big[v^{\prime 2}-\left(\partial_iv\right)^2-\left( \frac{\nu_s^2-\frac{1}{4}}{\tau^2} \right) v^2\Big],
\label{screfa}
\end{equation}
where
\begin{equation}
\nu_s \approx \frac{3}{2}+\epsilon+\frac{\eta}{2}
\end{equation}
\begin{equation}
\delta^{(2)}S_s = \int d\tau d^3x \Big[u_{ij}^{\prime 2}-\left(\partial_iu_{ij}\right)^2-\left( \frac{\nu_t^2-\frac{1}{4}}{\tau^2} \right) u_{ij}^2\Big],
\label{tensor-mode}
\end{equation}
and
\begin{equation}
\nu_t \approx \frac{3}{2}+\epsilon\,.
\end{equation}
We now promote the fields $v,\, u_{ij}$ and their corresponding conjugate momenta $\pi_s,\, \pi_{ij}$ to operators and define the canonical commutation relations as
\begin{equation}
\begin{aligned}
\Big[\hat{v}\left( \tau,\,\textbf{x} \right),\, \hat{\pi}_s\left( \tau,\, \textbf{x}^\prime \right)\Big] & = i\delta\left( \textbf{x}-\textbf{x}^\prime \right) \\
\Big[\hat{u}_{ij}\left( \tau,\,\textbf{x} \right),\, \hat{\pi}_{ij}\left( \tau,\, \textbf{x}^\prime \right)\Big] & = i\delta\left( \textbf{x}-\textbf{x}^\prime \right)
\label{cancom}
\end{aligned}
\end{equation}
We expand the field $\hat{v}\left( \tau,\, \textbf{x} \right)$ in terms of creation and annihilation operators $ a_\textbf{k},\, a^\dagger_{\textbf{k}} $ as
\begin{equation}
\begin{aligned}
\hat{v}\left( \tau,\, \textbf{x} \right) & = \frac{1}{\left( 2\pi \right)^{3/2}}\int d^3k \hat{v}_\textbf{k}\left( \tau \right) e^{i\textbf{k}\cdot \textbf{x}} \\
& = \frac{1}{\left( 2\pi \right)^{3/2}}\int d^3k\Bigg[ a_\textbf{k} {v}_k\left( \tau \right) e^{i\textbf{k}\cdot \textbf{x}} + a_\textbf{k}^\dagger {v}_k^\ast e^{-i\textbf{k}\cdot \textbf{x}} \Bigg]\,
\end{aligned}
\end{equation}
where $v_k$ is the mode function which satisfies Mukhanov-Sasaki (MS) equation
\begin{equation}
v_k^{\prime\prime} + \left( k^2-\frac{v_s^2-\frac{1}{4}}{\tau^2} \right) v_k = 0\,.
\label{MS-equation}
\end{equation}
and $\left[ a_\textbf{k},\, a^\dagger_{\textbf{k}^\prime}\right] = \delta\left( \textbf{k}-{\textbf{k}}^\prime \right)$
The generic solution of \eqref{MS-equation} by treating $\nu_s\approx\, {\rm const}$ is
\begin{equation}
\begin{aligned}
& v_k\left( \tau \right) \\ &= \dfrac{\sqrt{\pi}}{2} \left( -\tau \right)^{1/2} e^{i\left( 2\nu_s+1\right) \pi}\Bigg[A_k H^{(1)}_{\nu_s}\left( -k \tau \right)+ B_k H^{(2)}_{\nu_s}\left( -k \tau \right)\Bigg].
\end{aligned}
\end{equation}
where $A_k$ and $B_k$ are the Bogoliubov coefficients.
Due to the commutation relations \eqref{cancom}, the mode functions must satisfy
\begin{equation}
v_kv^{\ast\prime}_k - v_k^\ast v^{\prime}_k = i
\end{equation}
which implies
\begin{equation}
\vert A_k\vert^2 - \vert B_k\vert^2 =1\,.
\label{wrAB}
\end{equation}
From the above relation we can carry a parametrization $A_k = \cosh\alpha$ and $B_k =\sinh\alpha$. The case $\alpha\neq 0$ corresponds the well-known $\alpha-$ vacuum in the limit $\left( \epsilon,\,\eta \right)\to 0$ \cite{Allen:1985ux}.
The case $\alpha=0$ corresponds to the Bunch-Davies (BD) vacuum in which case the mode functions behave like Minkowski-one in the limit $\tau\to -\infty$ when they are deep inside the horizon i.e., when $k\gg aH$ the mode function of the BD vacuum becomes
\begin{equation}
v_k \simeq \sqrt{\frac{1}{2k}}e^{-ik\tau}
\end{equation}
Thus the BD vacuum is given by
\begin{equation}
a_\textbf{k} \vert 0\rangle =0\,.
\end{equation}
In the limit $\left( \epsilon,\, \eta \right)\to 0$ the mode functions can be approximated as
\begin{equation}
v_k\left( \tau \right) = \sqrt{\frac{1}{2k}} e^{-ik\tau}\left( 1-\frac{i}{k\tau} \right)
\end{equation}
The two point correlations of the fields can be computed as
\begin{equation}
\langle 0 \vert \hat{v}\left( \textbf{x},\, \tau \right) \hat{v}\left( \textbf{x}+\boldsymbol{\xi}, \, \tau \right)\vert 0\rangle = \frac{4\pi}{\left( 2\pi \right)^3} \int \frac{dk}{k} \frac{\sin k \xi}{k\xi} k^3 \vert v_k\vert^2\,.
\end{equation}
The power spectrum of the canonical scalar field $v$ is defined as
\begin{equation}
\begin{aligned}
\langle 0\vert \hat{v}_\textbf{k}\hat{v}_{\textbf{k}^\prime}\vert 0\rangle & = \vert v_k\vert^2 \delta\left( \textbf{k}+\textbf{k}^\prime \right)\, \\
& = P_v\left( k \right) \delta\left( \textbf{k}+\textbf{k}^\prime \right)
\end{aligned}
\end{equation}
The power spectrum of curvature perturbation is obtained by pulling back the rescaling we have performed in \eqref{rescaleeq}
\begin{equation}
\langle \zeta_\textbf{k}\zeta_{\textbf{k}^\prime}\rangle = \frac{1}{2a^2\epsilon} \times \langle v_\textbf{k} v_{\textbf{k}^\prime}\rangle\, = \frac{2\pi^2}{k^3} P_\zeta \left( k \right) \delta\left( \textbf{k}+\textbf{k}^\prime \right)\,.
\end{equation}
The curvature perturbation $\zeta$ freezes out on super-horizon scales and therefore we evaluate the spectrum of curvature perturbation at the time of horizon exit $k = a H$.
Thus, the dimensionless power spectrum of curvature perturbation becomes\footnote{Since the curvature perturbation is conversed on super-horizon scales the (scale dependent) magnitude of power spectrum is given by \eqref{Amppr} which is evaluated at Horizon exit.}
\begin{equation}
\begin{aligned}
P_\zeta\left( k \right) \Big\vert_{k=aH} & = \frac{k^3}{2\pi^2} \frac{1}{2a^2\epsilon}P_v\\
& = \frac{H^2}{8\pi^2\epsilon} \Big\vert_{k =a H}\,.
\end{aligned}
\label{Amppr}
\end{equation}
In the above derivation we substituted $\tau \approx -\frac{1}{aH}$.
The tilt of the power spectrum $P_\zeta$ can be computed as
\begin{equation}
n_s -1 = \frac{d\ln P_\zeta}{d\ln k} \approx -\frac{d\ln P_\zeta}{dN} \approx -2\epsilon-\eta\,.
\label{speci}
\end{equation}
Note that the above result \eqref{Amppr} and \eqref{speci} remains the same even without performing the approximation $\nu_s\approx 3/2\implies \left( \epsilon,\, \eta \right)\to 0$ \cite{Kinney:2009vz}. Since the power spectrum is obtained by evaluating it at $k =aH$, the small corrections to the parameter $\nu_s$ give subdominant results. In addition, the spectral index $n_s$ and $n_t $ are often written in terms of $\nu_s,\, \nu_t$, but such definitions are not precisely correct as it is explained in \cite{Kinney:2009vz}. One must indeed evaluate the power spectrum at $k = aH$ because beyond this point curvature perturbation is frozen. So, the super-horizon amplitude of the power spectrum can be written in terms of quantities evaluated at the horizon exit \cite{Leach:2000yw} by fixing the normalization factors \cite{Kinney:2009vz}.
Similarly, we can work out the mode function for the tensor fluctuations and imposing the BD vacuum we obtain
\begin{equation}
\begin{aligned}
& u_{ij} = e_{ij} \dfrac{\sqrt{\pi}}{2} \left( -\tau \right)^{1/2} e^{i\left( 2\nu_t+1\right) \pi} H^{(1)}_{\nu_t}\left( -k \tau \right)
\end{aligned}
\end{equation}
We can perform similar calculations for the tensor power spectrum and tilt and as a result we obtain
\begin{equation}
P_T = \frac{2H^2}{\pi^2}\Bigg\vert_{k= a H},\quad n_t = -\frac{d\ln P_T}{d\ln N} \Bigg\vert_{k= a H} = -2\epsilon\,.
\end{equation}
\section{New scheme of quantizing inflationary power spectra}
\label{sec:questions}
In the previous section, we have discussed the standard quantization scheme of inflationary fluctuations. In this section, we enlist some open questions in the context of quantization of inflationary fluctuations. Based on these, we propose a new scheme of quantization of the inflationary fluctuations, which not only target some of the open questions, but also leads to some observable features in the CMB and primordial gravitational wave spectra. Below we discuss point by point some of the aspects of standard calculations of inflationary quantum fluctuations in relation with our understanding of spacetime in quantum theory and general relativity.
\begin{itemize}
\item By quantizing inflationary fluctuations we actually deal with quantum field theory (QFT) in curved spacetime. But in standard QFT we have particles that propagate forward and backward in time (anti-particle). A natural question is what happens to particles and anti-particle states in a curved spacetime. This implies one should consistently define the role of time reversal in a curved spacetime.
\item QFT in Minkowski spacetime is mainly built on discrete symmetries such as $\mathcal{C}\mathcal{P}\mathcal{T}$ or $\mathcal{P}\mathcal{T}$ (in the context of real scalar field), but in curved spacetime we often do not pay attention to the notion of discrete symmetries. The open question here is whether $\mathcal{C}\mathcal{P}\mathcal{T}$ holds or not in curved spacetime?
\item
In classical GR time is a coordinate and in quantum theory time is a parameter \cite{Donoghue:2017pgk,Donoghue:2019ecz}. In the case of inflationary cosmology we usually quantize both gravitational and matter degrees of freedom \cite{Martin:2004um,Baumann:2018muz,Kinney:2009vz} taking a classical notion of time. However, the concept of time in quantum theory is very different from the classical one \cite{Rovelli:2004tv}.
\item Since we quantize gravitational degrees of freedom, we can interpret inflationary quantum fluctuations as features of (linearized) quantum gravity \cite{Martin:2004um}. Note that, in the canonical quantum gravity that emerges through Wheeler-de Witt equation, time does not appear explicitly which indicates a difficulty to define positive/negative frequencies, like we usually do following the standard Schr\"{o}dinger equation. This is famously known as the problem of time in quantum cosmology \cite{Kiefer:2007ria,Rovelli:2004tv}.
\item Quantum theory is always time symmetric. The arrow of time emerges only after we specify initial and final states \cite{Hartle:2013tm}. This implies that we could first formulate QFT in curved spacetime, in a time symmetric way, and then impose initial conditions.
\end{itemize}
Our new formalism for the inflationary quantum fluctuations is highly motivated from the above questions. Before we go into details, we write down here few guidelines
\begin{itemize}
\item We identify the expansion or contraction of Universe with the shrinking or growing size of the co-moving horizon (in the context of de Sitter and quasi de Sitter). Change of size of the horizon defines our classical or thermodynamical arrow of time.
\item We build the quantum theory based on discrete transformations of spacetime within the co-moving horizon. We completely detach ourselves from the notions of observers, Penrose diagrams and the corresponding regions of spacetime when we do quantization because all such concepts are not well-defined according to quantum theory. In fact, classical notion of region of spacetime cannot have any consistent meaning from the quantum mechanical point of view \cite{Donoghue:2021meq}.
\item
The way forward is to first understand quantum mechanics in a dynamical spacetime and then take the classical limit. To achieve this goal, we take a small step by distinguishing classical notion of time with the quantum mechanical notion of time. This small step allow us to create all possible quantum states in the given spacetime geometry.
\item In our formalism, quantum fluctuations always come in pair from two different vacua described by $\mathcal{P}\mathcal{T}$ or $\mathcal{C}\mathcal{P}\mathcal{T}$ transformations. By this scheme we have the quantum fluctuations that propagate forward in time as well as backward in time and both can exist within the horizon. As the horizon changes, both of these type of fluctuations either exit or enter the horizon depending on whether the spacetime is expanding or contracting.
\item These pairs of quantum fluctuations within the horizon evolve independently of each other which means the commutator of corresponding operators always vanish. This means even though we can create quantum states non-locally we still do not violate locality and causality in our formulation of QFT.
\end{itemize}
\section{Quantum fluctuations in an expanding de Sitter spacetime}
\label{sec:dsspace}
In this section, we focus on elucidating our proposal in the context of exact de Sitter (dS) spacetime in FLRW co-ordinates. Since inflationary spacetime is near dS there are several concepts we can take forward from the understanding of dS. To make our analysis simpler we consider quantization of a massless scalar field in dS spacetime ignoring the issues with IR singularities and defining dS invariant vacuum \cite{Higuchi:2009ew,Allen:1985ux}.
The dS spacetime in FLRW coordinates looks like
\begin{equation}
\begin{aligned}
ds^2 & = -dt^2 + a(t)^2d\textbf{x}^2
& = \frac{1}{H^2\tau^2}\left( -d\tau^2+ d\textbf{x}^2 \right)\,.
\end{aligned}
\label{dsmetric}
\end{equation}
where $d\tau= \frac{dt}{a}$ is the conformal time and the
scale factor is given by
\begin{equation}
a(t) = e^{Ht},\quad H^2 = \left(\frac{1}{a}\frac{da}{dt}\right)^2={\rm const}\,.
\end{equation}
Note that dS metric \eqref{dsmetric} is $\mathcal{P}\mathcal{T}: \tau\to -\tau,\, \textbf{x}\to -\textbf{x}$ symmetric.
The scalar curvature of dS spacetime is $R=12H^2$ which does not tell whether the Hubble parameter is positive or negative. Each point in dS is surrounded by a co-moving horizon given by the radius
\begin{equation}
r_H = \Big\vert \frac{1}{aH} \Big\vert
\label{horizonds}
\end{equation}
One very simple observation we can make from \eqref{dsmetric} is that the spatial size of the Universe $a(t)^2$ grows with coordinate time $t$ when $H>0$. We then interpret the metric \eqref{dsmetric} as modelling an expanding universe.
In this case the scale factor grows with time $t$ and the size of the horizon shrinks.
We note here that this observation stems from the convention that $t$ runs from $-\infty $ to $+\infty$ (which means conformal time $-\infty<\tau< 0$). Defining $\bar{t}= -t$, which corresponds to \say{$t$ running from $+\infty$ to $-\infty$ or $+\infty>\tau> 0$} we find the scale factor $a(\tau)$ growing with $\bar{t}$ for $H<0$, corresponding to expansion. Therefore, the expanding dS space can only characterized by shrinking horizon rather than fixing the notion of time being positive or negative. Since time is a parameter in quantum theory, we can have quantum states that are labeled by $\tau<0$ and/or $\tau>0$. In the case of standard QFT in dS one usually make a biased choice of either $\tau<0$ or $\tau>0$ which actually breaks the time reversal nature of dS spacetime and also standard scheme of quantization completely ignores existence of states that evolve backward in time.
Therefore,
in our scheme of quantization in dS spacetime, we can have quantum states that propagate forward in time $t: -\infty \to \infty$ with $H > 0$ and states that propagate backward in time following an arrow of time $t: \infty \to -\infty$ with $H< 0$. Note that both type of quantum states can co-exist in a single expanding Universe. We stress again, the expanding dS Universe is defined by the following arrows of time
\begin{equation}
\tau: \pm \infty \to 0
\end{equation}
where $-$ sign corresponds to a state propagating forward in time and $+$sign corresponds to a state propagating backward in time. In our new picture quantization of a (massless) scalar field in dS spacetime leads to a pair of quantum fields that are produced out of two vacua. Being more precise we define total vacuum as the direct sum of two vacua
\begin{equation}
\vert 0\rangle = \frac{1}{\sqrt{2}} \Bigg(\vert 0\rangle_{\rm I}\oplus \vert 0\rangle_{\rm II} \Bigg)\,.
\end{equation}
In the vacuum $\vert 0\rangle_{\rm I}$ we create quantum fields at the position $ \textbf{x} $ that evolve forward in time ($\tau <0 $) and in the vacuum $\vert 0\rangle_{\rm II}$ we create quantum fields at $ -\textbf{x}$ which evolve backward in time ($\tau >0$).
The action of a massless scalar field in dS space is given
\begin{equation}
S_{\phi} = -\frac{1}{2}\int d\tau d^3x a^2 \phi\left( \partial_\tau^2+2\mathcal{H} \partial_\tau + k^2 \right) \phi\,.
\end{equation}
First thing we can notice here is that the above action is invariant under $\mathcal{P}\mathcal{T}$ which means $\textbf{x}\to -\textbf{x}$ and
\begin{equation}\label{timeref}
t\to -t\quad \left(-\tau \to \tau\right) \implies H\to -H \quad \left( \mathcal{H}\to -\mathcal{H} \right)
\end{equation}
Rescaling the field $\phi\to a \phi$ gives the following action which leads to Mukhanov-Sasaki equation as
\begin{equation}
S_\phi = \frac{1}{2} \int d\tau d^3x \Big[ \phi^{\prime 2} - \left( \partial_i\phi \right)^2 - \frac{2}{\tau^2} \phi^2 \Big]\,.
\label{msaction}
\end{equation}
The above Lagrangian is again symmetric under $\mathcal{P}\mathcal{T}: \textbf{x}\to -\textbf{x},\, \tau\to -\tau$. When we quantize this scalar field
we propose that the scalar field operator $\hat{\phi}\left( \tau,\, \textbf{x} \right)$ is the direct sum of the pair of fields as
\begin{equation}
\hat{\phi}\left( \tau,\, \textbf{x} \right) = \frac{1}{\sqrt{2}} \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right) \oplus \frac{1}{\sqrt{2}}\hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right) \,.
\end{equation}
which can be expanded in terms of different creation and annihilation operators in the following way
\begin{equation}
\begin{aligned}
& \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right) \\ & = \frac{1}{\left( 2\pi \right)^{3/2}}\int d\tau d^3k\Bigg[ a_\textbf{k} {\varphi}_{\rm I\,k}\left( \tau \right) e^{i\textbf{k}\cdot \textbf{x}} + a_\textbf{k}^\dagger {\varphi}^\ast_{\rm I\,k}\left( \tau \right) e^{-i\textbf{k}\cdot \textbf{x}} \Bigg]\,
\end{aligned}
\label{vdfield}
\end{equation}
where the creation and annihilation operators $a_\textbf{k},\, a^\dagger_\textbf{k}$ satisfy the canonical commutation relations and
\begin{equation}
a_\textbf{k}\vert 0 \rangle_{\rm I} = 0\,.
\end{equation}
Here the mode function $\varphi_{\rm I\,,k}$ is the solution of Mukhanov-Sasaki equation (that comes from varying \eqref{msaction}) given by
\begin{equation}
\varphi_{\rm I,\,k} = \alpha_{1k} \frac{e^{-ik\tau}}{\sqrt{2k}}\left( 1-\frac{i}{k\tau} \right) +\beta_{1k} \frac{e^{ik\tau}}{\sqrt{2k}}\left( 1+\frac{i}{k\tau} \right)\,,
\label{f1}
\end{equation}
where the Bogoliubov coefficients can be fixed as $\left( \alpha_{1,\,k},\, \beta_{1,\,k}\right) = \left( 1,\,0 \right)$, which is compatible with the Wronskian condition $ \varphi_{\rm I,\,k} \varphi^{\prime\ast}_{\rm I,\,k}- \varphi^\ast_{\rm I,\,k} \varphi^\prime_{\rm I,\,k} = i $ that corresponds to the canonical commutation relation
\begin{equation}
\big[ \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right),\, \hat{\pi}_{\rm I}\left( \tau,\, \textbf{x}^\prime \right)\big] = i\delta\left( \textbf{x}-\textbf{x}^\prime \right).
\end{equation}
The choice $\left( \alpha_{\rm I,\,k},\, \beta_{\rm I,\,k}\right) = \left( 1,\,0 \right)$ defines the vacuum and $\hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right) \vert 0\rangle $ corresponds to the positive frequency modes in the limit $\tau\to -\infty$. Quantum mechanically $\hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right) \vert 0\rangle $ is the postive energy state that propagate forward in time at \textbf{x}.
Similarly, we can expand the second field operator as
\begin{equation}
\begin{aligned}
& \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right) \\ & = \frac{1}{\left( 2\pi \right)^{3/2}}\int d\tau d^3k\Bigg[ b_\textbf{k} \varphi_{\rm II\,k}\left( -\tau \right) e^{-i\textbf{k}\cdot \textbf{x}} + b_\textbf{k}^\dagger {\varphi}^\ast_{\rm II\,k}\left( -\tau \right) e^{i\textbf{k}\cdot \textbf{x}} \Bigg]\,
\end{aligned}
\label{vdfield2}
\end{equation}
where
\begin{equation}
\varphi_{\rm II,\,k} = \alpha_{2,\,k} \frac{e^{ik\tau}}{\sqrt{2k}}\left( 1+\frac{i}{k\tau} \right) +\beta_{2,\,k} \frac{e^{-ik\tau}}{\sqrt{2k}}\left( 1-\frac{i}{k\tau} \right)\,,
\label{f2}
\end{equation}
where the Bogoliubov coefficients can be fixed as $\left( \alpha_{2,\,k},\, \beta_{2,\,k}\right) = \left( 1,\,0 \right)$ which is compatible with the Wronskian condition $ \varphi_{\rm II,\,k} \varphi^{\prime\ast}_{\rm II,\,k}- \varphi^\ast_{\rm II,\,k} \varphi^\prime_{\rm II,\,k} = -i $ that corresponds to the canonical commutation relation \cite{Donoghue:2019ecz}
\begin{equation}
\big[ \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right),\, \hat{\pi}_{\rm II}\left( -\tau,\, -\textbf{x}^\prime \right)\big] = -i\delta\left( \textbf{x}-\textbf{x}^\prime \right).
\end{equation}
which describe the quantum fields that propagate backward in time.
Here the second vacuum is defined by
\begin{equation}
b_\textbf{k}\vert 0 \rangle_{\rm II} = 0\,.
\end{equation}
Here $ \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right)\vert 0\rangle_{\rm II}$ corresponds to positive energy state that evolves backward in time at -\textbf{x}.
We demand that these two quantum fluctuations evolve independently. This is manifest by
\begin{equation}
[ \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right),\, \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right) ] =0\,.
\end{equation}
which implies that the respective creation and annihilation operators commute
\begin{equation}
\big[a_\textbf{k},\, b_{\textbf{k}^\prime}\big] = 0,\quad \big[a^\dagger_\textbf{k},\, b^\dagger_{\textbf{k}^\prime}\big] = 0
\end{equation}
Since dS spacetime is perfectly $\mathcal{P}\mathcal{T}$ symmetric we have that the quantum fields $ \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right)\vert 0\rangle_{\rm I}$ and $ \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right)\vert 0\rangle_{\rm II}$ behave identically, which can be seen from the fact that their equal time correlations are the same
\begin{equation}
\begin{aligned}
& \frac{1}{a^2}{}_{\rm I}\langle 0\vert \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x} \right) \hat{\varphi}_{\rm I}\left( \tau,\, \textbf{x}^\prime \right)\vert 0
|
\rangle_{\rm I} = \\ & \frac{1}{a^2} {}_{\rm II}\langle 0\vert \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x} \right) \hat{\varphi}_{\rm II}\left( -\tau,\, -\textbf{x}^\prime \right)\vert 0\rangle_{\rm II} = \frac{H^2}{4\pi^2k^3}\,.
\end{aligned}
\label{eqcorr}
\end{equation}
This implies the correlations of quantum fields related by $\mathcal{P}\mathcal{T}$ transformations are identical in the case of dS spacetime.
This indicates our formulation of QFT in dS spacetime might be $\mathcal{P}\mathcal{T}$ or $\mathcal{C}\mathcal{P}\mathcal{T}$ (if we include charged fields) symmetric.\footnote{We note that the result in \eqref{eqcorr} is similar to what was derived in the context of elliptic dS spacetime \cite{SANCHEZ19871111,Schrodinger1956}. However, in our scheme of quantization we do not make any (classical) antipodal identification of spacetime and therefore we significantly deviate conceptually from the proposal of elliptic dS made by Erwin Schr\"{o}dinger \cite{Schrodinger1956}. }
However, this may not be true in a spacetime with less symmetries such as quasi-dS. Indeed, in the context of inflationary quantum fluctuations, it is often assumed, in the effective field theory (EFT) approach, that Lorentz symmetry (in other words $(\mathcal{C})\mathcal{P}\mathcal{T}$) must be spontaneously broken \cite{Cheung:2007st}. However, it is not known how $\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry can be broken in a curved spacetime.
In the next section, we provide a framework of quantization where we can explicitly see that $(\mathcal{C})\mathcal{P}\mathcal{T}$ is spontaneously broken in the sense that the qunatum fields defined by our new notion of $\mathcal{P}\mathcal{T}$ transformations evolve differently.
\section{Double vacuum inflationary spacetime and the power spectra}
In the previous section, we have studied the quantization of a massless scalar field in dS. Here we quantize the inflationary quantum fluctuations. There is a crucial difference here. In the previous section we quantized an arbitrary scalar field in dS spacetime which is completely fixed but in the context of inflation we quantize metric and matter degrees of freedom in order to find an effective quantum correction to the classical quasi-dS (inflationary) spacetime. To be more precise, the classical inflationary spacetime is given by an arrow of time and the background initial conditions are
\begin{equation}
t: -\infty \to \infty \implies \left( \epsilon,\, \eta \right): \left( \approx 0,\, \approx 0 \right) \to \left( \sim 1,\, \sim 1 \right)\,.
\end{equation}
Which means the slow-roll parameters are very small and positive during inflation and inflation ends when the slow-roll parameters evolve to the order of unity, which happens at some time $t_{\rm end}\gg 1$. The inflationary quantum fluctuations are produced in the phase when $\left( \epsilon,\, \eta \right) \ll 1$. As we discussed earlier, ``time'' in quantum theory has totally different role and one must be very careful in quantizing inflationary fluctuations. We emphasize that the quantum fluctuations do not necessarily follow arrow of time prescribed by the initial conditions in classical theory.
Unlike dS spacetime the quasi-dS metric does not have $\mathcal{P}\mathcal{T}$-symmetry, therefore naturally one would expect the $\mathcal{P}\mathcal{T}$-symmetry to be spontaneously broken at the quantum level. To see this, we apply the scheme of quantization similar to what we have done in the context of exact dS. Let us first focus on inflationary scalar fluctuations which are described by the action \eqref{scalar}. After the field redefinition $\hat{v}$ is the canonical variable which we need to quantize \eqref{screfa}. As we postulated before quantum fluctuations always come as pair from two different vacua related by $\mathcal{P}\mathcal{T}$ transformations. This means that unlike the standard quantization of inflationary fluctuations, where only one kind of quantum states are considered which only evolve forward in time, here we describe a pair of quantum fluctuations that evolve both forward and backward in time.
Similar to the exact dS case we write the canonical field operator as a direct sum of two quantum fields
\begin{equation}
\hat{v}\left( \tau,\, \textbf{x} \right) = \frac{1}{\sqrt{2}} \hat{v}_{\rm I}\left( \tau,\, \textbf{x} \right) \oplus \frac{1}{\sqrt{2}} \hat{v}_{\rm II}\left( -\tau,\, -\textbf{x} \right)\,.
\end{equation}
which emerges from two different vacua. The total vacuum is the direct sum of these two which can be written as
\begin{equation}
\vert 0\rangle_{\rm qdS} = \vert 0 \rangle_{\rm qdS_{I}} \oplus \vert 0\rangle_{\rm qdS_{II}}\,.
\end{equation}
The quantum field operators acting on the vacua $\hat{v}_{\rm I}\left( \tau,\, \textbf{x} \right) \vert 0\rangle_{\rm qdS_{II}} $ and $\hat{v}_{\rm I}\left( -\tau,\, -\textbf{x} \right)\vert 0\rangle_{\rm qdS_{II}}$ leads to the creation of the pair of quantum fluctuations at $ \textbf{x} $ and at $ -\textbf{x} $ which propagate forward in time and back ward in time respectively. We will make it clear shortly the meaning of quantum fluctuations that evolve forward and backward in time in the context of quasi-dS.
Going into the details
we expand the fields in terms of creation and annhilation operators as explained below.
\begin{equation}
\begin{aligned}
& \hat{v}_{\rm I}\left( \tau,\, \textbf{x} \right) \\ & = \frac{1}{\left( 2\pi \right)^{3/2}}\int d\tau d^3k\Bigg[ c_\textbf{k} {v}_{\rm I,\,k}\left( \tau \right) e^{i\textbf{k}\cdot \textbf{x}} + c_\textbf{k}^\dagger {v}_{\rm I,\,k}^\ast\left( \tau \right) e^{-i\textbf{k}\cdot \textbf{x}} \Bigg]
\end{aligned}
\label{vid}
\end{equation}
with $c_\textbf{k},\, c_\textbf{k}^\dagger$ being the creation and annihilation operators of quasi-dS vacuum-I defined by
\begin{equation}
c_\textbf{k}\vert 0\rangle_{\rm qdS_I} = 0\,,
\label{qds1}
\end{equation}
and $ {v}_{\rm I,\,k}$ is the mode function obtained by solving MS-equation for $v_{\rm I}\left(\tau \right)$
\begin{equation}
v_{\rm I,\,k}^{\prime\prime}+ \left( k^2-\frac{{\nu}_s^2-\frac{1}{4}}{\tau^2} \right) v_{\rm I,\,k}^2 =0\,.
\end{equation}
Here $\tau: -\infty \to 0$ and
the quasi-dS vacuum $\vert 0\rangle_{\rm qdS_I}$ corresponds to the creation of positive frequency modes in the limit $\tau\to -\infty$.
Notice that unlike the dS case the MS-equation \eqref{MS-equation} is not symmetric under time reversal because there is an additional time dependence enters through the nearly constant variable $\nu_s$ which contains the slow-roll parameters $\left( \epsilon,\, \eta \right)$ \eqref{sldef}. Assuming $0<\left( \epsilon,\, \eta\right) \ll 1$ and constant we obtain
\begin{equation}
\begin{aligned}
& {v}_{\rm I,\,k} \\ & = \frac{\sqrt{\pi}}{2} \left( -\tau \right)^{1/2} e^{\left( i\nu_s+1\right)} \Bigg[C_k H^{(1)}_{\nu_s}\left( -k \tau \right)+ D_k H^{(2)}_{\nu_s}\left( -k \tau \right)\Bigg]\,.
\label{new-vac1}
\end{aligned}
\end{equation}
The above mode function corresponds to the creation of positive frequency modes in the limit $\tau\to -\infty$ for the case $\left( \mathcal{C}_k,\, D_k \right) = \left( 1,\,0 \right) $ which corresponds to the standard BD state that satisfies the Wronskian $v_{I,k}v_{I,k}^{\prime\ast}-v_{I,k}^\ast v_{I,k}^{\prime}=i (\implies \vert C_k\vert^2-\vert D_k\vert^2=1)$. We can expand the mode function for the case of $\left( \mathcal{C}_k,\, D_k \right) = \left( 1,\,0 \right) $ in the order of slow-roll as
\begin{equation}
\begin{aligned}
{v}_{\rm I,\,k} & \approx \frac{\sqrt{\pi}}{2} \left( -\tau \right)^{1/2} e^{\left( i\nu_s+1\right)} H^{(1)}_{\nu_s}\left( -k \tau \right)\,\\
& \approx \sqrt{\frac{1}{2k}} e^{-ik\tau}\left( 1-\frac{i}{k\tau} \right) \\ & + \left( \epsilon+\frac{\eta}{2} \right) \frac{\sqrt{\pi}}{2\sqrt{k}} \sqrt{\frac{k}{aH}} \frac{\partial H^{(1)}_{\nu_s}\left( \frac{k}{aH}\right)}{\partial \nu_s}\Big\vert_{\nu_s=3/2}
\label{new-vac1BD}
\end{aligned}
\end{equation}
The second field $\hat{v}_{II\,,\textbf{k}}\left( -\tau,\, -\textbf{x} \right)$ is expanded as
\begin{equation}
\begin{aligned}
& \hat{v}_{\rm II}\left( -\tau,\, -\textbf{x} \right) \\ & = \frac{1}{\left( 2\pi \right)^{3/2}}\int d\tau d^3k\Bigg[ d_\textbf{k} {v}_{\rm II,\,k}\left( -\tau \right) e^{-i\textbf{k}\cdot \textbf{x}} + d_\textbf{k}^\dagger {v}_{\rm I,\,k}^\ast\left( -\tau \right) e^{i\textbf{k}\cdot \textbf{x}} \Bigg]
\end{aligned}
\label{v2d}
\end{equation}
where $d_\textbf{k},\ , d_\textbf{k}^\dagger$ are the creation and annihilation operators that satisfy the canonical commutation relations and the second vacuum is defined by\footnote{We note that $
\big[c_\textbf{k},\, d_{\textbf{k}^\prime}\big] = 0,\quad \big[c^\dagger_\textbf{k},\, d^\dagger_{\textbf{k}^\prime}\big] = 0 $ similar to the dS case which means the pair of quantum fluctuations are not causally connected.}
\begin{equation}
d_\textbf{k}\vert 0\rangle_{\rm qdS_{II}}=0\,.
\end{equation}
The evolution of the mode function $v_{\rm II,\,k}$ is determined by solving the following time reversed MS-equation
\begin{equation}
v_{\rm II,\,k}^{\prime\prime}+ \left( k^2-\frac{\bar{\nu}_s^2-\frac{1}{4}}{\tau^2} \right) v_{\rm II,\,k}^2 =0\,.
\end{equation}
Here $\tau: +\infty \to 0$ and
\begin{equation}
\bar{\nu}_s \approx \frac{3}{2}-\epsilon-\frac{\eta}{2}
\end{equation}
Here the time reversal transformations are
\begin{equation}
t\to -t \implies H\to -H,\quad \epsilon\to -\epsilon,\quad \eta\to -\eta\,.
\label{timerevqds}
\end{equation}
which can be understood in the following sense. Our goal here is to define how the quantum fluctuations propagate backward in time in an expanding Universe. To achieve this goal we need to formulate the meaning of time reversal operation. Logically, if the fluctuation propagates forward in time in a slow-roll background, the fluctuation that goes backward in time experience spacetime as a "slow-climb"\footnote{In the standard slow-roll we have scalar field rolling down the potential whose time reversal can be understood as a phantom field that is slowly climbing the potential \cite{Piao:2004tq}. } which is given by reversing the signs of the parameters $\left( \epsilon,\, \eta \right)$ as stated in \eqref{timerevqds}. We impose this time reversal operation in a completely quantum mechanical sense and this has no classical meaning i.e., our background (classical) dynamics are completely determined by Friedmann equations \eqref{Freq} and we do not apply at all time reversal to the classical background. Since we treat time differently at quantum level we restrain ourselves from any intuition from classical physics.\footnote{Notion of time is a very non-trivial concept in physics and its meaning varies in different contexts. We suggest the reader \cite{Rovelli:2004tv} for an extended physical discussion. Our statement, that quantum states evolving backward in time has no classical analog, is deeply rooted in the quantum gravity concept time as it is presented in p. 184 of \cite{Rovelli:2004tv}.} Since dynamics of quantum fields emerge from MS-equation \eqref{MS-equation} the functions $\left( \epsilon,\, \eta \right)$ are now treated along with time in as parameters to specify the nature of quantum state. This would encode a subtle difference between quantum fluctuations propagating forward and backward in time.
\begin{equation}
\begin{aligned}
& v_{\rm II,\,k}\left( -\tau \right) \\ &= \dfrac{\sqrt{\pi}}{2} \left( \tau \right)^{1/2} e^{i\left( 2\bar{\nu}_s+1\right) \pi}\Bigg[\bar{C}_k H^{(1)}_{\bar{\nu}_s}\left( k \tau \right)+ \bar{D}_k H^{(2)}_{\bar{\nu}_s}\left( k \tau \right)\Bigg].
\end{aligned}
\label{v2sol}
\end{equation}
Now imposing the Wronskian condition $v_{\rm II,\,k}v_{\rm II,\,k}^{\prime\ast}-v_{\rm II,\,k}^\ast v_{\rm II,\,k}^\prime = -i$ which corresponds to the canonical commutation relation for a reversed arrow of time \cite{Donoghue:2019ecz}
\begin{equation}
\Big[ \hat{v}_{\rm II}\left( -\tau,\,-\textbf{x} \right),\, \hat{\Pi}_{\rm II}\left( -\tau,\,-\textbf{x}^\prime \right) \Big] = -i \delta\left( \textbf{x}-\textbf{x}^\prime \right)\,.
\end{equation}
we obtain $\vert \bar{C}_k\vert^2-\vert \bar{D}_k\vert^2 = 1$. We consider $\left(\bar{C},\, \bar{D}_k\right) =0$ which corresponds postive energy states in the limit $\tau \to \infty$. Expanding \eqref{v2sol} up to the leading order for $\left(\bar{C},\, \bar{D}_k\right) =0$ we get
\begin{equation}
\begin{aligned}
{v}_{\rm II,\,k} & \approx \frac{\sqrt{\pi}}{2} \left( \tau \right)^{1/2} e^{\left( i\nu_s+1\right)} H^{(1)}_{\nu_s}\left( k \tau \right)\,\\
& \approx \sqrt{\frac{1}{2k}} e^{ik\tau}\left( 1+\frac{i}{k\tau} \right) \\ & -\left( \epsilon+\frac{\eta}{2} \right) \frac{\sqrt{\pi}}{2\sqrt{k}} \sqrt{k\tau} \frac{\partial H^{(1)}_{\nu_s}\left( k\tau \right)}{\partial \nu_s}\Big\vert_{\nu_s=3/2}
\label{new-vac2BD}
\end{aligned}
\end{equation}
Comparing \eqref{new-vac1BD} and \eqref{new-vac2BD} we can deduce that the two mode functions get different slow-roll (quantum) corrections.
This is because the parameter $\nu_s$ that enters in the MS-equation \eqref{MS-equation} contains a tiny time dependence. In the standard framework of quantization this is treated it to be nearly constant there is only one type of quantum fluctuation (see Sec.~\ref{sec:review}). But the fact that MS-equation has time dependence in terms of $\epsilon,\, \eta$ is crucial to probe the nature of quantum fluctuations. These quantities change sign under discrete spacetime transformation and this has to be understood in a completely quantum mechanical sense. Our scheme of quantization implies a pair of quantum fluctuations: those that propagate forward in time ($\tau: -\infty \to 0$) at spatial position $\textbf{x}$ which behave differently from those fluctuations that propagate backward in time ($\tau: +\infty \to 0$) at the spatial position $-\textbf{x}$. These pairs of fluctuations exit the horizon in the two opposite directions and now we observe the modes as they re-enter.
So far we applied our quantization method for the scalar mode $v\left( \tau,\,\textbf{x} \right)$. We can apply the similar method of quantization for the tensor modes \eqref{tensor-mode} by writing the tensor fluctuation as direct some fluctuations that are related by $\mathcal{P}\mathcal{T}$ transformations
\begin{equation}
\hat{u}_{ij} \left( \tau,\, \textbf{x} \right)= \frac{1}{\sqrt{2}} \hat{u}^{\rm I}_{ij}\left( \tau,\, \textbf{x} \right) \oplus \frac{1}{\sqrt{2}}\hat{u}^{\rm II}_{ij}\left( -\tau,\,-\textbf{x} \right)
\end{equation}
The above fields can be written in terms of creation and annihilation operators similar to what we have done for \eqref{vid} and \eqref{v2d}. The corresponding mode functions tensor modes can be straight forwardly derived as
\begin{equation}
\begin{aligned}
{u}^{\rm I}_{ij,\,k}
& \approx e_{ij}\sqrt{\frac{1}{2k}} e^{-ik\tau}\left( 1-\frac{i}{k\tau} \right) \\ & + e_{ij} \epsilon\frac{\sqrt{\pi}}{2\sqrt{k}} \sqrt{-k\tau} \frac{\partial H^{(1)}_{\nu_t}\left( -k\tau \right)}{\partial \nu_t}\Big\vert_{\nu_t=3/2}
\end{aligned}
\label{new-vac1TBD}
\end{equation}
and
\begin{equation}
\begin{aligned}
{u}^{\rm II}_{ij,\,k}
& \approx e_{ij}\sqrt{\frac{1}{2k}} e^{ik\tau}\left( 1+\frac{i}{k\tau} \right) \\ & - e_{ij} \epsilon\frac{\sqrt{\pi}}{2\sqrt{k}} \sqrt{k\tau} \frac{\partial H^{(1)}_{\nu_t}\left( k\tau \right)}{\partial \nu_t}\Big\vert_{\nu_t=3/2}
\end{aligned}
\label{new-vac2TBD}
\end{equation}
where $e_{ij}$ denotes the polarization tensor.
In the next section, we will compute the inflationary power spectra for scalar and tensor modes.
\section{Inflationary power spectra and predictions for hemispherical asymmetry}
As we learned in the previous section, inflationary quantum fluctuations now come in pair which exit the horizon on two opposite sides. The two point correlations of these fluctuations can be computed as
\begin{equation}
\begin{aligned}
& {}_{\rm qdS_{I}}\langle 0 \vert \hat{v}_{\rm I}\left( \textbf{x},\, \tau \right) \hat{v}_{\rm I}\left( \textbf{x}+\boldsymbol{\xi}, \, \tau \right)\vert 0\rangle_{\rm qdS_{I}} = \\ & \frac{4\pi}{\left( 2\pi \right)^3} \int \frac{dk}{k} \frac{\sin k \xi}{k\xi} k^3 \vert v_{\rm I,\,k}\vert^2\, \\
& {}_{\rm qdS_{II}}\langle 0 \vert \hat{v}_{\rm II}\left( -\textbf{x},\, -\tau \right) \hat{v}_{\rm II}\left( -\textbf{x}-\boldsymbol{\xi}, \, -\tau \right)\vert 0\rangle_{\rm qdS_{II}} = \\ &\frac{4\pi}{\left( 2\pi \right)^3} \int \frac{dk}{k} \frac{\sin k \xi}{k\xi} k^3 \vert v_{\rm II,\,k}\vert^2\,.
\end{aligned}
\label{power-spectrav}
\end{equation}
Substituting the expressions \eqref{new-vac1BD} and \eqref{new-vac2TBD} in the above expressions we learn that the above two correlations are not equal. In the case of exact dS we obtained the correlations to be equal (see \eqref{eqcorr}) but in the case of quasi-dS we obtained an asymmetry because the background spacetime is not $\mathcal{P}\mathcal{T}$ symmetric.
As we know that the curvature perturbation is frozen on super-horizon scales.
To calculate the two point correlations of curvature perturbation on super-horizon scales we re scale the canonical fields with the classical background quantities \eqref{rescaleeq}. This implies
\begin{equation}
\begin{aligned}
{}_{\rm qdS}\langle 0 \vert \zeta_\textbf{k} \zeta_{\textbf{k}^\prime} \vert 0\rangle_{\rm qdS} & = \left( \frac{1}{2a^2\epsilon}\right)\Bigg\vert_{\rm classical} \Big[\frac{1}{2}{}_{\rm qdS_I}\langle 0 \vert \hat{v}_{\rm I\,\textbf{k}} \hat{v}_{\rm I\,\textbf{k}^\prime} \vert 0\rangle_{\rm qdS_I}\\ & \quad +\frac{1}{2}{}_{\rm qdS_{II}}\langle 0 \vert \hat{v}_{\rm {II}\,\textbf{k}} \hat{v}_{\rm {II}\,\textbf{k}^\prime} \vert 0\rangle_{\rm qdS_{II}}\Big] \\
& = \frac{2\pi^2}{k^3}\left( P_{\zeta_1}+P_{\zeta_2} \right) \delta\left( \textbf{k}+\textbf{k}^\prime \right)\,,
\end{aligned}
\end{equation}
where $\zeta_1,\,\zeta_2$ are the curvature perturbations that exit the horizon on two opposite directions and they become frozen on super-horizon scales which follows from \eqref{power-spectrav}. With appropriate normalization we evaluate the power spectrum of these curvature perturbations at the moment of horizon exit. Computing the two power spectra at the horizon exit we obtain
\begin{equation}
\begin{aligned}
P_{\zeta1} & = \frac{k^3}{2\pi^2}\frac{1}{2a^2\epsilon} \vert v_{\rm I,\,k}\vert^2\Bigg\vert_{\tau = -\frac{1}{aH}} \\
& \approx \frac{H^2}{8\pi^2\epsilon} \left( 1+\left( \frac{k}{aH} \right)^2 \right) \\ &\quad + \frac{H^2}{4\pi\epsilon} \left( \epsilon+\frac{\eta}{2} \right) \left(\frac{k}{aH}\right)^3 H_{3/2}^{(1)} \left( \frac{k}{aH} \right) \frac{\partial H^{(1)}_{\nu_s}\left( \frac{k}{aH} \right)}{\partial \nu_s}\Big\vert_{\nu_s=3/2}\, \\
P_{\zeta2} & = \frac{k^3}{2\pi^2}\frac{1}{2a^2\epsilon} \vert v_{\rm II,\,k}\vert^2\Bigg\vert_{\tau_\ast = \frac{1}{aH}} \\
& \approx \frac{H^2}{8\pi^2\epsilon} \left( 1+\left( \frac{k}{aH} \right)^2 \right) \\ & - \frac{H^2}{4\pi\epsilon} \left( \epsilon+\frac{\eta}{2} \right) \left(\frac{k}{aH}\right)^3 H_{3/2}^{(1)} \left( \frac{k}{aH} \right) \frac{\partial H^{(1)}_{\nu_s}\left( \frac{k}{aH} \right)}{\partial \nu_s}\Big\vert_{\nu_s=3/2}\,.
\end{aligned}
\label{pw12}
\end{equation}
If we combine the above two expressions we obtain
\begin{equation}
P_\zeta = \frac{1}{2}P_{\zeta1}+ \frac{1}{2} P_{\zeta2} = \frac{H^2}{8\pi^2\epsilon}\Bigg\vert_{k=aH}
\end{equation}
which is the expression in the context of standard inflation (see Sec.~\ref{sec:review}). From \eqref{power-spectrav} we can deduce that the power spectrum $P_{\zeta 1}$ can be mapped to the two-point correlations in the direction $\hat{\textbf{n}}$ and $P_{\zeta 2}$ can be mapped to the two-point correlations in the opposite direction $-\hat{\textbf{n}}$ of the CMB sky. The difference between the two power spectra gives us the non-zero scale dependent contribution. Our formalism of producing inflationary correlations in pair naturally explains the hemispherical asymmetry and this can also be interpreted as an indication of spontaneous breaking of $\mathcal{P}\mathcal{T}$ symmetry in the inflationary background. In our context we can define the
the amplitude of dipolar modulation as (according to Eq.~\eqref{Akg})
\begin{equation}
A(k) = \frac{P_{\zeta1}-P_{\zeta2}}{4P_\zeta}
\label{Ako}
\end{equation}
From \eqref{pw12} we can notice that the two power spectra differ only by a small scale dependent correction of the order of slow-roll parameters. In addition, through CMB we can only probe very limited range of $k$ corresponding to initial $7-8$ e-foldings centered around the pivot scale \cite{Martin:2004um}.
At the leading order the first terms in the two power spectra dominate which gives us the tilt of the two power spectra nearly the same in the leading order in slow-roll approximation
\begin{equation}
\frac{d\ln P_{\zeta 1}}{d\ln k}\approx \frac{d\ln P_{\zeta 2}}{d\ln k} \approx n_s-1 \approx -2\epsilon-\eta\,.
\label{tilts2}
\end{equation}
The above result is in-line with the data from the Planck satellite \cite{Mukherjee:2015mma,Axelsson:2013mva}.
In Fig.~\ref{fig:fig1} we depict the power asymmetry amplitude of the scalar power spectra for $n_s = 0.963$. This plot remains the same for any single-field slow-roll inflation.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{Fig1}
\caption{In the above plot we depict the amplitude of CMB dipolar modulation or hemispherical asymmetry of scalar power spectra obtained from \eqref{Ako}. In the plot we choose the pivot scale $k_\ast=a_\ast H_\ast=0.05 {\rm Mpc}^{-1}$ and fix $n_s=0.963$. The blue dot with error bar in the plot corresponds to the observational constraint $\vert A \vert = 0.066\pm 0.021$ on HPA at large angular scales or at low-$\ell-2-64$ or $k\lesssim 10^{-1}k_\ast$. We can notice from the plot that the HPA ceases to exist for the small angular scales or high-$\ell$ which is compatible with the Planck satellite observations \cite{Planck:2013lks,Akrami:2014eta,Aiola:2015rqa}. }
\label{fig:fig1}
\end{figure}
Similar to the scalar perturbations we propose the tensor perturbations to be direct sum of two kinds of fluctuations related by our definition of $\mathcal{P}\mathcal{T}$ transformation.
\begin{equation}
\hat{u}_{ij}\left( \tau,\, \textbf{x} \right) = \frac{1}{\sqrt{2}} \hat{u}_{\rm I,\, ij}\left( \tau,\, \textbf{x} \right) \oplus \frac{1}{\sqrt{2}} \hat{u}_{\rm II,\, ij}\left( -\tau,\, -\textbf{x} \right)\,.
\end{equation}
Similar to the scalar power spectra we obtain two tensor power spectra which describe two point tensor correlations in the direction $\hat{\textbf{n}}$ and $-\hat{\textbf{n}}$ respectively. The two power spectra of tensor correlations are computed as
\begin{equation}
\begin{aligned}
P_{h1} & = \frac{k^3}{2\pi^2}\frac{4}{a^2} \vert u_{\rm I\,k}\vert^2\Bigg\vert_{\tau = -\frac{1}{aH}} \\
& \approx \frac{2H^2}{\pi^2} \left( 1+\left( \frac{k}{aH} \right)^2 \right) \\ &\quad + \frac{H^2}{\pi}\epsilon\left(\frac{k}{aH}\right)^3 H_{3/2}^{(1)} \left( \frac{k}{aH} \right) \frac{\partial H^{(1)}_{\nu_t}\left( \frac{k}{aH} \right)}{\partial \nu_t}\Big\vert_{\nu_t=3/2}\, \\
P_{h2} & = \frac{k^3}{2\pi^2}\frac{4}{a^2} \vert u_{\rm II,\,k}\vert^2\Bigg\vert_{\tau_\ast = \frac{1}{aH}} \\
& \approx \frac{2H^2}{\pi^2} \left( 1+\left( \frac{k}{aH} \right)^2 \right)\\ &\quad - \frac{H^2}{\pi} \epsilon \left(\frac{k}{aH}\right)^3 H_{3/2}^{(1)} \left( \frac{k}{aH} \right) \frac{\partial H^{(1)}_{\nu_t}\left( \frac{k}{aH} \right)}{\partial \nu_t}\Big\vert_{\nu_t=3/2}\,.
\end{aligned}
\label{pwt12}
\end{equation}
The hemispherical power asymmetry of the tensor-power spectrum can be defined as
\begin{equation}
T(k) = \frac{P_{h1}-P_{h2}}{4P_h}
\label{Tko}
\end{equation}
To quantify the above amplitude $T(k)$ we choose Starobinsky or Higgs inflationary potential in Einstein frame
\begin{equation}
V(\phi) = \frac{\lambda}{4} \left( 1- e^{-\sqrt{\frac{2}{3}}\phi} \right)^2\,.
\label{staro-in}
\end{equation}
which gives the slow-roll parameters in terms of e-folds as \cite{Kehagias:2013mya}
\begin{equation}
\epsilon = \frac{3}{4N^2},\quad \eta= \frac{2}{N}
\end{equation}
In Fig.~\ref{fig:fig2} we depict HPA amplitude of the tensor power spectra for the case of Starobinsky potential \eqref{staro-in}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\
|
}=l_z\sqrt{{\frac{-a_1+\sqrt{a_1^2-36\sigma_1a_2}}{18\sigma_1}}},
\label{5eq:27}\\
&&\hspace*{-1.3cm}V_{p}\equiv V_{p-}=l_z\sqrt{{\frac{-a_1-\sqrt{a_1^2-36\sigma_1a_2}}{18\sigma_1}}},
\label{5eq:28}\
\end{eqnarray}
where $a_1=-{6\sigma_1\alpha_2+6\sigma_1\alpha_3+9+\Lambda n_1}$ and $a_2=4\sigma_1\alpha_2\alpha_3+6\alpha_2+6\alpha_1\alpha_3\Lambda$.
The $x$ and $y$-components of the first-order momentum equations can be written as
\begin{eqnarray}
&&\hspace*{-1.3cm}u_{1x}^{(1)}=-\frac{3l_yV_p^2}{\Omega_{c}(3V_p^2-2\alpha_2l_z^2)}~\frac{\partial\psi^{(1)}}{\partial\xi},
\label{5eq:29}\\
&&\hspace*{-1.3cm}u_{1y}^{(1)}=\frac{3l_xV_p^2}{\Omega_{c}(3V_p^2-2\alpha_2l_z^2)}~\frac{\partial\psi^{(1)}}{\partial\xi},
\label{5eq:30}\\
&&\hspace*{-1.3cm}u_{2x}^{(1)}=-\frac{3l_yV_p^2}{\Omega_{c}(3V_p^2-2\alpha_3l_z^2)}~\frac{\partial\psi^{(1)}}{\partial\xi},
\label{5eq:31}\\
&&\hspace*{-1.3cm} u_{2y}^{(1)}=\frac{3l_xV_p^2}{\Omega_{c}(3V_p^2-2\sigma_3l_z^2)}~\frac{\partial \psi^{(1)}}{\partial\xi}.
\label{5eq:32}\
\end{eqnarray}
Now, by following the next higher-order terms, the equation of continuity, momentum equation, and Poisson's equation can be written as
\begin{eqnarray}
&&\hspace*{-1.3cm}\frac{\partial n_1^{(1)}}{\partial\tau}-V_p\frac{\partial n_1^{(2)}}{\partial\xi}+l_x\frac{\partial u_{1x}^{(1)}}{\partial\xi}+l_y\frac{\partial u_{1y}^{(1)}}{\partial\xi}
\nonumber\\
&&\hspace*{1.5cm}+l_z\frac{\partial u_{1z}^{(2)}}{\partial\xi}+l_z\frac{\partial}{\partial\xi}\big(n_1^{(1)}u_{+z}^{(1)}\big)=0,
\label{5eq:33}\\
&&\hspace*{-1.3cm}\frac{\partial u_{1z}^{(1)}}{\partial\tau}-V_p\frac{\partial u_{1z}^{(2)}}{\partial\xi}+l_zu_{1z}^{(1)}\frac{\partial u_{1z}^{(1)}}{\partial\xi}+\alpha_1l_z\frac{\partial\psi^{(2)}}{\partial\xi}
\nonumber\\
&&\hspace*{0.5cm}+\alpha_2l_z\frac{\partial }{\partial\xi}\bigg[\frac{2}{3}n_1^{(2)}-\frac{1}{9}(n_1^{(1)})^2\bigg]-\eta\frac{\partial^2u_{1z}^{(1)}}{\partial\xi^2}=0,
\label{5eq:34}\\
&&\hspace*{-1.3cm}\frac{\partial n_2^{(1)}}{\partial\tau}-V_p\frac{\partial n_2^{(2)}}{\partial\xi}+l_x\frac{\partial u_{2x}^{(1)}}{\partial\xi}+l_y\frac{\partial u_{2y}^{(1)}}{\partial\xi}
\nonumber\\
&&\hspace*{1.5cm}+l_z\frac{\partial u_{2z}^{(2)}}{\partial\xi}+l_z\frac{\partial}{\partial\xi}\big(n_2^{(1)}u_{2z}^{(1)}\big)=0,
\label{5eq:35}\\
&&\hspace*{-1.3cm}\frac{\partial u_{2z}^{(1)}}{\partial\tau}-V_p\frac{\partial u_{2z}^{(2)}}{\partial\xi}+l_zu_{2z}^{(1)}\frac{\partial u_{2z}^{(1)}}{\partial\xi}-l_z\frac{\partial\psi^{(2)}}{\partial\xi}
\nonumber\\
&&\hspace*{0.5cm}+\alpha_3l_z\frac{\partial }{\partial\xi}\bigg[\frac{2}{3}n_2^{(2)}-\frac{1}{9}(n_2^{(1)})^2\bigg]-\eta\frac{\partial^2u_{2z}^{(1)}}{\partial\xi^2}=0,
\label{5eq:36}\\
&&\hspace*{-1.3cm}\sigma_1\psi^{(2)}+\sigma_2{[\psi^{(1)}]}^2+n_2^{(2)}-\Lambda n_1^{(2)}=0.
\label{5eq:37}\
\end{eqnarray}
Finally, the next higher-order terms of Eqs. \eqref{5eq:6}$-$\eqref{5eq:9}, and \eqref{5eq:12}, with the help of
Eqs. \eqref{5eq:23}$-$\eqref{5eq:37}, can provide the Burgers' equation as
\begin{eqnarray}
&&\hspace*{-1.3cm} \frac{\partial\Psi}{\partial\tau}+A\Psi\frac{\partial\Psi}{\partial\xi}=C\frac{\partial^2\Psi}{\partial\xi^2},
\label{5eq:38}\
\end{eqnarray}
where $\Psi=\psi^{(1)}$ is used for simplicity. In Eq. \eqref{5eq:38}, the nonlinear coefficient ($A$) and dissipative coefficient ($C$) are given by the following expression
\begin{eqnarray}
&&\hspace*{-1.3cm}A=\frac{(M_1S_1^3-M_2S_2^3-2\sigma_2S_1^3S_2^3)}{M_3S_1S_2},~~\mbox{and}~~C =\frac{\eta}{2},
\label{5eq:39}\
\end{eqnarray}
where $M_1 =\Lambda(81\alpha_1^2V_p^2l_z^4-6\alpha_2\alpha_1^2l_z^6)$, $M_2=81V_p^2l_z^4-6\alpha_3l_z^6$, and $M_3=18V_pl_z^2{1+\alpha_1\Lambda}$.
Now, we look forward to the stationary shock wave solution of this Burgers' equation by
taking $\zeta =\xi-U_0\tau'$ and $\tau =\tau'$, where $U_0$ is the speed of the shock waves in the reference frame.
These allow us to represent the stationary shock wave solution as \cite{Karpman1975,Hasegawa1975,Hossen2017aa}
\begin{eqnarray}
&&\hspace*{-1.3cm}\Psi=\Psi_m \Big[1-\tanh\bigg(\frac{\zeta}{\Delta}\bigg)\Big],
\label{5eq:40}\
\end{eqnarray}
where $\Psi_m$ is the amplitude and $\Delta$ is the width. The expression of the amplitude and width can be given by the following equations
\begin{eqnarray}
&&\hspace*{-1.3cm}\Psi_m=\frac{U_0}{A},~~~~\mbox{and}~~~~\Delta=\frac{2C}{U_0}.
\label{5eq:41}\
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=80mm]{F1.eps}
\caption{The variation of nonlinear coefficient $A$ with $\alpha_4$ along with $\alpha_1=1.5$,
$\alpha_2 =0.05$, $\alpha_3=0.03$, $\lambda_p=1.5$, $\lambda_e=1.7$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\alpha=0.5$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{F2.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\eta$ under consideration of $\alpha_4>\alpha_{4c}$
along with $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_p=1.5$,
$\lambda_e=1.7$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{F3.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\eta$ under consideration of $\alpha_4<\alpha_{4c}$
along with $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=0.5$, $\lambda_p=1
|
.5$,
$\lambda_e=1.7$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{F4.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\delta$ under consideration of $\alpha_4>\alpha_{4c}$ along with $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_p=1.5$,
$\lambda_e=1.7$, $\lambda_d=0.05$, $\eta=0.3$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F4}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{F5.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\alpha$ under consideration of $\alpha_4>\alpha_{4c}$
along with $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_p=1.5$,
$\lambda_e=1.7$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\eta=0.3$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F5}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=80mm]{F6.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\lambda_p$ under consideration of $\alpha_4>\alpha_{4c}$
along with $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_e=1.7$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\eta=0.3$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F6}
\end{figure}
\section{Numerical Analysis and Discussion}
\label{5sec:Numerical Analysis and Discussion}
Now, we would like to observe the basic properties of DIASHWs in a magnetized PIP having inertial pair-ions, inertialess non-thermal
distributed electrons and positrons, and static negatively charged massive dust grains by changing the various plasma parameters,
viz., ion kinematic viscosity, oblique angle, non-thermality of electrons and positrons, mass, charge, temperature, and number density
of the plasma species. Equation \eqref{5eq:41} shows that under consideration $U_0>0$ and $C>0$, no shock wave will exist if $A=0$
as the amplitude of the wave becomes infinite which clearly violates the reductive perturbation method. So, $A$ can be positive (i.e., $A>0$)
or negative (i.e., $A<0$) according to the value of other plasma parameters. Figure \ref{5Fig:F1} illustrates the variation of
$A$ with $\alpha_4$, and it is obvious from this figure that $A$ can be negative, zero, and positive according to the values of $\alpha_4$
when other plasma parameters are $\alpha_1=1.5$, $\alpha_2 =0.05$, $\alpha_3=0.03$, $\lambda_p=1.5$, $\lambda_e=1.7$, $\lambda_d=0.05$,
$\delta=30^\circ$, and $\alpha=0.5$. The point at which $A$ becomes zero for the value of $\alpha_4$ is known as the critical value of
$\alpha_4$ (i.e., $\alpha_{4c}$). In our present analysis, the critical value of $\alpha_4$ is $\alpha_{4c}\equiv 01$. So, the negative (positive) shock profile can be exist for the value of $\alpha_{4}<\alpha_{4c}$ ($\alpha_{4}>\alpha_{4c}$).
\begin{figure}
\centering
\includegraphics[width=80mm]{F7.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\alpha_1$ under consideration of $\alpha_4>\alpha_{4c}$
along with $\alpha_2 =0.05$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_e=1.7$, $\lambda_p=1.5$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\eta=0.3$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F7}
\end{figure}
Figures \ref{5Fig:F2} and \ref{5Fig:F3} respectively represent the variation of the positive and negative shock profiles with ion kinematic
viscosity (via $\eta$) when other plasma parameters are remain constant. It is really interestingly that the steepness of both
positive and negative shock profiles declines with the increase of $\eta$ without affecting the height.
Figure \ref{5Fig:F4} describes the effects of the external magnetic field to the formation of the positive shock profile.
The increase in oblique angle rises the height of the positive shock profile and this result is analogous to the result of Ref. \cite{Shahmansouri2014}.
The height of the positive shock profile is so much sensitive to the change of non-thermality of the electrons and positrons which can be
seen in Fig. \ref{5Fig:F5}. There is a decrease in the amplitude of positive shock profile when electrons and positrons deviate from thermodynamic equilibrium, and this result is compatible with the result of Ref. \cite{Alinejad2010}. The variation of the DIASHWs with negative ion charge state, negative ion and positron number densities (via $\lambda_p$) can be observed in Fig. \ref{5Fig:F6}. It is clear from Fig. \ref{5Fig:F6} that as we increase the positron (negative ion) number density, the height of the positive shock wave increases (decreases) when the charge of the negative ion remains constant or the the height of the positive shock wave decreases with the charge of the negative ion for a fixed value of the number density of
positron and negative ion.
\begin{figure}
\centering
\includegraphics[width=80mm]{F8.eps}
\caption{The variation of $\Psi$ with $\zeta$ for different values of $\alpha_2$ under consideration of $\alpha_4>\alpha_{4c}$
along with $\alpha_1 =1.5$, $\alpha_3=0.03$, $\alpha_4=1.5$, $\lambda_e=1.7$, $\lambda_p=1.5$, $\lambda_d=0.05$, $\delta = 30^\circ$, $\eta=0.3$, $\alpha=0.5$, $U_0=0.01$, and $V_{p}\equiv V_{p+}$.}
\label{5Fig:F8}
\end{figure}
The charge and mass of the positive and negative ions are rigourously responsible to change the height of the positive shock profile.
The variation of the DIASHWs with $\alpha_1$ has been demonstrated in Fig. \ref{5Fig:F7}, and it is obvious from this figure that
the height of the positive shock profile increases (decreases) with increasing the value of positive (negative) ion mass for a fixed value of the
their charge state. But as we increase the charge state of the negative (positive) ion then the amplitude of the positive shock profile
increases (decreases) when their mass are constant. The effects of the temperature of electron and positive ion (via $\alpha_2$) can be seen in Fig. \ref{5Fig:F8}, and the amplitude of the shock profile enhances (diminishes) with electron (positive ion) temperature when other plasma parameters are invariant.
\section{Conclusion}
\label{5sec:Conclusion}
In our present investigation, we have considered a multi-component magnetized PIP having static dust grains, non-thermal electrons and positrons.
The Burgers' equation has been derived by employing reductive perturbation method \cite{C3} for studying DIASHWs. The results that we have found from this investigation can be summarized as follows:
\begin{itemize}
\item The negative (positive) shock profile can be exist for the value of $\alpha_{4}<\alpha_{4c}$ ($\alpha_{4}>\alpha_{4c}$).
\item The steepness of both positive and negative shock profiles declines with the increase of $\eta$ without affecting the height.
\item The increase in oblique angle rises the height of the positive shock profile.
\item The height of the positive shock wave increases with the number density of positron.
\item The temperature of the electrons enhances the amplitude of the shock profile.
\end{itemize}
The results are applicable in understanding the criteria for the formation of DIASHWs in astrophysical plasmas,
viz., cometary comae \cite{Chaizy1991}, upper regions of Titan's atmosphere \cite{Coates2007,Massey1976,Sabry2009},
plasmas in the D and F-regions of Earth's ionosphere \cite{Massey1976,Sabry2009,Abdelwahed2016}, and also in
laboratory environments, viz., ($K^+$, $SF_6^-$) plasma \cite{Song1991,Sato1994}, ($Ar^+$, $F^-$) plasma \cite{Nakamura1984}, plasma processing reactors \cite{Gottscho1986}, plasma etching \cite{Sheehan1988}, combustion products \cite{Sheehan1988}, ($Xe^+$, $F^-$) plasma \cite{Ichiki2002}, neutral beam sources \cite{Bacal1979}, ($Ar^+$, $SF_6^-$) plasma \cite{Wong1975,Cooney1991,Nakamura1997},($Ar^+$, $O_2^-$) plasma, and Fullerene ($C_{60}^+$, $C_{60}^-$) plasma \cite{Oohara2003,Hatakeyama2005}, etc.
|
\section{Introduction}
Throughout the paper we work over the field of complex numbers $\mathbb C$.
Motivated by applications, there has been a considerable amount of recent research on ranks
and border ranks of tensors, see, e.g., \cite{Ltensorbook,MR2535056} and
references therein.
In signal processing one is interested in
determining ranks of tensors, see, e.g., \cite{Como02:oxford} and
references therein. In computational complexity, one looks
for exotic algorithms via limits of tensors of a given rank,
see \cite{MR623057}. There are adequate tests
to determine the border ranks of tensors of small
border rank, however the possible ranks
of such tensors are not well understood.
In this article we present normal forms for tensors of
border rank three. Already in this case the problem becomes subtle. We work in
the more general setting of secant varieties.
\subsection{Definitions, notational conventions} For a projective variety $X\subset \mathbb P}\def\BT{\mathbb T V$ not contained in
a hyperplane, the
\emph{$X$-rank} of $p\in \mathbb P}\def\BT{\mathbb T V$, $R_X(p)$, is defined to be the smallest $r$ such that
there exist $x_1, \hdots , x_r\in X$ such that $p$ is in the span of
of $x_1, \hdots , x_r$,
and the \emph{$X$-border rank} $\brank}\def\uR{\brank_X(p)$ is defined to be the smallest
$r$ such that there exist curves $x_1(t), \hdots , x_r(t)\in X$ such that $p$ is in the span of
the limiting plane $\lim_{t\rightarrow 0}\langle x_1(t), \hdots , x_r(t)\rangle$.
Let $\sigma_r(X)\subset \mathbb P}\def\BT{\mathbb T V$ denote the set of points of
$X$-border rank at most $r$. When $X=Seg(\mathbb P}\def\BT{\mathbb T A_1\times \cdots\times \mathbb P}\def\BT{\mathbb T A_n)
\subset \mathbb P}\def\BT{\mathbb T (A_1{\mathord{\otimes\cdots\otimes}\;} A_n)$ is the set of rank one tensors in a space
of tensors, the $X$-rank and border rank agree with the usual
notions of tensor rank and border rank.
The set of points of $X$-rank $r$
contains a Zariski open subset of $\sigma_r(X)$ and we are interested
in the complement of this set.
We let $\sigma_r^0(X)$ denote the points of $\sigma_r(X)$ of
rank $r$. The tangential variety of a smooth variety $X\subset \mathbb P}\def\BT{\mathbb T V$, $\tau(X)\subset \mathbb P}\def\BT{\mathbb T V$, consists of all points
on all embedded tangent $\pp 1$'s. For varieties $X,Y\subset \mathbb P}\def\BT{\mathbb T V$, define
$$J(X,Y):=\overline{\{ p\in \mathbb P}\def\BT{\mathbb T V\mid \exists x\in X,\ y\in Y {\rm such\ that\ } p\in
\langle x,y\rangle\} },
$$
the {\it join} of $X$ and $Y$. Note that $J(X,X)=\sigma_2(X)$.
For a set $Z\subset \mathbb P}\def\BT{\mathbb T V$, $\hat Z\subset V$ denotes
the cone over it and $\langle Z\rangle$ its linear span. For a variety $Y\subset \mathbb P}\def\BT{\mathbb T V$, $Y_{sing}$ denotes the singular points of $Y$.
The affine tangent space to a variety $X\subset \mathbb P}\def\BT{\mathbb T V$ at a smooth point $x$ is denoted $\hat T_xX\subset V$.
Throughout the paper we assume $\fromto{A_1}{A_n}, A, B, C$ are complex vector spaces
of dimension at least $2$.
\subsection{Results on ranks and normal forms
for tensors}
The following proposition was probably \lq\lq known to the experts\rq\rq\ but we did not find it
in the literature, so we include a statement and proof.
\renewcommand{\theenumi}{(\alph{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\begin{proposition}\label{lastthm}
Let $X=Seg(\mathbb P}\def\BT{\mathbb T A_1\times \dots \times \mathbb P}\def\BT{\mathbb T A_n)\subset \mathbb P}\def\BT{\mathbb T (A_1{\mathord{\otimes\cdots\otimes}\;} A_n)$ be a Segre variety.
There is a normal form for points $x\in \hat\sigma_2(X)$:
\begin{enumerate}
\item $x=a_1^1{\mathord{\otimes\cdots\otimes}\;} a_1^n$ for a point of $X$, which has rank $1$,
\item $x=a_1^1{\mathord{\otimes\cdots\otimes}\;} a_1^n+ a_2^1{\mathord{\otimes\cdots\otimes}\;} a_2^n$ for a point on
a secant line to $X$ (here we require at least two of the $a^i_2$ to
be independent of the corresponding $a^i_1$), which has rank $2$,
\item and for
each $J\subseteq \setfromto{1}{n}$, $|J|>2$, the normal form
\begin{equation}\label{tprimevect}
x=\sum_{j \in J}a_1^1{\mathord{\otimes\cdots\otimes}\;} a_{1}^{j -1}{\mathord{ \otimes } } a_2^{j }{\mathord{ \otimes } } a_1^{j +1}{\mathord{\otimes\cdots\otimes}\;} a_1^n
\end{equation}
where each
$a^j_2$ is independent of the corresponding $a^j_1$. This case has rank $|J|$.
\end{enumerate}
In particular, all ranks from $1$ to $n$ occur for elements of $\sigma_2(X)$.
\end{proposition}
Our main result is the analogous classification for points in the third secant variety of the Segre product:
\renewcommand{\theenumi}{(\roman{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\begin{theorem}\label{s3nformthm}
Assume $n\geq 3$
and let $X := Seg(\mathbb P}\def\BT{\mathbb T A_1\times \cdots\times \mathbb P}\def\BT{\mathbb T A_n)$.
Let $p=[v]\in \sigma_3(X)\setminus \sigma_2(X)$.
Then $v$ has one of the following normal forms:
\begin{enumerate}
\item\label{item_main_thm_normal_form_honest_secant}
$v=x+y+z$ with $[x], [y], [z] \in X$,
\item\label{item_main_thm_normal_form_point_plus_tangent}
$v= x'+y$, with $[x],[y]\in X$ and $x'\in \hat T_{[x]}X$,
\item\label{item_main_thm_normal_form_third_order_pt}
$v = x'+x''$, where $[x(t)]\subset X$ is a curve and $x'=x'(0)$, $x''=x''(0)$, or
\item\label{item_main_thm_normal_form_two_tangents}
$v=x'+y'$, where $[x],[y]\in X$ are distinct points that
lie on a line contained in $X$,
$x'\in \hat T_{[x]}X$, and $y'\in \hat T_{[y]}X$.
\end{enumerate}
The points of type \ref{item_main_thm_normal_form_honest_secant}
contain a Zariski open subset of $\sigma_3(X)\setminus \sigma_2(X)$.
If ${\rm dim}\; A_i \ge 3$, then
those of type \ref{item_main_thm_normal_form_point_plus_tangent}
have codimension one in $\sigma_3(X)$,
those of type \ref{item_main_thm_normal_form_third_order_pt}
are contained in the closure of those of type \ref{item_main_thm_normal_form_point_plus_tangent}
and have codimension two in $\sigma_3(X)$,
those of type \ref{item_main_thm_normal_form_two_tangents}
are in the closure of the set of points of type \ref{item_main_thm_normal_form_third_order_pt}
and have codimension four in $\sigma_3(X)$.
There are $n$ distinct components of points of type \ref{item_main_thm_normal_form_two_tangents}.
A general point of each type is not a point of any of the other types.
\end{theorem}
When $n=2$, all points on $ \sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A_1\times \mathbb P}\def\BT{\mathbb T A_2))\backslash \sigma_2(Seg(\mathbb P}\def\BT{\mathbb T A_1\times \mathbb P}\def\BT{\mathbb T A_2))$ are of type \ref{item_main_thm_normal_form_honest_secant}.
The following result may also have been \lq\lq known to the experts\rq\rq\ but we did not find it
in the literature either:
\begin{theorem}\label{txsmooth}
A general point of $\tau(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$, i.e., a
point with the normal form \eqref{tprimevect} with $|J|=3$, is a smooth point
of $\sigma_2(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$.
In particular
\[
\text{codim}(\sigma_2(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))_{sing},
\sigma_2(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)))\geq 2.
\]
\end{theorem}
We prove an analogous result for $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$:
\begin{theorem}\label{s3sing}
Let $p \in \sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$.
If $p$ is a general point of type \ref{item_main_thm_normal_form_point_plus_tangent}
or \ref{item_main_thm_normal_form_third_order_pt},
or a general point of any component of points of type \ref{item_main_thm_normal_form_two_tangents},
then $p$ is a nonsingular point of $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$.
Moreover, if ${\rm dim}\; A, {\rm dim}\; B,{\rm dim}\; C \ge 3$,
and $p$ is a general point in the set of the points contained in some $\mathbb P(\mathbb C^2 \otimes \mathbb C^3 \otimes \mathbb C^3)$,
then $p$ is a nonsingular point of $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$,
and similarly for permuted statements.
In particular $\text{codim}(\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))_{sing},
\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)))\geq 2$.
\end{theorem}
Normal forms for Theorem \ref{s3nformthm} when $n=3$ are as follows:
\begin{enumerate}
\item $a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1+ a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2+ a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_3$
\item $a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1+ a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_3$
\item $a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2+ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_3 +a_1{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_1 +a_3{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1$
\item $ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2 + a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1+a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_3+a_1{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_1+a_3{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1$.
\end{enumerate}
For type \ref{item_main_thm_normal_form_two_tangents} there are two other normal forms,
where the role of $a$ is switched with that of $b$ and $c$.
These normal forms are depicted in terms of \lq\lq slices\rq\rq\ in
Table~\ref{table_orbits_general} on page~\pageref{table_orbits_general}.
(In the tensor literature, $3$-way tensors $T\in A{\mathord{ \otimes } } B{\mathord{ \otimes } } C$ are often studied by their
images $T(A^*)\subset B{\mathord{ \otimes } } C$ etc... and these images are studied in terms of bases, resulting
in a parametrized subspace of a space of matrices.
These parametrized spaces of matrices are called {\it slices}.)
Here $a_j,b_j,c_j$ need not be independent vectors, so
to parametrize the spaces, fix bases of each space and
write the $a_j,b_j,c_j$ as arbitrary linear combinations
of basis vectors. (However there are some independence requirements.)
Here are normal forms for all $n$:
\begin{align}
\label{nf1}& p_{(i)}= a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^n_{1}+ a^1_{2}{\mathord{\otimes\cdots\otimes}\;} a^n_{2}+
a^1_{3}{\mathord{\otimes\cdots\otimes}\;} a^n_{3} \\
&\label{nf2} p_{(ii)}= \sum_{i} a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2{\mathord{ \otimes } } a^{i+1}_{1}{\mathord{\otimes\cdots\otimes}\;}
a^n_{1}
+ a^1_{3}{\mathord{\otimes\cdots\otimes}\;} a^n_{3}\\
\label{nf3}
&
p_{(iii)}=\sum_{ i<j } a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2
{\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;}
a^{j-1}_{1}{\mathord{ \otimes } } a^j_2
{\mathord{ \otimes } } a^{j+1}_1
{\mathord{\otimes\cdots\otimes}\;}
a^n_{1}
\\ &\nonumber
+
\sum_{ i } a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3
{\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;}
a^n_{1}
\\
\label{nf4}
&p_{(iv)}=
\sum_{ s=2 }^{n} a^1_{2}{\mathord{ \otimes } } a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{s-1}_{1}{\mathord{ \otimes } } a^s_2
{\mathord{ \otimes } } a^{s+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1}.
\\
&\nonumber +
\sum_{ i=1 }^n a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3
{\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;}
a^n_{1}
\end{align}
Again, \eqref{nf4} has $n-1$ other normal forms, where the role of $a^1_{*}$ is exchanged with $a^i_{*}$.
Also, the vectors need not all be linearly independent.
\begin{remark}
In contrast to case (iv) above, already with four points on a three factor Segre spanning a three dimensional
vector space, one can obtain new limits by taking a second derivative, even when the limiting
points are distinct.
Consider the points $x_1=a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1$, $x_2=a_2{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1$,
$x_3=\frac 12(a_1+a_2){\mathord{ \otimes } }( b_1-b_2){\mathord{ \otimes } } c_1$, $x_4=\frac 12(a_1-a_2){\mathord{ \otimes } }( b_1+b_2){\mathord{ \otimes } } c_1$.
Note that $x_1=x_2+x_3+x_4$. Here both first and second derivatives of curves give
new points.
More generally, consider
\[
Seg(v_2(\pp 1)\times \underbrace{\pp 0\times \cdots\times \pp 0}_{(n-2)\text{ factors}}) \subset Seg(\mathbb P}\def\BT{\mathbb T A_1{\mathord{\otimes\cdots\otimes}\;} \mathbb P}\def\BT{\mathbb T A_n).
\]
Any four
points lying on
$
Seg(v_2(\pp 1)\times \pp 0\times \cdots\times \pp 0)
$
will be linearly dependent.
Exceptional limit points turn out to be important - an exceptional limit in
$\sigma_5(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ is used in Bini's
approximate algorithm to multiply $2\times 2$ matrices with an entry zero,
and an exceptional limit in $ \sigma_7(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)) $ is used in Sch\"onhage's
approximate algorithm to multiply $3\times 3$ matrices using $21$ multiplications,
see \cite[\S4.4]{jabu_ginensky_landsberg_Eisenbuds_conjecture}.
\end{remark}
\renewcommand{\theenumi}{(\arabic{enumi})}
\renewcommand{\labelenumi}{\theenumi}
Since there are only finitely many configurations of triples of points in $A_i$ up to the action of $GL(A_i)$, we conclude:
\begin{corollary}
There are only finitely many orbits of the action of
$GL(A_1) \times \dots \times GL (A_n)$ on $\sigma_3 (Seg(\mathbb P}\def\BT{\mathbb T A_1\times \dots \times \mathbb P}\def\BT{\mathbb T A_n))$.
\end{corollary}
In the three factor case, there are $39$ orbits, see \S\ref{sec_ranks_and_orbits}.
\begin{remark}
Points of the form $y+y'+y''$
where $y(t)$ is a curve on $\hat Seg(\mathbb P}\def\BT{\mathbb T A_1{\mathord{\otimes\cdots\otimes}\;} \mathbb P}\def\BT{\mathbb T A_n)$ have
rank at most $\binom {n+1}2$ because
all such points are of the form \eqref{nf3} (perhaps with linearly dependent variables).
The bound $R_{Seg(\mathbb P}\def\BT{\mathbb T A_1{\mathord{\otimes\cdots\otimes}\;} \mathbb P}\def\BT{\mathbb T A_n)}(y+y'+y'') \le \binom{n+1}{2}$ is not tight,
as for $n=3$ the following theorem
shows $R_{Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B \times \mathbb P}\def\BT{\mathbb T C)}(y+y'+y'')$ is at most five.
\end{remark}
\begin{theorem}\label{lastcor} The rank of a general point of the form $[y+y'+y'']$ of
$ \sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ as well as the
rank of a general point of the form $[x'+y']$ where $[x],[y]$ lie on a line in $Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)$,
is $5$.
All other points of $ \sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$
have rank less than five, so in particular, the maximum rank of any point of $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ is $5$.
\end{theorem}
\begin{remark} Theorem \ref{lastcor} seems to have been a \lq\lq folklore\rq\rq\ theorem
in the tensor literature. For example, in \cite{MR2535056}, Table 3.2 the result
is stated and refers to \cite{MR1088949}, but in that paper the result is stated
and a paper that never appeared is referred to. Also, there appear to have been privately circulating
proofs, one due to R. Rocci from 1993 has been shown to us.
We thank M. Mohlenkamp for these historical remarks.
\end{remark}
The Comon conjecture on ranks says that for
$T\in S^dV\subset V^{{\mathord{ \otimes } } d}$
the symmetric tensor rank of $T$ equals the tensor rank of $T$.
\begin{corollary}\label{comoncor} The Comon conjecture holds for
$T\in \hat \sigma_3(v_3(\mathbb P}\def\BT{\mathbb T V))$.
\end{corollary}
Corollary \ref{comoncor} follows by comparing the normal forms and ranks of this paper with those
of \cite{LTrank}.
\smallskip
In \S\ref{gencomin} we generalize Theorem \ref{s3nformthm} to generalized cominuscule varieties,
a class of homogeneous varieties which includes Grassmannians and spinor varieties.
See \S\ref{gencomin} for the definition of a generalized cominuscule variety, and \S 2 for the definition of the second fundamental form $II$.
\begin{theorem}\label{thm_points_in_Sigma} Let $X\subset \mathbb P}\def\BT{\mathbb T V$ be generalized cominuscule.
Then $[\inV{p}] \in \sigma_3(X)$
if and only if
at least
one of the following situations occurs:
\begin{itemize}
\item[\ref{item_main_thm_normal_form_honest_secant}]
$\inV{p} = \inV{\xi} + \inV{\eta} +\inV{\zeta}$
for some linearly independent $\inV{\xi}, \inV{\eta}, \inV{\zeta} \in \hat X$ ($\inV{p}$ is on an honest $3$-secant plane),
\item[\ref{item_main_thm_normal_form_point_plus_tangent}]
$\inV{p} = \inV{\xi'} +\inV{\eta}$ for some $\inV{\xi}, \inV{\eta} \in \hat X^0$
and $\inV{\xi'} \in \hat{T}_{[\inV{\xi}]} X$,
\item[\ref{item_main_thm_normal_form_third_order_pt}]
$\inV{p} = \inV{\xi'} + II( (\eta')^2)$ for some $\inV{\xi} \in \hat X$,
$\inV{\xi'} \in {\hat T}_{[\inV{\xi}]} X$, $\inV{\eta'} \in {T}_{[\inV{\xi}]} X$, or
\item[\ref{item_main_thm_normal_form_two_tangents}]
$\inV{p} = \inV{\xi'} + \inV{\eta'}$ for some $\inV{\xi},\inV{\eta} \in \hat X$,
$\inV{\xi'} \in { \hat T}_{[\inV{\xi}]} X$,
$\inV{\eta'} \in {\hat T}_{[\inV{\eta}]} X$ with the line $\mathbb P}\def\BT{\mathbb T \langle \xi,\eta\rangle$ contained in $X$.
\end{itemize}
To make sense of elements of the tangent and normal spaces as elements of $V$ we
have chosen a splitting $V=\hat x\oplus T\oplus N$ as described in
\S\ref{taylorsect}.
\end{theorem}
\subsection{Overview}
In \S\ref{diffgsect} we review facts from projective differential geometry.
In \S\ref{gencomin} we prove Theorem \ref{thm_points_in_Sigma}.
In \S\ref{s3exsect} we apply Theorem \ref{thm_points_in_Sigma} to cominuscule varieties,
including Grassmannians and spinor varieties.
In \S\ref{smallsecsegsect} we analyze the case of the Segre variety in detail,
and we give two proofs of Theorem~\ref{s3nformthm}, a short proof by computing the
Lie algebras of the stabilizers of the points $p_{(*)}$, and a longer proof that contains
more precise information which is of interest in its own right.
In \S\ref{sec_ranks_and_orbits} we restrict attention to the three-factor Segre variety,
and prove Theorems~\ref{txsmooth}, \ref{s3sing} and \ref{lastcor}.
\subsection{Acknowledgments}
We thank M. Mohlenkamp for pointing out
an error in an earlier version of this article,
related to the rank of
$y+y'+y''$ in Theorem~\ref{lastcor}.
This paper grew out of questions raised at the
2008 AIM workshop {\it Geometry and representation theory of tensors
for computer science, statistics and other areas}, and the authors
thank AIM and the conference participants for inspiration.
The mathematics in this paper was finally completed while the authors were
guests at the Mittag-Leffler Institut in Spring 2011 and we gratefully
thank the institute for providing a wonderful environment for
doing mathematics.
We truly appreciate the help of the referee,
his careful proof reading of the article,
and his many thoughtful comments.
\section{Curves in submanifolds of projective space}\label{diffgsect}
\subsection{Fubini forms, fundamental forms, and the prolongation property}\label{taylorsect}
Let $X^n\subset \mathbb P}\def\BT{\mathbb T V$ be a subvariety and let $\basept \in X$ be a smooth point.
We may choose a splitting
\begin{equation}\label{splitting}
V = \baseptline \oplus T \oplus N,
\end{equation}
such that $\baseptline \simeq \mathbb C$
is the one dimensional linear subspace corresponding to $\basept \in \mathbb P V$,
and $\baseptline \oplus T$ is the affine tangent space $\hat T_{\basept} X$.
We will abuse notation and identify $T$ with the Zariski tangent space
$T_\basept X= \hat o^*{\mathord{ \otimes } } (\hat T_{\basept}X/\hat o)$ and $N$ with the normal space
$N_\basept X:=T_{\basept}\mathbb P}\def\BT{\mathbb T V/T_{\basept}X$. Since we are working at a point, the twist by the line
bundle will not matter. Our choice of splitting will not effect the end
results of the calculations.
Any point $[\inV{v}] \in \mathbb P V$ has a lift to a point $\inV{v} \in V$
of the form $(\baseptaff, \prj{v}, \nrml{v})$ or $(0, \prj{v}, \nrml{v})$,
where $0$ and $\baseptaff$ are points in $\baseptline \simeq \mathbb C$,
and $\prj{v} \in T$, $\nrml{v} \in N$.
In an analytic neighborhood of $\basept$ we may write $X$ as a graph, that is, for $\inV{x} \in X$ near $\basept$,
the vector $\nrml{x}$ depends holomorphically on the vector $\prj{x}$ and we expand this holomorphic map into a Taylor series:
\begin{equation}\label{taylore}
\nrml{x}=\nrml{x}(\prj{x})=
II_{\basept}(\prj{x}^2) + F_{3, \basept}(\prj{x}^3) + F_{4, \basept}(\prj{x}^4)+\dotsb
\end{equation}
Here $\prj{x}\in T$ and $\prj{x}^s\in S^sT$.
Later we will study curves $x(t) \in X$, and express the whole curve using \eqref{taylore},
writing $\prj{x}(t)$ to be the curve in $T$, $\prj{x}^s(t) \in S^sT$.
Note that by our choice of splitting there is no constant or linear term in \eqref{taylore}.
The quadratic part $II_{\basept}= F_{2,\basept}$ gives rise to a well defined tensor in $S^2 T^*_{\basept}X{\mathord{ \otimes } } N_{\basept} X \simeq S^2 T^*{\mathord{ \otimes } } N$,
called the \emph{second fundamental form}.
Further, $F_{s,{\basept}} \in S^sT^*{\mathord{ \otimes } } N$
are called the {\it Fubini forms},
but they depend on the choice of splitting $V = \baseptline \oplus T \oplus N$.
See \cite[Chap. 3]{IvL} for more details.
\smallskip
One can extract tensors from the Fubini forms, called \emph{fundamental forms}.
Let
$$N_{s,{\basept}}:=
N_{\basept}X\operatorname{mod} \operatorname{Image}(F_{2,{\basept}}, \hdots , F_{s-1,{\basept}}),
$$
the tensor $\BF_{s,{\basept}}:= \bigl( F_{s,{\basept}} \operatorname{mod} \operatorname{Image}(F_{2,{\basept}}, \hdots , F_{s-1,{\basept}})\bigr)
\in S^sT^*_{\basept}X{\mathord{ \otimes } } N_{s,{\basept}}$
is well-defined (independent of the choice of splitting \eqref{splitting}) and called the {\it $s$-th fundamental form} of $X$ at~${\basept}$.
Fundamental forms satisfy a {\it prolongation property} (see \cite[Chap. 3]{IvL}):
if ${\basept}\in X$ is a general point, then for all $f_1\in S^{s_1} T$ and $f_2 \in S^{s_2} T$ we have
\begin{equation}\label{equ_prolongation}
\BF_{s_1,{\basept}}(f_1) = 0 \Longrightarrow \BF_{s_1+s_2,{\basept}}(f_1 f_2) =0.
\end{equation}
We write $III_{\basept}=\BF_{3,\basept}$.
If there is no risk of confusion, we will often omit the base point and write $II := II_{\basept}$, $F_s:= F_{s,\basept}$, etc.
\subsection{When taking limits, we may assume one curve is stationary}
\begin{lemma}\label{lem_one_curve_is_constant}
Let $G$ be a connected algebraic group and $P$ a parabolic subgroup.
Let $X=G/P\subset \mathbb P}\def\BT{\mathbb T V$ be a homogeneously embedded homogeneous
variety and let $p \in \sigma_r(X)$. Then there exist a point $\inV{\xi} \in \hat{X}$
and $r-1$ curves $\inV{y_j}(t) \in \hat X $ such that
$p \in \lim_{t \to 0} \langle \inV{\xi}, \inV{y_1}(t), \hdots , \inV{y_{r-1}}(t)\rangle$.
\end{lemma}
\begin{proof}
Since $p \in \sigma_r(X)$, there exist $r$ curves $\inV{x}(t), \inV{y_1}(t), \hdots , \inV{y_{r-1}}(t) \in \hat{X}$
such that
$$\inV{p} \in \lim_{t \to 0} \langle \inV{x}(t), \inV{y_1}(t), \hdots , \inV{y_{r-1}}(t)\rangle.$$
Choose a curve $g_t \in G$, such that $g_t(\inV{x}(t)) = \inV{x}_0 = \inV{x}(0)$ for all $t$ and $g_0=Id$.
We have \begin{align*}
\langle \inV{x}(t), \inV{y_1}(t), \hdots , \inV{y_{r-1}}(t)\rangle & =
{g_t}^{-1} \cdot \langle \inV{x}_0, g_t \cdot\inV{y_1}(t), \hdots , g_t \cdot\inV{y_{r-1}}(t)\rangle \text{ and}\\
\lim_{t \to 0} \langle \inV{x}(t), \inV{y_1}(t), \hdots , \inV{y_{r-1}}(t) \rangle & =
\lim_{t \to 0} \bigl({g_t}^{-1} \cdot \langle \inV{x}_0, g_t \cdot\inV{y_1}(t), \hdots , g_t \cdot\inV{y_{r-1}}(t)\rangle \bigr) \\
& {=}
\lim_{t \to 0} \langle \inV{x}_0, g_t \cdot\inV{y_1}(t), \hdots , g_t \cdot \inV{y_{r-1}}(t) \rangle . \\
\end{align*}
Set $\inV{\xi} = \inV{x_0}$ and appropriately modify the $\inV{y_j}(t)$
to complete the proof.
\end{proof}
We remark, that for non-homogeneous $X$, an analogous statement is rarely true.
If $r=2$, and $X$ is smooth, then it is true, see Proposition~\ref{prop_case_r=2}.
But already if $r=2$ and $X$ is singular, one often needs both curves moving
(a cuspidical rational curve embedded in $\mathbb P^3$ is an example).
Also if $r=3$, and $X$ has a trisecant line
(for example $X$ is a high degree rational normal curve projected from a general point on a trisecant plane),
then one also needs three curves moving to obtain some of the points on the third secant variety.
\subsection{Dimension counting and higher order invariants}
Since $\operatorname{dim} \sigma_r(X)\leq r\operatorname{dim} X+r-1$, one can use a parameter
count to see what one expects in choosing a point of the boundary.
Suppose $\operatorname{dim} X>1$, $X$ is not a cone and
the third fundamental form
is nonzero --- for example $X= Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)$.
One can predict that the third fundamental form does not arise when computing
a point of $\sigma_3(X)$ which is on a plane obtained as a limit of spans of $3$ points converging to the same general point of $X$.
This is because the third fundamental form is only well defined modulo the second osculating space,
which will have dimension greater than $\operatorname{dim} X$.
In the case of the three factor Segre variety
the second osculating space has dimension ${\mathbf{a}}{\mathbf{b}}+{\mathbf{a}}{\mathbf{c}}+{\mathbf{b}}{\mathbf{c}}$,
and the third fundamental form is only well defined modulo
the second osculating space.
So were there a term $III(v^3)$ appearing in an expression
for a point on $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$, with no restrictions
on $v$, then the resulting variety would
have to have dimension at least ${\mathbf{a}}{\mathbf{b}}+{\mathbf{a}}{\mathbf{c}}+{\mathbf{b}}{\mathbf{c}}$
for the term to be well defined.
If the dimensions of the vector spaces are sufficiently large, this contradicts the dimension count.
Such heuristics can be useful in calculations.
The following lemma will allow us to eliminate higher fundamental forms from our considerations
when studying $\sigma_3(X)$.
It illustrates the dimension counting principle.
\begin{lemma}\label{lem_vanishing_of_II_gives_vanishing_of_F_s}
Let $X\subset \mathbb P}\def\BT{\mathbb T V$ be a variety and let $\basept\in X$ be a general point. Adopt the notations
of \S\ref{taylorsect}.
Suppose $\prj{v}(t) \subset T$ is a curve such that $II(\prj{v}(t)^2)$ vanishes at $t=0$ up to order $m-1$,
that is $II(\prj{v}(t)^2) = t^m (\cdots)$.
If $m>0$ and $s\ge 2$, then $\mathbb F_s(\prj{v}(t)^s)$ vanishes at $t=0$ up to order $m+s-3$,
that is $\mathbb F_s(\prj{v}(t)^s)= t^{m+s-2} (\cdots)$.
\end{lemma}
\begin{proof}
Let $\ccI^d:=\set{f \in S^d T \mid \mathbb F_d(f) =0}$.
Since $\ccI^d$ is a linear subspace of $S^d T$,
the prolongation property \eqref{equ_prolongation} implies $\ccI^{d_1} \cdot S^{d_2} T \subset \ccI^{d_1+d_2}$.
Thus, if $S:=\bigoplus_{d=0}^{\infty} S^d T$ is the symmetric algebra,
and $\ccI := \bigoplus_{d=0}^{\infty} \ccI^d$, then $\ccI$ is a homogeneous ideal.
Consider $S[[t]]$, the power series ring with coefficients in $S$, and let $\ccJ_k$
be the ideal generated by $\ccI$ and $t^k$.
The curve $\prj{v} = \prj{v}(t) = \prj{v}_0 + t \prj{v}_1 + t^2 \prj{v}_2 +\dotsb$
is naturally an element in $S[[t]]$.
In this interpretation,
$\mathbb F_s(\prj{v}(t)^s) = t^k (\cdots)$ if and only if $\prj{v}(t)^s \in \ccJ_{k}$.
In particular, our assumptions are:
\begin{itemize}
\item $\prj{v}(t)^2 \in \ccJ_{m}$ and
\item the constant coefficient $\prj{v}_0^2 \in \ccI$ (because $m >0$), thus also $\prj{v}_0^s \in \ccI$ for $s \ge 2$.
\end{itemize}
To show that $\prj{v}(t)^s \in \ccJ_{m+s-2}$ for $s\ge 2$,
we argue by induction on $s$.
Consider $\frac{\partial}{\partial t} \left(\prj{v}(t)^s\right) = s \frac{\partial \prj{v}}{\partial t} \prj{v}^{s-1}$.
By the inductive assumption, $\prj{v}^{s-1}\in \ccJ_{m+s-3}$,
so $\frac{\partial}{\partial t} \left(\prj{v}(t)^s\right) \in \ccJ_{m+s-3}$.
Since the constant coefficient $\prj{v_0}^s \in \ccI$, it follows,
that $\prj{v}(t)^s \in \ccJ_{m+s-2}$ as claimed.
\end{proof}
\subsection{Points on $\sigma_2(X)$}
We reprove the standard fact that a point
on a secant variety to a smooth variety $X$ is either on $X$, on
an honest secant line, or on a tangent line to $X$.
The proof we present prepares the way for new results.
Recall that if a point of $\sigma_2(X)$ is not on an
honest secant line, it must arise from a point on a limiting $\pp 1$ which is obtained
by a curve of $\pp 1$'s, $\langle x(t), y(t)\rangle$ where $[x(0)]=[y(0)]$.
\begin{proposition}\label{prop_case_r=2}
Let $X\subset \mathbb P}\def\BT{\mathbb T V$ be a smooth variety and let $[z]\in \sigma_2(X)\backslash \sigma_2(X)^0$. Then $z$
may be obtained from first order information, that is,
$z=u'$ for some $[u]\in X$ and $u'\in \hat T_{[u]}X$.
\end{proposition}
\begin{proof} There exist curves
$[\inV{x}(t)],[\inV{y}(t)]\subset X$ with
$\inV{x}(0) = \inV{y}(0) =\baseptline \in \basept \setminus \set{0}$,
such that $[z]$ may be obtained
as a point of the limiting $\pp 1=\mathbb P}\def\BT{\mathbb T(\lim_{t\rightarrow 0} \langle \inV{x}(t),\inV{y}(t)\rangle)$.
Consider a splitting $V = \hat \basept \oplus T \oplus N$
and the curves $\prj{x}(t), \prj{y}(t) \in T$ as above.
Write:
\begin{align*}
\prj{x}(t) & = \prj{x_1} t + \prj{x_2} t^2 + \dotsb + \prj{x_{k-1}} t^{k-1} + \prj{x_k} t^{k} + \prj{x_{k+1}} t^{k+1} + \dotsb\\
\prj{y}(t) & = \prj{x_1} t + \prj{x_2} t^2 + \dotsb + \prj{x_{k-1}} t^{k-1} + \prj{y_{k}} t^{k} + \prj{y_{k+1}} t^{k+1} + \dotsb
\end{align*}
where $\prj{x_j},\prj{y_j}\in T$
and $k$ is the smallest integer such that $\prj{v_0} := \prj{y_k}- \prj{x_k} \ne 0$.
Let $\prj{v}(t) := t^{-k}(\prj{y}(t) -\prj{x}(t)) = (\prj{y_k}-\prj{x_k}) + (\prj{y_{k+1}}-\prj{x_{k+1}}) t + \dotsc$.
Then:
\begin{align*}
\inV{y}(t) &- \inV{x}(t)
= (\baseptaff + \prj{y}(t) + II(\prj{y}(t)^2) + F_3(\prj{y}(t)^3) + \dotsb)
- (\baseptaff + \prj{x}(t) + II(\prj{x}(t)^2) + F_3(\prj{x}(t)^3) + \dotsb) \\
& = t^k \prj{v}(t) + II\left(\prj{y}(t)^2 - \prj{x}(t)^2\right) + F_3\left(\prj{y}(t)^3 - \prj{x}(t)^3\right) + \dotsb \\
& = t^k \prj{v}(t) + II\left((\prj{y}(t) - \prj{x}(t))(\prj{x}(t)+\prj{y}(t))\right) + F_3\left((\prj{y}(t) - \prj{x}(t))(\prj{x}(t)^2 + \prj{x}(t)\prj{y}(t) + \prj{y}(t)^2)\right) + \dotsb \\
& = t^k \prj{v}(t) + II\left(t^k \prj{v}(t) (\prj{x}(t)+\prj{y}(t))\right) + F_3\left(t^k \prj{v}(t)(\prj{x}(t)^2 + \prj{x}(t)\prj{y}(t) + \prj{y}(t)^2)\right) + \dotsb
\end{align*}
Since $\prj{x}(t)$ and $\prj{y}(t)$ have no constant terms, we obtain:
\begin{align*}
\inV{y}(t) - \inV{x}(t) & = t^k \prj{v_0} + t^{k+1} (\dotsb) \text{ and} \\
\inV{x}(t) \wedge \inV{y}(t)
& = \inV{x}(t) \wedge (\inV{y}(t) - \inV{x}(t)) \\
& = \left( \baseptaff + t(\dotsc) \right) \wedge \left( t^k \prj{v_0} + t^{k+1} (\dotsb) \right)\\
& = t^k \left( \baseptaff \wedge \prj{v_0} \right) + t^{k+1}(\dotsc).
\end{align*}
Recall that $\prj{v_0} \wedge \baseptaff \ne 0$.
Thus the limiting affine plane $\lim_{t\rightarrow 0} \langle \inV{x}(t),\inV{y}(t)\rangle)$ is equal to
$\langle\baseptaff, \prj{v_0} \rangle$.
Set $\prj{z}(t) := t\prj{v}(t) \in T$ and $\inV{z}(t) := \baseptaff + t \prj{v}(t) + II(t^2 \prj{v}(t)^2) + \dotsb \in \hat X$.
Then the same affine plane can be obtained as
$\lim_{t\rightarrow 0} \langle \baseptaff,\inV{z}(t)\rangle$,
thus one point is fixed and the other approaches the first one from the direction of $\prj{v_0}$.
\end{proof}
\section{Generalized cominuscule varieties: proof of theorem \ref{thm_points_in_Sigma} }\label{gencomin}
Following \cite{LWtan}, a homogeneously embedded homogeneous variety $G/P\subset \mathbb P}\def\BT{\mathbb T V$ is called \emph{generalized
cominuscule} if there is a choice of splitting (at any point) such that the Fubini forms
reduce to fundamental forms, that is:
\begin{equation}\label{equ_splitting_for_gen_cominuscule}
V = \baseptline \oplus T \oplus N_2 \oplus N_3 \oplus \dotsb \oplus N_f
\end{equation}
with $F_s(S^s T) \subset N_s$ and thus $F_s = \mathbb F_s$ for all $s \in \setfromto{2}{f}$,
and $F_s = \mathbb F_s = 0$ for all $s > f$.
Generalized cominuscule varieties may be characterized intrinsically
as the homogeneously embedded $G/P$ where the unipotent radical of $P$ is abelian. A generalized cominuscule variety is cominuscule
if and only if $G$ is simple and the embedding is the minimal homogeneous one.
For those familiar with representation theory, a homogeneously embedded homogeneous variety $G/P\subset \mathbb P}\def\BT{\mathbb T V$
is cominuscule if $V$ is a fundamental representation $V_{\omega_i}$ where $\omega_i$ is a cominuscule weight, that is,
the highest root of $\mathfrak g$ has coefficient one on the simple root $\alpha_i$. Generalized cominuscule varieties
are Segre-Veronese embeddings of products of cominuscule varieties.
Grassmannians $G(k,W)$, projective spaces $\mathbb P^n$ and
products of projective spaces in any homogeneous embedding
(in particular, respectively, $G(k,W)$ in the Pl\"ucker embedding, Veronese varieties, and Segre varieties)
are generalized cominuscule.
Throughout this section we assume $X$ is generalized cominuscule.
When studying points of $\sigma_3(X)$, one has to take into account
curves limiting to points on a trisecant line of $X$. When $X$ is cut out
by quadrics, as with homogeneous varieties,
any trisecant line of $X$ will be contained in $X$.
Theorem~\ref{thm_points_in_Sigma}
shows such points are already accounted for
by curves with just one or two limit points, and that higher order differential invariants
do not appear,
as was hinted at in Lemma \ref{lem_vanishing_of_II_gives_vanishing_of_F_s}.
We commence the proof of Theorem \ref{thm_points_in_Sigma} with an observation about the freedom of choice of splitting as in \eqref{equ_splitting_for_gen_cominuscule}.
\begin{lemma}\label{lem_can_pick_splitting}
Let $X$ be generalized cominuscule and let $x$, $\fromto{y_1}{y_{r-1}}$ be $r$ points on $X$.
Then there exists a choice of splitting as in \eqref{equ_splitting_for_gen_cominuscule} (so $F_s(S^sT) \subset N_s$ for all $s$),
such that $x = \basept$ is the center of this splitting and none of the points $\fromto{y_1}{y_{r-1}}$
lies on the hyperplane $T\oplus N_2 \oplus N_3\oplus \dotsb$.
\end{lemma}
\begin{proof}
Let $G$ be the automorphism group of $X$ and $P \subset G$ be the parabolic subgroup preserving $x$.
Let $Y \subset X\times \mathbb P}\def\BT{\mathbb T V^*$ be the set of those pairs $(\basept, H)$, where $\basept \in X$ and $H \subset V$
is a hyperplane, such that $V = \baseptline \oplus H$
and there exists a splitting $H= T \oplus N_2 \oplus N_3\oplus \dotsb$,
making the splitting of $V$ as in \eqref{equ_splitting_for_gen_cominuscule}.
Since $X$ is generalized cominuscule, $Y$ is non-empty.
It is also $G$-invariant, under the natural action $g\cdot (x, H) = (g\cdot x, g \cdot H)$.
Let $Y_x \subset \mathbb P(V^*)$ be the fiber over $x$.
It is also non-empty, because $G$ acts on $X$ transitively, and it is $P$-invariant.
Since the Lie algebra of $P$ contains all positive root spaces,
and $\hat x$ is the highest weight space,
the line $\hat x$ is contained in every $P$-invariant
linear subspace of $V$ (see, e.g., \cite[Prop.~14.13]{FH}).
Fix $H_0 \in Y_x$ and consider
the intersection $B:= \bigcap_{p \in P} p \cdot H_0$.
This is a linear subspace of $V$, which is invariant under $P$.
So either $B=0$ or $\hat x \subset B$. The latter is however impossible, as $\hat x \cap H_0 =0$ by our assumptions.
So $B=0$.
The set of hyperplanes $\set{p \cdot H_0 \in \mathbb P V^* \mid p \in P}$ is non-empty, irreducible with trivial base locus,
so its dimension is positive and by a trivial instance of Bertini's Theorem there exists at least one hyperplane $H$
in this set that avoids all points $\fromto{y_1}{y_{r-1}}$.
\end{proof}
Since there are only finitely many non-zero Fubini forms, the parameterization:
\begin{align*}
\phi \colon T & \to \hat X\\
\prj{v} & \mapsto \baseptaff + \prj{v} + II(\prj{v}^2) + \dotsb
\end{align*}
is polynomial.
\begin{remark}\label{rem_every_pt_is_in_infty_or_in_image}
Suppose $X$ is the closure of the image of a map
\begin{align*}
\phi \colon T & \to \mathbb P V \\
\prj{v} & \mapsto \baseptaff + \prj{v} + \nrml{v}(\prj{v})
\end{align*}
with $V = \baseptline \oplus T \oplus N$, $\baseptaff \in \baseptline \setminus \set{0}$,
and a polynomial map $\nrml{v} : T \to N$.
Then every point $y \in X$ is either on the hyperplane $\mathbb P(T\oplus N)$,
or is in the image of the parameterization $\phi$.
\end{remark}
\begin{proof}
We use the following elementary topological statement:
Let $P$ be a topological space, let $I \subset U \subset P$
with $I$ closed in $U$, and let $\bar{I}$ be the closure of $I$ in $P$.
Then $\bar{I} \cap U = I$.
To prove this, let $J \subset P$ be a closed subset such that $U \cap J = I$,
which exists from the definition of subspace topology.
Then $\bar{I} \subset J$, from the definition of the closure, and so
\[
I \subset \bar{I} \cap U \subset J \cap U = I.
\]
We use the statement with $P = \mathbb P V$, $U$ the affine piece of $\mathbb P V$,
which is the complement of the hyperplane $\mathbb P(T\oplus N)$,
and $I = \phi(T)$.
Note that $\phi(T)$ is closed in $U \simeq T \oplus N$, because it is the graph of $\nrml{v}$
(which is a polynomial map by our assumption).
Moreover, $\bar{I} = X$, and so $X\cap U = I$, and $X \subset I \cup \mathbb P(T\oplus N)$ as claimed.
\end{proof}
This implies the following property of tangent spaces on $X$.
\begin{lemma}\label{lem_tangent_at_line}
Let $X$ be generalized cominuscule and let $\ell \subset X$ be a line.
Then the space $\ccT^{\ell}:= { \hat T}_{[\inV{\xi}]} X + {\hat T}_{[\inV{\eta}]} X$
for any $[\inV{\xi}], [\inV{\eta}] \in \ell$ is independent of the choice of $[\inV{\xi}], [\inV{\eta}]$.
Moreover, ${\rm dim}\; \ccT^{\ell}$ is constant over each irreducible component
of the space parameterizing lines on $X$.
\end{lemma}
\begin{proof}
Fix $\basept:=[\inV{\xi}] \in \ell$. By Lemma~\ref{lem_can_pick_splitting} we may
choose a splitting \eqref{equ_splitting_for_gen_cominuscule} such that $[\inV{\eta}] \notin T\oplus N$.
Thus $[\inV{\eta}]$ is in the image
of the parameterization by Remark~\ref{rem_every_pt_is_in_infty_or_in_image}.
Consider a curve $\inV{y}(t) \in \hat X$ with $\inV{y}(0) = \inV{\eta}$.
Note that $\prj{y}(0) \in T$ is in the tangent direction to $\ell$.
Then in the splitting \eqref{equ_splitting_for_gen_cominuscule}:
\begin{align*}
\inV{y}'(0) & = \frac{\ud}{\ud t} (\baseptaff + \prj{y}(t) + II(\prj{y}(t)^2) + III(\prj{y}(t)^3) + \dotsb)|_{t=0}\\
& = \prj{y}'(0) + 2 II(\prj{y}'(0) \prj{y}(0)) + 3 III(\prj{y}'(0) \prj{y}(0)^2) + \dotsb\\
& \stackrel{(\star)}{=} \prj{y}'(0) + 2 II(\prj{y}'(0) \prj{y}(0)).
\end{align*}
Here $(\star)$ holds by the prolongation property \eqref{equ_prolongation}, because $II( \prj{y(0)}^2) =0$.
Thus letting $\nu'$ be any non-zero vector in $T_{\xi}\ell \subset T$ we have:
\begin{equation}\label{equ_formula_for_ccTell}
\ccT^{\ell} = { \hat T}_{[\inV{\xi}]} X + {\hat T}_{[\inV{\eta}]} X =
\set{\inV{\xi'} + II( \zeta' \nu' ) \mid \text{for } \inV{\xi'} \in {\hat T}_{[\inV{\xi}]} X, \inV{\zeta'} \in {T}_{[\inV{\xi}]} X}.
\end{equation}
This formula is independent of $\inV{\eta}$, so we can vary $\inV{\eta}\in \ell$ freely.
Exchanging the roles of $\inV{\xi}$, and $\inV{\eta}$, we can also vary $\inV{\xi}$.
Thus, $\ccT^{\ell}$ is determined by the geometry of $\ell\subset X$.
But the group of automorphisms of $X$ acts transitively on each irreducible component of the space parameterizing lines on $X$.
When $X=G/P$ with $G$ simple, this is
\cite[Thm.~4.3]{LM0} and \cite{MR1443819}.
(This is true for any minimally embedded homogeneous variety $G/P_I$, with $G$ simple, where $I$ indexes the
deleted simple roots, as long as $I$ does not contain an \lq\lq exposed short root\rq\rq\ in the language of \cite{LM0}.)
When $X=Seg(v_{d_1}(G_1/P_1)\times \cdots\times v_{d_n}(G_n/P_n))$ is generalized cominuscule (with each $G_i/P_i$ cominuscule), the set of lines
on $X$ is the disjoint union of the variety of lines on each $G_i/P_i$ such that $d_i=1$.
Thus ${\rm dim}\; \ccT^{\ell}$ must be constant over these irreducible components.
\end{proof}
Lemma \ref{lem_tangent_at_line} allows an alternative interpretation of the points of type \ref{item_main_thm_normal_form_two_tangents}:
\begin{lemma}\label{lem_alternative_iv}
With the notation as in Theorem~\ref{thm_points_in_Sigma},
let $\ptstypeiv$ denote the set of points of type~\ref{item_main_thm_normal_form_two_tangents}.
Then $[\inV{p}]\in \ptstypeiv$ if and only if
\begin{enumerate}
\renewcommand{\theenumi}{(iv')}
\item \label{nitem_points_in_Sigma_case_closure_of_IIb}
$\inV{p} = \inV{\xi'} + II( \zeta' \nu' )$ for some $\inV{\xi} \in \hat X$,
$\inV{\xi'} \in {\hat T}_{[\inV{\xi}]} X$, $\inV{\zeta'}, \inV{\nu'} \in {T}_{[\inV{\xi}]} X$
with $II( (\nu')^2 )=0$, i.e., $\nu'$ is tangent to a line on $X$ through $\xi$.
\end{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
Furthermore, $\ptstypeiv$ is a closed subset of $\mathbb P V$.
\end{lemma}
\begin{proof}
The alternative description \ref{nitem_points_in_Sigma_case_closure_of_IIb}
follows from \eqref{equ_formula_for_ccTell}.
To see that $\ptstypeiv$ is a closed subset of $\mathbb P V$,
note $\ptstypeiv$ is the image of a projective space bundle over the variety parameterizing lines on $X$,
whose fiber over $\ell \subset X$ is $\mathbb P (\ccT^{\ell})$.
Since ${\rm dim}\; \ccT^{\ell}$ is locally constant by Lemma~\ref{lem_tangent_at_line},
this bundle is a projective variety,
and thus $\ptstypeiv$ is an image of a projective variety, hence projective.
\end{proof}
In the following lemma, we provide an uniform interpretation of the points of types
\ref{item_main_thm_normal_form_third_order_pt}--\ref{item_main_thm_normal_form_two_tangents}.
\begin{lemma}\label{lem_form_of_points_in_closure_of_II}
$[\inV{p}]$ is of type \ref{item_main_thm_normal_form_third_order_pt} or \ref{item_main_thm_normal_form_two_tangents},
if and only if
\begin{enumerate}
\renewcommand{\theenumi}{(iii--iv)}
\item \label{item_points_in_Sigma_case_closure_of_II}
$\inV{p} = \inV{\xi'} + u$ for some $\inV{\xi} \in \hat X^0$,
$\inV{\xi'} \in \hat{T}_{[\inV{\xi}]} X$,
and $u \in \overline{II} := \overline{\set{II(\prj{v}^2) : \prj{v} \in T}}$.
\end{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
Moreover, for $u \in V$, the following conditions are equivalent:
\begin{enumerate}
\item \label{item_form_of_u_in_II_bar}
$u \in \overline{II}$;
\item \label{item_form_of_u_with_curve}
There exist a curve $\prj{v}(t) \in T$ and an integer $m$,
such that $II(\prj{v}(t)^2) = t^m u + t^{m+1} (\dotsc)$;
\item \label{item_form_of_u_with_vectors}
There exist an integer $m$ and vectors $\fromto{\prj{v_0}, \prj{v_1}}{\prj{v_m}} \in T$,
such that
\[
II\left(\sum_{i=0}^d \prj{v_i} \prj{v_{d-i}} \right) =
\begin{cases}
0 & \text{if } d < m \\
u & \text{if } d = m
\end{cases}
\]
\end{enumerate}
\end{lemma}
Note that $\mathbb P}\def\BT{\mathbb T \overline{II}$ is the closure of the image of the rational map $ii: \mathbb P}\def\BT{\mathbb T T\dashrightarrow \mathbb P}\def\BT{\mathbb T N$
given by $[\prj v]\mapsto [II(\prj v^2)]$.
\begin{proof}[Proof of Lemma~\ref{lem_form_of_points_in_closure_of_II}]
The equivalence of \ref{item_form_of_u_in_II_bar}--\ref{item_form_of_u_with_vectors} is clear.
In the notation of \ref{item_form_of_u_with_vectors},
a point $\inV{p}$ is of type \ref{item_main_thm_normal_form_third_order_pt}
if and only if it is of type \ref{item_points_in_Sigma_case_closure_of_II} with $m=0$,
and it is of type \ref{nitem_points_in_Sigma_case_closure_of_IIb}
if and only if it is of type \ref{item_points_in_Sigma_case_closure_of_II} with $m=1$.
So suppose $\inV{p}$ is of type \ref{item_points_in_Sigma_case_closure_of_II} with $m>1$.
Then it is in the closure of $\ptstypeiv$, the set of points of type \ref{nitem_points_in_Sigma_case_closure_of_IIb}.
But $\ptstypeiv$ is closed by Lemma~\ref{lem_alternative_iv},
so $\inV{p}$ is of type \ref{item_main_thm_normal_form_two_tangents}.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm_points_in_Sigma}]
Suppose $\inV{p} \in \sigma_3(X)$, so
there exist $\inV{\xi}$ and $\inV{y}(t):=y_1(t), \inV{z}(t):=y_2(t)$ as in Lemma~\ref{lem_one_curve_is_constant}.
Write $\inV{\xi} = \baseptaff$,
and by Lemma \ref{lem_can_pick_splitting} we may choose the splitting \eqref{equ_splitting_for_gen_cominuscule} such that for small values of $t$,
we have $\inV{y}(t),\inV{z}(t)\not\in T\oplus N$.
So
$\inV{y}(t)=(\baseptaff ,\prj{y}(t),\nrml y(t))$
by Remark~\ref{rem_every_pt_is_in_infty_or_in_image}
and similarly for $\inV{z}(t)$.
Consider the curves $\prj{y}(t),\prj{z}(t) \in T$.
Exchanging the roles of $\inV{y}$ and $\inV{z}$ if necessary, pick maximal integers $k,l$,
with $l \ge k\geq 0$ and such that:
\begin{align*}
\prj{y}(t)& = t^k \prj{v}(t) \text{ and}\\
\prj{z}(t)& = t^k \lambda(t) \prj{v}(t) + t^l \prj{w}(t)
\end{align*}
for some holomorphic function $\lambda(t) \in \mathbb C$ and curves $\prj{v}(t), \prj{w}(t) \in T$.
From now on, we write $\inV{y}$ for $\inV{y}(t)$, etc.
We adopt the convention $l=\infty$ if $\prj w=0$.
If $l=0$, then $0, \prj{y}_0, \prj{z}_0$ are three distinct and non-collinear points in $T$.
This implies that $\inV{p}$ is on an honest $3$-secant,
and we are in case \ref{item_main_thm_normal_form_honest_secant}.
So from now on suppose $l >0$.
Our goal is to understand the leading term (in $t$) of
\begin{equation}\label{equ_xyz_after_gauss}
\baseptaff \wedge \inV{y} \wedge \inV{z} =\baseptaff \wedge (\inV{y}-\baseptaff)
\wedge (\inV{z} - \baseptaff - \lambda(\inV{y}-\baseptaff)).
\end{equation}
Expanding out terms we obtain:
\begin{align}
\inV{y}-\baseptaff &=t^k \prj{v} + t^{2k} II( \prj{v}^2) + t^{3k} III( \prj{v}^3) + \dotsb\nonumber\\
(\inV{z} - \baseptaff &- \lambda(\inV{y}-\baseptaff))=
t^l \prj{w} + \sum_{s=2}^{f} \mathbb F_s\Bigl(\prj{z}^s - \lambda \prj{y}^s\Bigr) \nonumber\\
& = t^l \prj{w} + \sum_{s=2}^{f} \mathbb F_s\biggl(\left(\lambda t^k \prj{v} + t^l \prj{w}\right)^s
- \lambda t^{sk} \prj{v}^s\biggr)\nonumber\\
&= t^l \prj{w}_0 + \sum_{s=2}^{f} \mathbb F_s\Bigl((\lambda^s - \lambda) t^{sk} \prj{v}^s
+ s \lambda^{s-1} t^{(s-1)k+l} \prj{v}^{s-1}\prj{w})\Bigr)
+ t^{l+1} (\dotsc)\nonumber\\
& = t^l \prj{w}_0 + \sum_{s=2}^f (\lambda^s - \lambda) t^{sk} \mathbb F_s\bigl(\prj{v}^s\bigr)
+ s \lambda^{s-1} t^{(s-1)k+l} \mathbb F_s\bigl(\prj{v}^{s-1}\prj{w})\bigr)
+ t^{l+1} (\dotsc).
\label{equ_expansion_of_normal_part}
\end{align}
First consider the case $k \ge 1$, so that the three limit points coincide: $\baseptaff =\inV{y}_0 =\inV{z}_0$.
In this case, the terms in \eqref{equ_expansion_of_normal_part} with $t^{(s-1)k+l}$ are of order higher than $l$.
By Lemma~\ref{lem_vanishing_of_II_gives_vanishing_of_F_s}, the higher fundamental forms $\mathbb F_s$ with $s\ge 3$
will always have higher degree leading term than $II$.
Thus:
\[
\baseptaff \wedge \inV{y} \wedge \inV{z} = \baseptaff \wedge t^k \prj{v_0} \wedge
\left(t^l\prj{w_0} + t^{2k}\lambda(\lambda-1) II(\prj{v}^2)\right) + \dotsb
\text{terms of higher order}.
\]
We conclude that any point $p$ in the limiting space, which is spanned by $\baseptaff$, $\prj{v_0}$, and
the leading term of $\left(t^l\prj{w_0} + t^{2k}\lambda(\lambda-1) II(\prj{v}^2)\right)$, is of the form
\ref{item_points_in_Sigma_case_closure_of_II}.
\smallskip
In the remainder of the argument assume $k=0$ and we still assume $l>0$.
If $\lambda_0 \ne 0,1$,
the three limit points $ 0, \prj{y}_0, \prj{z}_0$ are distinct,
but they lie on a line in $T$.
Also suppose that $II(\prj{v_0}^2) \ne 0$.
This means (e.g. by \eqref{taylore}) that the projective line from $\basept$ in the direction of $\prj{v_0}$ is not contained in $X$.
It follows that
$\baseptaff, \inV{y_0}, \inV{z_0}$ are linearly independent,
because any line trisecant to $X$ is entirely contained in $X$.
This leads to case \ref{item_main_thm_normal_form_honest_secant}.
Now say $\lambda_0 = 0$ or $1$, and $II(\prj{v_0}^2) \ne 0$.
If $\lambda_0= 0$, then $ \baseptaff=\inV{z}(0)$.
If $\lambda_0= 1$, then $\inV{y}(0)= \inV{z}(0)$.
Swapping the roles of $\inV{x}$ and $\inV{y}$ if necessary, we may assume $\lambda_0=0$
and write $\lambda = t^m \lambda_m + t^{m+1} (\dotsc)$ with $m\ge 1$ and $\lambda_m \ne 0$.
Note also $\prj{y} = \prj{v}$ in this case (because $k=0$).
Then the leading term of \eqref{equ_expansion_of_normal_part} is the leading term of
$t^l\prj{w}_0 + \sum_{s=2}^{f} (\lambda^s - \lambda) \mathbb F_s\bigl(\prj{y_0}^s\bigr)$
or it is of order at least $l+1$. Therefore:
\begin{align*}
\baseptaff \wedge \inV{y} \wedge \inV{z} &=
\baseptaff \wedge \inV{y_0} \wedge
\left(t^{l} \prj{w_0} + \sum_{s=2}^{f} (\lambda^s - \lambda) \mathbb F_s\bigl(\prj{y_0}^s\bigr)\right)
+ \text{ terms of higher order}\\
&= \baseptaff \wedge \inV{y_0} \wedge
\Bigg(t^{l} \prj{w_0} +
\underbrace{\sum_{s=2}^{f} \lambda^s \mathbb F_s\bigl(\prj{y_0}^s\bigr)}_{=t^{2m}\cdot(\dotsc)} -
\lambda \underbrace{\sum_{s=2}^{f} \mathbb F_s\bigl(\prj{y_0}^s\bigr)}_{=\inV{y_0} - \baseptaff - \prj{y}_0}\Bigg)
+ \text{ terms of higher order}\\
&= \baseptaff \wedge \inV{y_0} \wedge
\left(\lambda \prj{y_0} + t^{l} \prj{w_0} \right)
+ \text{ terms of higher order}\\
&= \baseptaff \wedge \inV{y_0} \wedge
\left(\lambda_m t^m \prj{y_0} + t^{l} \prj{w_0}\right) + \text{ terms of higher order}.
\end{align*}
Note $\inV{y_0}$ is linearly independent from $T$, because $II(\prj{y_0}^2) \ne 0$.
We cannot have $m = l$ and $\prj{w_0} = - \lambda_m \prj{y_0}$, because then the choice of $l$ would not
be maximal.
Thus we have non-zero terms of degrees $l$ or $m$,
and the limiting space is spanned by $\baseptaff, \inV{y_0}$ and a tangent vector to $\basept$
(which is a linear combination of $\prj{y_0}$ and $\prj{w_0}$).
Therefore we are in case \ref{item_main_thm_normal_form_point_plus_tangent}.
Finally, suppose $II(\prj{v_0}^2) = 0$
(so the line $\langle \basept, y(0)\rangle$ is contained in $X$).
Hence \eqref{equ_expansion_of_normal_part} becomes:
\[
t^l \prj{w}_0 + \sum_{s=2}^f \bigl( (\lambda^s - \lambda) \mathbb F_s\bigl(\prj{v}^s\bigr)
+ s \lambda^{s-1} t^{l} \mathbb F_s\bigl(\prj{v}^{s-1}\prj{w})\bigr)
+ t^{l+1} (\dotsc).
\]
We claim that the summands with $\mathbb F_s$ for $s \ge 3$ are irrelevant to the leading term.
First note for $s\ge 3$ the fundamental form $ \mathbb F_s (\prj{v}^{s-1}\prj{w})$ vanishes at $t=0$
by the prolongation property \eqref{equ_prolongation}.
So $ t^{l} \mathbb F_s\bigl(\prj{v}^{s-1}\prj{w})\bigr)$ has order of vanishing at least $l+1$, unless $s=2$.
Next we treat
\begin{align*}
\sum_{s=2}^f (\lambda^s - \lambda) \mathbb F_s\bigl(\prj{v}^s\bigr) &
= (\lambda^2 - \lambda) \sum_{s=2}^f (1 + \lambda + \dotsb + \lambda^{s-2}) \mathbb F_s\bigl(\prj{v}^s\bigr)
\end{align*}
By Lemma~\ref{lem_vanishing_of_II_gives_vanishing_of_F_s},
for $s\ge 3$ the leading term of $\mathbb F_s(\prj{v}^s)$
is of higher order than that of $II(\prj{v}^s)$.
Thus the leading term of \eqref{equ_expansion_of_normal_part} can only come from the leading term of
\begin{equation} \label{equ_leading_term_in_last_case}
t^l \prj{w}_0 + (\lambda^2 - \lambda) II\bigl(\prj{v}^2\bigr)
+ 2 \lambda t^{l} II \bigl(\prj{v}\prj{w}_0\bigr).
\end{equation}
Suppose $\mu$ is a holomorphic function in one variable, and $m$ is the maximal integer such that
$\lambda -1 = t^m \mu^2$ for sufficiently small values of $t$. Note that $\mu$ has invertible values near $t=0$.
If $m \ge l$, then only $t^l \prj{w}_0 + 2 \lambda t^{l} II (\prj{v}\prj{w}_0)$
contributes to the leading term of \eqref{equ_expansion_of_normal_part},
and $p$ is of type \ref{item_points_in_Sigma_case_closure_of_II}.
Suppose $m <l$, and rewrite \eqref{equ_leading_term_in_last_case}, up to terms of order $> l$:
\[
t^l \prj{w}_0 + \lambda t^{m}II \bigl((\mu \prj{v} + \frac{t^{l-m}}{\mu} \prj{w}_0)^2\bigr)
\]
Thus there exists $u \in \overline{II}$
(either $u=0$ or $u$ is the leading coefficient of
$II \bigl((\mu \prj{v} + \frac{t^{l-m}}{\mu} \prj{w}_0)^2\bigr)$ up to scale,
compare with Lemma~\ref{lem_form_of_points_in_closure_of_II}\ref{item_form_of_u_with_curve}),
such that the limiting space $\lim_{t\to 0} \langle \baseptaff, \inV{y}(t), \inV{z}(t)\rangle$
is spanned by either $\baseptaff, \inV{y_0}, u$ or $\baseptaff, \inV{y_0}, \prj{w_0} + u$.
Since $\inV{y_0} \in \hat\basept \oplus T$,
in either case we have $p= \xi' + u$ for some $\xi'\in \hat\basept \oplus T$,
a linear combination of $\baseptaff$, $\inV{y_0}$ and $\prj{w_0}$, and also after possible rescaling of $u$.
That is, $p$ is a point of type \ref{item_points_in_Sigma_case_closure_of_II}.
\smallskip
It remains to prove that any point $p$ of the form
\ref{item_main_thm_normal_form_honest_secant}, \ref{item_main_thm_normal_form_point_plus_tangent},
or \ref{item_points_in_Sigma_case_closure_of_II}
is in $\sigma_3(X)$.
Case \ref{item_main_thm_normal_form_honest_secant} is clear,
case \ref{item_main_thm_normal_form_point_plus_tangent}
follows as $\sigma_3(X)=J(X,\sigma_2(X))\supset J(X,\tau(X))$
and points on tangent lines are handled by Proposition \ref{prop_case_r=2}.
Finally, for case \ref{item_points_in_Sigma_case_closure_of_II},
take $\xi = \baseptaff$, and $\xi' = \baseptaff+ \prj{w_0}$ with $\prj{w_0} \in T$.
For $u \in \overline{II}$,
let $\prj{v}$ and $m$ be as in Lemma~\ref{lem_form_of_points_in_closure_of_II}\ref{item_form_of_u_with_curve}.
Set:
\begin{align*}
\inV{x}(t) &:= \baseptaff,\\
\inV{y}(t) &:= \baseptaff + t \prj{v} + t^2 II(\prj{v}^2) + \dotsb, \text{ and}\\
\inV{z}(t) &:= \baseptaff + 2t \prj{v} + 4t^2 II(\prj{v}^2) + \dotsb + 2 t^{m+2} \prj{w_0} + \dotsb
\end{align*}
i.e. $\prj{y}(t) = t \prj{v}$ and $\prj{z}(t) = 2t \prj{v} + 2 t^{m+2} \prj{w_0}$.
We calculate:
\[
\inV{x}(t) \wedge \inV{y}(t) \wedge\inV{z}(t) = \baseptaff \wedge t \prj{v} \wedge t^{m+2}(2 \prj{w_0} + 2 u)
+ \dotsb \text{terms of higher order}.
\]
Here $\xi' + u = \baseptaff+\prj{w_0} + u$ is in the limiting space.
\end{proof}
\section{Examples}\label{s3exsect}
In the next sections we treat the case of Segre product with at least $3$ factors in detail.
Here we briefly review some other cases.
\subsection{Known results}
We record the following known results:
\begin{example} Let $X\subset \mathbb P}\def\BT{\mathbb T V$ be one of
$v_2(\pp n)$ (symmetric matrices of rank one), $G(2,n)$
(skew-symmetric matrices of rank two), $Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B)$
(matrices of rank one), or the Cayley plane $\mathbb{O}}\def\BA{\mathbb{A}}\def\BF{\mathbb{F}}\def\BH{\mathbb{H}}\def\BZ{\mathbb{Z} \pp 2$. Then
any point on $\sigma_r(X)$ for any $r$ is on an honest secant
$\pp{r-1}$.
\end{example}
\begin{example}\cite{LTrank} Let $X=v_d(\pp n)$ for $d>2$.
Then any point in $\sigma_3(X)$ is of the form
\renewcommand{\theenumi}{(\roman{enumi})}
\renewcommand{\labelenumi}{\theenumi}
\begin{enumerate}
\item
$\inV{p} = \inV{\xi} + \inV{\eta} +\inV{\zeta}$
for some $\inV{\xi}, \inV{\eta}, \inV{\zeta} \in \hat X$ ($\inV{p}$ is on an honest $3$-secant plane), or
\item
$\inV{p} = \inV{\xi'} +\inV{\eta}$ for some $\inV{\xi}, \inV{\eta} \in \hat X$
and $\inV{\xi'} \in {T}_{[\inV{\xi}]} X$, or
\item
$\inV{p} = \inV{\xi'} + II(\eta',\eta')$ for some $\inV{\xi} \in \hat X$,
$\inV{\xi'}, \inV{\eta'} \in {T}_{[\inV{\xi}]} X$,
\end{enumerate}
Normal forms for $\sigma_3(v_d(\mathbb P}\def\BT{\mathbb T V))\backslash
\sigma_2(v_d(\mathbb P}\def\BT{\mathbb T V))$ of these types are respectively $x^d+y^d+z^d$, $x^{d-1}y+z^d$ and
$x^{d-1}y+ x^{d-2}z^2$, where $x,y,z\in V$.
Thus the points of type \ref{item_main_thm_normal_form_two_tangents} do not occur in this case.
\end{example}
\renewcommand{\theenumi}{(\arabic{enumi})}
\renewcommand{\labelenumi}{\theenumi}
The generalized cominuscule varieties with $\sigma_2(X)=\mathbb P}\def\BT{\mathbb T V$ are
$Seg(\pp 1 \times \pp n)$, $Seg(\pp 1\times \pp 1\times \pp 1)$, quadric hypersurfaces $Q$,
the Veronese varieties
$v_2(\pp 1)$, $v_3(\pp 1)$,
the Grassmannians $G(2,5)$ and $G(3,6)$, the spinor varieties $\BS_5$ and $\BS_6$,
the Lagrangian Grassmannian $G_{Lag}(3,6)$, $Seg(\pp 1\times Q)$, and the Freudenthal variety
$E_7/P_7$.
\subsection{Grassmannians in Pl\"ucker embedding}
Let $X:=G(k,n) \subset \mathbb P(\Wedge{k} \mathbb C^n)$, and suppose $3\leq k\leq n-k$ and $n-k>3$.
The tangent space at $E\in G(k,n)$
can be identified with the space of $k \times (n-k)$-matrices $\Wedge{k-1}E \otimes F \simeq E^* \otimes F$,
where $F = \mathbb C^n/E$.
The local parametrization in this case comes from a choice of splitting $\mathbb C^n \simeq E \oplus F$
and the determined splitting:
\begin{alignat*}{7}
\Wedge{k} (E\oplus F) &=&\ \Wedge{k} E \ & \oplus & \ \Wedge{k-1} E &\otimes F \ &&\oplus & \ \Wedge{k-2} E & \otimes \Wedge{2} F
&&\oplus \dotsb \oplus & & \phantom{\otimes } \ \Wedge{k} F \\
&\simeq& \baseptline \ &\oplus &E^*& \otimes F &&\oplus &\Wedge{2} E^* &\otimes \Wedge{2} F
&&\oplus \dotsb \oplus & \ \Wedge{k} E^* & \otimes \Wedge{k} F.
\end{alignat*}
The parametrization has the following form:
\[
T \simeq E^* \otimes F \ni M \stackrel{\varphi}{\mapsto}
[\underbrace{1,}_{\in \baseptline} \underbrace{M,}_{\in T} \underbrace{\Wedge{2} M,}_{=II(M^2)\in N_2} \dotsc,
\underbrace{\Wedge{k} M}_{\in N_k}],
\]
where $\mathbb F_s(M^s) = \Wedge{s} M \in \Wedge{s} E^* \otimes \Wedge{s} F$, expressed in linear coordinates, is the collection of all $s\times s$ minors of $M$.
In the normal forms of Theorem~\ref{thm_points_in_Sigma} we can take the first point $\xi = \baseptaff$,
for the second we have $k$ choices given the rank of $M$.
Let $\epsilon_i$ for $i \in \setfromto{1}{k}$
denote the matrix of rank $i$ with the block form
$
\begin{pmatrix}
\operatorname{Id}_i & 0\\
0 & 0
\end{pmatrix}
$.
The normal forms are:
\begin{itemize}
\item[\ref{item_main_thm_normal_form_honest_secant}]
$\inV{p} = \baseptaff + \varphi(\epsilon_i) + \varphi(M)$ for some $i$, $M$,
\item[\ref{item_main_thm_normal_form_point_plus_tangent}]
$\inV{p} = \baseptaff + M+ \varphi(\epsilon_i) $ or $\inV{p} = M + \varphi(\epsilon_i)$ for some $i$, $M$,
\item[\ref{item_main_thm_normal_form_third_order_pt}]
$\inV{p} = \baseptaff + M + \Wedge{2}\epsilon_i$ or $\inV{p} = M + \Wedge{2}\epsilon_i$ for some $i$, $M$,
\item[\ref{nitem_points_in_Sigma_case_closure_of_IIb}]
$\inV{p} = \baseptaff + M + \Wedge{2}\epsilon_{i+1} -\Wedge{2}\epsilon_{i}$
or $\inV{p} = M + \Wedge{2}\epsilon_{i+1} -\Wedge{2}\epsilon_{i}$
for some $i \ne k$, $M$.
\end{itemize}
In \ref{nitem_points_in_Sigma_case_closure_of_IIb}, $\nu = \epsilon_{i+1} -\epsilon_{i}$ is a rank $1$ matrix,
so $II(\nu^2)=0$, and $\Wedge{2}\epsilon_{i+1} -\Wedge{2}\epsilon_{i} = \frac{1}{2} II (u, \epsilon_i)$.
In all normal forms, we can pick $M$ to be in some normal form. For example, if $i=k=n-k$,
then $M$ may be
(at least)
assumed to be in Jordan normal form.
\subsection{Lagrangian Grassmannians}
Let $X$ be the Lagrangian Grassmannian $G_{Lag}(k,2k)=C_k/P_k \subset \mathbb P (V_{\omega_k})$
with $k>3$, where $V_{\omega} = \Wedge{k} \mathbb C^{2k} /\Wedge{k-2} \mathbb C^{2k}$ is the minimal homogeneous embedding.
In this case the local parametrization is identical, but with $T \simeq S^2 \mathbb C^k$
and $M$ a symmetric $k \times k$ matrix,
see \cite[\S 5]{boralevi_jabu_secants_to_LG}.
The normal forms are also identical.
\subsection{Spinor varieties}
Let $X$ be the spinor variety $\BS_{k}=D_k/P_k$ for $k\ge 7$
in its minimal homogeneous embedding $\mathbb P(\Wedge{even} \mathbb C^{k})$.
In this case $T\simeq \Wedge{2} \mathbb C^k$ and $M$ is a skew-symmetric $k\times k$ matrix,
and the parameterization is similar to the previous cases:
\[
M \stackrel{\varphi}{\mapsto}
[\underbrace{1,}_{\in \baseptline} \underbrace{M,}_{\in T} \ \underbrace{\Pf_4 M,}_{=II(M^2)\in N_2} \
\underbrace{\Pf_6 M,}_{=III(M^3)\in N_3}\dotsc],
\]
where $\Pf_{2s} M\in \Wedge{2s} \mathbb C^k$, expressed in linear coordinates, is the collection of all $2s\times 2s$ sub-Pfaffians of $M$.
Let $\epsilon^{skew}_i$ for $i \in \setfromto{1}{\lfloor \frac{1}{2} k \rfloor}$
denote the matrix of rank $2i$ with the block form
$
\begin{pmatrix}
0 & \operatorname{Id}_i & 0 \\
-\operatorname{Id}_i & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
$.
The normal forms are:
\begin{itemize}
\item[\ref{item_main_thm_normal_form_honest_secant}]
$\inV{p} = \baseptaff + \varphi(\epsilon^{skew}_i) + \varphi(M)$ for some $i$, $M$,
\item[\ref{item_main_thm_normal_form_point_plus_tangent}]
$\inV{p} = \baseptaff + M+ \varphi(\epsilon^{skew}_i) $ or $\inV{p} = M + \varphi(\epsilon^{skew}_i)$ for some $i$, $M$,
\item[\ref{item_main_thm_normal_form_third_order_pt}]
$\inV{p} = \baseptaff + M + \Pf_4\epsilon^{skew}_i$ or $\inV{p} = M + \Pf_4\epsilon^{skew}_i$ for some $i$, $M$,
\item[\ref{nitem_points_in_Sigma_case_closure_of_IIb}]
$\inV{p} = \baseptaff + M + \Pf_4\epsilon^{skew}_{i+1} -\Pf_4\epsilon^{skew}_{i}$
or $\inV{p} = M + \Pf_4\epsilon^{skew}_{i+1} -\Pf_4\epsilon^{skew}_{i}$
for some $i \ne \lfloor \frac{1}{2} k \rfloor$, $M$.
\end{itemize}
\section{The Segre product $Seg(\mathbb P}\def\BT{\mathbb T A_1\times \cdots\times \mathbb P}\def\BT{\mathbb T A_n))$}\label{smallsecsegsect}
Recall that for any smooth variety $X$, if $x\in \sigma_2(X)$, then either
$x\in X$, $x\in \sigma_2^0(X)$ or $x$ lies on an embedded tangent line to $X$, see Proposition~\ref{prop_case_r=2}.
\subsection{Proof of Proposition \ref{lastthm}}
All the assertions except for the rank
of $x$ in \eqref{tprimevect} are immediate.
The rank of $x$ is at most $|J|$ because there
are $|J|$ terms in the summation.
Assume without loss of generality $|J|=n$ and work by induction. The case $n=2$ is clear.
Now assume we have established the result up to $n-1$, and consider
$x(A_1^*)$. It is spanned by
$$
a_1^2{\mathord{\otimes\cdots\otimes}\;} a_1^n,
\sum_{j=2}^n a_1^2{\mathord{\otimes\cdots\otimes}\;} a_{1}^{j-1}{\mathord{ \otimes } } a_2^{j}{\mathord{ \otimes } } a_1^{j+1}{\mathord{\otimes\cdots\otimes}\;} a_1^n.
$$
By induction, the second vector has rank $n-1$, so the only way $x(A_1^*)$ could be spanned
by $n-1$ rank one elements would be if there were an expression of the second vector as a sum of $n-1$ decomposable tensors
where one of terms is a multiple of $a_1^2{\mathord{\otimes\cdots\otimes}\;} a_1^n$. Say there were such an expression, where $a_1^2{\mathord{\otimes\cdots\otimes}\;} a_1^n$
appeared with coefficient $\lambda$, then the tensor
$\sum_{j=2}^n a_1^2{\mathord{\otimes\cdots\otimes}\;} a_{1}^{j-1}{\mathord{ \otimes } } a_2^{j}{\mathord{ \otimes } } a_1^{j+1}{\mathord{\otimes\cdots\otimes}\;} a_1^n - \lambda a_1^2{\mathord{\otimes\cdots\otimes}\;} a_1^n$
would have rank $n-2$, but setting $\tilde a^2_2=a^2_2-\lambda a^2_1$
and $\tilde a^j_2=a^j_2$ for $j \in \setfromto{3}{n}$,
this would imply that
\[
\sum_{j=2}^n a_1^2{\mathord{\otimes\cdots\otimes}\;} a_{1}^{j-1}{\mathord{ \otimes } } \tilde a_2^{j}{\mathord{ \otimes } } a_1^{j+1}{\mathord{\otimes\cdots\otimes}\;} a_1^n
\]
had rank $n-2$,
a contradiction.
\begin{remark} The case $n=3$ was previously established by Grigoriev, Ja'Ja' and Teichert.
\end{remark}
\subsection{Parameterization in the Segre case}\label{sec_parameterization_Segre}
Suppose $X = Seg(\mathbb P}\def\BT{\mathbb T A_1\times \cdots\times \mathbb P}\def\BT{\mathbb T A_n)$.
Let $\baseptaff=a^1_1{\mathord{\otimes\cdots\otimes}\;} a^n_1$, and let $A_j'=a^1_1{\mathord{\otimes\cdots\otimes}\;} a^{j-1}_1{\mathord{ \otimes } } (A_j/a^j_1){\mathord{ \otimes } } a^{j+1}_1{\mathord{\otimes\cdots\otimes}\;} a^n_1
\simeq A_j/a^j_1
$.
Then $T = A_1'\oplus\cdots \oplus A_n'$ and $X$ is parametrized by
\[
(\fromto{a'_1}{a'_n}) \mapsto [\underbrace{1,}_{\in \baseptline} \underbrace{\fromto{a'_1}{a'_n}}_{\in T}, \ \underbrace{\fromto{a'_1\otimes a'_2}{a'_{n-1}\otimes a'_n}}_{=II((\fromto{a'_1}{a'_n})^2)\in N_2}, \dotsc,
\underbrace{a'_1\otimes a'_2 \otimes\dotsm\otimes a'_n}_{\in N_n}].
\]
Thus $II((\fromto{a'_1}{a'_n})\cdot(\fromto{b'_1}{b'_n}))
= \frac{1}{2}(\fromto{a'_1 \otimes b'_2 + b'_1 \otimes a'_2}{a'_{n-1} \otimes b'_n + b'_{n-1} \otimes a'_{n}})$.
In this case the base locus of $II$ is $\mathbb P}\def\BT{\mathbb T A_1'\sqcup \cdots \sqcup \mathbb P}\def\BT{\mathbb T A_n'\subset \mathbb P}\def\BT{\mathbb T (A_1'\oplus\cdots \oplus A_n') \simeq \mathbb P T$.
If $II(\prj v_0^2)=0$, then $\prj v_0\in A_i'$ for some $i$ and if
further $II(\prj v_0\prj v_1)=0$ then
$\prj v_1\in A_i'$ for the same $i$.
In particular, if a line $\ell \subset X$ contains $\basept$ and is tangent to $\prj v_0$,
then by \eqref{equ_formula_for_ccTell} we have:
\begin{equation}\label{equ_dim_ccT_ell}
{\rm dim}\; \ccT^{\ell} = 2 {\rm dim}\; X + 1 - {\rm dim}\; \ker II(\prj{v}_0 \ \cdot ) = 2 {\rm dim}\; X +2 - {\rm dim}\; A_i.
\end{equation}
Now we prove Theorem~\ref{s3nformthm}.
The normal forms follow from the discussion in the previous
sections.
Now suppose ${\rm dim}\; A_i \ge 3$.
To see that the general points of each type do not belong to the other types,
note that for any type and for any $i$,
in the normal forms \eqref{nf1}--\eqref{nf4}
either $a_1^i, a_2^i, a_3^i$ are linearly independent,
or the point is contained in a subspace variety, i.e., a closed subvariety consisting of tensors in some
$A_1 \otimes \dotsb \otimes A_{i-1} \otimes \mathbb C^2 \otimes A_{i+1} \otimes \dotsb \otimes A_n$.
Thus the general points of each type form a single orbit
(or $n$ orbits for type \ref{item_main_thm_normal_form_two_tangents}) of the action of $GL(A_1) \times \dotsm \times GL(A_n)$.
Therefore the only possible way that they could overlap, is if one of the orbits were equal to the other.
But the orbits are distinct by the dimension count below, which we present in two different forms.
\subsection{First proof of dimensions in Theorem~\ref{s3nformthm}}
We compute the Lie algebras of the stabilizers of each type of point.
Without loss of generality (for computing codimension), assume $\operatorname{dim} A_j=3$.
Write $\Gamma =(x_1, \hdots , x_n)$ where $x_\alpha=(x^i_{j,\alpha})$, $1\leq i,j\leq 3$.
We calculate the $\Gamma $ such that $\Gamma .p_{(*)}=0$ in each case $*=i,ii,iii,iv$ and denote this algebra
by $\mathfrak g_{p_{(*)}}$. In each case one has a system of $3^n=\operatorname{dim}(A_1{\mathord{\otimes\cdots\otimes}\;} A_n)$ linear
equations, many of which are zero or redundant.
$$
\mathfrak g_{p_{(i)}}=
\left\{ \Times_{\alpha=1, \hdots , n} \begin{pmatrix} x^1_{1,\alpha} & 0 & 0 \\ 0 & x^2_{2,\alpha} & 0 \\ 0 & 0 & x^3_{3,\alpha}\end{pmatrix}
\mid \sum_{\alpha}x^i_{i,\alpha}=0,\ i=1,2,3 \right\}.
$$
Note $\operatorname{dim} \mathfrak g_{p_{(i)}}=3n-3$.
$$
\mathfrak g_{p_{(ii)}}=
\left\{\Times_{\alpha=1, \hdots , n} \begin{pmatrix} x^1_{1,\alpha} & x^1_{2,\alpha} & 0 \\ 0 & -\sum_{\beta\neq \alpha} x^1_{1,\beta} & 0 \\ 0 & 0 & x^3_{3,\alpha}\end{pmatrix}
\mid \sum_{\alpha}x^3_{3,\alpha}=0, \ \sum_{\alpha}x^1_{2,\alpha}=0\right\}.
$$
Note $\operatorname{dim} \mathfrak g_{p_{(ii)}}=3n-2$.
$$
\mathfrak g_{p_{(iii)}}=
\left\{ \Times_{\alpha=1, \hdots , n} \begin{pmatrix} x^1_{1,\alpha} & x^1_{2,\alpha} & x^1_{3,\alpha}
\\ 0 & -\sum_{\beta\neq \alpha} x^1_{1,\beta} & -\sum_{\beta\neq \alpha}x^1_{2,\beta} \\ 0 & 0 & -\sum_{\beta\neq \alpha}x^1_{1,\beta}\end{pmatrix}
\mid \ \sum_{\alpha}x^1_{3,\alpha}=0\right\}.
$$
Note $\operatorname{dim} \mathfrak g_{p_{(iii)}}=3n-1$.
\begin{align*}
&\mathfrak g_{p_{(iv)}}=\\
&
\left\{
\begin{pmatrix} x^1_{1,1} & x^1_{2,1} & x^1_{3,1}
\\ x^2_{1,1} & x^2_{2,1} & -\sum_{\rho}x^1_{2,\rho}
\\ 0 & 0 & -\sum_{\rho}x^1_{1,\rho}\end{pmatrix}
,
\Times_{\rho=2, \hdots , n}\begin{pmatrix} x^1_{1,\rho} & x^1_{2,\rho} & x^1_{3,\rho}
\\ 0 & -\sum_{\sigma\neq \rho} x^1_{1,\sigma}-x^2_{2,1} & -x^1_{2,1} \\ 0 & -x^2_{1,1} & -\sum_{\beta\neq \rho}x^1_{1,\beta}\end{pmatrix}
\mid \ \sum_{\alpha}x^1_{3,\alpha}=0\right\}.
\end{align*}
Here the index ranges are $1\leq \alpha,\beta\leq n$, $2\leq \rho,\sigma\leq n$.
Note $\operatorname{dim} \mathfrak g_{p_{(iv)}}=3n+1$.
\subsection{Second Proof of dimensions in Theorem~\ref{s3nformthm}}
Throughout this section $X= Seg(\mathbb P A_1 \times \dotsm \times \mathbb P A_n)$.
We show the assertion about the codimensions of types (ii),(iii),(iv).
Type \ref{item_main_thm_normal_form_point_plus_tangent} is immediate as its closure is $J(X,\tau(X))$ which is easily seen to
have the expected dimension via Terracini's lemma.
We will use the following lemma:
\begin{lemma}\label{lem_scheme_deg_3}
Suppose $n \ge 2$ and ${\rm dim}\; A_i \ge 3$ for all $i \in \setfromto{1}{n}$.
Let $R$ be a degree $3$, zero dimensional subscheme of $X= Seg(\mathbb P A_1 \times \dotsm \times \mathbb P A_n)$.
Suppose moreover $R$ is in general position, that is, it is not contained in any
$Seg(\mathbb P A_1 \times \dotsm \times \mathbb P^1 \times \dotsm \times \mathbb P A_n)$.
Let $\langle R \rangle \simeq \mathbb P^2 \subset \mathbb P(A_1 \otimes \dotsb \otimes A_n)$
denote the smallest linear space containing $R$.
Then $X \cap \langle R \rangle = R$.
\end{lemma}
\begin{proof}
Any such $R$ is isomorphic either to $3$ distinct reduced points,
or a double point and a reduced point,
or one of the two kinds of triple points: $\operatorname{Spec} \mathbb C[x]/x^3$, or $\operatorname{Spec} \mathbb C[x,y]/\langle x^2, xy, y^2 \rangle$.
If $n=2$, without loss of generality, we may suppose ${\rm dim}\; A_1 = {\rm dim}\; A_2 = 3$.
We can write down explicitly $\langle R \rangle \subset \mathbb P (A_1 \otimes A_2)$ for each of the schemes as, respectively:
\[
\begin{pmatrix}
s & & \\
& t & \\
& & u
\end{pmatrix},
\begin{pmatrix}
t & s & \\
s & & \\
& & u
\end{pmatrix},
\begin{pmatrix}
u & t & s \\
t & s & \\
s & &
\end{pmatrix},
\begin{pmatrix}
t & s & u \\
s & & \\
u & &
\end{pmatrix}
\]
The claim may be verified explicitly for each case, by calculating the scheme defined
by $2 \times 2$ minors of each of the matrices.
If $n \ge 3$, let $B_i = A_1 \otimes\dotsb \otimes A_{i-1} \otimes A_{i+1} \otimes\dotsb \otimes A_n$.
Then $X = \bigcap_{i =1}^n\mathbb P A_i \times \mathbb P B_i$, and
the claim easily follows from the $n=2$ statement.
\end{proof}
\begin{lemma}\label{lem_line}
Suppose $n \ge 2$ and ${\rm dim}\; A_i \ge 3$ for all $i \in \setfromto{1}{n}$.
Let $X= Seg(\mathbb P A_1 \times \dotsm \times \mathbb P A_n)$
and let $\ell \subset X$ be a line spanned by $x,y\in X$.
Let $v \in \hat{T}_x X + \hat{T}_y X$ be general and consider $\mathbb P^2$ spanned by $\ell$ and $[v]$.
Then $\mathbb P^2 \cap X = \ell$.
\end{lemma}
\begin{proof}
Let $x = a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{n}_{1}$, $y =
a^1_{2}
{\mathord{ \otimes } } a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{n}_{1}$,
and $v$ be as in \eqref{nf4}.
Let $B:= A_2 \otimes\dotsb \otimes A_n$ and:
\begin{align*}
b_1 &:=a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{n}_{1}, \\
b_2 &:=\sum_{ i=2 }^n a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1},\\
b_3 &:=\sum_{ i=2 }^n a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1}.
\end{align*}
Then $x = a^1_1 \otimes b_1$, $y =
a^1_2 \otimes b_1$ and
$v = a^1_1 \otimes b_3 + a^1_2 \otimes b_2 + a^1_3 \otimes b_1 $.
Consider a linear combination $ sv + tx + uy$.
The intersection $\mathbb P^2 \cap X$ is contained in the zero locus of the $2 \times 2$ minors of the following matrix:
\[
\begin{pmatrix}
t & & s \\
u & s & \\
s & &
\end{pmatrix},
\]
which can be identified with the line $s=0$, that is the line spanned by $x$ and $y$.
\end{proof}
\smallskip
Let $\ptstypeiii$ be the closure of the set of points of type \ref{item_main_thm_normal_form_third_order_pt}.
Let $[p] \in \ptstypeiii$ be a general point.
We claim such $p$ uniquely determines $[x]$ such that $p = x+ x' + x''$.
Suppose without loss of generality ${\rm dim}\; A_1 =3$.
Write $p=p_{(iii)}$ of \eqref{nf3},
and consider the underlying map $p_{(iii)}:
{A_1}^* \to
A_{2} \otimes\dotsb \otimes A_n$:
\begin{align*}
p_{(iii)}({a^1_1}^*) & = \sum_{2 \le i<j}^n a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2 {\mathord{ \otimes } } a^{i+1}_1
{\mathord{\otimes\cdots\otimes}\;} a^{j-1}_{1}{\mathord{ \otimes } } a^j_2 {\mathord{ \otimes } } a^{j+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1} \\
& + \sum_{i=2}^n a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1},\\
p_{(iii)}({a^1_2}^*) & = \sum_{j=2}^n a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{j-1}_{1}{\mathord{ \otimes } } a^j_2 {\mathord{ \otimes } } a^{j+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1},\\
p_{(iii)}({a^1_3}^*) & = a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^n_{1}.
\end{align*}
The projectivization of the image is a $\mathbb P^2$ containing
a degree $3$ scheme $R \subset Seg(\mathbb P A_{2} \times\dotsb \times \mathbb P A_n)$
in general position,
which is isomorphic to the triple point $\operatorname{Spec} \mathbb C[x]/x^3$ point supported at $[p({a^1_3}^*)]$.
By Lemma~\ref{lem_scheme_deg_3},
$R$ is determined by $\langle R \rangle = \mathbb P( p({A_1}^*))$,
so it is independent of the choice of normal form.
Therefore $\langle {a^1_2}^*, {a^1_3}^* \rangle$, which is the linear span of the unique degree $2$ subscheme of $R$,
is determined by $p$, and so is $a^1_1$ (up to scale).
Similarly, $a^i_1$ are determined by $p$ up to scale.
Thus we have a rational dominant map
$\psi: \ptstypeiii \dashrightarrow X$,
$\psi(p) := [a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^n_{1}]$.
A general fiber over $\basept \in X$ is contained in the second osculating space
$\mathbb P(\baseptline \oplus T \oplus N_2)$,
and its closure is equal to the closure of points of the form $\baseptaff + \prj{\xi'} + II(\prj{v}^2)$.
Thus ${\rm dim}\; \ptstypeiii = 3 \sum ({\rm dim}\; A_i-1)$.
\smallskip
Finally consider $\ptstypeiv$, the set of points of type \ref{item_main_thm_normal_form_two_tangents}, which is
closed by Lemma \ref{lem_alternative_iv}.
Let $[p] \in \ptstypeiv$ be a general point of any of the irreducible components.
We claim $p$ uniquely determines the line $\mathbb P \langle x,y \rangle$
such that $p = x+ x' + y + y'$.
Suppose without loss of generality ${\rm dim}\; A_i =3$ for all $i$.
Possibly permuting the factors, write $p=p_{(iv)}$ of \eqref{nf4}.
First consider the underlying map $p_{(iv)}:{A_1}^*: \to A_{2} \otimes\dotsb \otimes A_n$:
\begin{align*}
p_{(iv)}({a^1_1}^*) & = \sum_{ i=2 }^n a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1},\\
p_{(iv)}({a^1_2}^*) & = \sum_{ i=2 }^{n} a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^n_{1},\\
p_{(iv)}({a^1_3}^*) & = a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^n_{1}.
\end{align*}
The projectivization of the image is a $\mathbb P^2$ containing
a degree $3$ scheme $R \subset Seg(\mathbb P A_{2} \times\dotsb \times \mathbb P A_n)$
in general position,
which is isomorphic to the triple point $\operatorname{Spec} \mathbb C[x,y]/\langle x^2, xy, y^2 \rangle$ point supported at $[p_{(iv)}({a^1_3}^*)]$.
By Lemma~\ref{lem_scheme_deg_3},
$[p_{(iv)}({a^1_3}^*)]$ is the unique reduced point in $\mathbb P( p_{(iv)}({A_1}^*)) \cap Seg(\mathbb P A_{2} \times\dotsb \times \mathbb P A_n)$,
so independent of the choice of normal form.
Therefore $\langle a^1_1, {a^1_2} \rangle \subset \mathbb P A_1$ is determined by $p_{(iv)}$.
Now consider $p_{(iv)}:{A_n}^*: \to A_{1} \otimes\dotsb \otimes A_{n-1}$:
\begin{align*}
p_{(iv)}({a^n_1}^*) & = \sum_{ i=1 }^{n-1} a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_3 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^{n-1}_{1}\\
& + \sum_{ i=2 }^{n-1} a^1_{2}{\mathord{ \otimes } } a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{i-1}_{1}{\mathord{ \otimes } } a^i_2 {\mathord{ \otimes } } a^{i+1}_1 {\mathord{\otimes\cdots\otimes}\;} a^{n-1}_{1}.\\
p_{(iv)}({a^n_2}^*) & = a^1_{2}{\mathord{ \otimes } } a^2_{1}{\mathord{\otimes\cdots\otimes}\;} a^{n-1}_{1},\\
p_{(iv)}({a^n_3}^*) & = a^1_{1}{\mathord{\otimes\cdots\otimes}\;} a^{n-1}_{1}.
\end{align*}
By Lemma~\ref{lem_line} the projective line $\mathbb P \langle p_{(iv)}({a^n_2}^*),p_{(iv)}({a^n_3}^*) \rangle$ is determined by $p_{(iv)}$.
Thus $a^n_1$ (and similarly $a^i_1$ for $i \in \setfromto{2}{n}$) is determined (up to scale) by $p_{(iv)}$.
Therefore, the line $\mathbb P(\langle a^1_1,
a^1_2 \rangle \otimes a^2_1 \otimes \dotsb \otimes a^n_1) \subset X$
is uniquely determined by $p_{(iv)}$.
The lines on $X$ are parametrized by $n$ irreducible varieties:
\[
L_i:= \mathbb P A_1 \times\dotsb \times \mathbb P A_{i-1} \times G(2, A_i) \times \mathbb P A_{i+1} \times \dotsb \times \mathbb P A_n.
\]
By the argument above we have a rational dominant map
$\chi: \ptstypeiv \dashrightarrow L_1\sqcup \dots\sqcup L_n$.
A general fiber over $\ell \in L_i$ is $\mathbb P \ccT^{\ell}$ in the notation of Lemma~\ref{lem_tangent_at_line},
the linear span of projective tangent spaces to $X$ at points of $\ell$.
By \eqref{equ_dim_ccT_ell} ${\rm dim}\; \ccT^{\ell} = 2 {\rm dim}\; X + 2 - {\rm dim}\; A_i$,
and the dimension of each irreducible component of $\ptstypeiv$ is equal to $3 \sum ({\rm dim}\; A_i-1) -2$.
\section{Orbits of tensors in $A\otimes B \otimes C$ of border rank at most $3$}\label{sec_ranks_and_orbits}
Let $A\simeq \mathbb C^{{\mathbf{a}}}$, $B\simeq \mathbb C^{{\mathbf{b}}}$, $C\simeq \mathbb C^{{\mathbf{c}}}$.
Let
\begin{multline*}
Sub_{{\mathbf{a}}',{\mathbf{b}}',{\mathbf{c}}'}= Sub_{{\mathbf{a}}',{\mathbf{b}}',{\mathbf{c}}'}(A{\mathord{ \otimes } } B{\mathord{ \otimes } } C)=\\
\{T\in A{\mathord{ \otimes } } B{\mathord{ \otimes } } C\mid \exists \mathbb C^{{\mathbf{a}} '}\subset A, \ \mathbb C^{{\mathbf{b}} '}\subset B,\ \mathbb C^{{\mathbf{c}} '}\subset C, \rm{\ such \ that \ }
T\in \mathbb C^{{\mathbf{a}} '}{\mathord{ \otimes } } \mathbb C^{{\mathbf{b}} '}{\mathord{ \otimes } } \mathbb C^{{\mathbf{c}} '}\}
\end{multline*}
This {\it subspace variety} admits a desingularization as follows. Let $\mathcal E\rightarrow G({\mathbf{a}}',A)\times G({\mathbf{b}}',B)\times G({\mathbf{c}}',C)$
be $\mathcal E={\mathcal S}}\def\cN{{\mathcal N}_{A} \boxtimes {\mathcal S}}\def\cN{{\mathcal N}_{B}\boxtimes {\mathcal S}}\def\cN{{\mathcal N}_{C}$, where ${\mathcal S}}\def\cN{{\mathcal N}_{A}\rightarrow G({\mathbf{a}}',A)$ is the tautological rank ${\mathbf{a}}'$ subspace bundle and similarly
for $B,C$.
|
Then $\mathbb P}\def\BT{\mathbb T \mathcal E \rightarrow Sub_{{\mathbf{a}}',{\mathbf{b}}',{\mathbf{c}}'}(A{\mathord{ \otimes } } B{\mathord{ \otimes } } C)$ is a desingularization and using it one can see that
$Sub_{{\mathbf{a}}',{\mathbf{b}}',{\mathbf{c}}'}(A{\mathord{ \otimes } } B{\mathord{ \otimes } } C)_{sing}=
Sub_{{\mathbf{a}}'-1,{\mathbf{b}}',{\mathbf{c}}'} \cup Sub_{{\mathbf{a}}',{\mathbf{b}}'-1,{\mathbf{c}}'} \cup Sub_{{\mathbf{a}}',{\mathbf{b}}',{\mathbf{c}}'-1}
$, whenever ${\mathbf{a}}' < {\mathbf{b}} {\mathbf{c}}$, and similarly for permuted statements.
In \cite[\S 6]{BLtensor}, normal forms for tensors in $Sub_{233}\cup Sub_{323}\cup Sub_{332}$ are given. There are
$33$ such.
We present the list of remaining orbits in $\sigma_3(Seg(\mathbb P A \times \mathbb P B \times \mathbb P C))$
under the action of $GL(A) \times GL(B) \times GL(C)$.
Each orbit is uniquely determined by its closure, which is an algebraic variety listed in the second column of the table.
The orbit itself is an open dense subset of this variety.
The dimension of the algebraic variety is in the third column.
The fourth column is the normal form of the underlying tensor,
the distinct variables are assumed to be linearly independent.
The normal form is also given as a slice.
The border rank and rank are given in the next columns.
\begin{table}[htb]
$$
\begin{array}{|r|c|c|ll|c|c|}
\hline
\hline
\#&\text{orbit closure}& {\rm dim}\; &\text{normal form} & \text{slice} & \brank}\def\uR{\brank & \mathbf{R} \\
\hline
\hline
34& \ptstypeiv_{A} & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} - 11 & \begin{matrix}
a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2 + b_2{\mathord{ \otimes } } c_1) +a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1\\
+ a_3{\mathord{ \otimes } } (b_3{\mathord{ \otimes } } c_1 + b_1{\mathord{ \otimes } } c_3)
\end{matrix}&
\left(\begin{smallmatrix} t & s& u\\ s & & \\ u & & \end{smallmatrix} \right)
& 3&5\\
\hline
35& \ptstypeiv_{B} & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} -11 & \begin{matrix}
a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2 + b_2{\mathord{ \otimes } } c_1 + b_3 {\mathord{ \otimes } } c_3)\\
+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1 + a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_1
\end{matrix}&
\left(\begin{smallmatrix} t & s&\\ s & & \\ u& & s\end{smallmatrix} \right)
& 3&5\\
\hline
36& \ptstypeiv_{C} & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} -11 & \begin{matrix}
a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2 + b_2{\mathord{ \otimes } } c_1 + b_3 {\mathord{ \otimes } } c_3)\\
+a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1 + a_3{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_3
\end{matrix}&
\left(\begin{smallmatrix} t & s& u\\ s & & \\ & & s\end{smallmatrix} \right)
& 3&5\\
\hline
\hline
37& \ptstypeiii & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} -9 & \begin{matrix}
a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_3+ b_2{\mathord{ \otimes } } c_2 + b_3{\mathord{ \otimes } } c_1) \\
+ a_2{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2+ b_2 c_1) + a_3{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1
\end{matrix}&
\left(\begin{smallmatrix} u & t&s\\ t&s & \\ s & & \end{smallmatrix} \right)
& 3&5 \\
\hline
\hline
38& \ptstypeii & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} -8 & \begin{matrix}
a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2+ b_2{\mathord{ \otimes } } c_1)\\
+ a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1+ a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_3
\end{matrix}&
\left(\begin{smallmatrix} t & s&\\ s & & \\ & & u\end{smallmatrix} \right)
& 3&4 \\
\hline
\hline
39& \sigma_3(X) & 3{\mathbf{a}} + 3{\mathbf{b}}+3{\mathbf{c}} -7 & \begin{matrix}
a_1 {\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1+a_2 {\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_2
+ a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_3
\end{matrix}
& \left(\begin{smallmatrix} s & & \\ & t & \\ & & u\end{smallmatrix} \right)
& 3&3\\
\hline
\hline
\end{array}
$$
\caption{Orbits of border rank $3$ in $A\otimes B \otimes C$
that are not contained in a $Sub_{233}$, $Sub_{323}$, or $Sub_{332}$.
Orbits $34$--$36$ are identical up to permutations of $A$, $B$, $C$.}
\label{table_orbits_general}
\end{table}
$\ptstypeiv_A$, $\ptstypeiv_B$, $\ptstypeiv_C$
denote the three components of $\ptstypeiv$, the set points of type \ref{item_main_thm_normal_form_two_tangents}
in Theorem~\ref{s3nformthm}.
$\ptstypeiii$ denotes the the closure of the set points of type \ref{item_main_thm_normal_form_third_order_pt},
while $\ptstypeii$ denotes the the closure of the set points of type \ref{item_main_thm_normal_form_point_plus_tangent}.
The ranks of cases $34$--$37$ in Table~\ref{table_orbits_general} are calculated in \S\ref{sec_calculating_ranks}.
The rank of case $39$ is obvious, while the rank of case $38$ is at most $4$, due to the normal form expression.
If it were $3$, then a general point of type \ref{item_main_thm_normal_form_point_plus_tangent},
would be expressible as a point of type \ref{item_main_thm_normal_form_honest_secant}, a contradiction with
Theorem~\ref{s3nformthm}.
\subsection{Proof of Theorem \ref{lastcor}}\label{sec_calculating_ranks}
The {\it rank} of a linear subspace $U\subset \mathbb C^k{\mathord{ \otimes } } \mathbb C^l$ is the smallest $r$ such that $U$ is contained in a linear
space of dimension $r$ spanned by rank one elements. The rank of a tensor $T\in A{\mathord{ \otimes } } B{\mathord{ \otimes } } C$ equals the
rank of the linear subspace $T(A^*)\subset B{\mathord{ \otimes } } C$, see, e.g., \cite[Thm. 3.1.1.1]{Ltensorbook}.
\begin{proposition} \label{lastprop}
The ranks of the spaces parametrized by
$
\begin{pmatrix} u&t&s \\ t&s&0 \\ s & 0 & 0\end{pmatrix}$,
and by
$
\begin{pmatrix}
t & s &u\\
s & 0 & 0\\
u & 0 & 0
\end{pmatrix}
$
are both $5$.
\end{proposition}
\begin{proof}
We first show the rank is at most $5$: in the second case, it is immediate. In the first case the rank of
$
\begin{pmatrix}
0&t&s\\
t&s&0\\
s&0&0
\end{pmatrix}$
is $4$ (see\cite[\S 6]{BLtensor}), and the rank of
$$
\begin{pmatrix} u &0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}
$$
is one.
To see the ranks are at least five,
were it four in the first case, we would be able to find
a $3 \times 3$ matrix $T=
\begin{pmatrix}
f_1g_1& f_1g_2 & f_1g_3\\
f_2g_1& f_2g_2 & f_2g_3\\
f_3g_1& f_3g_2 & f_3g_3
\end{pmatrix}$
of rank 1,
such that the $4$-plane spanned by:
\[
T_1:=\begin{pmatrix}
0&0&1\\
0&1&0\\
1&0&0
\end{pmatrix},
T_2:=\begin{pmatrix}
0&1&0\\
1&0&0\\
0&0&0
\end{pmatrix},
T_3:=\begin{pmatrix}
1&0&0\\
0&0&0\\
0&0&0
\end{pmatrix},
T
\]
is spanned by matrices of rank $1$.
In particular,
$T_1$ would be in the span of $T_2, T_3, T$ and another matrix of rank $1$.
Thus we would be able to find constants
$\beta,\gamma,f_1,f_2,f_3,g_1,g_2,g_3$,
such that the rank of
$$
\begin{pmatrix}
\gamma& \beta & 1\\
\beta&1&0\\
1&0&0
\end{pmatrix}
+
\begin{pmatrix}
f_1g_1& f_1g_2 & f_1g_3\\
f_2g_1& f_2g_2 & f_2g_3\\
f_3g_1& f_3g_2 & f_3g_3
\end{pmatrix}
$$
is one.
There are two cases: if $g_3\neq 0$, then we can subtract
$ \frac{g_1}{g_3}$ times the third column from the first, and
$ \frac{g_2}{g_3}$ times the third column from the second to obtain
$$
\begin{pmatrix}
* & * & 1+f_1g_3\\
* & 1 & f_2g_3\\
1 & 0 & f_2g_3
\end{pmatrix}
$$
which has rank at least two.
If $g_3=0$ the matrix already visibly has rank at least two.
Thus it is impossible to find such constants $\beta, \gamma ,f_i,g_i$
and the rank in question is necessarily at least $5$.
The second case is more delicate.
Write all $2 \times 2$ minors of
\[
\begin{pmatrix}
t & s &u\\
s & 0 & 0\\
u & 0 & 0
\end{pmatrix} + x
\begin{pmatrix} f_1g_1& f_1g_2 & f_1g_3\\ f_2g_1& f_2g_2 & f_2g_3\\ f_3g_1& f_3g_2 & f_3g_3 \end{pmatrix}
\]
and consider $f_i$ and $g_j$ as parameters of degree $0$, and remaining variables $\alpha_1 , \alpha_2, \alpha_3, x$ of degree $1$.
We claim $(s f_3 - u f_2)^2$ and $(s g_3 - u g_2)^2$ are in the ideal $\ccI$ generated by minors.
This can be verified by patient calculation, or using a computer algebra system, such as Magma \cite{magma}.
Thus $f_2=f_3=g_2=g_3 =0$, for otherwise we have a degree $1$ equation in the radical ideal $\sqrt{I}$,
and then the rank $1$ matrices do not span the four dimensional linear space.
But in such a case ${u}^2$ and ${s}^2$ are among the minors, giving $u$ and $s$ as linear equations in $\sqrt{I}$,
a contradiction.
\end{proof}
\subsection{Singularities}\label{s3smooth}
In this subsection we prove Theorems~\ref{txsmooth} and \ref{s3sing}.
The strategy is uniform to most cases:
using the desingularization $\mathbb P}\def\BT{\mathbb T {\mathcal E}}\def\cT{{\mathcal T} \to Sub_{i,j,k}$ as in the beginning paragraph of \S\ref{sec_ranks_and_orbits},
which is birational away from the locus
$Sub_{i-1,j,k} \cup Sub_{i,j-1,k} \cup Sub_{i,j,k-1}$,
we reduce statements to properties of secant varieties of low dimensional Segre products.
\begin{proof}[Proof of Theorem~\ref{txsmooth}]
First note that $\sigma_2(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))=Sub_{2,2,2}$. In
particular, any point of $\sigma_2(Seg(\pp 1\times \pp 1\times \pp 1))=\pp 7$ is a smooth point.
Now just observe that $[a_1{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_2+ a_1{\mathord{ \otimes } } b_2{\mathord{ \otimes } } c_1 + a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1]$ is a smooth point of
$Sub_{2,2,2}$, because it is not contained in $Sub_{2,2,1} \cup Sub_{2,1,2} \cup Sub_{1,2,2}$.
\end{proof}
Similarly, in Theorem~\ref{s3sing} if ${\rm dim}\; A = 2$, then $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))=Sub_{2,3,3}$.
A general point of each type \ref{item_main_thm_normal_form_honest_secant}--\ref{item_main_thm_normal_form_two_tangents}
is not contained in any of the smaller subspace varieties, so the same argument works.
So we will assume $ {\rm dim}\; A, {\rm dim}\; B, {\rm dim}\; C \ge 3$.
\begin{lemma}
Suppose $ {\rm dim}\; A = {\rm dim}\; B = {\rm dim}\; C = 3$.
Then a general point of each component of points of type \ref{item_main_thm_normal_form_two_tangents}
is a smooth point of $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$.
\end{lemma}
\begin{proof}
The only defining equations of $\sigma_3(Seg(\pp 2\times\pp 2\times \pp 2))$
are the $27$ (degree four) Strassen equations.
If we write
$T=a_1{\mathord{ \otimes } } X+ a_2{\mathord{ \otimes } } Y+a_3{\mathord{ \otimes } } Z$, then $9$ of the equations are the entries of the $3\times 3$ matrix
\begin{equation}\label{strpoly}
P(T)^s_t=
\sum_{j,k}(-1)^{j+k}(\operatorname{det}}\def\tpfaff{\operatorname{Pf}}\def\thc{\operatorname{HC} X^{\hat j}_{\hat k})
(Y^j_tZ^s_k-Y^s_kZ^j_t)
\end{equation}
where $X^{\hat j}_{\hat k}$ is $X$ with its $j$-th row and
$k$-th column removed.
The remaining equations come from permuting the roles of $X, Y, Z$, see, e.g. \cite{LWsecseg}.
Take
$T=a_1{\mathord{ \otimes } } (b_1{\mathord{ \otimes } } c_2 + b_2{\mathord{ \otimes } } c_1 + b_3 {\mathord{ \otimes } } c_3) +a_2{\mathord{ \otimes } } b_1{\mathord{ \otimes } } c_1 + a_3{\mathord{ \otimes } } b_3{\mathord{ \otimes } } c_1 $
as in Table~\ref{table_orbits_general} row $35$.
Writing $T=
a_1{\mathord{ \otimes } } X+a_2
{\mathord{ \otimes } } Y+a_3{\mathord{ \otimes } } Z$, we have
$$
X=\begin{pmatrix} 0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix}, \ \
Y=\begin{pmatrix} 1&0&0\\ 0&0&0\\ 0&0&0\end{pmatrix}, \ \
Z=\begin{pmatrix} 0&0&0\\ 0&0&0\\ 1&0&0\end{pmatrix}.
$$
Then
$$
dP_T=
\begin{pmatrix}
-dx_{2,3}+ dy_{1,3}- dz_{1,2} +dz_{2,1} & -dz_{2,2} & dz_{2,3}\\
dy_{2,3}-dz_{2,2} & 0 & 0\\
dx_{2,2}- dy_{2,1}+ dy_{3,3}- dz_{3,2} & -dy_{2,2} & -dy_{2,3}\\
\end{pmatrix}
$$
which indeed has six linearly independent differentials.
To argue for the other components, i.e., when $T$ is of the form $34$ or $36$ in Table~\ref{table_orbits_general},
one can permute the factors $A$, $B$, and $C$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{s3sing}]
Assume $ {\rm dim}\; A, {\rm dim}\; B, {\rm dim}\; C \ge 3$.
Since the map $\mathbb P({\mathcal E}}\def\cT{{\mathcal T}) \to Sub_{3,3,3}$ is an isomorphism
near a general point of type \ref{item_main_thm_normal_form_two_tangents},
the Lemma implies that such a point is a smooth point of
$\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ for any $A$, $B$, $C$ (each of dimension at least $3$).
But orbits $34$--$36$ from Table~\ref{table_orbits_general} are in the closure of orbits $37$ and $38$.
So $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ is non-singular at a general point of each type \ref{item_main_thm_normal_form_point_plus_tangent}--\ref{item_main_thm_normal_form_two_tangents}.
The final thing to prove is that $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$
is non-singular at a general point of $Sub_{233}$.
Let $p$ be such a point. Since $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))\subset Sub_{333}$, we may assume $\operatorname{dim} A=\operatorname{dim} B=\operatorname{dim} C=3$.
First note that $Sub_{233}$ is not contained in $\ptstypeii$, as they are both irreducible, have the same dimension
and $\ptstypeii\not\subset Sub_{233}$.
So $p$ is not in $\ptstypeii$.
By Theorem~\ref{s3nformthm}, this implies that there exists an open neighborhood
$U \subset \sigma_3(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C)$ of $p$,
such that in this neighborhood all points are of type~\ref{item_main_thm_normal_form_honest_secant}.
Consider the dominant rational map
\begin{align*}
\phi: (A \times B \times C)^{\times 3} & \dashrightarrow \hat \sigma_3(Seg(\mathbb P A \times \mathbb P B \times \mathbb P C)) \\
(a_1,b_1, c_1),(a_2,b_2, c_2),(a_3,b_3, c_3) &
\mapsto a_1 \otimes b_1 \otimes c_1 +a_2 \otimes b_2 \otimes c_2 + a_3 \otimes b_3 \otimes c_3
\end{align*}
Let $W := \phi^{-1} (U)$.
Then $\phi|_{W}: W \to U$ is a regular surjective map.
The aim is to calculate the tangent map at any point in $\phi^{-1}(p)$.
We commence with identifying $\phi^{-1}(p)$.
Since $R_X(p) =3$, any point in $\phi^{-1} (p)$ will be contained in a fixed $(A' \times B' \times C')^{\times 3}$
with ${\rm dim}\; A' =2$, ${\rm dim}\; B' ={\rm dim}\; C' = 3$ by \cite[Cor.~2.2]{BLtensor}.
Write $p = [a_1 \otimes b_1 \otimes c_1 + (a_1 +a_2) \otimes b_2 \otimes c_2 + a_2 \otimes b_3 \otimes c_3]$
(see \cite[\S 6]{BLtensor}).
We claim that this normal form is unique up to trivialities such as $7$-dimensions worth of rescalings,
and permutations of summands.
By writing $p:(A')^* \to B' \otimes C'$,
we obtain the slice
$\left(\begin{smallmatrix}
s & & \\
& s+t & \\
& & t
\end{smallmatrix} \right)$.
The set of rank $2$ elements in this linear space is given by the determinant of the matrix.
This set consists of three lines in $(A')^*$ spanned by $a_1^*$, $a_1^* - a^*_2$, and $a_2^*$.
Thus the triple $a_1, (a_1 + a_2), a_2$ is (up to order and scale) determined by $p$.
In a similar way we consider the other slices, and $2\times 2$ minors of the resulting matrices, to conclude,
that triples $b_1, b_2, b_3$ and $c_1, c_2, c_3$ are determined by $p$, up to order and scale.
It is easy to see, that any meaningfully different choice of orders, or scaling will give a different tensor,
so the preimage of $p$ consists of $6$ components, each of dimension $7$, isomorphic to $(\mathbb C^*)^7$.
\begin{table}[tb]
$$
\begin{array}{|c|r|r|}
\hline
\hline
A\otimes B \otimes C & {\rm dim}\; \sigma_3 & {\rm dim}\; Sing \le \\
\hline
\hline
\mathbb C^2\otimes \mathbb C^2 \otimes \mathbb C^2 & 7 & -1 \\
\hline
\mathbb C^2\otimes \mathbb C^2 \otimes \mathbb C^3 & 11 & -1 \\
\hline
\mathbb C^2\otimes \mathbb C^2 \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{c}}+2 & 2{\mathbf{c}}+3 \\
\hline
\mathbb C^2\otimes \mathbb C^3 \otimes \mathbb C^{3} & 17 & -1 \\
\hline
\mathbb C^2\otimes \mathbb C^3 \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{c}}+8 & 3{\mathbf{c}}+4 \\
\hline
\mathbb C^2\otimes \mathbb C^{{\mathbf{b}}} \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{b}}+3{\mathbf{c}}-1 &
\begin{matrix} \max \{ 3{\mathbf{b}}+3{\mathbf{c}}-9,\\ 2{\mathbf{b}}+3{\mathbf{c}}-2\} \end{matrix} \\
\hline
\mathbb C^3\otimes \mathbb C^3 \otimes \mathbb C^{3} & 20 & 18 \\
\hline
\mathbb C^3\otimes \mathbb C^3 \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{c}}+11& 3{\mathbf{c}}+9 \\
\hline
\mathbb C^3\otimes \mathbb C^{{\mathbf{b}}} \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{b}}+3{\mathbf{c}}+2& 3{\mathbf{b}}+3{\mathbf{c}} \\
\hline
\mathbb C^{{\mathbf{a}}}\otimes \mathbb C^{{\mathbf{b}}} \otimes \mathbb C^{{\mathbf{c}}} & 3{\mathbf{a}}+3{\mathbf{b}}+3{\mathbf{c}}-7 & 2{\mathbf{a}}+3{\mathbf{b}}+3{\mathbf{c}}-6 \\
\hline
\hline
\end{array}
$$
\caption{Singularities of $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$.
In the first column we list the tensor space, assuming $4 \le {\mathbf{a}} \le {\mathbf{b}} \le {\mathbf{c}}$.
In the second column we write the dimension of the secant variety.
In the third column we present the upper bound on the dimension of the singular locus of the secant variety,
which follows from our results in this section.}
\label{table_sings}
\end{table}
Next, we calculate the image of tangent map of $\phi$ at any $q \in \phi^{-1} (p)$,
say $q= [(a_1,b_1,c_1), (a_1 +a_2,b_2,c_2), (a_2,b_3,c_3)$.
This image is spanned by the following tensors, all considered modulo $p$,
as we look at a subspace of $T_p \mathbb P(A\otimes B \otimes C) \simeq (A\otimes B \otimes C)/p$:
\begin{align*}
a_i\otimes b_1 \otimes c_1 && a_i\otimes b_2 \otimes c_2 && a_i\otimes b_3 \otimes c_3 &&
\text{for any } i \in\setfromto{1}{{\rm dim}\; A}, \\
a_1\otimes b_j \otimes c_1 && (a_1+a_2)\otimes b_j \otimes c_2 && a_2\otimes b_j \otimes c_3 &&
\text{for any } j \in\setfromto{1}{{\rm dim}\; B}, \\
a_1\otimes b_1 \otimes c_k && (a_1+a_2)\otimes b_2 \otimes c_k && a_2\otimes b_3 \otimes c_k &&
\text{for any } k \in\setfromto{1}{{\rm dim}\; C}.
\end{align*}
This space is independent of the choice of the order or scalings in $q$.
Also the linear space above has dimension
\[
3 ({\rm dim}\; A + {\rm dim}\; B + {\rm dim}\; C) -7 = {\rm dim}\; \sigma_3(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C),
\]
because there are $3 ({\rm dim}\; A + {\rm dim}\; B + {\rm dim}\; C)$ tensors listed above,
and each $a_1 \otimes b_1 \otimes c_1$, $(a_1 +a_2) \otimes b_2 \otimes c_2$, $a_2 \otimes b_3 \otimes c_3$
is listed three times and $p$ is a sum of those three tensors.
One can check there no other linear dependencies.
Thus $\phi: W \to \mathbb P (A\otimes B \otimes C)$ is a map with constant rank on an open subset containing $\phi^{-1}(p)$.
Therefore the image is non-singular at $p$ as claimed.
\end{proof}
We summarize our results in Table~\ref{table_sings}. In particular, it follows that $\sigma_3(Seg(\mathbb P}\def\BT{\mathbb T A\times \mathbb P}\def\BT{\mathbb T B\times \mathbb P}\def\BT{\mathbb T C))$ is always non-singular in codimension $1$, that is, codimension of the singular locus is at least $2$.
Moreover, it is of codimension $2$ if and only if, one of the factors is $\mathbb C^3$, and the others have dimension at least $3$.
\bibliographystyle{amsplain}
|
\section{\label{intro} Introduction}
The recent potential discovery of the standard model Higgs boson with
a comparatively low mass of $m_{\text{H}}\simeq 125$GeV \cite{Aad:2012tfa} has
stimulated renewed interest in Higgs mass bounds within the standard
model itself
\cite{Maiani:1977cg,Krasnikov:1978pu,Lindner:1985uk,Altarelli:1994rb,Schrempp:1996fb,Hambye:1996wb}
and beyond
\cite{Cabibbo:1979ay,Espinosa:1991gr,Chen:2012faa,Lebedev:2012zw}. In
particular, arguments based on vacuum stability
\cite{Krive:1976sg,Hung:1979dn,Linde:1979ny,Politzer:1978ic,Sher:1988mj,Lindner:1988ww,Ford:1992mv}
(or sufficient metastability
\cite{Arnold:1989cb,Sher:1993mf,Bergerhoff:1999jj,Isidori:2001bm})
appear to give rise to a lower bound for the Higgs mass
\cite{Ellis:2009tp,EliasMiro:2011aa,Degrassi:2012ry,Alekhin:2012py,Masina:2012tz,Buttazzo:2013uya}. The
measured value for the mass of the discovered scalar boson is either
close to or on top of the bound or might even violate the bound,
depending on various other parameters, most notably the value of the
top mass (in the appropriate scheme) and the strong coupling constant.
The consequences of the true Higgs mass satisfying or violating the
bound can be rather dramatic, ranging from measured constraints on the
underlying UV theory structure, via an upper bound for the scale of new
physics to the prediction of the decay of the universe as we know
it. Therefore, a thorough understanding of Higgs mass bounds within
the standard model is clearly mandatory.
Even though typical computations of mass bounds are often done with
perturbative (RG-resummed) methods, the problem is generically
nonperturbative. This is obvious for the upper Higgs mass bound -- the
so-called unitarity or triviality bound -- which is, in principle,
related to a strongly coupled Higgs sector in the UV. In perturbation
theory, this becomes manifest from the vicinity to the Landau pole,
indicating the failure of perturbation theory.\footnote{In fact, the
upper bound is often motivated from the requirement that the
standard model {\it per definitionem} should be describable within
perturbation theory in the UV. Since this is if at all an aesthetic
but not a physical criterion, we rely on the criterion of triviality
in the present work.} But also the lower bound involves
nonperturbative information for two reasons: first, the prediction of
infrared (IR) quantities such as Higgs and top masses involve a proper
description of threshold effects. These are nonperturbative, as such
mass scales are related to the couplings. Second, an investigation of
stability issues requires the computation of a full effective potential
for arbitrary field amplitudes.
In a series of works, Higgs mass bounds have therefore recently been studied
within lattice quantum field theory both for a simple $\mathds{Z}_2$
Higgs-Yukawa model \cite{Holland:2003jr,Holland:2004sd,Fodor:2007fn}, as well
as for a Higgs-Yukawa model more similar to and significant for the standard
model \cite{Fodor:2007fn,Gerhold:2007yb}. In particular, the lower Higgs mass
bound arises from the mere criterion of starting from a physically meaningful
bare UV theory on the lattice. No reference to low-energy stability issues had
to be made, and no indications for an instability have been observed. Most
prominently, the simulations of \cite{Gerhold:2010wv,Bulava:2012pb}
essentially rule out or put strong constraints on the existence of a fourth
flavor generation for the measured Higgs boson mass; similar conclusions have
been drawn from analytic considerations \cite{Djouadi:2012ae}.
In the present work, we revisit the Higgs mass bounds by analytic
means using the functional renormalization group (RG). Within a
consistent systematic derivative expansion, the functional RG provides
for a tool to analyze the problem nonperturbatively and allows to
estimate errors of the approximation scheme. In order to concentrate
on the basic mechanisms for the mass bounds, we confine ourselves to
the simple $\mathds{Z}_2$ Higgs-Yukawa model, as it avoids intricate
questions arising from the gauge-Higgs interplay in the full standard
model \cite{Frohlich:1981yi,Maas:2012tj}, while at the same time
maintaining the standard model property that no Goldstone bosons arise
in the broken phase. First functional RG studies of Higgs mass bounds
have already been performed in \cite{Branchina:2005tu,Gneiting:2005}.
In the present work, we particularly concentrate on the influence of
generic UV actions on the Higgs mass bounds. In fact, we find a rather
substantial influence of the precise form of the bare scalar potential
on the lower bound of the Higgs boson. At first sight, this seems to
be at odds with common wisdom of renormalizable field theories that IR
observables should be independent of the details of the microscopic UV
theory. This statement (formulated under suitable mild assumptions)
is, of course, left untouched by our work. However, the main point is
that the notion of a Higgs mass bound is strictly speaking not a pure
IR observable. Higgs mass bounds are typically formulated as a
function of the UV cutoff $\Lambda$, i.e.,
$m_{\text{H}}{}_{,\text{bound}}=m_{\text{H}}{}_{,\text{bound}}(\Lambda)$. Hence, in
order to quantify this dependence, we have to make certain assumptions
about the system at and near the cutoff. This includes the choice of a
regularization scheme, specifying the details of the UV regularization
at the cutoff; in this sense, part of the scheme-dependence of the Higgs
mass bounds is actually physical. And this includes dynamical properties of
the flow near the cutoff which can be rather strongly influenced by
the bare theory. Quantitatively, we find that rather mild
modifications of the bare potential can have a significant impact
on the lower Higgs mass bound.
This article is organized as follows: in Sect.~\ref{sec:model}, we
briefly introduce our simple toy model. Section \ref{sec:floweq}
summarizes the concepts of the functional RG applied to this model and
presents the resulting flow equations. As a warm-up, a simple
mean-field analysis already illustrating many of the properties of
the Higgs mass bounds is given in
Sect.~\ref{sec:meanfield}. Incidentally, these mean-field properties
do actually not require the functional RG framework, but could equally
well be derived within a large-$N$ type of reasoning. Our main results
based on the nonperturbative RG flow equations are summarized in
Sect.~\ref{sec:flow}.
\section{$\mathds{Z}_2$-symmetric Higgs-Yukawa toy model}
\label{sec:model}
Many of the fluctuation-induced features of Higgs mass bounds in the
standard model can already be studied in a greatly simplified model
involving a Dirac fermion flavor $\psi$ (the top quark) and a real scalar
boson $\phi$. The model is defined by the Euclidean classical action
\begin{equation}
S=\int d^4x \left[ \frac{1}{2}(\partial_\mu \phi)^2 +
\frac{\bar{m}}{2} \phi^2 + \frac{\bar{\lambda}}{8} \phi^4
+ \bar{\psi}\, {i} \fss{\partial} \psi + {i} \bar{h}
\phi\bar{\psi}\psi \right]. \label{eq:bareaction}
\end{equation}
For later purposes, we allow the top quark to appear in $N_{\mathrm{f}}$ flavor
copies. We use $N_{\mathrm{f}}$ merely as an ordering parameter of the
calculation, but not as a physical parameter mimicking the generations
of the standard model. For quantitative statements, we will use
$N_{\mathrm{f}}=1$. The model is invariant under a
discrete ``chiral'' symmetry,
\begin{equation}
\psi\to \text{e}^{{i} \frac{\pi}{2} \gamma_5} \psi, \quad
\bar{\psi}\to \bar{\psi}\, \text{e}^{{i} \frac{\pi}{2} \gamma_5}, \quad
\phi\to -\phi,\label{eq:discsym}
\end{equation}
which protects the fermions against acquiring a direct mass
term. Since the symmetry is only discrete, its spontaneous breaking
owing to a nonzero expectation value for the scalar field
$v=\langle\phi\rangle$ does not give rise to massless Goldstone
bosons. This feature mimics the property of the standard model that
the Goldstone modes are eaten by the massive electroweak gauge bosons.
The quantum theory corresponding to \Eqref{eq:bareaction} has to be
defined with a finite ultraviolet (UV) cutoff $\Lambda$ which, together
with a specified regularization prescription, remains an implicit
physical parameter of the theory. This is because triviality inhibits
an ultraviolet extension to arbitrarily high scales while keeping the
physical low-energy parameters fixed \cite{Wilson:1973jj}; in
perturbation theory, this feature is reflected by the existence of a
Landau pole in the running coupling.
Solving the so defined quantum theory provides for a mapping from the
microscopic bare parameters $\bar{m}^2,\bar{\lambda}, \bar{h},
\Lambda$ and possibly further RG irrelevant bare couplings to the
set of physical parameters which are given by the top mass $m_{\text{top}}$,
the Higgs mass $m_{\text{H}}$, the vacuum expectation value $v$ and still the
cutoff $\Lambda$. These physical parameters are directly related to
renormalized couplings in the quantum effective action, such as the
renormalized Yukawa coupling $h$, see below, and the effective potential
$U(\rho)$, where $\rho=\frac{\phi^2}{2}$. Denoting the minimum of the
effective potential by $\rho_0$, we identify
\begin{equation}
v={Z_\phi^{1/2}}\langle\phi\rangle=\sqrt{2{Z_\phi}\rho_0},
\,\, m_{\text{top}}^2={v^2} h^2, \,\,
m_{\text{H}}^2={v^2 \frac{U''(\rho_0)}{Z_\phi^2}} , \label{eq:physpar}
\end{equation}
where the wave function renormalization $Z_\phi$ is introduced below.
In this work, we consider the vacuum expectation value and the top
mass as given, $v\simeq246$GeV and $m_{\text{top}}\simeq 173$GeV.\footnote{We
use here the value for the top mass measured by kinematically
reconstructing its decay products and comparing these to Monte Carlo
simulations. For Higgs mass bounds, actually the pole mass is
considered to be the appropriate quantity, which could significantly
differ from the experimentally quoted value
\cite{Alekhin:2012py}. As a rule of thumb, an uncertainty of $\sim
1$GeV in the top mass leads to a $\pm2$GeV variation of the lower
Higgs mass bound for large cutoffs $\Lambda$. In any case,
quantitative results of the present toy model should anyway only be
considered as an illustrative example.} Furthermore, choosing a
fixed cutoff $\Lambda$ leaves only $m_{\text{H}}$ as a free parameter which
becomes a function of the whole set of microscopic bare parameters.
Constraints on the Higgs mass are now obtained if the region of
attainable Higgs masses is bounded for any given combination of bare
parameters. These bare parameters are essentially unconstrained, as
they are provided by a yet unknown underlying microscopic theory (UV
completion). Only a stable bare scalar potential bounded from below is
required in order to facilitate a meaningful definition of the quantum
theory. In the present work, we start with the standard class of
initial bare $\bar\lambda \phi^4$ potentials. UV stability then
implies that $\bar\lambda\geq 0$ for this class of potentials. We then
extend our considerations to more general potentials. For
instance, also a negative $\bar\lambda$ is permitted if the potential
is stabilized for large $\phi$, e.g., by positive $\phi^6,\phi^8,
\dots$ terms in the bare potential. We emphasize that these
higher-order terms cannot be excluded by referring to
renormalizability criteria. This is because we consider them to be
present in the microscopic UV potential at a fixed (possibly physical)
UV cutoff $\Lambda$. Presently no experiment can impose relevant
constraints on such terms which could arise from an underlying UV
completion of the standard model. Renormalizability rather tells us
that the IR is dominated by the power-counting ``renormalizable''
operators in the standard model, provided that the UV theory starts
near the perturbative Gau\ss ian fixed point.
\section{Renormalization Flow}
\label{sec:floweq}
As an alternative to the functional-integral definition of continuum
quantum field theory, we use a differential formulation provided by
the functional RG. A convenient version is given by the flow equation
for the effective average action $\Gamma_k$, which interpolates
between the bare action $\Gamma_{k=\Lambda}= S$ at the UV cutoff
$\Lambda$ and the full quantum effective action $\Gamma=\Gamma_{k=0}$
\cite{Wetterich:1992yh}. The latter corresponds to the generator of
fully-dressed proper vertices. The variation of the effective action
with respect to the scale $k$ is given by the Wetterich equation
\begin{equation}
k\,\partial_{k}\Gamma_{k}
\equiv\partial_{t}\Gamma_{k}
=\frac{1}{2}\mathrm{STr} \left[(\partial_{t}R_{k})
(\Gamma_{k}^{(2)} +R_{k})^{-1} \right]
, \,\, t=\ln\frac{ k}{\Lambda}.
\label{eq:flow_eq1}
\end{equation}
Here, $\Gamma_{k}^{(2)}$ denotes the second functional derivative with respect
to the fluctuating fields $\Phi=(\phi,\psi,\bar{\psi})$, and the super-trace also
includes a minus sign for the fermions. The regulator $R_{k}$ in the
denominator is chosen such that it suppresses IR modes below the scale $k$, and its
derivative $k\partial_{k}R_{k}$ establishes ultraviolet (UV) finiteness; as a
consequence, the flow of $\Gamma_k$ is dominated by fluctuations with momenta
$p^2\simeq k^2$, implementing the concept of smooth momentum-shell
integrations, for reviews see~\cite{Berges:2000ew,Aoki:2000wm,Pawlowski:2005xe,Gies:2006wv,
Delamotte:2007pf,Kopietz:2010zz,Braun:2011pp}.
As we are working with an explicit finite cutoff $\Lambda$, also the
choice of the regularization scheme strictly speaking belongs to the
definition of the model. This scheme is here specified in terms of the
regulator function $R_{k}$, more precisely in terms of the regulator
shape functions $r(p^2/k^2)$, $r_{\text{F}}(p^2/k^2)$ introduced in
the appendix. From the viewpoint of the model definition, these shape
functions determine how the modes are physically cut off in the
UV. Since a change of the regularization scheme such as a change of
the shape functions can be mapped onto a change of the initial
conditions for the bare couplings, we keep the regulator fixed in the
present work and vary the bare couplings.
In addition to perturbative expansions, nonperturbative approximation schemes
can be devised for the flow equation. Systematic and consistent
expansion schemes which do not rely on a perturbative coupling ordering are,
for instance, the vertex expansion or the derivative expansion.
In this work, we study the renormalization flow of the Yukawa system
nonperturbatively within the following truncation based on the derivative
expansion:
\begin{equation}
\Gamma_{k} = \int_x \left[
\frac{Z_{\phi,k}}{2} \, ( \partial_{\mu} \phi )^2 + U_{k}(\rho)
+ Z_{\psi,k} \, \bar{\psi}\, {i} \fss{\partial}
\psi + i \, \bar{h}_{k} \, \phi \bar{\psi} \psi \right], \label{eq:trunc}
\end{equation}
where $\rho=\frac{1}{2} \phi^2$, and the potential $U_k$ generally
includes arbitrary powers of the field.
In fact, the accuracy of the derivative expansion for scalar theories
has been verified quantitatively in many contexts. Here, we actively
study its convergence by comparing leading-order (LO) results (obtained for
$Z_{\phi,k}=1, Z_{\psi,k}=1$) to next-to-leading order (NLO) results. We find no
signatures of a failure of this expansion even at comparatively strong
coupling, see below.
Inserting this {\it ansatz} \eqref{eq:trunc} into the flow equation
\eqref{eq:flow_eq1} provides us with the RG flows of $\bar{h}_{k},U_k$ and the
wave function renormalizations $Z_{\phi,k}$ and $Z_{\psi,k}$; the latter flows
will be followed in terms of the anomalous dimensions
\begin{equation}
\eta_\phi=-\partial_t \ln Z_{\phi,k}, \quad \eta_\psi=-\partial_t \ln
Z_{\psi,k}. \label{eq:etadef}
\end{equation}
The flow equation for the effective potential reads
\begin{eqnarray}
\partial_{t} U_{k} & = & 2 \, v_{d} \, k^{d} \, \Big[ l_{0}^{d} \left(
k^{-2} \, Z_{\phi,k}^{-1} \, \left[ 2 \, \rho \,
U_{k}'' + U_{k}' \right];\eta_{\phi} \right)
\nonumber \\
& & \qquad - N_{\mathrm{f}} d_{\gamma} \, l_{0}^{(F)\,d} \left(
2 \, k^{-2} \, Z_{\psi,k}^{-2} \,
\bar{h}_{k}^{2} \, \rho;\eta_{\psi} \right)
\Big], \label{eq:potflow}
\end{eqnarray}
where the primes denote derivatives with respect to $\rho$, and
$v_{d}^{-1} = 2^{d+1} \, \pi^{d/2} \, \Gamma(d/2)$. For generality, we
work in $d$ dimensions and with a $d_\gamma$ dimensional
representation of the Dirac algebra. We will later specialize to $d=4$
and $d_\gamma=4$. The threshold functions $l_{0}^{d}$ and
$l_{0}^{(F)\,d}$ {arise from the integration over the
loop momentum and carry the non-universal regulator dependence. For
any physically admissible regulator, they} approach finite constants
for vanishing argument and decrease to zero for large first argument,
describing the decoupling of massive modes; details can be found in
Appendix \ref{appA}.
It is useful to introduce renormalized fields
\begin{equation}
\tilde{\phi} =Z_{\phi,k}^{1/2} \phi, \quad
\tilde{\psi} = Z_{\psi,k}^{1/2} \psi, \label{eq:renfields}
\end{equation}
as well as the dimensionless renormalized Z${}_2$ invariant quantity
\begin{equation}
\tilde{\rho}= Z_{\phi,k} \, k^{\, 2-d} \rho. \label{eq:trhodef}
\end{equation}
The dimensionless renormalized Yukawa coupling is defined by
\begin{equation}
h_{k}^{2} = Z_{\phi,k}^{-1} \, Z_{\psi,k}^{-2} \, k^{\, d-4} \,
\bar{h}_{k}^{2}, \label{eq:hqdef}
\end{equation}
and the dimensionless potential simply is:
\begin{equation}
u_{k} = U_{k} \, k^{\, -d}. \label{eq:potdef}
\end{equation}
The flow of $u_k$ for fixed $\tilde{\rho}$ is given by
\begin{widetext}
\begin{eqnarray}
\partial_{t} \, u_{k}
& = & -d \, u_{k} + (d-2+\eta_{\phi}) \,
\tilde{\rho} \, u'_{k} + 2 \, v_{d} \, \Big[ l_{0}^{d}\left(u_{k}' + 2 \,
\tilde{\rho} \, u_{k}''; \eta_{\phi} \right)
- N_{\mathrm{f}} d_{\gamma} \, l_{0}^{(F)\,d}\left(2 \, \tilde{\rho} \,
h_{k}^{2} ; \eta_{\psi} \right) \Big],
\label{PotentialFlowEquation}
\end{eqnarray}
where primes now denote derivatives with respect to $\tilde{\rho}$. The flow
of the Yukawa coupling is of the form
\begin{eqnarray}
\partial_{t} \, h_{k}^{2}
& = & \left[ \eta_{\phi} + 2 \, \eta_{\psi} + d - 4 \right] \,
h_{k}^{2}
+ 8 \, h_{k}^{4} \,v_{d} \,
l_{1,1}^{(FB)\,d}\left(\omega_{1},\omega_{2};
\eta_{\psi},\eta_{\phi}\right) \label{eq:flowhq} \\
&& - \left[ 48 \, \kappa_{k} \, u_{k}''(\kappa_{k})
+ 32 \, \kappa_{k}^{2} \, u_{k}'''(\kappa_{k}) \right]
\, h_{k}^{4} \, v_{d} \, l_{1,2}^{(FB)\,d}\left(\omega_{1},\omega_{2};
\eta_{\psi},\eta_{\phi} \right)
- 32 \, h_{k}^{6} \, \kappa_{k} \, v_{d} \,
l_{2,1}^{(FB)\,d}\left(\omega_{1},\omega_{2};
\eta_{\psi},\eta_{\phi}\right), \nonumber
\end{eqnarray}
with
\begin{eqnarray}
\omega_{1} & = & 2 \, \kappa_{k} \, h_{k}^{2}, \quad
\omega_{2} = u_{k}'(\kappa_{k}) + 2 \kappa_{k} u_{k}''(\kappa_{k})
\, , \label{eq:omegadef}
\end{eqnarray}
and $\kappa_k=\tilde{\rho}_{\text{min}}$ denotes the minimum of the
potential; i.e., if $\kappa_k\neq 0$ then $u_k'(\kappa_k)=0$. Finally,
the anomalous dimensions are determined by
\begin{eqnarray}
\eta_{\phi} & = & 8 \, \frac{v_{d}}{d}
\bigg[ \kappa_{k} \left[3 \, u_{k}''(\kappa_{k})+2 \, \kappa_{k} \,
u_{k}'''(\kappa_{k}) \right]^{2} m_{4,0}^{d}\left(2 \, \kappa_{k}
\, u_{k}''(\kappa_{k})+u_{k}'(\kappa_{k}),0;\eta_{\phi}\right)
\nonumber \\
& & \qquad\;\;\; + N_{\mathrm{f}} d_{\gamma} \, h_{k}^{2} \Big[
m_{4}^{(F)\,d}\left(2 \, \kappa_{k} \,
h_{k}^{2};\eta_{\psi}\right)
-2 \, \kappa_{k} \, h_{k}^{2} \, m_{2}^{(F)\,d}\left(2 \, \kappa_{k}
\, h_{k}^{2};\eta_{\psi}\right) \Big] \bigg], \label{eq:etaphi}\\
\eta_{\psi} &=& 8 \, h_{k}^{2} \, \frac{v_{d}}{d} \,
m_{1,2}^{(FB)\,d}\left(2 \, \kappa_{k} \, h_{k}^{2},2 \, \kappa_{k} \,
u_{k}''(\kappa_{k}) +
u_{k}'(\kappa_{k});\eta_{\psi},\eta_{\phi}\right), \label{eq:etapsi}
\end{eqnarray}
where the threshold functions are again discussed in Appendix
\ref{appA}. These flow equations can be compared to those of similar
investigations in the literature
\cite{Hofling:2002hj,Gies:2009hq,Braun:2010tt} within different
physical contexts. Once the flow equations have been solved for
suitable initial conditions, we can read off the fully renormalized
long-range quantities in the limit $k\to0$. For instance, the physical
quantities defined in \Eqref{eq:physpar} require the renormalized
Yukawa coupling $h=h_{k\to0}$ and the wave function renormalization
$Z_\phi=Z_{\phi,k\to0}$. The renormalized vacuum expectation value is
obtained from $v^2=\lim_{k\to0} 2k^2 \kappa_k$.
\end{widetext
\section{Mean-field analysis}
\label{sec:meanfield}
Let us first perform a mean-field analysis, corresponding to a
one-loop approximation of the effective potential including fermion as
well as boson fluctuations, while keeping the wave function renormalizations
and the Yukawa coupling fixed,
\begin{equation}
Z_{\phi,k},Z_{\psi,k}\to 1, \quad h_k \to h_\Lambda.
\end{equation}
The mean-field effective potential $U^{\text{MF}}$ could, of course, be
calculated directly from a Gau\ss ian approximation of the generating
functional, yielding the standard $\log\det$-formula. Nevertheless, we
derive it from the flow equation, since it provides direct access to
the use of an arbitrarily shaped regulator function, which can be used
to model the physical UV cutoff mechanism.
The standard mean-field (MF) approximation is equivalent to the large-$N_{\mathrm{f}}$
approximation, taking only fermionic fluctuations into account. The
corresponding mean-field effective potential is obtained from the flow
equation \eqref{eq:potflow} by integrating the fermion contributions $\simN_{\mathrm{f}}$
from $k=\Lambda$ to $0$, while keeping the potential on the right-hand side
fixed at $U_k \to U_{\Lambda}$. We obtain for the mean-field effective potential
\begin{eqnarray}
U^{\text{MF}}_k(\rho) &=& U_{\Lambda}(\rho) \label{eq:MFU}\\
&& +\frac{N_{\mathrm{f}} d_\gamma}{2} \int_p \ln
\left(
\frac{p^2(1+r_{\text{F}}(p^2/\Lambda^2))^2 +2\bar{h}_{\Lambda}^2\rho}
{p^2(1+r_{\text{F}}(p^2/k^2))^2 +2\bar{h}_{\Lambda}^2\rho}
\right),
\nonumber
\end{eqnarray}
where $\int_p= \int \frac{d^d p}{(2\pi)^d}$. The extended mean-field
(EMF) approximation is obtained by including also the scalar fluctuations on
the same Gau\ss ian level. Introducing the abbreviation
\begin{equation}
M_{\Lambda}^2(\rho)=U_{\Lambda}'(\rho)+2\rhoU_{\Lambda}''(\rho),
\end{equation}
we find,
\begin{eqnarray}
U^{\text{EMF}}_k(\rho) &=& U^{\text{MF}}_k(\rho) \label{eq:EMFU}\\
&& -\frac{1}{2} \int_p \ln
\left[
\frac{p^2(1+r(p^2/\Lambda^2)) + M_{\Lambda}^2(\rho)}
{p^2(1+r(p^2/k^2)) + M_{\Lambda}^2(\rho)}
\right].
\nonumber
\end{eqnarray}
Whereas the mean-field approximation becomes exact in the strict
large-$N_{\mathrm{f}}$ limit, no such anchoring to an exact limit is known for
the extended-mean-field approximation. Moreover, further subtleties
arise in the extended-mean-field case from convexity violations and
complex solutions for the potential as discussed in
\cite{Weinberg:1987vp}. These subtleties of the extended mean-field
approximation are however irrelevant for the nonperturbative functional
RG solution discussed below. Hence, we will mainly stay within the
standard mean-field approximation in the following for the purpose of
illustration.
For both approximations, the momentum integration can be done
analytically for a suitable choice of the regulator shape functions
$r(x),r_{\text{F}}(x)$. For instance, for the linear regulator
(cf. App.~\ref{appA}) we obtain in the limit $k\to0$
and in $d=4$ dimensions (where $\bar{h}_{\Lambda}=h_{\Lambda}$)
\begin{widetext}
\begin{eqnarray}
U^{\text{EMF}}(\rho)&=&U_{\Lambda}(\rho) + \frac{1}{64 \pi^2} \Bigg\{ [M_{\Lambda}^2(\rho) -M_{\Lambda}^2(0)
-2N_{\mathrm{f}} d_\gamma h_\Lambda^2\rho] \Lambda^2 +4N_{\mathrm{f}} d_\gamma h_\Lambda^4 \rho^2 \ln
\frac{\Lambda^2 +2h_\Lambda^2\rho}{2h_\Lambda^2\rho} \nonumber\\
&&\qquad\qquad \qquad\qquad
-M_{\Lambda}^4(\rho) \ln \frac{\Lambda^2+M_{\Lambda}^2(\rho)}{M_{\Lambda}^2(\rho)}
+M_{\Lambda}^4(0) \ln \frac{\Lambda^2+M_{\Lambda}^2(0)}{M_{\Lambda}^2(0)} \Bigg\},
\label{eq:umf}
\end{eqnarray}
\end{widetext}
where we have normalized $U^{\text{EMF}}(\rho)$ such that
$U^{\text{EMF}}(0)=0$. In the following we will show that \Eqref{eq:umf} can
be used to illustrate the appearance of a lower bound for the Higgs mass.
\subsection{Bare potentials of $\phi^4$-type}
Let us confine ourselves to bare potentials of
$\phi^4$-type,
\begin{equation}
U_{\Lambda}(\rho)=m_\Lambda^2 \rho + \frac{\lambda_\Lambda}{2} \rho^2. \label{eq:conf}
\end{equation}
For a given UV cutoff $\Lambda$, two out of the three bare parameters
$m_\Lambda^2,\lambda_\Lambda,h_\Lambda$ can be fixed by fixing the top mass and the vacuum
expectation value; more precisely, fixing $h_\Lambda=m_{\text{top}}/v$ and
determining $m_\Lambda^2$ from the transcendental equation
\begin{equation}
U^{\text{EMF}}{}'(\rho_0=v^2/2)=0,\label{eq:umft}
\end{equation}
leaves us with the Higgs mass as a function of the bare scalar
coupling, $m_{\text{H}}=m_{\text{H}}(\lambda_\Lambda)$. In the standard mean-field approximation, it
is easy to see that $m_{\text{H}}=m_{\text{H}}(\lambda_\Lambda)$ increases monotonically with $\lambda_\Lambda$,
therefore a lower bound on the Higgs mass is obtained from the lowest
possible value of $\lambda_\Lambda$, which is $\lambda_{\Lambda,\text{min}}=0$ for
potentials of the form of \Eqref{eq:conf}. (In the extended mean-field
approximation, the same conclusion holds unless $\lambda_\Lambda$ approaches the
strong-coupling value $\lambda_\Lambda\to \frac{8}{3}h_{\Lambda}^2$ where an
EMF-artifact induces singular behavior).
Equation~\eqref{eq:umft} can easily be solved numerically. For an
analytical estimate, let us stay within the mean-field approximation and keep
only the terms $\simN_{\mathrm{f}}$. Determining $m_\Lambda^2$ from the condition
$U^{\text{MF}}{}'(\rho_0=v^2/2)=0$ for fixed values of $m_{\text{top}}$ and
$v$, we find (setting $N_{\mathrm{f}}=1$, $d_\gamma=4$)
\begin{eqnarray}
m_\Lambda^2(\Lambda,\lambda_\Lambda
|
)&=&- \frac{\lambda_\Lambda}{2} v^2+\frac{h_\Lambda^2}{8\pi^2} \Lambda^2
\label{eq:MFmL}\\
&&- \frac{h_\Lambda^4
v^2}{8\pi^2} \left[2\ln\left( 1+ \frac{\Lambda^2}{m_{\text{top}}^2} \right) -
\frac{\Lambda^2}{\Lambda^2 + m_{\text{top}}^2} \right] .
\nonumber
\end{eqnarray}
This fixes the effective mean-field potential as a function of $\lambda_\Lambda$ and
$\Lambda$, yielding the Higgs mass
\begin{eqnarray}
m_{\text{H}}^2(\Lambda,\lambda_\Lambda)&=& v^2 U^{\text{MF}}{}''(v^2/2) \nonumber\\
&=&\frac{m_{\text{top}}^4}{4\pi^2 v^2}\! \left[2 \ln \left(\! 1+
\frac{\Lambda^2}{m_{\text{top}}^2} \right)\! - \frac{
3\Lambda^4+2m_{\text{top}}^2\Lambda^2}{(\Lambda^2+m_{\text{top}}^2)^2}
\!\right]\nonumber\\
&& + v^2 \lambda_\Lambda. \label{eq:MFmH}
\end{eqnarray}
This renders explicit that the lower bound for UV
potentials of the form of \Eqref{eq:conf} is given by
$m_{\text{H}}(\Lambda,\lambda_\Lambda=0)$.
The mean-field analysis performed here gives a first insight into how
lower bounds for the Higgs mass follow from the mapping from bare to
renormalized quantities. It also exemplifies that the mere existence
of a lower bound on the Higgs mass for bare potentials of
$\phi^4$-type is essentially a consequence of top fluctuations that
drive the curvature of the effective potential at its nontrivial
minimum to finite values. This statement will also hold on the
nonperturbative level. We plot the mean-field results for the Higgs
mass as a function of $\Lambda$ for various values of $\lambda_\Lambda$ in
Fig.~\ref{fig:mHiggsMF} as solid lines.
The plot also shows corrections from bosonic fluctuations as described by
extended mean-field theory $U^{\text{EMF}}(\rho)$ as dashed lines for the same
values of $\lambda_\Lambda$. We observe that scalar fluctuations tend to decrease the
Higgs mass values. This agrees with the fact that scalar fluctuations drive
the effective potential towards the symmetric regime, thus depleting also the
curvature near the minimum. However, the lower bound of the Higgs mass remains
unaffected by the scalar fluctuations, because the scalar field is
non-interacting for $\lambda_\Lambda=0$ in the EMF approximation.
\begin{figure}
{\centering
\includegraphics[width=8.5cm]{LowerHiggsBoundMF.eps}
}
\caption{Extended mean-field analysis of the lower bound for the Higgs
mass $m_{\text{H}}$ versus the UV cutoff $\Lambda$, based on a bare potential
$U_{\Lambda}$ of $\phi^4$-type for $N_{\mathrm{f}}=1$. For an initial potential which
is flat apart from a mass term $U_{\Lambda}=\frac{1}{2} m_\Lambda^2 \phi^2$, the
fermionic fluctuations drive the Higgs mass to a finite minimal
value. The solid lines correspond to standard mean-field theory
accounting only for top fluctuations, cf. \Eqref{eq:MFmH}, whereas
the dashed lines also include scalar fluctuations on the Gau\ss ian
level (extended mean-field). The four different line sets correspond
to increasing values of the initial $\phi^4$ coupling of
$\lambda_\Lambda=0,\frac{1}{6},\frac{1}{3},\frac{2}{3}$ from bottom to top.}
\label{fig:mHiggsMF}
\end{figure}
\subsection{Generalized bare potentials}
The lower Higgs mass bound determined above arises from the fact that
the values for the bare quartic coupling $\lambda_{\Lambda}$ are
bounded from below. This is necessary in order to start with a
well-defined theory in the UV for our confined bare potentials
\eqref{eq:conf} of $\phi^4$-type. Such a restriction on the bare
potential is typically also required in perturbation theory because
higher-order operators are perturbatively non-renormalizable. By
contrast, the Wetterich equation provides us with a nonperturbative
tool, so we can study also the influence of RG irrelevant higher-order
operators on the flow of the effective average
action. {Alternatively, this could also be studied
with perturbative methods in an effective-theory approach.}
In the following we address the question how modifications $\Delta
U_{\Lambda}(\rho)$ of the quartic bare potential can exert an influence on the
lower Higgs mass bound. The bare potential can in principle be an arbitrary
function of the scalar field. The only constraint which we impose is that the
potential is bounded from below in order to start from a well-defined quantum
field theory at the cutoff. We emphasize that no further experimental
constraints exist. The simplest extention of the standard potential has an
additional operator of the form $\phi^6$.
\begin{align}
U_{\Lambda}(\rho) &= m_{\Lambda}^2\rho+\frac{\lambda_{\Lambda}}{2}\rho^2 + \Delta U_{\Lambda}(\rho)\notag\\
&= m_{\Lambda}^2\rho + \frac{\lambda_{\Lambda}}{2}\rho^2 + \frac{\lambda_{3,\Lambda}}{6\Lambda^2}\rho^3.
\end{align}
Again, in the mean-field case \Eqref{eq:umft} can be solved explicitly for $m_{\Lambda}^2$,
yielding the Higgs mass as a function of $\lambda_{\Lambda}$ and $\lambda_{3,\Lambda}$ for a given
cutoff, $m_{\text{H}}=m_{\text{H}}(\lambda_{\Lambda},\lambda_{3,\Lambda})$. With $\lambda_{3,\Lambda}$ positive
we can study a wider range of values for the bare quartic coupling. The Higgs mass reads
\begin{widetext}
\begin{align}
m_{\text{H}}^2(\Lambda,\lambda_{\Lambda},\lambda_{3,\Lambda}) &= \frac{m_{\mathrm{top}}^4}{4\pi^2v^2} \left[ 2\ln{\left(1+\frac{\Lambda^2}{m_{\mathrm{top}}^2}\right)} - \frac{3\Lambda^4+2m_{\mathrm{top}}^2\Lambda^2}{(\Lambda^2+m_{\mathrm{top}}^2)^2} \right] + v^2\lambda_{\Lambda} + \frac{v^4}{2\Lambda^2}\lambda_{3,\Lambda}.
\label{eq:mHmfep}
\end{align}
\end{widetext}
Obviously, we are able to construct a theory with a Higgs mass below the
previous lower bound if the contribution of the term $\sim \lambda_\Lambda$ for $\lambda_\Lambda<0$
exceeds that of the positive term $\sim \lambda_{3,\Lambda}$.
The same mechanism works in the extended mean-field analysis but
there it requires a solution to the transcendental \Eqref{eq:umft} in order to
determine $m_{\Lambda}^2$. A numerical solution is plotted in
Fig.~\ref{fig:mHiggsMF2} for different values of $\lambda_{\Lambda}$
and $\lambda_{3,\Lambda}$. Furthermore, we have checked that for the
given masses no additional minimum appears in the effective potential
besides the one at $v=246\,$GeV.
\begin{figure}
{\centering
\includegraphics[width=8.5cm]{BelowLowerHiggsBoundMF.eps}
}
\caption{Extended mean-field analysis of the lower bound for the Higgs mass
$m_{\text{H}}$ versus the UV cutoff $\Lambda$, based on a bare potential $U_{\Lambda}$ of
$\phi^6$-type for $N_{\mathrm{f}}=1$. We have plotted the lower bound in the $\phi^4$
theory ($\lambda_\Lambda=\lambda_{3,\Lambda}=0$) as solid black line. Theories with bare
couplings $\lambda_\Lambda=-\frac{1}{30}$ and $\lambda_{3,\Lambda}=\frac{2}{3}$ are
depicted as red dashed line, and $\lambda_\Lambda=-\frac{1}{15}$ and
$\lambda_{3,\Lambda}=2$ as blue dotted line.}
\label{fig:mHiggsMF2}
\end{figure}
Let us finally remark that upper bounds cannot meaningfully be studied
in the mean-field approximation; this is because ``RG improvement'' is
necessary to observe the nonperturbative approach to triviality (reflected by
the Landau-pole behavior within RG-improved perturbation theory).
\section{Nonperturbative Higgs mass bounds}
\label{sec:flow}
The mean-field approximation has turned out to be remarkably accurate by
direct comparison with nonperturbative lattice simulations for the present
model \cite{Holland:2003jr,Holland:2004sd}. As lattice simulations are
typically limited as far as the separation of the UV scale from the physical
scales is concerned, a nonperturbative continuum analysis of beyond mean-field
theory seems indispensable in order to appropriately account for scalar
fluctuations and the mutual back-reactions between fermionic and scalar
fluctuations on a wide range of scales.
For the solution of the flow equations, we use the formulation in terms
of dimensionless renormalized quantities as introduced in
Sect.~\ref{sec:floweq}. To leading-order in the derivative expansion,
we solve the flow equations for the effective potential
$u_k$ and for the Yukawa coupling $h_k$. At next-to-leading order, we
include the wave-function renormalizations $\eta_\phi$ and
$\eta_\psi$.
Since we are mainly interested in the properties of the effective
potential near its minimum, we use a polynomial expansion of the
potential. The stability and convergence of this expansion will be
checked explicitly. In the symmetric regime (SYM) where the minimum of
the potential occurs at $\kappa_k=0$,
we use the truncated expansion
\begin{equation}
u_k=\sum_{n=1}^{N_{\text{p}}} \frac{\lambda_n}{n!} \, \tilde{\rho}^n, \label{eq:usym}
\end{equation}
such that the potential is parameterized by $N_{\text{p}}$ couplings
$\lambda_n$ (the mass term is related to $\lambda_1$ and we identify
the $\phi^4$ interaction as $\lambda\equiv\lambda_2$). In
the symmetry-broken regime (SSB), we instead use
\begin{equation}
u_k=\sum_{n=2}^{N_{\text{p}}} \frac{\lambda_n}{n!} \, (\tilde{\rho}-\kappa_k)^n. \label{eq:ussb}
\end{equation}
The flows of $\lambda_1, \dots, \lambda_{N_{\text{p}}}$ (SYM), or $\kappa_k,\lambda_2, \dots
\lambda_{N_{\text{p}}}$ (SSB), can directly be derived from
\Eqref{PotentialFlowEquation}.
For small bare scalar coupling $\lambda_\Lambda\equiv \lambda_{2,\Lambda}$, a physical
flow typically starts in the SYM regime. Near the electroweak scale, fermionic
fluctuations drive the system into the SSB regime at a scale $k_{\text{SSB}}$,
where we have to switch from the SYM flow to the SSB flow. Here, a nonzero
minimum builds up, inducing masses for the fermions and the Higgs scalar. This
leads to a decoupling of the modes, and the flow freezes out completely; i.e.,
all right-hand sides of the flow equations go to zero for $k\to 0$. For large
bare scalar coupling $\lambda_\Lambda$, the physical flow starts already in the SSB regime
with a small value for $\kappa_k$. The flow can still run over many scales
until $\kappa_k$ grows large near the electroweak scale, implying again the
decoupling of all modes.
\subsection{$\phi^4$-type bare potentials}
Let us again start with the restricted class of bare potentials of
$\phi^4$-type,
\begin{equation}
u_\Lambda=\lambda_{1,\Lambda} \tilde{\rho} +
\frac{\lambda_{\Lambda}}{2} \tilde{\rho}^2, \label{eq:conf2}
\end{equation}
where $\lambda_{1,\Lambda}\equivm_\Lambda^2/\Lambda^2$ for a wave function
renormalization $Z_{\phi,\Lambda}=1$. For a given cutoff $\Lambda$,
the flow equations map the bare parameters $m_\Lambda^2$, $\lambda_\Lambda$, $h_\Lambda$ onto
the physical parameters $v$, $m_{\text{top}}$, $m_{\text{H}}$. In practice, we tune
$m_\Lambda^2$ to establish the correct vacuum expectation value
$v\simeq246$GeV for a given cutoff $\Lambda$. This is, in fact, a
fine-tuning problem, corresponding to the problem of separating the
scale hierarchies in the standard model. At the same time, $h_\Lambda$ is
varied until the flow ends at the right value of $m_{\text{top}}$. This leaves
us with the Higgs mass as a function of $\lambda_\Lambda$ for a given cutoff
$\Lambda$, $m_{\text{H}}=m_{\text{H}}(\Lambda,\lambda_\Lambda)$, where $\lambda_\Lambda$ is allowed to be an a
priori arbitrary non-negative real number for the class of bare
potentials \eqref{eq:conf2}.
\begin{figure}
{
\centering\includegraphics[width=8.5cm]{mhiggsasfuncoflam.eps}
}
\caption{Higgs mass values versus the bare scalar coupling $\lambda_\Lambda$ for a
cutoff $\Lambda=10^7$GeV. The dashed lines denote the results within
LO derivative expansion; the NLO deviates from the LO result by at
most 10 \% for large coupling, demonstrating the satisfactory
convergence of the derivative expansion. Also the convergence of the
polynomial expansion is shown: green lines with squares are obtained
within the lowest nontrivial order with $N_{\text{p}}=2$, blue lines with
circles denote the $N_{\text{p}}=4$ result; even higher orders $N_P=6,8$ show no
further deviation from the $N_{\text{p}}=4$ curves. }
\label{fig:lambdamap}
\end{figure}
In Fig.~\ref{fig:lambdamap}, we depict this function $m_{\text{H}}(\lambda_\Lambda)$ for a
cutoff $\Lambda=10^7$GeV for various approximations. For $\lambda_\Lambda\lesssim
0.01$, the Higgs mass becomes rather independent of $\lambda_\Lambda$ approaching
its lower bound. {This observation is in perfect agreement with
lattice simulations
\cite{Holland:2003jr,Holland:2004sd,Gerhold:2007yb,Gerhold:2010wv,Bulava:2012pb}.}
For larger bare coupling $\lambda_\Lambda$, the Higgs mass increases and
approaches a regime of saturation for $\lambda_\Lambda\gg 1$. This is reminiscent
to RG-improved perturbation theory, where the bare coupling hits the
Landau pole $\lambda_\Lambda\to \infty$ already at a finite cutoff $\Lambda$.
Whereas the Landau pole in perturbation theory in the first place
signals the breakdown of the perturbative expansion, our truncation of
the RG flow does neither rely on perturbative ordering nor require a
weak coupling. Instead, our derivative expansion is organized in terms
of field operators with increasing number of derivatives. In order to
check the convergence of this expansion, we can compare the results
for the Higgs mass to leading order (LO) and next-to-leading order
(NLO) in this expansion. To leading order, we drop the running of the
kinetic terms in \Eqref{eq:trunc} by setting the anomalous dimensions
to zero, $\eta_{\psi,\phi}\to 0$. The resulting Higgs masses are
plotted as dashed lines in Fig.~\ref{fig:lambdamap}. We observe that
the difference to the NLO result (solid lines) is rather small for the
lower Higgs mass bound for $\lambda_\Lambda\to 0$; even for the largest accessible
couplings, we observe a maximum deviation of 10\%, confirming that the
derivative expansion constitutes a satisfactory approximation for our
purpose for the whole range from weak to strong coupling.
\begin{figure}
{
\centering\includegraphics[width=8.5cm]{BoundsWithinPhi4.eps}
}
\caption{Higgs mass bounds versus cutoff $\Lambda$. The thick green/solid
line denotes the lower bound for the Higgs mass derived within the
class of bare $\phi^4$ potentials. The thin red/solid line shows the
lower bound as derived within mean-field approximation. The
turquois/dashed lines mark upper bounds if the bare scalar coupling
is allowed to start maximally from $\lambda_\Lambda=1,10,50,100$ from bottom to top,
respectively. An artificial restriction to the perturbative domain
$\lambda_\Lambda\lesssim 1$ underestimates the upper bound by a factor
$\gtrsim\mathcal{O}(1)$. }
\label{fig:boundsphi4}
\end{figure}
Furthermore, we study the convergence of the polynomial expansion of
the scalar potential in Fig.~\ref{fig:lambdamap}. To lowest nontrivial
order $N_{\text{p}}=2$ (green lines with squares), we obtain already a complete
picture of the physics of Higgs mass bounds. For the next order
$N_{\text{p}}=4$ (blue lines with circles), though the upper Higgs mass bound
is already approached for smaller bare couplings $\lambda_\Lambda$, the value of
the upper bound changes by at most 5\%. For even higher
orders, the corresponding results lie on top of the $N_{\text{p}}=4$
curves. Within our numerical accuracy we find no significant
difference for $N_{\text{p}}=4,6,8$.
In Fig.~\ref{fig:boundsphi4}, we show the resulting Higgs mass bounds, arising
within the class of $\phi^4$ bare potentials. The thick solid/green line
characterizes the lower bound resulting from the RG flow for a wide range of
cutoffs $\Lambda=10^4 \dots 10^{8}$GeV. Also shown is the lower bound as
derived within the mean-field approximation in the previous section (thin
solid/red line), which
neglects the running of the Yukawa coupling, of the anomalous dimension, and
RG improvement of the scalar potential. In the full flow, we observe
nontrivial cancelations among these terms, such that the mean-field result
represents a surprisingly good approximation over a wide range of cutoff
scales. The turquois/dashed lines depict upper bounds for the Higgs mass for
bare couplings $\lambda_\Lambda=1,10,50,100$, respectively. In particular, we find that if
we limited ourselves to a perturbative domain, choosing $\lambda_\Lambda=1$, we would
artificially underestimate the upper bound by a factor
$\gtrsim\mathcal{O}(1)$.
\subsection{Generalized bare potentials}
Let us now study extensions of the initial bare potential beyond the
$\phi^4$-type. Motivated by the results of the mean-field
approximation, we concentrate on potentials with a negative
$\lambda_{2,\Lambda}$ where the UV stability is guaranteed by a
positive $\lambda_3\phi^6$. It is possible to construct bare
potentials which give rise to Higgs masses below the lower bound
within the class of $\phi^4$ bare potentials, similar to the
mean-field approach. Fig.~\ref{fig:negphi4} shows the lower bound
within $\phi^4$ theory (black solid line) in comparison to Higgs mass
values for an example flow which starts with
$\lambda_{2,\Lambda}=-0.1$ and $\lambda_{3,\Lambda}=3$ in the UV (red
solid line). This example clearly illustrates that the lower bound
within $\phi^4$-like initial potentials does no longer hold, if higher
dimensional operators are also permitted.
This phenomenon can be understood from the RG flow itself: first we
note that in both cases ($\phi^4$-like as well as the beyond-$\phi^4$
example above) the flow starts in the symmetric regime. In the
beyond-$\phi^4$ example, the quartic coupling $\lambda_2$ runs quickly
to positive values, whereas $\lambda_3$ becomes very small as is
expected in the vicinity of the Gau\ss ian fixed point. As a
consequence, this particular system flows back into the class of
$\phi^4$-type potentials. The decisive difference, however, is that
the scale $k_{\text{GFP}}$ where the system is again near the
Gau\ss{}ian fixed point is now lower than the initial UV scale
$\Lambda$. Loosely speaking, some ``RG time'' is required to run from
the beyond-$\phi^4$ form of the potential back to the $\phi^4$
Gau\ss{}ian type.
From another viewpoint, the RG flow can map an initial bare action with
$\lambda_2<0$ and $\lambda_3>0$ at an initial UV scale $\Lambda$ to a theory
with $\lambda_2\geq 0$ and $\lambda_3\approx 0$ at a smaller scale
$k_{\text{GFP}}<\Lambda$. Therefore, the red curve (beyond-$\phi^4$) in
Fig.~\ref{fig:negphi4} can also be viewed as a horizontally displaced version
of the black curve ($\phi^4$-like) to effectively larger cutoff values. We
emphasize that the present example has neither been specifically designed or
fine-tuned, nor does it represent an exhaustive study of admissible initial
potentials. A wide range of beyond-$\phi^4$ potentials initiating the flow at
$\Lambda$ leads to Higgs masses below the bound of the $\phi^4$-type
class. Still, the mechanism observed above starting from stable potentials with
$\lambda_2<0$ and globally stabilizing higher-order terms appears rather
generic. We have also checked for more involved initial conditions that
the results for the Higgs masses do not change for higher-order $N_P\geq 4$
polynomial expansions of the scalar potential.
In fact, the influence of higher dimensional operators has also been studied
in recent lattice simulations in a chiral Higgs-Yukawa model
\cite{Bulava:2012pb}, by adding a positive $\lambda_3 \phi^6$ term to the bare
potential. No lowering of the Higgs mass bound has been observed in this
study. This is indeed in agreement with our observations, because merely
adding this term has barely any effect on the Higgs mass bound and rather
leads to an increase of the Higgs mass. Our mechanism for lowering the mass
bound works particularly well for initial potentials with $\lambda_2< 0$. In
other words, the $\lambda_2< 0$ deformation requires a comparatively long RG
time to run the potential back to the $\phi^4$ Gau\ss{}ian type. A lattice
study with such (or even more general) bare potentials could hence put our
mechanism to test.
\begin{figure}
{
\centering\includegraphics[width=8.5cm]{BelowLowerHiggsBound2.eps}
}
\caption{Higgs mass versus cutoff $\Lambda$. The black
line denotes the lower bound for the Higgs mass derived within the
class of bare $\phi^4$ potentials. The red line shows how we can construct
Higgs masses below the lower bound by giving up the restriction to quartic
bare potentials. The masses are derived for $\lambda_2=-0.1$ and $\lambda_3=3$.}
\label{fig:negphi4}
\end{figure}
Having put the significance of the lower bound of the Higgs mass derived for
$\phi^4$-type bare potentials into perspective, let us address the issue of
stability: while the standard approach to vacuum stability in the present
simple model based on RG-improved perturbation theory has been questioned by
lattice simulations \cite{Holland:2003jr,Holland:2004sd} and functional RG
methods \cite{Branchina:2005tu} (in turn critically assessed by
\cite{Einhorn:2007rv}), a full stability analysis would require to follow the
RG flow of arbitrary physically admissible initial potentials. In particular,
the RG evolution of potentials with multiple local minima would have to be
dealt with quantitatively. While this is indeed possible with appropriate
numerical solvers
\cite{Gneiting:2005,Hofling:2002hj,Adams:1995cv,Bohr:2000gp,Fischbacher:2012ib},
we here confine ourselves to the validity region of the polynomial expansion
of the effective potential about a local minimum.
Since high-order polynomials typically have multiple local minima, we have to
estimate the radius of convergence of our expansion in field space. A new
local minimum showing up within this convergence region could then be
interpreted as a signature of instability. If such minima only occur outside
the convergence radius, we consider them as an artifact of the polynomial
expansion.
A rough estimate for the radius of convergence is given by comparing
the quotients of successive couplings $\lambda_n/\lambda_{n+1}$ for
large $n$ in the infrared. In practice we solve the system of coupled
differential equations for $N_P=20$, switching back to dimensionful
quantities at a scale where the flows are frozen out, e.g. $U_k=u_k
k^4=\sum_n a_n({Z_\phi}\rho- v^2/2)^n$ with
$a_n=\frac{\lambda_n}{n!}k^{4-2n}$, and computing the dimensionful
radius of convergence by comparing $\frac{a_n(k)}{a_{n+1}(k)}$ for
$k\rightarrow 0$. The results expressed in units of $10^3$GeV for
various initial conditions
are plotted in Fig. \ref{fig:ratioofcouplings}.
Our primary observation is that this estimate for the radius of
convergence appears to stabilize at a universal value rather
independent of the chosen initial conditions. The resulting value near
$\simeq 23 000$GeV$^2$ is of the order of the vacuum expectation value
$v^2/2=30258\,$GeV$^2$ for large $n$. We still observe a slight drift
in our data even at high order, which might be due to the fact that
the inner region of the effective potential owing to its convexity
cannot be resolved within a polynomial expansion as a matter of
principle. Restricting the field amplitudes to values of the order of
the ratio of the highest couplings in the truncation,
$Z_\phi\rho_{\text{max}}\simeq (v^2/2) + |\frac{a_{N_P-1}}{a_{N_P}}|$,
we find in all studied cases that the effective potential is a convex
monotonically rising function in the outer region ($\phi > v$). No
evidence for an instability within this radius of convergence is
found.
These observations agree with solutions of the RG flow for the full
effective potential beyond the polynomial expansion as worked out in
\cite{Gneiting:2005} using pseudo-spectral methods (Chebyshev
expansion). Both methods lead to equivalent results for both, the
Higgs mass bounds for $\phi^4$-type initial potentials as well as the
absence of any indication for an instability.
\begin{figure}[t]
{
\centering\includegraphics[width=8.5cm]{ratioofcouplings3.eps}
}
\caption{Estimate for the radius of convergence in units of $10^3\text{GeV}^2$
of the polynomial expansion of the effective potentials in terms of the
absolute values of the ratios of expansion coefficients. The red filled
circles are derived for a theory which starts at $\Lambda=10^7\,$GeV with
all couplings set to zero apart from the mass term. The black empty circles
are for the case $\Lambda=10^7\,$GeV and $\lambda_{2}=1$ and $\lambda_{n}=0$
($n\geq 3$).}
\label{fig:ratioofcouplings}
\end{figure}
\section{Conclusions}
We have determined Higgs mass bounds in a simple Higgs-Yukawa toy model
sharing some similarities with the standard model Higgs--top-quark sector. Our
study is based on the functional renormalization group which can keep track of
threshold phenomena, has better access to strong coupling regimes and
automatically accounts for ``RG improvement''.
In agreement with the standard literature, the existence of an upper Higgs
mass bound is a consequence of triviality of the scalar sector. As such, it is
inherently non-universal. In this work, we have also emphasized the
non-universality of the lower Higgs mass bound. In addition to the
regularization scheme dependence which the lower bound shares with the upper
bound in any UV incomplete theory, we have discovered that the lower mass
bound can depend sensitively on the microscopic details of the bare effective
potential for the Higgs field.
This observation does not contradict Wilsonian renormalizability arguments
stating that IR observables should be independent of the details of the UV
theory. The reason is that a Higgs mass bound given in the form
$m_{\text{H,bound}}=m_{\text{H,bound}}(\Lambda)$ as a function of the UV
cutoff $\Lambda$ does not constitute a pure IR observable. By contrast, it
should be understood as a mapping of initial conditions at the microscopic UV
scale onto the set of possible IR observables. As the UV initial conditions are
typically not accessible by low energy measurements, they are unconstraint. A
statement about Higgs mass bounds therefore often goes along with (typically
only implicit) constraints on the UV initial conditions, i.e., bare actions or
bare potentials.
In the conventional discussions of Higgs mass bounds, the IR measured
observables are taken from experiment and the RG flow is run to higher
scales. This procedure lacks any control over RG irrelevant operators, as
their influence on the IR observables is exponentially small. Therefore, their
high-energy behavior is simply ignored or implicitly fixed by computational
recipes such as RG-improved perturbation theory. Latest results along this
line of reasoning show that the measured mass of the Higgs boson is close to
the ``vacuum stability'' bound or even in the ``metastable region'' (with the
biggest uncertainty arising from the exact value of the top mass, to be
specified in an appropriate scheme)
\cite{Degrassi:2012ry,Alekhin:2012py,Masina:2012tz,Buttazzo:2013uya}. From
this viewpoint, the fact that the Higgs mass together with the whole standard
model is close to a phase transition is a remarkable result of the LHC,
requiring an explanation of this ``near-criticality'' property
\cite{Buttazzo:2013uya}. Since this running-up of the perturbative RG cannot
access the large field regime, where a new vacuum is expected to occur, a full
resolution of this near-criticality puzzle either requires nonperturbative
complements or even calls for beyond-standard-model explanations.
Our results offer a different viewpoint: as we have hardly any information
about the bare action at an initial scale $\Lambda$, bounds on particle masses
can only arise from the mapping of {\it all admissible} bare initial conditions
onto the IR observables as is provided by the RG. Of course, the resulting
bounds will depend on the criteria of {\it admissibility} which we may
impose. In this work, we have demonstrated that strict Higgs mass bounds arise
if we restrict the initial conditions to $\phi^4$-type potentials. We
emphasize, however, that this restriction is somewhat arbitrary: it cannot be
justified by Wilsonian renormalizability arguments, as they simply do not
apply to bare actions. Hence, if we lift this artificial restriction, we can
easily discover initial conditions that lead to Higgs masses substantially
smaller than the Higgs mass bound within the $\phi^4$ class. This is already
the case for initial potentials with comparatively small higher-order
operators. Nonperturbatively large deformations of the initial potential are
not required.
From this viewpoint, the near-criticality property of the standard model
remains nevertheless remarkable, as it may provide for a first handle on the
microscopic action at some high (GUT-like or Planck) scale that has to emerge
from an underlying theory (``a UV completion''). The top-down analog of this
reasoning has been used in a model with asymptotically safe gravity that
predicted the value of the Higgs mass \cite{Shaposhnikov:2009pv} (see also
\cite{Bezrukov:2012sa}), based on the fact that asymptotically safe gravity
interactions are likely to put the Higgs mass onto its ``conventional'' lower
bound. Already earlier, arguments for putting the standard model onto this
conventional lower bound lead to similar predictions \cite{Froggatt:1995rt}.
By contrast, if the Higgs mass turns out to lie below this conventional lower
bound, this may not be a sufficient reason for concern regarding vacuum
stability or metastability. Stability might simply be provided by higher-order
operators in the initial bare action. Rather generically, we find that models
with a negative $\lambda_{2,\Lambda}$ being stabilized by higher-order
operators yield Higgs mass values below the conventional lower bound. Of
course, the presence and magnitude of these higher order operators eventually
has to be explained by a (more) UV complete underlying theory. In fact,
models with a negative $\lambda_{2,\Lambda}$ have recently been discussed from
a string-theory perspective \cite{Hebecker:2013lha}. A UV complete example for
models with a potentially smaller Higgs mass has recently been given within
pure quantum field theory in the context of an asymptotically safe gauged
Higgs-Yukawa model \cite{Gies:2013pma}.
\section*{Acknowledgments}
We thank Tobias Hellwig, Karl Jansen, Stefan Lippoldt, Axel
Maas, Jan Pawlowski and Luca Zambelli for interesting and enlightening
discussions. HG and RS acknowledge support by the DFG under grants
GRK1523, Gi 328/5-2 (Heisenberg program).
|
\section*{Author Summary}
\section{Introduction\label{intro}}
During the last decade there has been an increasing evidence of presence of quantum structures in
processes that find their origin in human behaviour and cognition,
more specifically, in situations of decision making and in the structure of language (see, e.g., \cite{aa1995,aabg2000,ac2004,vr2004,ag2005a,ag2005b,a2009a,a2009b,pb2009,k2010,as2011,Aerts2011b,Aerts2012,bpft2011,bb2012,abgs2013,ags2013,hk2013,pb2013,wbap2013,as2014,ast2014,s2014a,pnas,plosone,s2014b,as2014b}).
The success of this quantum modeling is interpreted as
due to `descriptive effectiveness of the mathematical apparatus of quantum theory as formal instrument to model cognitive dynamics and structures in situations where classical set-based approaches are problematical', without
a priori a direct or precise connection with the validity of quantum laws in the microscopic world,
although also recently a reflection connecting the quantum modeling in the micro-world with these new quantum cognition approaches has been put forward \cite{as2014b}. In particular, the mathematics of quantum theory in Hilbert space has proved very successful in modeling conceptual combinations,
i.e. conjunctions and disjunctions of two concepts (see, e.g.,
\cite{ag2005a,ag2005b,a2009a,a2009b,as2011,abgs2013,ags2013,as2014,s2014a,s2014b}).
The `combination problem', that is, the problem of how the combination of two or more natural concepts can be represented starting from the representation of the component concepts, has been studied experimentally and within classical concept theories in great detail in the last thirty years. More specifically:
(i) The `Guppy effect' in concept conjunction, also known as the `Pet-Fish problem' \cite{os1981,os1982}. If one measures the typicality of specific exemplars with respect to the concepts {\it Pet} and {\it Fish} and their conjunction {\it Pet-Fish}, then one experimentally finds that an exemplar such as {\it Guppy} is a very typical example of {\it Pet-Fish}, while it is neither a very typical example of {\it Pet} nor of {\it Fish}.
(ii) The deviation from classical (fuzzy) set-theoretic membership weights of exemplars with respect to pairs of concepts and their conjunction or disjunction \cite{h1988a,h1988b}. If one measures the membership weight of an exemplar with respect to a pair of concepts and their conjunction (disjunction), then one experimentally finds that the membership weight of the exemplar with respect to the conjunction (disjunction) is greater (less) than the membership weight of the exemplar with respect to at least one of the component concepts.
(iii) The existence of `borderline contradictions' in sentences expressing vague properties \cite{bovw1999,ap2011}. Roughly speaking, a borderline contradiction is a sentence of the form ${\mathscr P}(x) \land \lnot {\mathscr P}(x)$, for a vague predicate ${\mathscr P}$ and a borderline case $x$, e.g., the sentence ``John is tall and John is not tall''.
What one typically finds in the above situations is a failure of set-theoretic approaches (classical set, fuzzy set, Kolmogorovian probability) to supply satisfactory theoretic models for the experimentally observed patterns. Indeed, all traditional approaches to concept theory (mainly, `prototype theory' \cite{r1973,r1978,r1983}, `exemplar theory' \cite{n1988,n1992} and `theory theory' \cite{mm1985,rn1988}) and concept representation (mainly, `extensional' membership-based \cite{z1982,r1995} and `intensional' attribute-based \cite{h1988b,m1975,h1997})
have structural difficulties to cope with the experimental data exactly where
the `graded', or `vague' nature of these data violates (fuzzy) set theoretic structures abundantly
\cite{os1982,z1982,z1965}, indicating that this violation of set theoretic structures is the core of the problem.
More specifically, none of the traditional approaches,
when representing the membership weights and typicalities expressing such gradeness in a classical (fuzzy) set-theoretic model, where conceptual conjunctions are represented by logical conjunctions and conceptual disjunctions are represented by logical disjunctions,
puts forward even moderately successful models with respect to the experimental data.
This situation is experienced as one of the major problems in the domain of traditional concept theories and an obstacle for progress
\cite{r1995,h1997,k1992,f1994,kp1995,os1997}.
Important results in concept research and modeling have been obtained in the last decade within the approach of `quantum cognition' in which our research group has contributed substantially. The fundamentals of our approach can be resumed in the following progressive steps.
(a) The structural aspects of the approach rest on the results of older
research on the foundations of quantum theory \cite{a1999}, the origins of quantum probability \cite{a1986,p1989} and the identification of typically quantum aspects outside the microscopic domain of quantum physics \cite{aa1995,aabg2000}.
(b) A first major step was taken in considering a concept as an `entity in a specific state changing under the influence of a context', rather than as a `container of instantiations', and this led to the development of a {\it `State Context Property' formalism} ({\it SCoP}), and allowed the authors to provide a quantum representation of the guppy effect \cite{ag2005a,ag2005b}.
(c) Continuing in this approach to concepts representation, the mathematical formalism of quantum theory was employed to model the overextension and underextension of membership weights measured in \cite{h1988a,h1988b}. More specifically, the overextension for conjunctions of concepts measured in \cite{h1988a} was described as an effect of quantum interference,
due to quantum superposition \cite{a2009a,ags2013}, which also plays a primary role in the description of both overextension and underextension for disjunctions of concepts \cite{h1988b}.
(d) A two-sector Fock space structure enabled a complete representation of data on conjunctions and disjunctions of two concepts
\cite{a2009a,ags2013}.
(e) Specific conceptual combinations experimentally revealed the presence of further genuine quantum effects, namely, entanglement \cite{a2009b,as2011,abgs2013,ags2013,as2014} and quantum-type indistinguishability \cite{IQSA1}.
(f) This quantum-theoretic framework was successfully applied to describe borderline vagueness \cite{s2014a} and the effects of negation on conceptual conjunction \cite{s2014b}.
(g) Other phenomena related to concept combination, such as `Ellsberg and Machina decision making paradoxes \cite{e1961,m2009} were sucessfully modeled in the same quantum-conceptual framework \cite{Aerts2011b,Aerts2012}.
In the present paper we extend the collection of data in \cite{s2014b} with the aim of further exploring the use of negation in conceptual combinations and, more generally, the underlying logical structures being at work in human thought in
the course of cognitive processes \cite{IQSA2}. Let us first put forward a specific comment with respect to the `negation of a concept'. From the perspective of prototype theory, for quite some concepts the negation of a concept can be considered as a `singular concept', since it does not have a well defined prototype. In fact, while it is plain to determine the
non-membership of, e.g., {\it Fruit}, this does not seem to lead to the determination involving a similarity with some prototype of {\it Not Fruit}. Some authors maintain, for this reason, that single negated concepts have little meaning and that conceptual negations can be evaluated only in conjunctions of the form {\it Fruits Which Are Not Vegetables} \cite{h1997}. We agree that there is an asymmetry between the way subjects estimate the membership of an exemplar, e.g., {\it Apple}, with respect to a positive concept, e.g., {\it Fruits}, and the way subjects estimate the membership of the same exemplar with respect to its negative counterpart, e.g., {\it Not Fruits}. Notwithstanding this, we believe it is meaningful to explicitly introduce the concept {\it Not Fruits} in our research. First of all, because we do not confine our concept modeling to prototype theory, on the contrary, our approach is more general, the basic structure of prototype theory can be recovered if we limit the concepts to be in their ground states \cite{ag2005a,ag2005b}. Secondly, we will see that the quantum modeling elaborated in the present paper sheds light exactly on this problem, namely, the `negated concept' only appears as a full concept in `one half of the representation', while
is treated as `non-membership with respect to the positive concept' in the other half. Hence, quantum cognition, in the form of the model we work out copes with this problem in a natural way.
There has been very little research on how human beings interpret and combine negated concepts. In a seminal
|
study, Hampton \cite{h1997} considered in a set of experiments both conjunctions of the form {\it Games Which Are Also Sports} and conjunctions of the form {\it Games Which Are Not Sports}.
His work confirmed overextension in both types of conjunctions, also showing a violation of Boolean classical logical rules for the negation, which has recenly been confirmed by ourselves \cite{s2014b}. These results were the starting point for our research in this paper, whose content can be summarized as follows.
In Section \ref{experiment} we illustrate design (Section \ref{participants}) and procedure (Section \ref{procedure}) of the four cognitive experiments we performed. In the first experiment $e_{AB}$, we tested the membership weights of four different sets of exemplars with respect to four pairs $(A,B)$ of concepts and their conjunction `$A \ {\rm and} \ B$'. In the second experiment $e_{AB'}$, we tested the membership weights of the same four sets of exemplars with respect to the same four pairs $(A,B)$ of concepts, but negating the second concept, hence actually considering $A$, $B'$ and the conjunction `$A \ {\rm and} B'$'. In the third experiment $e_{A'B}$, we tested the membership weights obtained considering $A'$, $B$ and the conjunction `$A' \ {\rm and} \ B$'. Finally, in the fourth experiment $e_{A'B'}$, we considered the membership weights obtained by negating both concepts, hence actually considering $A'$, $B'$ and the conjunction `$A'$ and $B'$'.
We investigate the representability of the collected data, reported in Appendix \ref{tables}, in a `unique classical Kolmogorovian probability space' \cite{k1933}. Basic notions and results on probability measures and classical modeling are briefly reviewed in Appendix \ref{classical}. We prove theorems providing necessary and sufficient conditions for the modeling of the conceptual conjunctions `$A$ and $B$', `$A$ and $B'$', `$A'$ and $B$', and `$A'$ and $B'$' in such a single classical Kolmogorovian framework in Section \ref{experimentalresults}. Then, we observe in Section \ref{res-experiments} that the data in Appendix \ref{tables} significantly violate our theorems.
One usually identifies four `types of deviations from classicality' when conceptual conjunctions and negations are considered together as specified in our experiments, namely, (1) `conjunction minimum rule deviation', or `overextension', (2) `Kolmogorovian conjunction factor deviation', (3) `double overextension', (4) `conceptual negation deviation'. Our experimental data exhibit deviations from classicality of type (1), (3) and (4), while (2) is not present. Moreover, type-(4) deviations are generally weaker than deviations of type-(1). However, our general analysis of classicality for the presence of conjunction and negation together (Section \ref{experimentalresults}), leads to five classicality conditions to be satisfied of the data to fit into one classical probability setting together. When we analyse the deviations of our data with respect to these five conditions we find a very strong systematic deviation, which is very stable, i.e. gives rise to the same numerical values for the deviation even over the different pairs of concepts that we have experimented on. Analysing this regular pattern of violation, we show how it constitutes a very strong evidence for the
presence and dominance of what we have called `conceptual emergence' in our earlier work on the Fock space model for the combination of concepts \cite{a2009a,ags2013,abgs2013,s2014a,s2014b}. That the violation is numerically the same independent of the considered pair of concepts indicates that we have identified a non-classical mechanism in human thought which is linked to the depth of concept formation itself, independent of the specific meaning for a specific pair of concepts and a specific set of considered exemplars. This was for ourselves the most surprising and unexpected result of the investigation that we present in the present article, and we also consider it as one of the most important results of this investigation. In our opinion it inescapably proves that, whenever uncertainty is involved, human thought does not follow the rules of classical probability, and that this deviation of classical probability is strong and takes place on a deep structural conceptual level. Let us stress in this respect that this overwhelming systematics of deviation of classicality was `not' able to be seen or detected in the foregoing studies of conjunctions of pairs of concepts, since a focus on `overextension' or `Kolmogorovian conjunction factor deviation', which was the focus in all earlier investigations \cite{h1988a,a2009a,ags2013,abgs2013,s2014a,s2014b}, could structurally `not' reveal the systematic deviation we find here now, by lack of symmetry. It was necessary to experiment on conjunction and negation together and derive the five classicality conditions containing the necessary symmetry, to be able to identify this strong and stable pattern of deviation of classicality. The second most important finding of the research we present in this article is related to the same identification of the `deviation of classicality pattern' over all the considered pairs of concepts and their negations. Indeed, like we will show in Section \ref{experimentalresults}, not only do we find a strong and stable numerical deviation independent of the considered concepts and exemplars, additionally, the numerical size of the deviation is `almost' equal to the deviation of our five classicality conditions `if we would substitute the theoretical values for an average quantum model by means of our Fock space model of the situation'. Hence, as a second, equally unexpected and for ourselves surprising result, the data indicate in a very strong way that the deviation is exactly the one that would theoretically be found in case the situation is modelled quantum mechanically. Hence, we believe our finding to be a strong support for the validity of our quantum Fock space model, where this conceptual emergence is described in the first sector of this Fock space.
In Section \ref{brussels} we elaborate this quantum-theoretic modeling in Fock space for conceptual negations and conjunctions which naturally extends the modeling in \cite{a2009a} and follows the general lines traced in \cite{s2014a,s2014b}. It is however important to remark that the simultaneous modeling of conjunction and negation requires the introduction of two
new conceptual steps which were not needed for the modeling of conjunction pairs: (i) the introduction of entangled states in the second sector, which enables formalizing the situation where probabilities in second sector of Fock space can be formed by a `product procedure', even if they are not independent -- this is an aspect of the Fock space model we had not understood in our earlier modeling, hence we could consider it a further new surprising finding of the investigation presented in this paper, i.e. that we can model dependent probabilities keeping the procedure for the conjunction of concepts to be a product procedure in the second sector, and it are the natural possibility for the conjunction to be in an entangled state that allows us to do so; (ii) the handling of `negation' in the second sector by `logical inversion', similarly like we handled conjunction in the second sector by `product', more concretely, an experiment with `negation' with respect to a concept is treated by `negating logically' an experiment on the concept itself. This is also the way in which the Fock space model naturally copes with the general non-prototypicality of a negated concept.
When elaborating our Fock space model for the four pairs, conjunction and negation, we proceed as follows. We firstly inquire into the representability of the data in only one sector of Fock space (individual Hilbert space) in Section \ref{onesector}. Secondly, we allow the possibility of representing the data in a larger two-sector Fock space in Section \ref{twosector}. In Section \ref{solconditions} we discuss the conditions that should be satisfied by a collection of experimental data to be modeled in this two-sector Fock space. In Section \ref{entanglement} we show that all the conditions in Section \ref{experiment} for a classical representation hold in second sector of Fock space (tensor product Hilbert space) if a suitable entangled state is chosen in this sector. This entails that typical logical rules are satisfied, though in a probabilistic setting or, equivalently, a specific quantum logical structure can be detected, in this sector.
We see in Section \ref{model} that a large amount of data fall outside these classicality conditions and illustrate some specific cases that are classically problematical and that can instead be faithfully represented in our quantum-mechanical framework. The detailed representation is provided in the Supplementary Material attached to the present paper. Finally, in Section \ref{results} we comment on our results, by extensively discussing novelties and confirmations of our quantum-theoretic approach, and we clarify in Section \ref{quantumlogic} how `standard formal quantum logic' may appear in second sector of Fock space, and we explore its connections with classical probabilistic logic in the same sector.
\section{Description of experiments and classicality analysis\label{experiment}}
James Hampton identified in his cognitive tests systematic deviations from classical set (fuzzy set) predictions of conjunctions and disjunctions of two concepts, and named these deviations `overextensions' and `underextensions' \cite{h1988a,h1988b}. Cases of `double overextension' were also observed. More explicitly, if the membership weight of an exemplar $x$ with respect to the conjunction `$A \ {\rm and} \ B$' of two concepts $A$ and $B$ is higher than the membership weight of $
|
.0,\,0.4,\,0.2$ are shown. For reference, the standard CDM
power spectrum \cite{MA} is also displayed. In the three descending
panels, the individual spectra are shown with the Peacock and Dodds
\cite{PeacockDodds} (PD) reconstruction of the linear power spectrum.
For $\Omega < 1$ the reconstructed spectrum has been scaled as $\propto
\Omega^{-0.3}$ (see equation 41 of \cite{PeacockDodds}) for
comparison.
We first consider structure formation by strings with HDM. Based on the
normalization of $\mu$ obtained by \cite{AllenEtAl}, we see from Fig.
\ref{figure3} that the power spectrum approximately fits the shape of
the PD spectrum on large scales. As a gauge of the string+HDM model
for low-$\Omega$, we have computed the variance of the excess mass
fluctuation in a ball of radius $R = 8 h^{-1}$ Mpc,
\begin{equation}
\sigma_8^2 = \int |w(k R)|^2 4 \pi k^2 P(k) dk, \qquad
w(x)=3(\sin{x} - x\cos{x})/x^3
\end{equation}
which is observed to be around unity
\cite{PeacockDodds,DavisPeebles,WhiteEtAl}.
An excellent fit to our results is given by the empirical formula
\begin{eqnarray}
&& \sigma_8(\Omega, h) = 0.25(\pm{0.1}) \, \times
\Big( \mu_6(\Omega)\,
{g(\Omega) \over \Omega } \,
{\Gamma(1 + 2.6 \Gamma - 1.6 \Gamma^2)
\over 1 + (10 \Gamma)^{-2}} \Big)\cr\cr
&& {\rm with} \qquad g(\Omega) \equiv {5 \over 2} \Omega/
\Big[ 1 + {1 \over 2} \Omega + \Omega^{4/7}\Big]
\label{signorm0}
\end{eqnarray}
which is valid to within $\sim 10\%$ for $0.1 \le \Omega \le 1$ and $0.4 < h <
0.8$. The error bars on $\sigma_8$ are estimated based on the quoted
uncertainties in the string parameters
\cite{BennettBouchet,BennettBouchetProc,AllenShellard,AllenShellardProc} and the
uncertainty in the CMB normalization of $\mu$ \cite{AllenEtAl} included in
$G\mu(\Omega)$, which we repeat are probably too small for low-$\Omega$.
Evaluating (\ref{signorm0}) for various values of the cosmological parameters,
we predict $\sigma_8(1.0,0.5) = 0.25\pm{0.1}$ and $\sigma_8(0.2,0.5) =
0.05\pm{0.02}$. For $\Omega = 1$ the string+HDM scenario requires a modest boost
or bias in the power in order to achieve $\sigma_8 \sim 0.57 - 0.75$
\cite{PeacockDodds,WhiteEtAl}. These results are in agreement with past work by
Colombi \cite{ColombiThesis}, based on the Bennett-Bouchet simulations. We
pause to note that the non-linear dynamics of wakes and filaments
\cite{AlbrechtStebbins,VVach,TVach,AguirreBrand,AvelinoShellard,MoessnerBrand,ZLBrand,SornBrand,VollickA,VollickB,MahonenHYMMA,MahonenHYMMB,Mahonen}
may produce such a bias sufficient to reproduce the observed clustering of
objects on large scales. However, in an open universe the peak amplitude of
$k^3 P(k)$ drops and shifts to larger scales, so that some sort of
scale-dependent boost would be required to produce more power for $k \gtrsim 1\,
\Omega h^2/$Mpc. Hence, string+HDM in an open universe does not appear to be a
viable model for structure formation.
Structure formation by strings with CDM in a flat, $\Omega=1$ universe, when
normalized on large scales, suffers from producing too much power on small
scales. As pointed out by \cite{Spergel,PenSpergel,Ferreira} this problem may
be overcome, as for standard CDM, in a low density, $\Omega < 1$ universe.
Examining Fig. \ref{figure3}, we see that the string+CDM power spectrum ``bends
over'' on small scales as we lower $\Omega$. Hence, for $\Gamma \equiv \Omega
h \sim 0.1 - 0.2$, the spectrum approximately fits the shape of the PD
reconstruction. The variance of the mass fluctuation is given by the empirical
formula
\begin{equation}
\sigma_8(\Omega, h) = 0.9(\pm 0.5) \, \times
\Big( \mu_6(\Omega) \,{g(\Omega) \over \Omega} \,
{\Gamma (1 - 0.36 \Gamma)
\over 1 + (50 \Gamma)^{-2}} \Big)
\label{signorm1}
\end{equation}
which is valid to within $\sim 10\%$ for $0.1 \le \Omega \le 1$ and $0.4 < h <
0.8$. Evaluating (\ref{signorm1}), we predict $\sigma_8(0.4,0.7) = 0.4\pm{0.2}$
and $\sigma_8(0.2,0.5) = 0.2\pm{0.1}$. Hence, for $\Gamma \sim 0.1 - 0.2$, the
range of values of the mass fluctuation excess falls well below the estimate of
$\sigma_8 = 0.6{}^{+32\%}_{-24\%}
\exp[({-0.36-0.31\Omega+0.28\Omega^2})\log\Omega]$
\cite{VianaLiddle} by a factor of $\sim 2 - 4$. Within the uncertainties quoted
in (\ref{signorm1}), a bias as low as $b \sim 1.5$ may be tolerated. Recent work
by Sornborger {\it et al} \cite{SornBrand} on the structure of cosmic string
wakes has shown that the ratio of the baryon to CDM density in wakes is
enhanced. For a single wake formed near radiation-matter equality, the baryon
enhancement at late times is $\sim 2.4$ in a region of thickness $\sim 0.3$Mpc.
These results, which suggest that structure formation by strings is biased,
complement our conclusion that the string+CDM model may be a viable candidate
for the formation of large scale structure in an open universe.
\section{Cosmological Constant} \label{seccosmoconst}
In this section we briefly consider the effect of a cosmological
constant on the cosmic string scenario. The background cosmology in this
case is a spatially flat, FRW space-time with a cosmological fluid
composed of vacuum- and matter-components such that
$\Omega_{m} + \Omega_{\Lambda} = 1$. The expansion
scale factor is given by the expression
\begin{equation}
a(t) = \Big[ {1 - \Omega_\Lambda \over \Omega_\Lambda}
\sinh^2\Big( {3 \over 2} H_o t \sqrt{\Omega_\Lambda}\Big) \Big]^{1/3}
\end{equation}
where $H_o,\,\Omega_\Lambda$ are the present-day Hubble constant and
vacuum-matter density parameter. We may now follow a similar procedure as
outlined in section \ref{secevol} to study the evolution of the long string
length scale $L$ and velocity $v$ by taking the spatially flat, $\Omega \to 1$
limit in equations (\ref{Levolutioneqn}-\ref{vevolutioneqn}). In this case we
find that for a comparable matter density as in an open universe, the dilution
of the string energy density and the damping of string motion is much weaker in
the cosmological constant universe. Note that the argument of the $\sinh$ in
the scale factor, evaluated at the present-day, is ${1\over 2}\log|(1 +
\sqrt{\Omega_\Lambda})/(1 - \sqrt{\Omega_\Lambda})|$. Hence, for
small-$\Omega_\Lambda$ the scale factor behaves to leading order as $a(t) \sim
t^{2/3}$, just as for matter-dominated expansion. Only when $\Omega_\Lambda \to
1$ are the effects of the exponential expansion important, damping the string
motion. For example, in the case of $\Omega_m = 0.3$, the ratio $L/t$ is only
$\sim 5\%$ larger and the velocity is only $\sim 5\%$ smaller than the $\Omega_m
= 1$, spatially flat value. For the open universe with $\Omega = 0.3$, the ratio
$L/t$ has grown by $\sim 15\%$ and the velocity has dropped by $\sim 30\%$ from
their $\Omega=1$ values.
We may estimate the CMB normalization of the mass per unit length as a
function of $\Omega_m$ following the methods of section \ref{seccmb}. However,
there is no correction for the geometry, since the spatial sections are
flat. Hence, we find the empirical formula
\begin{equation}
G \mu(\Omega_m)_\Lambda = 1.05{}^{+0.35}_{-0.20}
\times 10^{-6} \, \Omega_m^{-0.05}
\label{munorm2}
\end{equation}
fits our results to within $5\%$ for $0.1 \le \Omega_m \le 1$. The subscript
$\Lambda$ is used to differentiate the above normalization from the case of an
open universe, in equation (\ref{munorm}). We see that the normalization is
relatively insensitive to the presence of a cosmological constant.
Finally, we may consider the properties of the cosmic string scenario for
structure formation with CDM in the presence of a cosmological constant. We may
adapt equation (\ref{stringpower}) for the string+CDM power spectrum by setting
|
$\Omega = \Omega_m$ and using the appropriate growth factor \cite{GFactor}.
The variance of the mass excess on length scales $R = 8 h^{-1}$Mpc is fit by
the empirical formula
\begin{eqnarray}
&& \sigma_8(\Omega_m,h) = 0.9 (\pm 0.5) \, \times
\Big( \mu_6(\Omega) \, {g(\Omega_m,\Omega_\Lambda) \over \Omega_m}
\, {\Gamma (1 - 0.36 \Gamma)
\over 1 + (50 \Gamma)^{-2}} \Big)
\cr\cr
&& {\rm with} \qquad
g(\Omega_m,\Omega_\Lambda) \equiv {5 \over 2} \Omega_m /
\Big[\Omega_m^{4/7} - \Omega_\Lambda +
(1 + {1 \over 2} \Omega_m)(1 + {1 \over 70} \Omega_\Lambda)\Big]
\label{signorm2}
\end{eqnarray}
which is valid to within $\sim 10\%$ for $0.1 \le \Omega \le 1$ and $0.4 < h <
0.8$. We find that the amplitude of the string+CDM power spectrum with a
cosmological constant is higher than in an open universe with the same matter
density, as demonstrated in Fig. \ref{figure3}. Evaluating (\ref{signorm2}), we
predict $\sigma_8(0.2,0.5) = 0.3\pm{0.2}$ and $\sigma_8(0.4,0.7) = 0.5\pm{0.3}$.
Comparing to observations, based on the estimate
$\sigma_8 = 0.6{}^{+32\%}_{-24\%}
\exp[({-0.59-0.16\Omega+0.06\Omega^2})\log\Omega]$
\cite{VianaLiddle} for a spatially flat universe, we find that
a slightly lower bias than in an open universe, $b \sim 1.5 - 4$, is required.
Hence, the string+$\Lambda$CDM scenario may be viable if the strings generate a
sufficient boost to explain the biased clustering on $8 h^{-1}$Mpc scales.
\section{Conclusion}
\label{secend}
In this paper we have laid out many of the tools necessary to study cosmic
strings in an open universe. We have first derived the equations of motion and
energy conservation in an $\Omega < 1$ FRW space-time. We have extended the MS
model of quantitative string evolution \cite{MartinsShellard} to the case of an
open, $\Omega < 1$ universe. We believe this extrapolation is reasonable for
the range of values of $\Omega$ of interest. We have found that with the onset
of curvature dominated expansion, the long string energy density and mean
velocity decay rapidly. We have shown that the resulting effect on the large
angle CMB temperature fluctuations induced by cosmic strings is a lower level
of anisotropy than in a critical, $\Omega = 1$ universe, for the same $\mu$.
Constructing a semi-analytic model for the generation of CMB anisotropy in an
open universe, based in part on the AS numerical simulation
\cite{AllenShellard,AllenShellardProc}, we found that comparison with the
COBE-DMR observations \cite{COBE} leads to a higher normalization of the cosmic
string mass per unit length. To the extent that the CMB anisotropy induced by
realistic cosmic strings has been accurately simulated in Ref.
\cite{AllenEtAl}, we believe our results, equations
(\ref{munorm},\ref{munorm2}), are reliable within the errors discussed. The
new normalization of $\mu$, the first estimate of the normalization of $\mu$ in
a low density universe (as far as we are aware), is consistent with all other
observational constraints on cosmic strings, including the bound on a
stochastic gravitational wave background arising from pulsar timing
\cite{CaldwellBattyeShellard}.
Finally, we have demonstrated the effect of an open, $\Omega < 1$ universe on
the power spectrum of density fluctuations produced by cosmic strings with HDM
and CDM. As we mentioned in section \ref{secintro}, the power spectrum $P(k)$
does not completely specify the cosmic string structure formation scenario.
Fluctuations generated by string wakes and filaments are non-gaussian, so that
knowledge of $P(k)$ alone is insufficient to specify all the properties of the
density field. Although the linear power spectrum (\ref{stringpower}) is in
agreement with the results of Avelino \cite{Avelino} and Colombi
\cite{ColombiThesis} on a limited range of scales, we are unable to make finely
detailed comparisons with observations without more knowledge of the
distribution of cosmic string seeded density perturbations. For example, it is
not clear whether the estimates of the rms linear fluctuation in the mass
distribution \cite{PeacockDodds,WhiteEtAl} obtained from the various galaxy
redshift surveys, which depend strongly on the gaussianity of the initial
density field, are directly applicable to a theory with a non-gaussian
fluctuation spectrum. Nevertheless, we have found that the string+CDM spectrum
fits the shape of the PD reconstruction of the linear power spectrum
\cite{PeacockDodds} for cosmological parameters in the range $\Gamma \sim 0.1 -
0.2$. We have computed the variance of the mass fluctuation in a sphere of
radius $R = 8 \, h^{-1}$Mpc, requiring a bias $b \gtrsim 2$ for consistency with
the inferred $\sigma_8$ of the linear density field. In the case of a
cosmological constant, a slightly lower bias is required than for an open
universe string+CDM spectrum with the same matter density. These findings are
similar to Ref. \cite{Ferreira}, in which the product $b G\mu$ was estimated in
order to fit the string+CDM spectrum to the 1-in-6 IRAS QDOT survey
\cite{FeldmanEtAl}, and to Ref. \cite{PenSpergel}, in which the effects of an
open universe on global defects, including global strings and textures, were
considered. The results of Ref. \cite{SornBrand} indicate that the density of
baryonic matter is enhanced in CDM wakes by a factor of $\sim 2.4$, suggesting
that a bias $b \sim 2$ may be possible. It is clear that high resolution
simulations, as Ref. \cite{SornWake}, are necessary to further develop of the
cosmic string structure formation scenario.
The results presented in this paper provide excellent motivation to
continue investigation of the cosmic string scenario, which should be
possible with the equations of motion for strings and the normalization
of $\mu$ for $\Omega < 1$.
\acknowledgements
We would like to thank Chung-Pei Ma, Paul Shellard, Andrew
Sornborger, and Albert Stebbins for
useful conversations.
P.P.A. is funded by JNICT (Portugal) under
`Programa PRAXIS XXI' (grant no. PRAXIS XXI/BPD/9901/96).
The work of R.R.C. is supported by the DOE at
Penn (DOE-EY-76-C-02-3071).
C.M. is funded by JNICT (Portugal) under
`Programa PRAXIS XXI' (grant no. PRAXIS XXI/BD/3321/94).
\eject
\begin{figure}
\caption{
The evolution of a circular cosmic string loop formed
at $t=t_{eq}$ with an initial radius $R_{loop}=10\,t_{eq}$, in a universe with
$\Omega=1,\, 0.2$ given by the solid and dotted lines respectively.
The top panel shows the evolution of the radius in units of $t_{eq}$
versus $\log_{10}(a/a_{eq})$. The bottom two panels show the evolution
of the velocity. An expanded scale shows the first oscillations
as the loop enters the horizon, after which we show only the maximum
velocity in each period of oscillation.
}
\label{figure0}
\end{figure}
\epsfxsize=2.5cm \epsfbox[0 700 100 800]{fig1.ps}
\eject
\begin{figure}
\caption{
The evolution of the average string velocity and the characteristic
length scale of long strings in an open FRW space-time with $\Omega =
1.0,\, 0.6,\, 0.2$ given by the solid, dashed, and dotted curves. The
horizontal axis is the log of the cosmological time $t$. As the
expansion becomes curvature-dominated, the average velocity decays and
the characteristic string length scale grows. As a result, the number
of long strings in a box of linear dimension $t$ decreases, although
the string energy density relative to the background energy density
grows.
}
\label{figure1}
\end{figure}
\epsfxsize=2.5cm \epsfbox[0 700 100 800]{fig2.ps}
\eject
\begin{figure}
\caption{
The CMB-normalized power spectrum $P(k)$ of density fluctuations
produced by cosmic strings with HDM (left) and CDM (right) are
presented for $\Omega = 1.0,\,0.4,\, 0.2$, given by the thick solid,
long-dashed, and short dashed curves. For all cases, we have used
$h=0.7$. In the top panels, the thin solid line is the standard CDM
spectrum normalized to COBE following [34]. In the lower panels, the
data points are the PD reconstruction of the linear power spectrum,
with the amplitude rescaled $\propto \Omega^{-0.3}$. In the bottom two
$\Omega < 1$ string+CDM panels, the thin solid line shows the
CMB-normalized power spectrum for the case of a cosmological constant
with the same matter density. A bias $b \sim 2-4$ is necessary to
obtain $\sigma_8 \sim 1$. In the presence of a cosmological
constant, a smaller bias is required.
}
\label{figure3}
\end{figure}
\epsfxsize=2.5cm \epsfbox[0 700 100 800]{fig3.ps}
\eject
\eject
|
\section{Introduction} \label{S-Intro}
Solar eruptive events, such as flares and Coronal Mass Ejections (CMEs), release huge amounts of energy. The profound effects of such explosive processes are not confined to their parent active regions, but also causes disturbances over a wide spatial range in the forms of fast magnetosonic waves and shocks \citep[e.g.,][]{Cliver95,Kwon13}. The existence of these coronal waves has long been established based on metric type II radio bursts \citep[e.g.,][ hereafter type IIs]{Wild50, Cliver99}. Such disturbances have also been observed in optical observation as arc-shaped bright fronts of H$\alpha$ wings, indicative of propagating disturbances in the chromospheric layer \citep{Moreton60,MR60}; such phenomena are termed Moreton or Moreton–Ramsey waves. Moreton–Ramsey waves were initially interpreted as the chromospheric response to the fast magnetosonic wave traveling in the solar corona \citep{Uchida1968}. Thus, they were thought to be remote-sensing observations of the shocks responsible for type IIs \citep{Uchida1974}. When arc-shaped bright fronts were also observed with Extreme Ultraviolet Imaging Telescope \citep[EIT;][]{Del95} onboard Solar and Heliospheric Observatory \citep[SOHO;][]{Domingo95} spacecraft, they were immediately interpreted as the {\it coronal} waves that are the origin of the {\it chromospheric} Moreton--Ramsey waves \citep{Moses97,Thompson98}. They were first named for the instrument that discovered this phenomenon, i.e., EIT waves, but later came to be called EUV waves since they are generally identified in EUV observations. EUV waves have a broad speed distribution, from several tens of km s$^{-1}$to more than 1000 km s$^{-1}$ \citep[see review by][]{Warmuth15, Long2017a}.
Due to the lack of coronagraphic observations in the 1960s and 1970s, the agent of the coronal shocks observed as type IIs and Moreton--Ramsey waves was thought to be flares; however, since 1995, the regular space-based coronagraphic observations of SOHO LASCO \citep{Brueckner95} have established that they have a stronger association with CMEs than flares. \citet{Gopalswamy05, Gopal09} have shown that metric type II emission is usually driven by CMEs. Support for the CME-driven shock scenario is given by observations of broadening and intensity changes in the UV emission lines ahead of the CME front, attributed to shocks associated with type IIs \citep{Mancuso02, Cia05}. Similarly, analyzing 173 EUV waves observed between 1997 and 1998 by SOHO EIT, \citet{Bie02} found an {\it unambiguous} relationship between EUV waves and CMEs, while EUV waves with {\it bright} and {\it sharp} fronts also have a strong relationship with flares.
It has been often thought that EUV waves and type IIs are two observational aspects of a single coronal shock wave, but physical inconsistencies between them hampers understanding of their different or common nature. The inconsistency is two-fold based on their one-to-one correspondence and speed. \citet{Muhr14} analyzed 60 strong EUV wave events from January 2007 to February 2011 and found a 22\% association of EUV waves with type IIs. \citet{Nitta13} presented a statistical analysis of 138 events, finding a 54\% association. \citet{Nitta14} presented examples in which a type II is not associated with an EUV wave and {\it vice versa}. They suggested that neither EUV waves nor type II bursts serve as a necessary condition for coronal shock waves. More recently, \citet{Long17} found that out of 164 events, 40\% of the EUV waves are associated with type IIs.
Few studies have compared their speeds between EUV waves and type IIs. \citet{kl00} found that they are strongly associated (90\%), but their speeds remain poorly correlated. Recently, \citet{Long17} found that the median speeds for individual EUV wave event correlate closely with their accelerations, indicating that the median speed is a physically meaningful characteristic. We interpret that the random error in speed has less effect on the median values. While systematic errors may not be removed by taking the median value, the correlation would have less of an effect on the systematic error. Interestingly, however, they found that even their median speeds are not correlated with the speeds of type IIs. Such inconsistency was also found by \citet{Warmuth10}.
Supposing that EUV waves are the imaging observations of fast magnetosonic shock waves, which are themselves observed as type IIs in radio observations, several possible explanations may account for this inconsistency. First, speed inconsistencies could result from different propagating directions. As indicated by their slow frequency drift, type IIs are likely to be a shock driven by the CME radial motion. On the other hand, it is widely accepted that EUV waves are driven by the lateral expansion of their associated CME. Thus, the directionality naturally explains their different speeds. However, it is also well-known that CME radial speeds and lateral expansion speeds are strongly correlated \citep{Dal03, Schwenn05}, suggesting that the directionality itself should not affect their correlations. For instance, \citet{Warmuth10} showed that the speeds of Moreton--Ramsey waves closely correlate with those of type IIs. This has provided the basis for various non-wave interpretations of EUV waves \citep[see][for reviews]{Warmuth15, Long2017a}; in contrast, Moreton--Ramsey waves have been widely accepted to be a type II-related wave phenomenon.
Secondly, it may result from uncertainties inherent to measuring their speeds. In case the same fronts are given, the speeds measured by various people with different methods will strongly correlate one another, and thus these differences in the speeds will not significantly affect the correlation when comparing them with other parameters. Because the fronts to track for type IIs and CME noses are clearly given, the largest uncertainties in correlations should arise from the fact that the EUV observations show not only the wave fronts but also non-wave components \citep{Warmuth15, Long2017a}. Taking the occurrence of slower non-wave components following a true wave front into consideration \citep[e.g.,][]{Chen11}, one may consider only the faster one when determining the wave speed. However, EUV waves are also subject to the projection effect, and thus the projected fronts could appear faster than the actual ones \citep{Kwon13, Lario14, Downs21}. Such complicated appearance in an event inhibits the measurements of their accurate speeds and errors. Alternatively, we analyze EUV waves listed in \citet{Nitta13}, in which the speeds are used for statistical studies \citep{Nitta13,Nitta14,Long17}. We take advantage of these previous speed measurements to find out if there exists a physically meaningful trend of tendency between EUV waves and type IIs.
In this paper, we re-examine the physical relationship between EUV waves and type IIs. We have identified 60 EUV wave events associated with type IIs from a catalog compiled by \citet{Nitta13}. In Section \ref{sec:res}, we show comparisons among these EUV waves, type IIs, and parent CMEs to evaluate whether the inconsistency is due to speeds uncertainties. The speeds determined by \citet{Nitta13} and \citet{Long17} are also presented for the comparisons. Finally, in Section \ref{sec:DC}, we present discussions about our findings and conclusions.
\section{Results} \label{sec:res}
Since errors in speeds of type IIs and CMEs would have small effects on the correlations as discussed in Section \ref{S-Intro}, we used the speeds previously reported in the LASCO CDAW catalog \citep{y04} and the NOAA list for CMEs and type IIs, respectively.
Knowing the uncertainties described in Section \ref{S-Intro} and thus to be compared with previous measurements, we analyzed the EUV wave events listed in \citet{Nitta13} which are already extensively studied \citep{Nitta13,Nitta14,Long17}. We used the Solar Dynamics Observatory \citep[SDO;][]{Pesnell12}/ Atmospheric Imaging Assembly \citep[AIA;][]{Lemen12} AIA 193 \AA\ images to re-examine the properties of the EUV waves. We compared the timing of these events with CMEs and type IIs in the LASCO CDAW catalog and the NOAA list, respectively. As a result, we found 60 EUV wave events associated with type IIs. The list of the events investigated in this study and the physical parameters are given in Table 1.
Figure \ref{fig:hist}(a) shows that out of the 60 events studied, 28 featured the occurrence of type IIs within a time interval of 5 min after the generation of the EUV waves. The maximum delay preceeding type IIs was found to be 25 min after the appearance of the EUV waves. Interestingly, type IIs appeared before EUV waves in five events. A possible physical explanation is that the fast radial motion of the CME generates a shock wave prior to the lateral expansion that produces the EUV wave in the low corona. While these cases can be due to mismatches, their small number is expected to have a minimal effect on our statistical study, which uses a large number of samples.
Previous works have indicated that CMEs are a common agent of EUV waves and type IIs \citep{Bie02, Mancuso02, Cia05, Gopalswamy05, Gopal09}. In this regard, the properties of CMEs may provide indications as to the link between the EUV waves and type IIs. Figure \ref{fig:hist}(b) and \ref{fig:hist}(c) show the timing difference of the CME onsets from EUV waves and type IIs, respectively. The CME onset times have been taken from the CDAW catalog, and thus the onset time represents the first appearance of CMEs in the LASCO C2 field of view. The larger time delay of CMEs seen in these panels than in Figure \ref{fig:hist}(a) is due to the transit time of the CMEs as they propagate into the C2 field of view. There are few questionable events in which the CME appearance is earlier than that of EUV waves or type IIs, but this should not affect our statistical results.
We derived the speeds of the 60 EUV waves. Since EUV waves are observed most clearly in 193 \AA\ and 211 \AA\, we selected AIA 193 \AA\ images. Figure \ref{fig:method} shows an example of distance-time maps used to track the EUV wave fronts and determine their speeds. The distance-time map is constructed by stacking cut taken along a great circle passing the source region (the white curve in panel (a) in Figure \ref{fig:method}) on images taken during the EUV wave is in progress.
To determine the speed, one may use a manual or automatic method \citep[see review by][]{Long2017a}. Note that these methods can be applied only after an EUV wave front is identified by visual inspection. While the different approaches for the same front will result in different speeds, it is very likely that these speeds are correlated each other, and thus these differences may not affect greatly the correlation when comparing with other physical parameters. In this sense, we can speculate that the largest uncertainties in the correlation occur at the selection stage of the fronts due to interference by non-wave components and projection effects (Section \ref{S-Intro}).
Taking sources of speed uncertainty into consideration, we used a slightly different method, expecting that comparisons of the given type II speeds with EUV wave speeds determined with various methods may provide indications to find a trend. For every event, the distance-time maps revealed fast as well as slow EUV wave components. We used visual inspection to track these EUV wave-related fronts and computed their average speeds. We found that EUV waves propagate with speeds as low as $\approx$170 km s$^{-1}$ and as high as $\approx$1500 km s$^{-1}$. Note that \citet{Nitta13} and \citet{Long17} used the same catalog and determined the speeds of EUV waves, and thus our investigation can be made together with their speed measurements (Figure \ref{fig:comparisons}).
Figure \ref{fig:comparisons}(a) shows comparisons between the speeds of EUV waves and type IIs. Closed dots denote values determined in this study and show no correlation between the two speeds. Since EUV wave speeds could be uncertain, we also present the values in Figure 6 of \citet{Long17} with cross symbols to determine whether the low correlation is due to the uncertainty of our calculated speeds. Furthermore, the speeds for EUV waves also studied in \citet{Nitta13} and \citet{Long17} are shown in Figure \ref{fig:comparisons}(b) as ex and plus symbols, respectively. These comparisons made with three independent measurements do not provide any indication for a trend. This implies that the inconsistency is not merely due to uncertainties in measuring EUV wave speeds.
Instead, we found a tendency from these three independent studies, in which the type II speeds easily exceed a higher speed, e.g., 1000 km s$^{-1}$, whereas the EUV waves rarely do so.
The same tendency is also found in the speeds of CMEs and EUV waves, as shown in Figure \ref{fig:comparisons}(c). Their weak correlation was also reported in \citet{Nitta13}. It seems natural that EUV waves are waves propagating with their wave-mode speed while CME speeds depend on kinetic energy. Figure \ref{fig:comparisons}(d) shows that type II speeds are better correlated with those of CMEs (correlation coefficient of 0.29) than with EUV waves (correlation coefficient of 0.17). In addition, the full speed ranges of type IIs and CMEs are identical ($\sim$ 2500 km s$^{-1}$). It also seems natural if type IIs arise due to CME--driven shocks. Note that the CME speeds presented in this paper were determined at the noses. The weak correlation coefficient values between type IIs and CMEs can be understood considering the various source locations of type IIs. \citet{Ramesh12} studied 41 type IIs and found that they could originate from any location of the CME. There are equal possibilities for type II generation from the nose and flank.
\section{Discussion and Conclusion} \label{sec:DC}
We analyzed 60 EUV wave events associated with type IIs. We found no strong correlation between the speeds of EUV waves and type IIs; however, we invoke a tendency that has been regarded as an inconsistency, i.e., that the speeds of EUV waves and their range are much less than those of type IIs (Figure \ref{fig:comparisons}). Type IIs easily exceed higher speeds, e.g., 1000 km\,s$^{-1}$, whereas EUV waves rarely do so. The average and standard deviation (the full range and median value) are 480 $\pm$ 250 km\,s$^{-1}$ (170 -- 1570 km\,s$^{-1}$ and 450 km\,s$^{-1}$), i.e., approximately two times less than those of type IIs, which are 970 $\pm$ 470 km\,s$^{-1}$ (360 -- 2350 km\,s$^{-1}$ and 830 km s$^{-1}$).
The same tendency has been noted in two other independent studies \citep{kl00,Long17}, suggesting that differences in speed cannot be described only to uncertainties. \citet{Long17} showed that the speed and range of type IIs are three times greater than those of EUV waves. The average speed and standard deviation (the full range and its median value) of EUV waves and type IIs are 430 $\pm$ 170 km\,s$^{-1}$ (100 -- 790 km\,s$^{-1}$ and 410 km\,s$^{-1}$) and 1150 $\pm$ 670 km\,s$^{-1}$ (360 - 4520 km\,s$^{-1}$ and 1040 km\,s$^{-1}$), respectively. According to Figure 4 in \citet{kl00}, the values for EUV waves are 280 $\pm$ 70 km\,s$^{-1}$ (100 -- 500 km\,s$^{-1}$ and 250 km\,s$^{-1}$), whereas those for type IIs are 740 $\pm$ 230 km\,s$^{-1}$ (200 - 1300 km\,s$^{-1}$ and 750 km\,s$^{-1}$). These results also show that the type II speeds and the range are three times greater than those of EUV waves. Meanwhile, we did not see the same tendency between the speeds of the type IIs and CMEs. The average speed and the standard deviation of CMEs are 980 $\pm$ 610 km\,s$^{-1}$, approximately identical to those of the type IIs (970 $\pm$ 470 km\,s$^{-1}$).
This tendency or inconsistency has been expected from the three-dimensional speed distribution of coronal shocks \citep{Kwon17}. \citet{Kwon17} found the same tendency in two shock speeds taken from two different directions in an event, nearly along the radial direction and the direction of the EUV waves. They analyzed three halo CMEs with significantly differing radial speeds in the CDAW catalog. They considered that halo fronts are the observation of coronal shocks \citep{Sheeley00,Kwon15}, and thus the coronal counterparts of EUV waves \citep{Kwon13}. The 3D geometry and kinematics of coronal shocks were determined using the ellipsoid model \citep{Kwon14}. While the shock leading-edge speeds of the three CMEs vary from 1355 to 2157 km\,s$^{-1}$, speeds close to the solar surface, i.e., nearly in the direction of EUV waves, are identical and consistent with the local fast magnetosonic wave speeds (see Figure 16 in \citealp{Kwon17}). It is well-known that type IIs from due to CME-driven shocks propagating with the driver CMEs. Since EUV waves are decaying shock waves decoupled from the driver in the low corona where the driver had left, their speeds should be largely affected by the local fast magnetosonic wave speeds. In this regard, it is not necessary that the speeds of EUV waves and type IIs are correlated.
Considering that EUV waves are a coronal counterpart of Moreton–Ramsey waves, our interpretation above seems contradictory to the strong correlation between the speeds of Moreton–Ramsey waves and type IIs \citep{kl00,Warmuth10}. An explanation for this can be found in the fact that Moreton–Ramsey waves are observed only in very energetic events \citep{Francile16, Cab19, Long19}, and their traveling distances are short compared with those of EUV waves \citep{Warmuth01}. This leads to the conjecture that Moreton-Ramsey waves are strictly related to either the shock driving phase or the initial strong shock phase \citep{Kwon13}, and thus they are a driven shock whose speeds are identical to those of the drivers. On the other hand, EUV waves can be observed even in later phases, when the speeds have reached the local fast magnetosonic wave speed \citep{Warmuth01}. This feature was clearly observed in the simultaneous observation of a Moreton-Ramsey wave and EUV wave in \citet{Asai12}, where they initially propagate cospatially. The Moreton–Ramsey wave disappears first while the EUV wave continues to propagate. Since both Moreton-Ramsey and EUV waves decelerate, the simple average speed over time will result in a slower speed for the one that travels a greater distance \citep{Warmuth01}. In this sense, the speeds of Moreton-Ramsey waves and type IIs,
|
i.e., CME-driven shock, should reflect the driver's radial or expansion speeds, and thus their speeds are very likely to be correlated, as apposed to the cases of EUV waves.
Our interpretation also provides indications for the association problem between EUV waves and type IIs. Type IIs are often observed without EUV waves and {\it vice versa} \citep{Bie02, Thompson09, Muhr14, Nitta13, Long17}. This may depend on the direction of shock wave propagation and whether driven or non-driven shocks are present. Type IIs are observed as a slow frequency drift in the dynamic spectra, and their frequency corresponds to the local electron density. Therefore, shocks strong enough to cause radio emission should travel in the direction of the electron density gradient (nearly radial) and along a sufficiently long distance to be identified as a frequency drift in the dynamic spectra. If a super-magnetosonic CME expansion in the initial phase serves as a piston, but its traveling distance in the direction of the electron density gradient is insufficient, there will be no observable frequency drift. However, once a shock wave is generated in all directions by the 3D piston, a fast magnetosonic wave is decoupled from the piston and can propagate to greater distances as a decaying shock wave, i.e., an EUV waves.
On the other hand, when piston-driven shocks travel to greater distances, both type IIs and EUV waves will be observed; however, once the EUV wave becomes freely propagating and decouples from the driver, it will be subject to refraction due to the local wave-mode speed gradient \citep{Uchida1968,Wang00,Af11,Kwon13,Kwon17}. Figure 4 in \citet{Kwon17} indicates that the EUV wave cannot be observed due to refraction toward the upper corona in which the local fast magnetosonic speed decreases with height. Additionally, when super-magnetosonic radial motion of a CME occurs with sub-magnetosonic expansion, the shock wave will be driven only in the radial direction. In this case, the EUV wave will not be observed.
In summary, we have confirmed the inconsistency between the speeds of EUV waves and associated type IIs. The correlation coefficient is 0.17. However, we also identified a tendency for the speeds and range of type IIs to significantly exceed those of the EUV waves. Type IIs can easily exceed 1000 km s$^{-1}$, whereas EUV wave speeds rarely do so. The average and standard deviation of the EUV wave speeds are also significantly less than those of the type IIs and CMEs. This tendency can be also found in \citet{kl00,Warmuth10,Long17}. Despite these differences, the average and standard deviation of type IIs and CMEs are identical and better correlated (correlation coefficient of 0.29).
These results indicate that type IIs can be as fast as CMEs, as opposed to EUV waves. As indicated by the close relationship between Moreton–Ramsey waves and type IIs \citep{kl00,Warmuth10}, the directionality itself is insufficient to account for this tendency. There must be an additional factor lowering the EUV wave speeds, local fast magnetosonic wave speeds.
We conclude that the inconsistency between the speeds of EUV waves and type IIs is an intrinsic tendency. Type IIs are the consequence of a driven shock in the extended solar corona, and their speeds are thus dependent on those of the driver, i.e., CME. On the other hand, the EUV waves are a non-driven shock wave, propagating in the low corona where the driver had already left; they evolve into a linear wave. The simple average speed of an EUV wave over time could be largely affected by the local fast magnetosonic wave speed and thus be independent of the speeds of both the CME and the accompanied type II. Since the corona Alfv\'en speed and sound speed on average may not vary greatly, the average speed of EUV waves must not show large event-to-event variation relative to type IIs whose speeds are dependent on the speed of the driver CMEs. Our results indicate that this is the case. While our analysis may contain significant uncertainties in identifying EUV waves and type IIs and measuring their speeds, our conclusions lead clearly to a conjecture that the observed inconsistency should remain even if all uncertainties were removed.
\begin{table}
\medskip
\centering
\scalebox{0.77}{
\begin{tabular}{cccccccccccc}
\hline
S. No. & Date & EUV Wave & GOES & Flare Start & Wave & CME Start & CME & Type II & Shock & Type II \\
& & Time (UT) & Flare & Time (UT) & Velocity & Time (UT) & Velocity & Time (UT) & Height (R$_{\odot}$) & Velocity\\
\hline
1. & 27-01-2012 & 18:08 & X1.7 & 17:37 & 800 & 18:27 & 2508 & 18:10 & 1.15 & 1523 \\
2. & 07-03-2012 & 00:05 & X5.4 & 00:02 & -- & 00:24 & 2684 & 00:17 & 1.15 & 2273 \\
3. & 07-03-2012 & 01:07 & X1.3 & 01:05 & 608 & 01:30 & 1825 & 01:09 & 1.15 & 1329 \\
4. & 09-03-2012 & 03:39 & M6.3 & 03:22 & 314 & 04:26 & 950 & 03:43 & 1.37 & 1285 \\
5. & 13-03-2012 & 17:15 & M7.9 & 17:12 & 493 & 17:36 & 1884 & 17:15 & 1.15 & 1366 \\
6. & 17-03-2012 & 20:37 & M1.3 & 20:32 & 520 & -- & -- & 20:38 & 1.15 & 1140 \\
7. & 05-04-2012 & 20:52 & C1.5 & 20:49 & 234 & 21:25 & 828 & 21:08 & 1.68 & 360 \\
8. & 09-04-2012 & 12:13 & C3.9 & 12:12 & -- & 12:36 & 921 & 12:28 & 1.59 & 478 \\
9. & 23-04-2012 & 17:39 & C2.0 & 17:38 & 534 & 18:24 & 528 & 17:42 & 1.42 & 1605 \\
10. & 17-05-2012 & 01:28 & M5.1 & 01:25 & 439 & 01:48 & 1582 & 01:31 & 1.15 & 645 \\
11. & 06-06-2012 & 19:57 & M2.1 & 19:54 & 556 & 20:36 & 494 & 20:03 & 1.21 & 1148 \\
12. & 02-07-2012 & 10:44 & M5.6 & 10:43 & 298 & 11:24 & 313 & 10:47 & 1.27 & 1063 \\
13. & 04-07-2012 & 16:38 & M1.8 & 16:33 & 381 & -- & -- & 16:42 & 1.15 & 807 \\
14. & 06-07-2012 & 23:04 & X1.1 & 23:01 & 417 & 23:24 & 1828 & 23:09 & 1.15 & 1771 \\
15. & 13-08-2012 & 12:35 & C2.8 & 12:33 & 638 & 13:25 & 435 & 12:41 & 1.15 & 736 \\
16. & 21-11-2012 & 06:47 & M1.4 & 06:45 & 387 & -- & -- & 06:54 & 1.15 & 720 \\
17. & 21-11-2012 & 15:20 & M3.5 & 15:10 & 453 & 16:00 & 529 & 15:33 & 1.15 & 1618 \\
18. & 11-01-2013 & 09:07 & M1.2 & 08:43 & 309 & -- & -- & 09:14 & 1.17 & 537 \\
19. & 13-01-2013 & 08:35 & M1.7 & 08:35 & 316 & -- & -- & 08:40 & 1.15 & 649 \\
20. & 18-01-2013 & 17:05 & C5.8 & 16:50 & 173 & -- & -- & 17:10 & 1.15 & 1695 \\
21. & 06-02-2013 & 00:03 & C8.7 & 00:04 & 421 & 00:24 & 1867 & 00:13 & 1.15 & 548 \\
22. & 11-04-2013 & 07:02 & M6.5 & 06:55 & 728 & 07:24 & 861 & 07:02 & 1.18 & 1059 \\
23. & 18-04-2013 & 18:01 & C6.5 & 17:56 & 318 & 18:24 & 495 & 18:23 & 1.57 & 1273 \\
24. & 23-04-2013 & 18:11 & C3.0 & 18:10 & 188 & 18:48 & 403 & 18:23 & 1.49 & 820 \\
25. & 02-05-2013 & 05:02 & M1.1 & 04:58 & 1570 & 05:24 & 671 & 05:06 & 1.15 & 703 \\
26. & 13-05-2013 & 02:00 & X1.7 & 01:53 & 405 & 02:00 & 1270 & 02:10 & 1.15 & 2347 \\
27. & 15-05-2013 & 01:42 & X1.2 & 01:25 & 283 & 01:25 & 296 & 01:37 & 1.15 & 501\\
28. & 30-08-2013 & 02:15 & C8.3 & 02:04 & -- & 02:48 & 949 & 02:12 & 1.27 & 1318 \\
29. & 11-10-2013 & 07:10 & M1.5 & 07:01 & 220 & 07:24 & 1200 & 07:11 & 1.15 & 510 \\
30. & 13-10-2013 & 00:30 & M1.7 & 00:04 & 496 & 01:25 & 478 & 00:38 & 1.29 & 798 \\
31. & 22-10-2013 & 21:20 & M4.2 & 21:15 & 998 & -- & -- & 21:21 & 1.08 & 1955 \\
32. & 25-10-2013 & 07:58 & X1.7 & 07:53 & 793 & 08:12 & 587 & 07:59 & 1.15 & 1240\\
33. & 28-10-2013 & 04:35 & M5.1 & 04:32 & 262 & 04:48 & 1201 & 04:37 & 1.29 & 508 \\
34. & 02-11-2013 & 04:44 & C8.2 & 04:40 & 275 & 04:48 & 828 & 04:46 & 1.15 & 584 \\
35. & 05-11-2013 & 22:10 & X3.3 & 22:07 & 514 & 22:36 & 562 & 22:13 & 0.9 & 1380 \\
36. & 08-11-2013 & 04:25 & X1.1 & 04:20 & 431 & -- & -- & 04:24 & 1.42 & 834 \\
37. & 10-11-2013 & 05:10 & X1.1 & 05:08 & 1164 & 05:36 & 682 & 05:13 & 1.15 & 1012 \\
38. & 07-12-2013 & 07:18 & M1.2 & 07:17 & 508 & 07:36 & 1085 & 07:27 & 1.15 & 691 \\
39. & 12-12-2013 & 03:15 & C4.6 & 03:11 & 271 & 03:36 & 1002 & 03:17 & 1.54 & 511 \\
40. & 07-01-2014 & 18:05 & X1.2 & 18:04 & 452 & 18:24 & 1850 & 18:17 & 1.49 & 1064 \\
41. & 08-01-2014 & 03:46 & M3.6 & 03:39 & 200 & 04:12 & 643 & 03:48 & 1.15 & 697 \\
42. & 11-02-2014 & 03:25 & M1.7 & 03:22 & 610 & -- & -- & 03:33 & 0.9 & 873 \\
43. & 20-02-2014 & 07:43 & M3.0 & 07:26 & 449 & 08:00 & 948 & 07:45 & 1.42 & 915 \\
44. & 25-02-2014 & 00:44 & X4.9 & 00:39 & 211 & 01:25 & 2147 & 00:56 & 1.15 & 909 \\
45. & 20-03-2014 & 03:45 & M1.7 & 03:42 & 306 & -- & -- & 03:52 & 1.16 & 572 \\
46. & 28-03-2014 & 19:11 & M2.0 & 19:04 & 372 & -- & -- & 19:18 & 1.26 & 528 \\
47. & 02-04-2014 & 13:18 & M6.5 & 13:18 & 516 & 13:36 & 1471 & 13:23 & 1.28 & 903 \\
48. & 04-04-2014 & 13:43 & C8.3 & 13:34 & 410 & 14:12 & 467 & 13:39 & 1.15 & 1803 \\
49. & 25-04-2014 & 00:20 & X1.3 & 00:17 & 555 & 00:48 & 456 & 00:22 & 1.15 & 753 \\
50. & 08-07-2014 & 16:14 & M6.5 & 16:06 & 483 & 16:36 & 773 & 16:14 & 1.15 & 949 \\
51. & 25-07-2014 & 06:57 & C2.2 & 06:57 & -- & -- & -- & 07:12 & 1.42 & 1090 \\
52. & 22-08-2014 & 10:16 & C2.2 & 10:13 & 399 & 11:12 & 600 & 10:17 & 1.19 & 465 \\
53. & 22-08-2014 &15:43 & C6.2 & 15:40 & 568 & -- & -- & 15:44 & 1.16 & 449 \\
54. & 24-08-2014 & 00:11 & C1.6 & 00:08 &655 & -- & -- & 00:15 & 1.15 & 471 \\
55. & 02-10-2014 & 18:51 & M7.3 & 18:49 & -- & 19:12 & 513 & 18:58 & 1.15 & 713 \\
56. & 30-10-2014 & 13:05 & C9.0 & 13:04 & -- & 13:36 & 285 & 13:08 & 1.15 & 1379 \\
57. & 03-11-2014 & 03:51 & C4.2 & 03:47 & 600 & -- & -- & 03:48 & 1.15 & 644\\
58. & 03-11-2014 & 22:32 & M6.5 & 22:15 & 499 & 23:12 & 638 & 22:33 & 1.15 & 601\\
59. & 06-11-2014 & 03:38 & M5.4 & 03:32 & 501 & 04:00 & 641 & 03:46 & 1.26 & 732\\
60. & 07-11-2014 & 17:04 & X1.6 & 16:53 & 404 & -- & -- & 17:19 & 1.15 & 602 \\
\end{tabular}
}
\vspace*{1cm}
\caption{Timeslice Analysis}
\end{table}
\section*{Acknowledgements}
We thank the referee for his/her constructive comments that helped in improving the manuscript. We acknowledge the SDO team who made AIA data available. SDO is a mission for NASA's Living With a Star (LWS) program. We also thank the online data center NOAA for the type II list and LASCO CDAW Catalog for the list of CMEs. R.-Y.K acknowledges support by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT; project No. 2019-2-850-09).
\bibliographystyle{aasjournal}
|
\section{Introduction}
The ideas discussed here were motivated by the following observation. Consider the simple two-coupling version of Yang-Mills theory, whose Minkowski-space Lagrangian is
\begin{eqnarray}
{\mathcal L}=\frac{1}{2(g_{0}^{\prime})^{2}}{\rm Tr}\,\,F_{01}^{2}+
\frac{1}{2g_{0}^{2}}\, {\rm Tr}\,\,F_{02}^{2}- \frac{1}{2g_{0}^{2}}{\rm Tr}\,\,F_{12}^{2},
\label{cont-lag-2+1}
\end{eqnarray}
where $F_{\mu\nu}$ is the Yang-Mills field strength $F_{\mu\nu}=\partial_{\mu}A_{\nu}
-\partial_{\nu}A_{\mu}-i [A_{\mu}, A_{\nu}]$ and the gauge field $A_{\mu}$ is in the Lie algebra of
SU($N$). When regularized on a lattice, this theory can be shown to confine quarks and have a mass gap, when $g_{0}^{\prime}$ is sufficiently small, for any $g_{0}$ \cite{orland2+1}.
Using simple arguments, the mass gap and quark-antiquark potential were found for
$(g_{0}^{\prime})^{2} \ll \frac{1}{g_{0}}e^{-4\pi/(g_{0}^{2}N)}$. This was done by exploiting the connection between the lattice anisotropic gauge theory and
the ${\rm SU}(N)\times {\rm SU}(N)$-symmetric sigma model in $1+1$ dimensions. Subleading corrections were found to these physical quantities \cite{hor-string, mass-spectrum,vert-string}, using the exact S-matrix \cite{abda-wieg} and form factors \cite{kar-wiesz}. It is noteworthy that these are entirely weak-coupling
methods. The main technical problem is the presence of a dimensional cross-over between
$1+1$-dimensional behavior and that of the isotropic gauge theory \cite{hor-string}.
In fact,
(\ref{cont-lag-2+1}) is the high-energy center-of-mass form of the $2+1$-dimensional gauge field
theory Lagrangian. The high-energy limit can be obtained by a longitudinal rescaling $x^{0, 1}\rightarrow \lambda x^{0,1}$,
$x^{2}\rightarrow \lambda x^{2}$, with $\lambda\ll1$ and $g_{0}^{\prime}=\lambda g_{0}$. In the more physical case of $3+1$-dimensional QCD, such a rescaling is
$x^{0,3}\rightarrow \lambda x^{0,3}$,
$x^{1,2}\rightarrow x^{1,2}$ and the effective Lagrangian is \cite{VV, MV}
\begin{eqnarray}
{\mathcal L}=\frac{1}{2g_{0}^{2}}\sum_{j=1,2}{\rm Tr}\,\,F_{0j}^{2}+\frac{1}{2(g_{0}^{\prime})^{2}}{\rm Tr}\,\,F_{03}^{2}
-\sum_{j=1,2}\frac{1}{2g_{0}^{2}}{\rm Tr}\,\,F_{j3}^{2}- \frac{1}{2(g_{0}^{\prime\prime})^{2}}{\rm Tr}\,\,F_{12}^{2}, \label{cont-lag-3+1}
\end{eqnarray}
where $g_{0}^{\prime\prime}=\lambda^{-1}g_{0}$. The structure of hadrons in a lattice version of this effective theory
was discussed in Reference \cite{orland-3+1}. The actual high-energy effective action must include anomalous powers of $\lambda$ in the coefficients of the field strength, which have been found to one loop \cite{orland-xiao}. In the limit of small $\lambda$, this theory can also be shown to confine
quarks \cite{orland-3+1}, but one of the couplings, namely $g_{0}^{\prime\prime}$, is
large. This means that the $3+1$ dimensional theory can only be studied by mixed
weak/strong-coupling
methods, unlike in $2+1$ dimensions, where no strong-coupling assumption is needed.
Some of the ideas discussed here were anticipated by others. The lattice formulation of
gauge theories as coupled sigma models was discussed more than three decades ago
in the light-cone gauge (in contrast to our use of the axial gauge)
\cite{bardeen-pearson}. There were attempts by Griffin to use integrability in $1+1$ dimensions, in this context, to understand Yang-Mills fields \cite{griffin}. Durhuus and
Fr\"{o}hlich estimated the potential energy of a quark and antiquark separated in the $2$-direction
\cite{durhuus-fr} (though not the $1$-direction), which agrees with
our leading result in powers of $g_{0}^{\prime}$ \cite{orland2+1}. See also Reference
\cite{koma}. The respect in which our work is different is
that we have successfully used this strategy to
study physical quantities, namely
the string tension and the mass spectrum, in detail.
Perturbing away from integrability to study field-theoretic and many-body systems is an active
sub-field of
statistical and condensed-matter physics (for reviews, see \cite{bhaseen-tsvelik}). The first
work in this field
is that of McCoy and Wu for the Ising
model \cite{mccoy-wu} and by Affleck and Weston \cite{affleck-weston} for spin chains.
\section{Gauge theories as coupled sigma models}
The formulation of the Yang-Mills theory as a collection of coupled sigma models has been discussed in detail elsewhere
\cite{orland2+1,hor-string,mass-spectrum}, so we will only present the result here. For simplicity we discuss the $2+1$-dimensional Hamiltonian (in which $x^{2}$ is discrete) in this section, but
the $3+1$-dimensional case \cite{orland-3+1} (for which $x^{1}$ and $x^{2}$ are discrete) is similar. We use axial gauge $A_{1}=0$, or $U_{1}=1$. The remaining lattice gauge field is $U_{2}(x^{0},x^{1}, x^{2})$, and we drop the subscript $2$. The coordinates $x^{0}$ and $x^{1}$ are continuous, but
$x^{2}$ is discrete (as mentioned earlier in this paragraph). The left-handed and right-handed currents are
$j^{\rm L}_{\mu}(x)_{b}={\rm i}{\rm Tr}\,t_{b} \, \partial_{\mu}U(x)\, U(x)^{\dagger}$ and
$j^{\rm R}_{\mu}(x)_{b}={\rm i}{\rm Tr}\,t_{b} \, U(x)^{\dagger}\partial_{\mu}U(x)$, respectively,
where $\mu=0,1$. The Hamiltonian obtained from (\ref{cont-lag-2+1}) is $H_{0}+H_{1}$, where
\begin{eqnarray}
H_{0}\!=\!\sum_{x^{2}}\int dx^{1} \frac{1}{2g_{0}^{2}}\{ [j^{\rm L}_{0}(x)_{b}]^{2}+[j^{\rm L}_{1}(x)_{b}]^{2}\}
\;,\label{HNLSM}
\end{eqnarray}
and
\begin{eqnarray}
H_{1}\!\!&\!\!=\!\!&\!\! \sum_{x^{2}} \int dx^{1} \,
\frac{(g_{0}^{\prime})^{2}a^{2}}{4}\,\partial_{1}\Phi(x^{1},x^{2})\partial_{1}\Phi(x^{1},x^{2}) \nonumber \\
\!\!&\!\!-\!\!&\!\!
\left(\frac{g_{0}^{\prime}}{g_{0}}\right)^{2}\,\,\sum_{x^{2}=0}^{L^{2}-a} \int dx^{1} \!\!
\left[ j^{\rm L}_{0}(x^{1},x^{2})\Phi(x^{1},x^{2}) -j^{\rm R}_{0}(x^{1},x^{2}) \Phi(x^{1},x^{2}+a) \right]
\nonumber \\
&+&\;\;\;\;\;(g_{0}^{\prime})^{2}q_{b}\Phi(u^{1},u^{2})_{b} -(g_{0}^{\prime})^{2}
q^{\prime}_{b}\Phi(v^{1},v^{2})_{b} \; ,
\label{continuum-local}
\end{eqnarray}
where $-\Phi_{b}=A_{0\,\,b}$ is the temporal gauge field, and
where in the last term
we have inserted two color charges - a quark with charge $q$ at site $u$
and an anti-quark with charge $q^{\prime}$ at site $v$. Some gauge invariance remains
after the axial-gauge fixing, namely that
for each $x^{2}$
\begin{eqnarray}
\left\{ \int d x^{1}\left[ j^{L}_{0}(x^{1},x^{2})_{b}-j^{R}_{0}(x^{1},x^{2}-a)_{b}\right] - g_{0}^{2}Q(x^{2})_{b} \right\}\Psi=0\;,
\label{physical}
\end{eqnarray}
on wave functionals $\Psi$,
where $Q(x^{2})_{b}$ is the total color charge from quarks at $x^{2}$ and $\Psi$ is any physical
state. To derive the constraint (\ref{physical}) more precisely, we started with open boundary
conditions in the $1$-direction and periodic boundary conditions in
the $2$-direction, meaning that the two-dimensional space is a cylinder.
The unperturbed Hamiltonian (\ref{HNLSM}) is a discrete sum of principal-chiral nonlinear sigma model Hamiltonians with ${\rm SU}_{\rm L}(N)\times {\rm SU}_{\rm R}(N)$ symmetry. This sigma model is asymptotically free and has been argued to have a mass gap. Its basic excitations are soliton-like
particles, labeled by index $r=1$, which can form bound states, labeled by an index
$r=2,\dots, N-1$ (the $r=N-1$ excitation, a bound state of $N-1$ ``elementary" $r=1$ particles, is the antiparticle of the of the $r=1$ particles). These elementary excitations of the
sigma model are color dipoles, and can be thought of as bound pairs of
chiral-Gross-Neveu Fermions.
The S-matrix of two elementary excitations of the sigma model is \cite{abda-wieg}
\begin{eqnarray}
{\mathfrak S}_{11}(\theta)
=
\frac{\sin (\theta/2-\pi{\rm i}/N)}{\sin(\theta/2+\pi{\rm i}/N)}\;S_{\rm CGN}(\theta)\otimes
S_{\rm CGN}(\theta) ,\label{s-matrix
|
}
\end{eqnarray}
where $S_{\rm CGN}$ is the S-matrix of two elementary excitations of the chiral Gross-Neveu
model \cite{cgn}:
\begin{eqnarray}
S_{\rm CGN}(\theta)\!\!=\!\!\frac{\Gamma({\rm i}\theta/2\pi+1)\Gamma(-{\rm i}\theta/2\pi-1/N) }{\Gamma({\rm i}\theta/2\pi+1-1/N) \Gamma(-{\rm i}\theta/2\pi)}
\left( 1-\frac{2\pi{\rm i}}{N\theta}P\right), \label{cgn}
\end{eqnarray}
where $P$ switches the colors of the elementary Gross-Neveu particles. Other S-matrix elements can be found by crossing and by fusion techniques. The mass spectrum of the bound states of the elementary particles is
$m_{r}=m_{1}\frac{\sin\pi r/N}{\sin \pi/N}$.
The physical interpretation is that transverse electric flux consists of the massive particles
of the sigma model. These are joined by a lighter longitudinal electric flux (to satisfy Gauss' law) which is essentially that of the $1+1$ dimensional gauge theory. For finite $N$, the bound states are not free strings, but scatter nontrivially.
\section{Confinement in $2+1$-dimensions}
The exact S-matrix and current form factor were used to study the dependence of the mass spectrum and the quark-antiquark potential on $g_{0}^{\prime}$ for gauge group SU($2$).
From the current form factor, the string tension in the longitudinal \cite{hor-string} was found to be
\begin{eqnarray}
\sigma_{\rm long}=
\frac{3 (g_{0}^{\prime})^{4} }{8 K} \;, \label{hor-string-tension}
\end{eqnarray}
where the factor $K$ is given by
\begin{eqnarray}
K=\frac{(g_{0}^{\prime})^{2}a^{2}}{4}+
\frac{1}{3m^{2}\pi^{2}}\left(\frac{g_{0}^{\prime}}{g_{0}}\right)^{4}
\exp\left[-2\int_{0}^{\infty} \frac{d\xi}{\xi} {e^{-\xi}}{\tanh^{2}\frac{\xi}{2}}\right] \;,
\label{constant-of-field-squared}
\end{eqnarray}
$m$ is the mass gap and $a$ is the lattice spacing. The string tension in the transverse direction \cite{vert-string} is
\begin{eqnarray}
\sigma_{\rm trans}=\frac{m}{a}-\frac{2{\sqrt 3}}{\pi}\frac{g_{0}^{\prime}}{g_{0}^{2}a^{2}}\;,
\label{vert-string-tension}
\nonumber
\end{eqnarray}
where $m$ is the sigma-model mass gap,
respectively. The terms of order $(g_{0}^{\prime})^{4}$
in (\ref{hor-string-tension}) come from transverse oscillations of the string. The second term in
(\ref{vert-string-tension}) comes from the smearing of color of each transverse string constituent (that is sigma-model excitation) over a region of size $m^{-1}$.
It is possible to find the spectrum of glueball states \cite{mass-spectrum}. The bound states consist of an elementary sigma-model particle and antipartcle bound by two lines of longitudinal electric flux (it has the topology of a ring). The role of the S-matrix is to determine the
matching condition of the bound-state wave
function at the origin. The spectrum
can be worked out using the WKB method. The result is
\begin{eqnarray}
M_{n}=2m+E_{n}=2m+\left[
\epsilon_{n}^{1/3}- \frac{3(3-2\ln2)\sigma_{\rm long} }{ 4\pi m} \epsilon_{n}^{-1/3}
\right]^{2},\;\;n=0,1,2,\dots \label{bound-state-spectrum}
\end{eqnarray}
where
\begin{eqnarray}
\epsilon_{n}=\frac{3\pi\sigma_{\rm long}( n+\frac{1}{2} ) }{4m^{1/2}}
+\left\{
\left[
\frac{3\pi\sigma_{\rm long}}{4m^{1/2}(n+\frac{1}{2})}
\right]^{2}
+\frac{1}{8}
\left[
\frac{3(3-2\ln2)\sigma_{\rm long}}{2\pi m}
\right]^{3}
\right\}^{1/2}.
\label{epsilon-definition}
\end{eqnarray}
This was found for the SU($2$) gauge group, but it is elementary to generalize the result to any
SU($N$).
What is interesting about these results is that they show there is no deconfining phase transition as $g_{0}^{\prime}$ is increased from zero. The leading terms in $g_{0}^{\prime}$ in all these expressions are just those found from elementary arguments \cite{orland2+1,durhuus-fr}. There is, however, a dimensional cross-over \cite{hor-string}. The
cross-over is a change from $1+1$-dimensional behavior
to $2+1$-dimensional behavior as $g_{0}^{\prime}$ is increased. This problem is still under investigation. In the case of the ${\mathbb Z}_{2}$ gauge theory, dual to the three-dimensional Ising model, the same type of cross-over has been overcome using the density-matrix renormalization group
\cite{konik} (in fact, the formulation of the three-dimensional Ising model as coupled two-dimensional Ising models is very similar to the formulation of the $2+1$-dimensional gauge theory
as coupled $1+1$-dimensional sigma models discussed here). Realistic results for the
correlation-length critical exponent $\nu$ were obtained this way.
\section{Longitudinal rescaling in $3+1$ dimensions}
As it happens, the naive classical rescaling of coordinates $x^{0,3}\rightarrow \lambda x^{0,3}$, $x^{1,2}\rightarrow x^{1,2}$, mentioned
in the introduction, is not how quantum field theories actually rescale. There are anomalous dimensions, just as there are for dilatations.
There is a straightforward Wilsonian procedure to determine how the Lagrangian changes under a longitudinal
rescaling. Suppose that there is a spherical momentum cut-off $\Lambda$ on the Fourier components of the gauge field (counterterms are
needed to restore gauge invariance). Let $\tilde \Lambda$ be less than or equal to
$\Lambda$. Consider the ``fast"
degrees of freedom, whose momenta outside of a four-dimensional ellipsoid with two major axes of
$2{\tilde \Lambda}$ and two minor axes of
$2{\tilde \Lambda}\lambda$, but inside the four-dimensional sphere of diameter $2\Lambda$. These fast degrees of freedom are integrated out
of the functional integral. Finally a longitudinal rescaling by $\lambda$ restores a spherical
cut-off, but now of diameter $2\tilde \Lambda$. The one-loop result is \cite{orland-xiao}
\begin{eqnarray}
{\mathcal L}_{\rm eff}=\frac{1}{ 4g_{\rm eff}^{2} }
\,{\rm Tr}\,\left(
{F}_{01}^{2}+{F}_{02}^{2}-{F}_{13}^{2}-{F}_{23}^{2}
+\lambda^{-2+\frac{17C_{N}}{48\pi^{2}}{\tilde g}_{0}^{2}}{F}_{03}^{2}-
\lambda^{2+\frac{7C_{N}}{48\pi^{2}}{\tilde g}_{0}^{2}}{F}_{12}^{2}
\right)+\cdots
\;,\label{effective-lag}
\end{eqnarray}
where
\begin{eqnarray}
\frac{1}{g_{\rm eff}^{2}}=\frac{1}{g_{0}^{2}}
-\frac{11C_{N}}{48\pi^{2}} \ln \frac{\Lambda}{\tilde \Lambda}
+\frac{C_{N}\ln {\lambda}}{32\pi^{2}}
\;. \label{effective-coupling}
\end{eqnarray}
The corrections to (\ref{effective-lag}) are of order $\ln \lambda^{2}$.
What (\ref{effective-lag}) shows is that one cannot really trust perturbation theory to find the
effective longitudinally-rescaled action at {\em very} high energies. The point is that
(\ref{effective-lag}) is really only valid at weak coupling, as the corrections from the anomalous dimension are not significant. Large rescalings generate a
large coupling associated with ${\rm Tr} F_{12}^{2}$. The coefficient of this term is small, meaning that the longitudinal magnetic field is wildly fluctuating. We cannot take this action too seriously for small $\lambda$, just as we cannot take strong-coupling approximations in
lattice or AdS-type QCD theories too seriously. A ``good" strongly-coupling theory is one in which
we have somehow correctly integrated out the high-momentum degrees of freedom over very large ranges of momentum. For further discussion of this point, see Reference \cite{orland-xiao}.
Since we cannot rigorously extend the expression (\ref{effective-lag}) to $\lambda \ll1 $, the best
we can do is guess the form of an effective theory at high energy. Probably the most sensible approach is to replace the sigma model of Reference \cite{orland-3+1} by an arbitrary $1+1$-dimensional interacting massive
field theory with global ${\rm SU}(N)_{\rm L}\times {\rm SU}(N)_{\rm R}$ symmetry, coupling its currents together to $A_{0}$. Though not QCD, this theory
has local SU($N$) gauge symmetry. Such an effective
theory certainly exhibits a forward peak, and may be a good model of soft scattering.
\section{Some new directions}
In $2+1$ dimensions, the main problem is to overcome the cross-over, to understand the isotropic
case. As mentioned already, this has been accomplished for the ${\mathbb Z}_{2}$ case
\cite{konik}. The problem should perhaps be easier for SU($N$) theories, as the critical point
is the same for both the isotropic and anisotropic theories; it is simply at
$g_{0}=g_{0}^{\prime}=0$.
Though string tensions have been studied for arbitrary SU($N$)
\cite{arbN} and the mass spectrum is possible to determine (though this has not been published yet) the corrections to the string tensions in powers of $g_{0}^{\prime}$ cannot be found until the form factors of the sigma model are determined. We have made some progress on the $1/N$-expansion of these form factors (the bound-state structure is much simpler for large $N$). This should make it possible to study string dynamics more generally, as the sigma S-matrix becomes trivial in this limit.
It seems worthwhile to study effective SU($N$) gauge theories discussed at the end of the last section in $3+1$ dimensions. These are effective parton models of soft scattering. In particular, it appears that the AGK cutting rules are valid
\cite{AGK}. Whether
other useful results can be obtained is an open question.
|
\section{Introduction and setting}
The notion of persistent homology arises when one has a filtration of spaces; viz.\ a sequence (or continuum) of spaces
${\mathcal Z}_1\subseteq{\mathcal Z}_2\subset\dots$ (or ${\mathcal Z}_t$, with ${\mathcal Z}_s\subseteq{\mathcal Z}_t$ whenever $s\leq t$) and one is interested in how homology changes as one moves along the sequence. For a typical example, suppose that ${\mathcal Z}$ is a nice space, and let $f:{\mathcal Z}\to\mR$ be a smooth function. Denote by ${\mathcal Z}_u$ the filtration of excursion, or upper-level, sets
\begin{eqnarray}
\label{Au-defn}
{\mathcal Z}_u \ \stackrel{\Delta}{=} \{z\in {\mathcal Z} : f(z) \in [u,\infty)\} \
\equiv\ f^{-1}([u,\infty)).
\end{eqnarray}
A useful way to describe persistent homology is via the notion of barcodes.
Assuming that $\dim ({\mathcal Z})=D$, the smoothness of $f$ implies
that, if ${\mathcal Z}_u$ is non-empty, then $\dim ({\mathcal Z}_u)$ will typically also be $D$. A barcode for the excursion sets of $f$ is then a collection of $D+1$ diagrams,
one for each collection of homology groups of common order. A bar in the
$k$-th diagram, starting at $u_1$ and
ending at $u_2$ ($u_1\geq u_2$) indicates the existence of a generator
of $H_k({\mathcal Z}_u)$ that appeared at level $u_1$ and disappeared at level $u_2$.
A different, and visually helpful, representation of a bar is as a point $(d,b)$ in the plane. Each bar has a `birth time' $b$ and `death time' $d$, where $d<b$ since, as described above, the filtration is for upper level sets, and we index these by levels descending from $+\infty$. The collection of points $(d,b)$ corresponding to all the bars is called the persistence diagram.
(We shall assume that the reader is familiar with these concepts. Recent excellent and quite different books and reviews by Carlsson \cite{Carlsson-review,CarlssonReview},
Edelsbrunner and Harer \cite{EdelsShortCourse,EdelsHarerSurvey,EdelsHarerBook}, Zomorodian \cite{Afra},
Oudot \cite{Oudot} and Ghrist
\cite{ghrist2014elementary} not only give
give broad expositions of homology, but also treat the much newer subject of persistent homology. (For a description of the history of persistent homology see the Introduction to \cite{EdelsHarerSurvey}.))
Persistence diagrams almost always arise as topological summaries of some underlying phenomenon, and, having been constructed, are typically subject to some kind of analysis. This can be thought of as a path
\begin{equation}
\label{eq:to}
phenomenon \ \to\ persistence\ diagram \ \to \ analysis.
\end{equation}
The analysis can be of various forms. Wasserman \cite{wasserman-review} gives a comprehensive and up to date review on topological data analysis from the viewpoint of statistics, but there are also non-statistical approaches, many of which involve summarizing the diagram with either a low dimensional vector of numerical descriptors, a large dimensional vector, or a real valued function. Many of these approaches adopt techniques such as
principal component analysis and support vector machines to analyse the summary data.
What is common to all these approaches, however, is the need for multiple instances of the persistence diagram, which in practical situations, is typically
not a trivial requirement. Although, in some scenarios, multiple observations of the `phenomenon' of \eqref{eq:to} may be available, it is more common that only one observation of the phenomenon is available, and so only one diagram. In those cases, the standard method to effectively increase the number of instances is via resampling, either of the phenomenon or the diagram. Virtually all of the above approaches have examples of this method.
In a previous paper \cite{adleragamprat} we entered the diagram \eqref{eq:to} at the intermediate step, by suggesting a new approach to providing multiple instances of a persistence diagram when, perhaps, only one such original diagram is available. This was done via probabilistic modeling of persistence diagrams.
We shall briefly recall the basic ideas of \cite{adleragamprat} in Sections 2 and 3 below, and then develop them, in terms of more sophisticated examples than were treated there, in Section 4.
\section{Parametric model}
\subsection{The basic setup}
\indent As above, let ${\mathcal Z}$ be a compact subset of $\mathbb R^D$, typically a sub-manifold or stratified sub-manifold, and suppose that we observe a sample ${\tilde Z}_n=\{Z_1,\dots,Z_n\}$ drawn from a distribution $P$ supported on ${\mathcal Z}$.
Based on this sample, we define a kernel density estimator, $\hat f_n$, given by
\begin{eqnarray}
\hat f_n(p) \ = \ \frac{1}{n(\sqrt{2\pi}\eta)^D} \sum_{i=1}^{n} e^{{-\|p-z_i\|^2}/{2\eta^2}},\qquad p\in\mathbb R^D,
\end{eqnarray}
where $\eta >0$ is a bandwidth parameter for the Gaussian kernel defining $\hat f_n$.
Our interest is in the persistence diagram generated by the upper-level set filtration generated by $\hat f_n$; viz.\ by the sets of \eqref{Au-defn} as $u$ decreases from $+\infty$.
The death and birth points in the diagram are denoted by ${(d_i,b_i)}_{i=1}^{N_k}$, where $N_k$ is the number of points in the diagram for the homology of order $k$. Typically, we shall treat only one order at a time, and drop the subscript on $N_k$.
Define a new set of $N$ points ${\tilde x}_N =\{x_i\}_{i=1}^N$, with $x^{(1)}_i=d_i$ and
$x^{(2)}_i=b_i-d_i$. That is, ${\tilde x}_N$ is a set of $N$ points in ${\mathcal X}=\mR\times\mR_+$.
This (invertible) transformation has the effect of moving the points in the original persistence diagram downwards, so that the diagonal line projects onto the horizontal axis, but still leaves a visually informative diagram, called the projected persistence diagram, or PPD, in \cite{adleragamprat} . The first step towards the goal of a statistical analysis of the persistence diagram is to develop a parametric, probabilistic model for ${\tilde x}_N$.
\subsection{The model}
\label{sec:Gibbs}
Following \cite{adleragamprat}, as a first step of building a model for ${\tilde x}_N$, consider the Gibbs distribution,
\begin{eqnarray}
\varphi_\Theta({\tilde x}_N) = \frac{1}{Z_\Theta} \exp ( - H_\Theta ({\tilde x}_N)),
\label{eq:Gibbs}
\end{eqnarray}
where $\Theta$ is a multi-dimensional parameter, $H_\Theta :{\mathcal X}\to\mR$ is a `Hamiltonian' that
describes the `energy' of ${\tilde x}_N$, and $Z_\Theta$ is the normalizing `partition function' required to make $\varphi_\Theta$ a probability density.
The next step is to choose the Hamiltonian, which needs to take into account the spread of the $N$ points, along with interactions between neighboring
points, which we shall think of as belonging to clusters. Regarding the spread, define
\begin{eqnarray*}
\sigma_H^2 = \sum_{x\in{\tilde x}_N} \big(x^{(1)} - \bar x^{(1)}\big)^2,\ \ \
\sigma_V^2 = \sum_{x\in{\tilde x}_N} \big(x^{(2)} \big)^2,
\end{eqnarray*}}%\pagestyle{plain
where $\bar x^{(1)}=N^{-1} \sum_{i=1}^N x_i^{(1)}$.
Then $\sigma_H^2$ is the (un-normalized) variance of the horizontal points, and $\sigma_V^2$ is the $L_2$ power of the vertical points (not centered because of the non-negativeness of $x^{(2)}$). As for the local interaction, for $x\in{\mathcal X}$ and for $k\geq 1$ let
$x^{nn}(k) \in {\mathcal X}$ be the $k$-th nearest neighbor of $x$, and set
\begin{eqnarray*}
\mathcal L_{\delta,k}({\tilde x}_N) = \sum_{x\in{\tilde x}_N} \|x-x^{nn}(k) \| \mathbbm{1}_{\{\|x-x^{nn}(k) \|\leq \delta \} }.
\end{eqnarray*}}%\pagestyle{plain
Then, as in \cite{adleragamprat}, we choose the Hamiltonian
\begin{eqnarray}
\label{eq:hamiltonian}
H_{\delta,\Theta}^K({\tilde x}_N)
= \theta_H \sigma^2_H
+\theta_V \sigma^2_V
+\sum_{k=1}^K \delta^{-2}\theta_k \mathcal L_{\delta,k}({\tilde x}_N),
\end{eqnarray}
where $\Theta=(\theta_H,\theta_V,\theta_1,\dots,\theta_K)$, and $K$ is the maximal cluster size. The inclusion of the normalising parameter $\delta^{-2}$ allows for the $\theta_k$ to be interpreted as energy densities, and improves the numerical stability of parameter estimation.
There are a number of reasons for this choice of
Hamiltonian, among them:
\begin{itemize}
\item[(i)] Cluster expansions of this form have been successfully employed in Statistical Mechanics for the best part of a century as a basic approximation tool in the study of particle systems. More specifically, for the model to be rich enough for TDA,
one needs to choose the Hamiltonian from a parameterised family that comes close to spanning all `reasonable' functions on PPDs. Since \cite{GunnarRing} showed that the ring of algebraic functions on the space of PPDs is spanned by a family of monomials
closely related to functions of the form \eqref{eq:hamiltonian}, this choice of Hamiltonian is an effective one for our setting.
\item[(ii)] These distributions are often used not as exact models for PPDs, but rather as a tool in a perturbative analysis. In these cases, the convenience of the models is more important than whether or not they provide a perfect fit to PPD data. For more details see \cite{adleragamprat}, SI (Sec.\ 2.2).
\end{itemize}
Finally, $\delta$ is determined by
\begin{eqnarray}
\label{eq:delta}
\delta =\frac{\delta^{*}}{N^{\alpha_{k,d}}}
\max\left( \max |x^{(1)}_i - x^{(1)}_j|,\ \max |x^{(2)}_i - x^{(2)}_j| \right),
\end{eqnarray}
where $\alpha_{0,d}=1/d$, $\alpha_{k,d}={k/(k+1)d}$, for $k\geq 1$, $d$ is the dimension of the data underlying the persistence diagram,
and $\delta^{*} $ is a data independent tuning parameter. Although we typically optimize over $\delta^{*}$, it can be taken to be $N^{-1/2}$, as a global default.
\subsection{Estimation and model specification}
Since there is no analytic form for $Z_\Theta$ and it is impossible to compute it numerically in any reasonable time, we cannot estimate $\Theta$ by maximum likelihood. A standard way around this, adopted in \cite{adleragamprat}, is to use pseudolikelihood estimation \cite{Besag,chalmond}; viz.\ to maximize the pseudolikelihood
\begin{eqnarray}
\label{eq:pseudo}
L^K_{\delta,\Theta}({\tilde x}_N) \stackrel{\Delta}{=}
\prod _{x\in{\tilde x}_N} f_\Theta \left(x\big| {\mathcal N}_{\delta,K}(x) \right).
\end{eqnarray}
Here
${\mathcal N}_{\delta,K}(x)$ denotes the collection of the $K$ nearest neighbours of $x$ in ${\tilde x}_N$ whose distance from $x$ is no greater than $\delta$, and
\begin{eqnarray}
f_\Theta\left(x\big| {\mathcal N}_{\delta,K}(x) \right)=\frac{
\exp \left(-H^K_{\delta,\Theta}\left(x\big| {\mathcal N}_{\delta,K}(x) \right)\right)
}{
\int _{\mR}\int _{\mR_+} \exp \left(-H^K_{\delta,\Theta}\left(z\big| {\mathcal N}_{\delta,K}(x) \right)\right)\,dz^{(1)}dz^{(2)},
}
\label{eq:conditionalham}
\end{eqnarray}
with
\begin{eqnarray*}
H^K_{\delta,\Theta}\left(x\big| {\mathcal N}_{\delta,K}(x)\right)=\theta_H\left[x^{(1)}-\bar x^{(1)}\right]^2+\theta_V(x^{(2)})^2 +\sum_{k=1}^K\delta^{-2} \theta_k{\mathcal L}_{\delta,k}\left({\mathcal N}_{\delta,K}(x)\right).
\end{eqnarray*}}%\pagestyle{plain
Optimal values of $K$ can be chosen via standard, automated, statistical procedures such as AIC, BIC, etc \ (cf.\ \cite{burnham}). However, considerable experimentation, much of it reported in \cite{adleragamprat}, leads to the conclusion that it suffices to take $K=2$ or $K=3$, so that the largest cluster size is 3 or 4.
\section{Replicating persistence diagrams}
\subsection{MCMC}
\label{Sec:MCMC}
Once the parametric distribution of the points of the persistence diagram is available, as in the pseudolikelihood \eqref{eq:pseudo}, simulated replications of the diagram can be generated using a standard Metropolis-Hastings MCMC algorithm \cite{RobertCasella,Handbook}.
Firstly, given a ${\tilde x}_N$, define a `proposal distribution'
$q(\cdot |{\tilde x}_N)$ as the bivariate Gaussian density, with mean vector and covariance matrix identical to the empirical mean and covariance of the points in ${\tilde x}_N$, but restricted to $\mR\times\mR_+$.
Next, for two points $x,x^*\in\mR\times\mR_+$ define an `acceptance probability', according to which $x\in{\tilde x}_N$ is replaced by $x^*$, leading to the updated PPD ${\tilde x}_N^*$, as
\begin{eqnarray*}
\rho \left(x, x^*\right)
= \min \left\{1,\frac{f_\Theta\left(x^*| N_{\delta,K}(x)\right) \cdot q(x|{\tilde x}^*_N)}{f_\Theta \left(x| N_{\delta,K}(x)\right) \cdot q(x^*|{\tilde x}_N)}\right\}.
\end{eqnarray*}}%\pagestyle{plain
\noindent Then the algorithm is Algorithm \ref{MCMC:algorithm}.
\begin{algorithm}
\caption{MCMC step updating diagram for ${\tilde x}_N$}
\label{MCMC:algorithm}
\begin{algorithmic}[1]
\State $k =0$
\State $k \gets k+1$
\State Choose $x^*$ according to $q(\cdot | {\tilde x}_N)$
\State Compute $\rho(x_k,x^*)$
\State Choose $U$ a standard uniform variable on $[0,1]$
\If{$U<\rho(x_k,x^*)$} set $x_k =x^*$
\EndIf
\If{$k<N$} go to Step 2
\EndIf
\end{algorithmic}
\end{algorithm}
To obtain $M$ approximately independent PPD's, \cite{adleragamprat}
adopt a procedure dependent on three parameters, $n_b$, $n_r$ and $n_R$, as follows.
Starting with the original PPD, run the algorithm for a burn in period. Then, starting with the final PPD from the burn in, run the algorithm a further $n_b$ times, choosing the last output of this block of $n_b$ iterations as the first simulated PPD. Repeat this procedure $n_r$ times, each time starting with the most recently simulated PPD; viz.\ the output of the previous block. Finally, replicate the entire procedure $n_R$ times, for a total of
$n= n_r\times n_R$ simulated PPDs.
The optimal choice of $n_b$, $n_r$ and $n_R$ typically depends on the specific problem, and is discussed in \cite{adleragamprat} SI (Sec.\ 2.1).
The burn in period is determined, empirically, via the evaluation of the distance between the MCMC simulations and the original persistence diagrams.
Recall that, for two diagrams $D_1$ and $D_2$, the Wasserstein $p$-distance, $W_p(D_1,D_2)$, $p>0$, is defined as
\begin{eqnarray}
\label{eq:SIWass}
W_p\left(D_1,D_2\right) \ =\ \inf_\gamma \big(
\sum_{u\in D_1} \|u-\gamma (u)\|_\infty^p\big)^{1/p}
\end{eqnarray}
where $\gamma$ ranges over all matchings between the points of $D_1$ and $D_2$, the latter having been augmented by adding all points on the diagonal. In the limit case of $p=\infty$ the Wasserstein distance is known as the bottleneck distance, which is the length of the longest edge in the best matching.
The distances between the MCMC simulations and the original persistence diagrams are measured via the bottleneck and Wasserstein distances as the MCMC progresses, and the value of the burn in period is chosen as the point at which the initial rapid growth of the distance functions ceases. Given the collection of $M$ simulated PPDs, each PPD is converted back to a regular persistence diagram with the mapping {$x\to ( x^{(1)},x^{(1)}+x^{(2)})=(d,b)$ }of its component points.
\subsection{Identification of topological signals}
In many situations, which include all the examples that we shall treat in this paper, the most prominent features of the persistence diagram are generally deemed most likely to represent true features of the underling space, rather than artifacts of sampling or noise. By `prominent' we mean those points in the diagrams which are furthest from the diagonal. These are typically called `topological signals', while the points closer to the diagonal are considered to be `topological noise'.
One of the key challenges in persistent homology is to separate the signal from the noise.
The replicated persistence diagrams can be used to identify the topological signals by providing information about statistical variation. As an example, consider the order statistics of the distances of the points of the persistence diagram to the diagonal. That is, given the points $(d_i,b_i)$ of the persistence diagram, the order statistics are $T_j$, the $j-$th largest among the differences $|b_i-d_i|$, $j=1,...,N$.
Denote by $\hat{T}$ the value of the relevant statistic based on the true persistence diagram, and denote by $\hat{T}^{*} $ the value of the relevant statistic based on the simulated persistence diagram.
Then one can calculate an empirical confidence interval (percentile bootstrap) $[c_1,c_2]$ and define a point on the persistence diagram to be a signal if $\hat{T}<\hat{c}_{1} $ or $\hat{T}>\hat{c}_{2} $.
Note that since here $T$ is non-negative, we generally only consider one-sided confidence intervals $[0,\hat{T}+\hat{c}_{2} ]$. In addition, one can calculate a one-side $p$-value as $\frac{1}{M} \sum _{m=1}^{M}{\mathbbm{1}}_{\left(\hat{T}_{m}^{*} \ge \hat{T}\right)} $, where $\hat{T}_{m}^{*}$ is $\hat{T}^{*}$ of the $m$-th simulated persistence diagram.
\section{Examples}
We now turn to the three examples which make up the new material of this paper. In each of these we show how to use the methodology described in the previous sections to identify the homology of spaces $\mathcal Z$, when all that is available is the persistence diagram generated by the upper level sets of a smoothed empirical density from a sample. The examples that we shall treat are those of a 2-sphere, a 2-torus and a collection of three concentric circles in the plane. Each of these will teach us something different about the behaviour of our methodology in practice.
\subsection{The two dimensional sphere}
\subsubsection{The data and fitting the model}
We start with a random sample of $n=1,000$ points from the uniform distribution on the sphere $S^2$ in $R^3$ with radius $r=1$, and smooth the data with a kernel density estimator of bandwidth of $\eta=0.1$. These are shown as Panels (a) and (b) of Figure \ref{fig:sphere}.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{sphere_plot_shrink}\ \ \ \
}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{sphere3_plot_shrink} \ \ \ \
}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{SpherePD01n1000}
}
\caption{\footnotesize
(a) Points sampled from a unit sphere. (b) The corresponding kernel density estimator, shown, for visual clarity, at only a few quantized levels. (c) The corresponding persistence diagram for the upper level sets of the kernel density estimate on the full sphere. Black circles are $H_0$ persistence points, red triangles are $H_1$ points, and the blue diamond is the $H_2$ persistence point. Birth times are on the vertical axis.}
\label{fig:sphere}
\end{figure}
The corresponding persistence diagram of the upper level sets filtration of $\hat f_N$ is shown in Panel (c) of Figure \ref{fig:sphere}. This diagram contains $N_0=110$ points corresponding to the zeroth homology $H_0$, represented by the black circles, $N_1=74$ points for the first homology $H_1$, represented by the red triangles, and $N_2=1$ point for the second homology $H_2$, represented by the blue diamond. As described above, each point in the diagram is a `death-birth' pair $(d,b)$. Since we know that the upper level sets of $\hat f_N$ are characterized by having a single connected component and a single void, we expect to have one black circle somewhat isolated from the other points in the diagram and one blue diamond. The void does not have to be isolated from the other points due to a short lifetime of high dimensional homologies. This is in fact the case.
While the persistence diagram in Figure \ref{fig:sphere} performs as expected, and it is easy to identify the points that, a priori, we knew had to be there, there are many other points in the diagram which, were we not in the situation of knowing ahead of time, and we would have difficulty in knowing how to discount.
Note that there are more than enough $H_0$ and $H_1$ points in Figure \ref{fig:sphere} to fit a spatial model to each of the two homologies.
Adopting the approach described in the first three sections of the paper, and working first with the $H_0$ persistence diagram without including the `point at infinity\footnote{In all our persistence diagrams, the `point at infinity' is the highest, leftmost point in the $H_0$ diagram. In essence, removing it from the analysis is much like working with reduced rather than standard homology, and has the effect of removing one generator from the $H_0$ diagram. Thus, in the statistical analysis to follow, it needs to be added, at the end, to all significant points found in the diagram.}', we estimated the parameters for a Gibbs distribution for the model with pseudolikelihood \eqref{eq:pseudo}, taking $K=3$. The estimate of $\delta$ was 0.0051. For this $\delta$, the estimates of $\Theta$ were $\theta _{1} = -0.0339$, $\theta _{2} = -0.0210$, $\theta _{3} = -0.0120$, $\theta _{H} =72.80$, and $\theta _{V} =39.50$.
In order to test how well the estimated model matches the persistence diagram, we followed the procedure described in \cite{adleragamprat}. We generated 100 collections of samples from the 2-sphere according the same procedure that generated the original data, and for each we fitted the model we found for the original data set; viz.\ the model that includes the parameters $\theta _{1}$, $\theta _{2}$, $\theta _{3}$, $\theta _{H}$, and $\theta _{V}$.
The blue plot in Figure \ref{fig:resPDestsph1} shows the (smoothed) empirical densities of the resulting parameter estimates\footnote{In some of these simulations the sums $L_{\delta,k}$ were identically zero for all $k=1,2,3$ simultaneously, since there were no $k$-th nearest neighbours at distance less than $\delta$. Consequently, the parameters $\theta_1$, $\theta_2$, and $\theta_3$ are all meaningless, and so these simulations (33 of them) were deleted from this part of the analysis. We shall do the same later on, in similar cases, without further comment.}.
(We will discuss the other two plots only later, when considering different replication procedures.)
Overall, the results indicate that the estimation procedure is stable, with an acceptable spread.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[scale=.18]{sphr3_h0_good_thet1}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr3_h0_good_thet2}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr3_h0_good_thet3}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr3_h0_good_thet4}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr3_h0_good_thet5}
}
\caption{\footnotesize
Smoothed empirical densities for the five parameter estimates in the Hamiltonian \eqref{eq:hamiltonian} for the
$H_0$ persistence diagram coming from the simulations of 2-sphere and from resampling, see text for details. (a) $\theta_1$, (b) $\theta_2$, (c) $\theta_3$, (d) $\theta_H$, (e) $\theta_V$.}.
\label{fig:resPDestsph1}
\end{figure}
\subsubsection{Replicating the persistence diagram}
For the calculation of the replicated persistence diagrams, we first need to determine the burn in period which we shall use for them. Following the procedure described in Section \ref{Sec:MCMC}
we calculated the bottleneck and Wasserstein distances using the 100 simulated persistence diagrams of the previous subsection. The results are shown in blue in Figure \ref{sphburn}.
The first row in Figure \ref{sphburn} shows the bottleneck distances, while the second row shows the $W_2$ differences. The first column shows the results of the first 50 steps of the MCMC algorithm on a linear scale. The second and third columns go out to 2,000 steps, first on a linear scale and then on a logarithmic scale. While the initial growth of the distances is rapid, they eventually approach their asymptotes at exponential rates. The rapidity is clear in Panels (a) and (d), and the exponential rate is clear from the linear behavior of the plot in logarithmic scales. The point where the initial rapid growth of the distance functions ceases is approximately 44 for the bottleneck distance and 47 in the Wasserstein case. At 44 steps, therefore, the results of Figure \ref{sphburn}
indicate that the dependence of the MCMC on the initial persistence diagram has dropped significantly, while at the same time the MCMC has produced persistence diagrams remaining close to the true distribution.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[scale=.18]{sphr_h0_burn3_bottle50}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn3_bottle2000}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr_h0_burn3_bottlelog2000}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr_h0_burn3_wasser50}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn3_wasser2000}
}
\subfigure[]
{
\includegraphics[scale=.18]{sphr_h0_burn3_wasserlog2000}
}
\caption{\footnotesize
Growth of the bottleneck (a) and Wasserstein (d) differences of MCMC simulations from a specific persistence diagram (vertical axis), as a function of the number of steps $n_b$ (horizontal axis, $1\leq n_b\leq 50$) averaged over 100 independent persistence diagrams. Panels (b) and (e) take $1\leq n_b\leq 2,000$, while (c) and (f) show the same data but on a logarithmic scale.}.
\label{sphburn}
\end{figure}
In addition we considered summary statistics of the 100 persistence diagrams as the MCMC progressed, to see how well the simulations replicate the statistical properties of the persistence diagrams. The results are presented in Figure \ref{fig:mcmcsphr}. Overall, the best fits are at burn in of 10, 25 and 50, which is consistent with the results of Figure \ref{sphburn}.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[scale=.09]{
|
sphr_h0_burn10_N1good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn25_N1good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn50_N1good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn1000_N1good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn10_N2good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn25_N2good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn50_N2good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn1000_N2good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn10_N3good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn25_N3good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn50_N3good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h0_burn1000_N3good}
}
\caption{\footnotesize
Summary statistics of average interaction strengths for 100 persistence diagrams. From left to right: cluster sizes 2, 3, and 4. From top to bottom, after 10, 25, 50, and 1,000 MCMC steps. See text for details.}
\label{fig:mcmcsphr}
\end{figure}
\subsubsection{Resampling}
The replicated persistence diagrams described in the previous subsection were based on knowing, a priori, that the original data was generated by sampling from a 2-sphere. The typical real-life situation is that one does not know the space generating the persistence diagram. (If one did, it would hardly be necessary to estimate its homology by sampling.) Consequently, we now look at resampling as a method for generating replications of the persistence diagram.
There are two natural approaches based on resampling. One is to resample from the original persistence diagram (``Setting I"), and another is to resample from the original data (``Setting II"). We examined both these alternatives, repeating them 100 times.
The results of these approaches are the other two plots of Figure \ref{fig:resPDestsph1}. The red (dot dashed) plot is the smoothed empirical density for the parameter estimates based on resampled sets from the original persistence diagram, and the yellow (dashed) plot corresponds to the resampled sets from the original data.
In order to assess the fit of the simulated data to the original, we computed, as previously, the bottleneck and the Wasserstein distances between the MCMC simulations and the data itself. The results are in Figure \ref{sphburn}, in addition to the results based on the 100 simulated persistence diagrams. The red (dot dashed ) plot shows the results for the 100 resampled sets from the original persistence diagram, and the yellow (dashed) plot corresponds to the 100 resampled sets from the original data. The point where the initial rapid growth of the distance functions ceases is approximately 22 and 46, respectively, in Setting I and Setting II for the bottleneck distance, and approximately 20 and 48 in the Wasserstein case.
This suggests taking a burn in period of 50 for generating the replicated persistence diagrams for $H_0$.
\subsubsection{$H_1$ persistence diagram}
We now turn to the analysis of the $H_1$ persistence diagram. Again estimating the parameters for the Giibs pseudolikelihood \eqref{eq:pseudo}, taking $K=3$, the estimate of $\delta$ was 0.0047. For this $\delta$, the estimates of $\Theta$ were $\theta _{1} = -0.0331$, $\theta _{2} = 0$, $\theta _{3} = 3.3842$, $\theta _{H} =60.00$, and $\theta _{V} =110.00$.
To check the match between the estimated model and the $H_1$ persistence diagram, we used the same 100 simulated sets of the 2-sphere used for the $H_0$ diagram, following the same procedure that we adopted then, this time restricting to a model with only
$\theta _{1}$, $\theta _{3}$, $\theta _{H}$, and $\theta _{V}$ non-zero.
The blue plot in Figure \ref{fig:resPDestsph1_h1} shows the smoothed empirical densities for the parameter estimates generated by these simulations.
As for the $H_0$ case, the results indicate that the estimation procedure is stable.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_thet1good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_thet3good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_thet4good}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_thet5good}
}
\caption{\footnotesize
Smoothed empirical densities for the four parameter estimates of
$H_1$ persistence diagram coming from the simulations of 2-sphere, see text for details. (a) $\theta_1$, (b) $\theta_3$, (c) $\theta_H$, (d) $\theta_V$.}.
\label{fig:resPDestsph1_h1}
\end{figure}
\subsubsection{Replicating the $H_1$ persistence diagram}
As for the analysis of the $H_0$ diagram, we calculated bottleneck and the Wasserstein distances between the original persistence diagram using those corresponding to 100 MCMC simulated diagrams of the previous section. The results are shown by the blue plots in Figure \ref{fig:burnsphr2}.
The first row in Figure \ref{fig:burnsphr2} shows the bottleneck distances, while the second row shows the $W_2$ differences. The first column shows the results of the first 50 steps of the MCMC algorithm on a linear scale. The second and third columns go out to 2,000 steps, first on a linear scale and then on a logarithmic scale. The point where the initial rapid growth of the distance functions ceases, is approximately 44 for the bottleneck distance and 47 in the Wasserstein case.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_bottle50}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_bottle2000}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_bottlelog_2000}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_wasser50}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_wasser2000}
}
\subfigure[]
{
\includegraphics[scale=.09]{sphr_h1_burn3all_wasserlog2000}
}
\caption{\footnotesize
Growth of the bottleneck (a) and Wasserstein (d) differences of MCMC simulations from a specific persistence diagram (vertical axis), as a function of the number of steps $n_b$ (horizontal axis, $1\leq n_b\leq 50$) averaged over 100 independent persistence diagrams. Panels (b) and (e) take $1\leq n_b\leq 2,000$, while (c) and (f) show he same data but on a logarithmic scale.}
\label{fig:burnsphr2}
\end{figure}
In addition we considered summary statistics of the 100 simulated persistence diagrams as the MCMC progressed, to ensure that the simulations reliably replicate the statistical properties of the persistence diagrams. Here the best fits were for a burn in of 50, which is consistent with the results of Figure \ref{fig:burnsphr2}.
\subsubsection{Resampling $H_1$}
As for the $H_0$ case, we again examine the performance of resampling from the original persistence diagram (Setting I) and from the original data (Setting II), repeating each procedure 100 times.
The results are summarised in Figure \ref{fig:resPDestsph1_h1}. The red (dot dashed) plots are the smoothed empirical densities for the parameter estimates in Setting I, while the yellow (dashed) plot correspond to Setting II.
In order to assess the fit of the simulated data to the original, we computed, as previously, the bottleneck and the Wasserstein distances between the MCMC simulations and the data itself. The results are presented in Figure \ref{fig:burnsphr2}, in addition to the results based on the 100 simulated persistence diagrams. The red (dot dashed ) plot shows the results for the 100 resampled sets from the original persistence diagram, and the yellow (dashed) plot shows the same thing, but for the 100 resampled sets from the original data. The point where the initial rapid growth of the distance functions ceases, is approximately 22 and 46 in Setting I and Setting II, respectively, for the bottleneck distance, and approximately 20 and 48 in the Wasserstein case.
This suggests taking a burn in period of 50 for generating the replicated persistence diagrams for $H_1$.
\subsubsection{Statistical inference}
We are now finally in a position to carry out a simulation study to test how well we can identify the homology of 2-sphere, using the methodology described earlier. To do so, we
generated 1,000 persistence diagrams from the fitted model, via MCMC, with a burn in period of 50 iterations and with $(n_b,n_r,n_R)$ given by (500,10,100), (500,20,50), (500,40,25), or (500,100,10).
Using these four sets of simulations, we computed the maximum statistics $T_1$, its confidence interval and its $p$-value, for both the $H_0$ and $H_1$ persistence diagrams. Table \ref{table:spherestat} summarizes the results.
\begin{table}[h!]
\begin{center}
\fontsize{8.5}{0.9}\selectfont
\begin{tabular}{llccccc}
homology&statistic&real PD &$(n_b, n_r, n_R)$ & CI & $p$-value& significance\\
\\
\\
\\
\\
$H_0$&$T_1$& 0.4769 &(500,10,100)&[0, 0.4769]&0.0990 & no\\
\\
& & &(500,20,50)&[0, 0.4769]&0.0520 & no\\
\\
& & &(500,40,25)&[0, 0.3273]&0.0320 & yes\\
\\
& & &(500,100,10)&[0, 0.2616]&0.0100 &yes\\
\\
$H_1$&$T_1$& 0.1673 &(500,10,100)&[0, 0.2140]&0.4060 & no\\
\\
& & &(500,20,50)&[0, 0.2069]&0.3780 & no\\
\\
& & &(500,40,25)&[0, 0.2065]&0.3550 & no\\
\\
& & &(500,100,10)&[0, 0.1995]&0.3270 &no\\
\\
\\
\\
\\
\end{tabular}
\end{center}
\caption{{ {\footnotesize Maximum statistic $T_1$ for the real $H_0$ and $H_1$ persistence diagram and the simulated $H_0$ and $H_1$ persistence diagrams of the 2-sphere. The CI is a one-sided confidence interval at a $5\%$ confidence level. The $p$-value is also one-sided. Both the CI and the $p$-value are based on 1,000 simulated persistence diagrams.}}}
\label{table:spherestat}
\end{table}
\normalsize
The results for the $H_0$ persistence diagram show that $T_1$, in two first scenarios, was statistically insignificant, and in the two other scenarios was significant. In other words, the evidence is split between one connected component (represented by the `point at infinity' not included in the analysis) and two components. The fact that the correct result occurs in the cases of a larger number of shorter MCMC runs is consistent with earlier findings in \cite{adleragamprat}.
As for the $H_1$ topology, all four scenarios showed that $T_1$ was insignificant for all
MCMC parameter, implying, correctly, a trivial $H_1$ homology.
\subsection{2-torus}
We now turn to our second example, that of the two-dimensional torus.
Since the analysis will be similar in approach to that for the two-dimensional sphere, we will give fewer details, concentrating primarily on the more important differences in the results.
\subsubsection{The data and fitting the model}
This example includes a sample of $n=1,000$ points from the 2-torus $T^2=S^1\times S^1$ in $R^3$, chosen uniformly with respect to the natural Riemannian metric induced on it as a subset on $\mathbb R^3$. This leads to the high density of points in the `interior' of the torus, obvious from Figure \ref{fig:torus}. (For more details on sampling from tori and other manifolds, see \cite{Persi}.)
More specifically, the torus was taken to be the rotation about the `$z$ axis' in $\mathbb R^3$ of a circle of radius $1.8$ with center in the `$(x,y)$ plane' at distance 2 from the origin.
Panel (a) in Figure \ref{fig:torus} shows the sample superimposed on the torus, and Panel (b) shows the corresponding
kernel density estimator based on a bandwidth of $\eta=0.2$.
The corresponding persistence diagram of the upper level set filtration of $\hat f_N$ is Panel (c). This diagram contains $N_0=216$ points of the zeroth homology $H_0$, represented by the black circles, $N_1=160$ points of the first homology $H_1$, represented by the red triangles, and $N_2=216$ points of the second homology $H_2$, represented by the blue diamonds. Since we know that the upper level sets of $\hat f_N$ are characterized by having a single connected component, two holes, and a single void, we expect to have one black circle and two red triangles somewhat isolated from the other points in the diagram, and one blue diamond. In fact, we can see the one isolated black circle point, but it is not clear which are the two main red triangles.
\begin{figure}[h!]
\centering
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{torus_plot_shrink} \ \ \ \
}
\subfigure[]
{
\includegraphics[width=1.25in, height=1.25in]{torus3_plot_shrink} \ \ \ \
}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{torusPD02}
}
\caption{\footnotesize
(a) Points sampled from a torus. View from above the torus. (b) The corresponding kernel density estimator, shown, for visual clarity, at only a few quantized levels, and from a different angle to (a). (c) The corresponding persistence diagram for the upper level sets of the kernel density estimate on the full torus. Black circles are $H_0$ persistence points, red triangles are $H_1$ points, blue diamonds are $H_2$ persistence points. Birth times are on the vertical axis.}
\label{fig:torus}
\end{figure}
Adopting the approach described above separately for the $H_0$ and $H_1$ persistence diagrams, we estimated the parameters for a Gibbs distribution for the model with pseudolikelihood \eqref{eq:pseudo}, taking $K=3$.
For the $H_0$ diagram, working without the point at infinity, the estimate of $\delta$ was 0.0010, and the estimates of $\Theta$ were $\theta _{1} = -0.0034$, $\theta _{2} = -0.0026$,
$\theta _{3} = -0.0032$, $\theta _{H} =1.08E+04$, and $\theta _{V} =4.19E+03$.
For the $H_1$ persistence diagram, the estimate of $\delta$ was 0.0007, and the estimates of $\Theta$ were $\theta _{1} = -0.0044$, $\theta _{2} = -0.0059$,
$\theta _{3} = -0.0036$, $\theta _{H} =1.29E+05$, and $\theta _{V} =1.50E+04$.
\subsubsection{Replicating the persistence diagram}
The determination of the burn in period in this example, for both $H_0$ and $H_1$, was only heuristic. Figure \ref{fig:torusmcmc} presents the original persistence diagrams of $H_0$ and $H_1$ and their MCMC with burn in periods of 10, 25, 50 and 1000. The best fits for both $H_0$ and $H_1$ occur for burn in periods in the range $[10,50]$.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.9in, height=0.9in]{torus_h0_original}
\includegraphics[width=0.9in, height=0.9in]{torus_h0_burn10}
\includegraphics[width=0.9in, height=0.9in]{torus_h0_burn25}
\includegraphics[width=0.9in, height=0.9in]{torus_h0_burn50}
\includegraphics[width=0.9in, height=0.9in]{torus_h0_burn1000}
\includegraphics[width=0.9in, height=0.9in]{torus_h1_original}
\includegraphics[width=0.9in, height=0.9in]{torus_h1_burn10}
\includegraphics[width=0.9in, height=0.9in]{torus_h1_burn25}
\includegraphics[width=0.9in, height=0.9in]{torus_h1_burn50}
\includegraphics[width=0.9in, height=0.9in]{torus_h1_burn1000}
\end{center}
\caption{\footnotesize
The first row shows the persistence diagrams of $H_0$, and the second row shows the persistence diagrams of $H_1$.
At each row, the left plot is the original persistence diagram, and the four other plots are simulated persistence diagrams based on an MCMC simulation with burn in of 10, 25, 50 and 1000. }
\label{fig:torusmcmc}
\end{figure}
\subsubsection{Statistical inference}
We generated 1,000 replicated persistence diagrams from the fitted model with a burn in period of 10 iterations and with $(n_b,n_r,n_R)$ given by (500,10,100), (500,20,50), (500,40,25), or (500,100,10).
Using these four sets of simulations, we computed the maximum statistics $T_j$, $j=1,...,4$, their confidence intervals and their $p$-values. Table \ref{table:torus} summarizes the results.
\begin{table}[h!]
\begin{center}
\fontsize{8.5}{0.9}\selectfont
\begin{tabular}{llccccc}
homology&statistics&real PD &$(n_b, n_r, n_R)$ & CI & $p$-value& significance\\
\\
\\
\\
$H_0$&$T_1$& 0.0295 &(500,10,100)&[0, 0.0371]&0.3360 &no\\
\\
& & &(500,20,50)&[0, 0.0362]&0.2770 & no\\
\\
& & &(500,40,25)&[0, 0.0359]&0.2350 & no\\
\\
& & &(500,100,10)&[0, 0.0322]]&0.2720 & no\\
\\
\\
\\
\\
$H_1$&$T_1$& 0.0136 &(500,10,100)&[0, 0.0123]&0.0340 & yes\\
\\
&& &(500,20,50)&[0, 0.0118]&0.0320 & yes\\
\\
&& &(500,40,25)&[0, 0.0107]&0.0220 & yes\\
\\
&& &(500,100,10)&[0, 0.0134]]&0.0490 & yes\\
\\
\\
\\
\\
$H_1$&$T_2$& 0.0118 &(500,10,100)&[0, 0.0103]&0.0030 & yes\\
\\
&& &(500,20,50)&[0, 0.0102]&0.0060 & yes\\
\\
&& &(500,40,25)&[0, 0.0102]&0 & yes\\
\\
&& &(500,100,10)&[0, 0.0101]]&0.0020 & yes\\
\\
\\
\\
\\
\\
$H_1$&$T_3$& 0.0103 &(500,10,100)&[0, 0.0100]&0.0360 & yes\\
\\
&& &(500,20,50)&[0, 0.0098]&0.0060 & yes\\
\\
&& &(500,40,25)&[0, 0.0099]&0.0060 & yes\\
\\
&& &(500,100,10)&[0, 0.0099]]&0.0180 & yes\\
\\
\\
\\
\\
\\
\label{table:torus}
\end{tabular}
\end{center}
\caption{{ {\footnotesize Maximum statistics $T_1$, $T_2$, $T_3$ for the real $H_0$ and $H_1$ persistence diagrams and the simulated $H_0$ and $H_1$ persistence diagrams of the 2-torus. The CI is a one-sided confidence interval at a $5\%$ confidence level. The $p$-value is also a one-sided. Both the CI and the $p$-value are based on 1000 simulated persistence diagrams.}}}
\label{table:torus}
\end{table}
The results for the $H_0$ diagram, for all scenarios, showed that $T_1$ was insignificant (the lowest $p$-value
reached in any of the six cases was 0.235). Thus, adding the `point at infinity' back into the diagram, we have evidence for exactly one connected component, as we hoped to find.
For the $H_1$ diagram, the results for all scenarios showed that $T_1$ and $T_2$ were significant (the highest $p$-value
reached in any of the 8 cases was 0.049). That is, two significant holes, as we hoped to find.
Unfortunately, however, $T_3,...,T_8$ were also statistically significant, leading to a significantly over-estimation of the complexity of the $H_1$ homology. This seems to be due to the sparsity of points in the sample, which is clear from the first two panels in Figure \ref{fig:torus}. Our conclusion here, therefore, is a need for either a larger sample size or, perhaps, a larger bandwidth for the kernel density estimator.
\subsection{Three circles}
Our final example is that of three concentric circles in $\mathbb R^2$. We describe the main results, skimping on detail.
\subsubsection{The data and fitting the model}
For this example we start with a random sample of $n=1,200$ points from three circles, of diameters 6, 4 and 1, as shown in the Panel (a) of Figure \ref{fig:threecirc}.
In total, 600 points were chosen from the largest circle, 400 from the middle circle, and 200 from the smallest one. Panel (b) shows the corresponding kernel density estimate, for which we took the bandwidth $\eta =0.1$. Panel (c) displays the corresponding persistence diagram of the upper level set filtration of $\hat f_N$, containing $N_0=80$ points of the zeroth homology $H_0$, represented by the black circles, with the red triangles corresponding to the first homology $H_1$. Since we know that the upper level sets of $\hat f_N$ are characterized by having three main components, each of which contains a single 1-cycle (hole) we expect to see three black circles and three red triangles somewhat isolated from the other points in the diagram, which is in fact the case.
\begin{figure}[h!]
\begin{center}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{Circles4new}
}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{KDENewV4}
}
\subfigure[]
{
\includegraphics[width=1.3in, height=1.3in]{PD3circNew1}
}
\end{center}
\caption{\footnotesize
(a) A random sample from three circles, 600 points from the larger circle, 400 points from the middle circle, and 200 from the smaller one, with a kernel density estimate (b) and the persistence diagram (c) for its upper level sets. Black circles are $H_0$ persistence points, red triangles are $H_1$ points. Birth times are on the vertical axis.}
\label{fig:threecirc}
\end{figure}
Note firstly that while there are quite a few (black, circular) points corresponding to the $H_0$ homology, there are only four (red, triangular) for $H_1$. Since the methodology described in the previous sections requires the estimation of a sophisticated model with a number of parameters, it follows that it is not appropriate for modelling the $H_1$ part of the diagram. However, there are more than enough $H_0$ points in Figure \ref{fig:threecirc} to fit a spatial model to them.
Adopting the approach described above, and working only with the $H_0$ persistence diagram without including the `point at infinity', we estimated the parameters for a Gibbs distribution for the model with pseudolikelihood \eqref{eq:pseudo}, taking $K=3$. The estimate of $\delta$ was 0.0012. For this $\delta$, the estimates of $\Theta$ were $\theta _{1} = -0.0105$, $\theta _{2} = 0$,
$\theta _{3} = 0$, $\theta _{H} =394.6$, and $\theta _{V} =147.7 $.
\subsubsection{Statistical inference}
We computed, as previously, the bottleneck and the Wasserstein distances between the MCMC simulations and the data itself, based on 100 simulated sets that behave the same as our original data. The point where the initial rapid growth of the distance functions ceases was approximately 10 for the bottleneck distance and was approximately 15 in the Wasserstein case.
Based on these results, we
generated 1,000 persistence diagrams from the fitted model with a burn in period of 10 iterations and with $(n_b,n_r,n_R)$ given by (500,20,50), (500,40,25), or (500,100,10).
Using these three sets of simulations, we computed the maximum statistics $T_j$, $j=1,2,3$, their confidence intervals and their $p$-value. Table \ref{table:circles} summarizes the results.
\begin{table}[h!]
\begin{center}
\fontsize{8.5}{0.9}\selectfont
\begin{tabular}{llcccc}
statistics&real PD &$(n_b, n_r, n_R)$ & CI & $p$-value& significance\\
\\
\\
\\
\\
$T_1$& 0.2145 &(500,20,50)&[0, 0.1848]&0.0010 & yes\\
\\
& &(500,40,25)&[0, 0.1750]&0.0020 & yes\\
\\
& &(500,100,10)&[0, 0.1713]]&0 & yes\\
\\
\\
\\
\\
$T_2$& 0.1740 &(500,20,50)&[0, 0.1428]&0.0030 & yes\\
\\
& &(500,40,25)&[0, 0.1402]&0.0010 & yes\\
\\
& &(500,100,10)&[0, 0.1424]&0.0020 & yes\\
\\
\\
\\
\\
$T_3$& 0.1180 &(500,20,50)&[0, 0.1250]&0.1150 & no\\
\\
& &(500,40,25)&[0, 0.1232]&0.0990 & no\\
\\
& &(500,100,10)&[0, 0.1244]&0.1170 &no\\
\\
\\
\\
\\
\end{tabular}
\end{center}
\caption{{ {\footnotesize Maximum statistics $T_1$, $T_2$ and $T_3$ for the real persistence diagram and the simulated persistence diagrams of the three circles. The CI is a one-sided confidence interval at a $5\%$ confidence level. The $p$-value is also one-sided. Both the CI and the $p$-value are based on 1,000 simulated persistence diagrams.}}}
\label{table:circles}
\end{table}
The results, for all three scenarios, showed that $T_1$ and $T_2$ were highly significant (the largest $p$-value
reached in any of the six cases was 0.003). In none of the three scenarios was $T_3$ significant, with $p$-values in the range (0.099,\, 0.117). That is, we found that the two points in the $H_0$ persistence diagram (as well as the `point at infinity', which, recall, we removed from the analysis) are significant. Therefore we have three connected components, as we hoped to find.
|
\section{Introduction}
\begin{figure}
\centering
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/17a.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/353b.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/735a.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/914b.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/461c.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.32\linewidth}
\centering
\includegraphics[width=.98\linewidth]{challenging/30c.jpg}
\caption{}
\end{subfigure}%
\caption{
Challenging scenarios present in our dataset. (a) Dazzle light (b) Shadow (c) Rain (d) Night (e) Occlusion (f) Deteriorated road markings}
\label{fi:challenging_scenarios}
\vspace{-1em}
\end{figure}
\begin{table*}[t]
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|}
\hline
\textbf{Dataset} & \textbf{Year} & \textbf{Images} & \textbf{Classes} & \textbf{Location} & \textbf{Annotation Format} \\
\hline\hline
Road Marking \cite{ananth} & 2012 & 1443 & 11 & USA & Bounding Box annotations (TXT) \\
\hline
TRoM \cite{TROM} & 2017 & 712 & 19 & China & Pixel-level annotations (PNG) \\
\hline
VPGNet \cite{VPGNet} & 2017 & 21097 & 17 & Korea & Pixel-level and Grid-level annotations (MAT) \\
\hline
\end{tabular}
\vspace{-0em}
\caption{Summary of existing road marking detection datasets.}
\label{Tab:summary}
\end{center}
\vspace{-1em}
\end{table*}
Understanding traffic regulations imposed by traffic symbols such as traffic signs, traffic lights, lane and road markings can be considered as a fundamental perception task involved in the development of advanced driver assistance systems (ADAS) and autonomous vehicles. Road markings refer to the symbols and text painted on the road surface, which assist the drivers to safely navigate on roads by regulating the traffic. Developing robust road marking detection algorithms is a challenging task due to occlusions, illumination changes, shadows, varying weather conditions and deterioration of road signs with time.
Though the detection and recognition of road markings is a vital task, it is often a less researched area, mainly due to the lack of publicly available datasets and limitations present in the existing datasets. The dataset introduced by \cite{ananth} is used in most of the earlier work done on road marking detection \cite{chen2015road,Kheyrollahi,IPMbased12,Srikanthan}. Though their dataset consists of 1443 images, its diversity is limited since many adjacent frames covering the same scenario have been annotated. Undefined train-test split and unavailability of an evaluation script can be identified as further issues. Some of the recent deep learning based methods \cite{VPGNet,TROM} tackle the lane detection and road marking detection as a single segmentation task. Therefore, the datasets such as VPGNet \cite{VPGNet} and TRoM \cite{TROM} contain annotations for both lane and road markings together. However, more focus is given to the lane detection task and the frequency of instances for road marking classes are much less than that for lanes. Moreover, the limited annotation formats available and the lack of proper evaluation metrics have made it difficult to accomplish novel developments and to compare with existing road marking detection approaches.
Having identified the requisite for a common benchmark for road marking detection, we introduce the CeyMo road marking dataset consisting of 2887 images and 4706 instances belonging to 11 road marking classes. As illustrated in Figure \ref{fi:challenging_scenarios}, the dataset covers a wide variety of challenging urban, sub-urban and rural road scenarios, and the test set is divided into six categories: normal, crowded, dazzle light, night, rain and shadow. The image annotations are provided as polygons, bounding boxes and segmentation masks, such that it will encourage a broad range of research in the road marking detection domain. Furthermore, we provide two evaluation metrics along with an evaluation script for the dataset, facilitating direct comparison of diverse road marking detection approaches.
Most of the existing work done on road marking detection \cite{wacv,Greenhalgh,chen2015road,Kheyrollahi,ding2020comprehensive,Suhr2015FastSR} generate candidate regions first, and then recognize the regions using machine learning based algorithms. End-to-end deep learning based instance segmentation and semantic segmentation networks have been used in recent works, \cite{TROM} and \cite{VPGNet}. The use of end-to-end object detector models to detect road markings is faster and more efficient, yet a less researched approach. We investigate the effectiveness of both instance segmentation and object detection based approaches for detecting road markings in our dataset. We use two Mask R-CNN \cite{mask_rcnn} based network architectures under the instance segmentation based approach and two SSD \cite{SSD} based object detector models, along with inverse perspective transform (IPT) are used under the object detection based approach. The inference speeds and the class-wise, scenario-wise and overall accuracy values of the four models are provided as a performance baseline. In summary, the contributions of this paper are as follows:
\begin{itemize}
\item We introduce the CeyMo road marking dataset covering a wide variety of challenging scenarios and addressing the limitations present in existing publicly available datasets. The dataset is provided with three annotation formats, and an evaluation script to facilitate subsequent research on road marking detection.
\item We evaluate the approaches of utilizing both instance segmentation and object detection based network architectures for the road marking detection task and provide results in terms of speed and accuracy for a set of selected models on our benchmark dataset.
\end{itemize}
The rest of the paper is organized as follows: Section \ref{sec:related_works} presents related work. In Section \ref{sec:dataset}, we provide details of our benchmark dataset, while the proposed detection pipelines and employed methods are discussed in Section \ref{sec:method}. The experimental details and results are presented in Sections \ref{sec:Experiments} and \ref{sec:results}, while Section \ref{sec:conclusion} draws important conclusions.
\section{Related Work}
\label{sec:related_works}
In this section, we analyze the existing publicly available road marking detection datasets and different algorithms and implementations carried out for the road marking detection task.
\subsection{Datasets}
\label{ssec:Datasets}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.25\linewidth}
\centering
\includegraphics[width=.98\linewidth]{annotationFormats/147a.jpg}
\caption{Image (JPG)}
\end{subfigure}%
\begin{subfigure}[b]{0.25\linewidth}
\centering
\includegraphics[width=.98\linewidth]{annotationFormats/new/147a_poly_thick.jpg}
\caption{Polygon (JSON)}
\end{subfigure}%
\begin{subfigure}[b]{0.25\linewidth}
\centering
\includegraphics[width=.98\linewidth]{annotationFormats/new/147a_bbox_2.jpg}
\caption{Bounding Box (XML)}
\end{subfigure}%
\begin{subfigure}[b]{0.25\linewidth}
\centering
\includegraphics[width=.98\linewidth]{annotationFormats/new/147a_seg_mask.jpg}
\caption{Segmentation Mask (PNG)}
\end{subfigure}%
\caption{
Annotation formats provided with our dataset.}
\label{fi:annotations}
\vspace{-0.5em}
\end{figure*}
As the first publicly available dataset for road marking detection, \cite{ananth} introduces a road marking dataset that contains 1443 annotated images covering 11 different road marking classes. The images have a relatively low resolution of $800 \times 600$ and the bounding box annotations for all road marking instances in all images are provided in a single text file. As their main focus relies on image processing based approaches, they do not provide separate train and test sets, and a clear evaluation metric is not specified.
Tsinghua Road Marking Dataset (TRoM) \cite{TROM} contains 712 images covering 19 different lane and road marking classes. Pixel-level semantic segmentation based annotations are provided in the PNG format. The dataset consists of 512 images in the train set, 100 images in the validation set and 100 images in the test set. Less number of images and road marking instances in this dataset might be insufficient for recent deep learning based network architectures. Furthermore, the annotation format and the evaluation metric of this dataset are designed for semantic segmentation based approaches, which limits its usability for diverse non-lane road marking detection algorithms.
VPGNet \cite{VPGNet} is a large dataset which consists of about 20000 images covering 17 lane and road marking classes. Pixel-level and grid-level annotations of the lanes and road markings including the vanishing point of the lanes are presented in the MAT file format. As the evaluation metric, the intersection over union (IoU) of the predicted cells with the ground truth grid cells are considered. However, only 8 road marking classes are covered and 52975 instances are annotated as a single class named ``other road markings". Their main focus lies on vanishing point guided lane detection, and this dataset is not widely used for road marking detection in subsequent research.
Most of the other implementations \cite{Greenhalgh,Suhr2015FastSR,Kheyrollahi,ding2020comprehensive} use their own datasets, which are not publicly available. A summary of the three main publicly available road marking detection datasets are presented in Table \ref{Tab:summary}.
\subsection{Algorithms}
Most of the work done on road marking detection rely on classical image processing techniques combined with simple machine learning algorithms. The usual detection pipeline includes image pre-processing, regions of interest (ROI) generation, feature extraction, and classification using machine learning algorithms. A review of non-lane road marking detection and recognition algorithms is present in \cite{review}. Rectifying the original image using inverse perspective transform (IPT) \cite{ananth,Greenhalgh,IPMbased12,Kheyrollahi} is a commonly used pre-processing technique. As an alternative to IPT, \cite{Suhr2015FastSR} suggests that the search area can be reduced using the lane information. However, this may result in poor performance since road marking detection accuracy directly depends on lane detection accuracy.
Maximally stable extremal regions (MSER) \cite{MSER} are used as possible candidate regions in \cite{ananth}, and histogram of oriented gradients (HOG) feature descriptors are used to build a template pool for each class. At inference time, each image is compared with all template images to assign the classes. However, supervised learning methods usually perform better than template matching methods, especially in complex scenarios. MSER \cite{MSER} regions and HOG features have also been used with a support vector machine (SVM) classifier in \cite{Greenhalgh}, to recognize symbol based road markings. A separate optical character recognition (OCR) algorithm is used to recognize text based road markings. However, having different approaches for different road markings may result in a computational redundancy. Both of these methods include HOG feature extraction, which is a time consuming process.
Binarized normed gradients (BING) \cite{cheng2014bing} objectiveness estimation algorithm is used to generate possible road marking region proposals in \cite{chen2015road}. They use PCANet \cite{chan2015pcanet} and SVM integrated classifier to recognize the road markings. The main drawback of this method is the lower localization accuracy since BING \cite{cheng2014bing} usually results in larger proposal regions. Logistic regression has been used with PCANet \cite{chan2015pcanet} in \cite{wacv}, to improve the classification accuracy. A shallow convolutional neural network (CNN) is also introduced as an alternative classifier for road marking recognition. After identifying MSER \cite{MSER} regions, a density based clustering algorithm is used to merge them to obtain road marking proposal regions for the classifier. However, they use many pre-processing techniques to obtain region proposals and the PCANet \cite{chan2015pcanet} or the shallow CNN classifier is only used for the recognition part.
End-to-end deep learning based networks have not been widely used in the domain of road marking detection. A convolutional neural network model which combines ResNet-101 \cite{resnet} and a pyramid pooling ensemble, is used in \cite{TROM}, to obtain lanes and road markings as semantic segmentation outputs. Their architecture achieves average results on the TRoM \cite{TROM} dataset which can be considered as a performance baseline. VPGNet \cite{VPGNet} is a CNN based architecture for detecting lanes and road markings simultaneously. They address the road marking detection as a grid regression task followed by grid sampling and box clustering as post-processing techniques for merging grid cells. However, their focus is more on lane detection and vanishing point prediction tasks, and quantitative results are only provided for four road marking classes.
\section{Benchmark Dataset}
\label{sec:dataset}
\begin{figure*}
\centering
\begin{subfigure}{.7\textwidth}
\centering
\includegraphics[width=.9\linewidth]{figures/classwise_shrinked.jpg}
\vspace{-2mm}
\caption{}
\label{fig:datastat_class}
\end{subfigure}%
\begin{subfigure}{.3\textwidth}
\centering
\vspace{3mm}
\includegraphics[width=\linewidth]{figures/scenario_pie.PNG}
\vspace{-1mm}
\caption{}
\label{fig:datastat_scene}
\end{subfigure}
\caption{(a) Frequency of each class in the dataset. (b) Proportion of each scenario in the test set.}
\label{fig:datastat}
\end{figure*}
In this section, we present the CeyMo road marking benchmark dataset including the data collection and annotation processes, dataset statistics and the evaluation metrics.
\subsection{Data Collection}
Modern traffic sign and road marking datasets are created using footage from vehicles with specifically mounted cameras \cite{BDD, Cordts2016Cityscapes} or frames taken from street view services such as Google or Tencent \cite{Zhe_2016_CVPR}. We collect video footage from two cameras mounted inside of four vehicles to capture a wide range of scenarios including urban, sub-urban and rural areas under different weather and lighting conditions. The frames which contain road markings are then extracted from the recorded video footage.
\subsection{Data Annotation}
Road markings belonging to 11 classes are manually annotated as polygons using the labelme \cite{labelme2016} annotation tool. Each image has a JSON file which contains the coordinates of the polygons enclosing the road markings present in that image. In addition to the polygon annotations in the JSON format, we also provide bounding box annotations in the XML format as well as pixel-level segmentation masks in the PNG format to facilitate different road marking detection approaches. The three annotation formats provided with the dataset are visualized in Figure \ref{fi:annotations}.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
\textbf{Scenario} & \textbf{No. of Images} &\textbf{Percentage} \\
\hline\hline
Normal & 303 & 38.45\% \\ \hline
Crowded & 109 & 13.83\% \\ \hline
Dazzle Light & 56 & 7.11\% \\ \hline
Night & 110 & 13.96\% \\ \hline
Rain & 136 & 17.26\% \\ \hline
Shadow & 74 & 9.39\% \\ \hline
\end{tabular}
\end{center}
\vspace{-1em}
\caption{Number of images for each scenario in the test set.}
\vspace{-1em}
\label{tab:scenarios}
\end{table}
\subsection{Dataset Statistics}
Our new benchmark consists of 2887 total images having a resolution of $1920 \times 1080$. The dataset is divided into the train set and the test set, which comprises 2099 images and 788 images, respectively. The benchmark covers 11 different road marking classes and the number of instances present in each class is shown in Table \ref{tab:instances}. There is an inherent class imbalance in the dataset which is highlighted in Figure \ref{fig:datastat_class}.
The 788 images in the test set are divided into 6 categories including normal and 5 challenging scenarios: crowded, dazzle light, night, rain and shadow. The number of images and the proportion of each category are shown in Table \ref{tab:scenarios} and Figure \ref{fig:datastat_scene}. It can be observed that the 5 challenging scenarios account for the majority (61.55\%) of the test set.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}
\hline
\textbf{Road Marking Class} & \textbf{Train Set} & \textbf{Test Set} & \textbf{Total} \\
\hline\hline
Straight Arrow & 1088 & 352 & 1440 \\ \hline
Left Arrow & 118 & 44 & 162 \\ \hline
Right Arrow & 260 & 92 & 352 \\ \hline
Straight-Left Arrow & 180 & 61 & 241 \\ \hline
Straight-Right Arrow & 58 & 22 & 80 \\ \hline
Diamond & 770 & 277 & 1047 \\ \hline
Pedestrian Crossing & 611 & 228 & 839 \\ \hline
Junction Box & 128 & 44 & 172 \\ \hline
Slow & 72 & 28 & 100 \\ \hline
Bus Lane & 61 & 21 & 82 \\ \hline
Cycle Lane & 142 & 49 & 191 \\ \hline
\textbf{Total} & \textbf{3488} & \textbf{1218} & \textbf{4706} \\ \hline
\end{tabular}
\end{center}
\vspace{-1em}
\caption{Number of instances for each class in the dataset.}
\vspace{-1em}
\label{tab:instances}
\end{table}
\begin{figure*}
\centering
\begin{subfigure}[b]{1\linewidth}
\centering
\input{arch}
\caption{}
\label{roadarch:sf1}
\vspace{2ex}
\end{subfigure}%
\begin{subfigure}[b]{1\linewidth}
\centering
\input{arch_mask}
\caption{}
\label{roadarch:sf2}
\end{subfigure}%
\caption{ (a) Proposed object detection based network architecture. The inverse perspective transform is used to obtain the bird's eye view of the road which will be fed to the object detector for detecting road markings as bounding boxes. The detections are mapped to 4-sided polygons in the original image using the inverse of the IPT matrix. SSD-MobileNet-v1 \cite{SSD,mobilenet} and SSD-Inception-v2 \cite{SSD,inception} are evaluated as the object detectors. (b) Proposed instance segmentation based network architecture. Two Mask R-CNN \cite{mask_rcnn} based networks with Inception-v2 \cite{inception} and ResNet-50 \cite{resnet} backbones are evaluated for detecting road markings on the input images as segmentation masks.}
\label{fi:roadarch}
\vspace{-0.7em}
\end{figure*}
\subsection{Evaluation Metrics}
We use $F_{1}$-score and Macro $F_{1}$-score as the evaluation metrics of our road marking dataset. Intersection over union values between the predictions and the ground truth are calculated and if IoU is greater than 0.3, the corresponding prediction is considered as a true positive. The total number of true positives ($TP$), false positives ($FP$) and false negatives ($FN$) are used to calculate the precision, recall and $F_{1}$-measure as follows:
\begin{equation}
precision = \frac{TP}{TP+FP}
\end{equation}
\begin{equation}
recall = \frac{TP}{TP+FN}
\end{equation}
\begin{equation}
F_{1}\mbox{-}score= \frac{2\times precision \times recall}{precision+recall}
\end{equation}
\vspace{0.5em}
Macro $F_{1}$-score is calculated as the mean of the individual $F_{1}$-scores of the 11 classes present in our dataset as follows:
\vspace{-1em}
\begin{equation}
Macro\mbox{-}F_{1}\mbox{-}score =\dfrac{1}{C} \sum_{i=1}^{C} F_{1}\mbox{-}score_{i}
\end{equation}
Macro $F_{1}$-score gives same importance for all classes regardless of the frequencies they appear in the dataset. Therefore, it will be low for models which only perform well on common classes.
The evaluation script which calculates class-wise, scenario-wise and overall precision, recall and $F_{1}$-score values, and the Macro $F_{1}$-score, will be made publicly available facilitating direct comparison of different road marking detection algorithms.
\section{Methodology}
\label{sec:method}
In this section, we explain our two detection pipelines under object detection and instance segmentation based approaches, that are used to detect road markings on the CeyMo road marking dataset.
\begin{table*}[t!]
\small
\begin{subtable}{\textwidth}
\begin{center}
\begin{tabular}{|p{3cm}|c|c|c|c|}
\hline
\textbf{Category} & \textbf{SSD-MobileNet-v1} & \textbf{SSD-Inception-v2} & \textbf{Mask-RCNN-Inception-v2} & \textbf{Mask-RCNN-ResNet50} \\
& \cite{SSD,mobilenet} & \cite{SSD,inception} & \cite{mask_rcnn,inception} & \cite{mask_rcnn,resnet} \\
\hline\hline
Normal & 86.57 & 87.10 & 93.20 & \textbf{94.14} \\ \hline
Crowded & 79.45 & 82.51 & 82.04 & \textbf{85.78} \\ \hline
Dazzle light & 84.97 & 85.90 & 86.06 & \textbf{89.29}
|
\\ \hline
Night & 83.08 & 84.85 & \textbf{92.59} & 91.51 \\ \hline
Rain & 73.68 & 81.87 & 87.50 & \textbf{89.08} \\ \hline
Shadow & 85.25 & 86.53 & 85.60 & \textbf{87.30} \\ \hline
\textbf{Total} & 82.90 & 85.16 & 89.04 & \textbf{90.62} \\ \hline
\textbf{Speed (FPS)} & \textbf{83} & 61 & 42 & 13 \\ \hline
\end{tabular}
\caption{}
\label{tab:results_scenario}
\end{center}
\end{subtable}
\vspace{1em}
\begin{subtable}{\textwidth}
\begin{center}
\begin{tabular}{|p{3cm}|c|c|c|c|}
\hline
\textbf{Class} & \textbf{SSD-MobileNet-v1} & \textbf{SSD-Inception-v2} & \textbf{Mask-RCNN-Inception-v2} & \textbf{Mask-RCNN-ResNet50} \\
& \cite{SSD,mobilenet} & \cite{SSD,inception} & \cite{mask_rcnn,inception} & \cite{mask_rcnn,resnet} \\
\hline\hline
Straight Arrow & 73.51 & 77.39 & 86.00 & 88.33 \\ \hline
Left Arrow & 66.67 & 73.97 & 59.70 & 74.36 \\ \hline
Right Arrow & 75.64 & 81.93 & 84.75 & 90.40 \\ \hline
Straight-Left Arrow & 65.22 & 65.93 & 84.55 & 89.47 \\ \hline
Straight-Right Arrow & 62.50 & 58.06 & 74.29 & 66.67 \\ \hline
Diamond & 87.82 & 88.58 & 92.05 & 91.05 \\ \hline
Pedestrian Crossing & 94.95 & 95.44 & 96.72 & 96.86 \\ \hline
Junction Box & 82.50 & 90.70 & 92.13 & 96.63 \\ \hline
Slow & 88.46 & 90.20 & 92.59 & 94.34 \\ \hline
Bus Lane & 98.00 & 100.00 & 93.33 & 91.26 \\ \hline
Cycle Lane & 95.00 & 89.47 & 87.18 & 92.31 \\ \hline
\textbf{Macro F1-Score}& 80.93 & 82.88 & 85.75 & \textbf{88.33} \\ \hline
\end{tabular}
\caption{}
\label{tab:results_class}
\end{center}
\end{subtable}
\caption{Road marking detection results. (a) For each of the detection models scenario-wise F1-scores, overall F1-score, and inference speed in frames per second (FPS) are listed. The inference speed is measured by taking the average FPS value for the 788 test images. (b) For each of the detection models class-wise F1-scores and the Macro F1-score are listed.}
\vspace{-1em}
\end{table*}
\subsection{Object Detection Approach}
Our object detection based model architecture for the road marking detection task is shown in Figure \ref{roadarch:sf1}. Each image is first transformed using the inverse perspective transform (IPT) to obtain a bird's eye view of the road area. IPT reduces perspective deformation of the captured images and it also removes a larger area of the background and the road markings become more prominent in the resultant image. The inverse perspective transform is a homography transformation given by the following equations, where $M$ is the relevant homography matrix.
\begin{equation}
\label{eq:IPT}
Destination[\hat{x},\hat{y},:] = Source[x,y,:]
\end{equation}
where,
\begin{equation}
\hat{x}=\frac{M_{11}x+M_{12}y+M_{13}}{M_{31}x+M_{32}y+M_{33}}
\end{equation}
\begin{equation}
\hat{y}=\frac{M_{21}x+M_{22}y+M_{23}}{M_{31}x+M_{32}y+M_{33}}
\end{equation}
\vspace{0.5em}
An end-to-end object detector model is then used to detect road markings on the inverse perspective transformed images. We evaluate the performance of two object detector models for the road marking detection task,
SSD \cite{SSD} with MobileNet-v1 backbone \cite{ mobilenet} and SSD \cite{SSD} with Inception-v2 \cite{inception} as the backbone. The input resolution of both the models is set to $ 500 \times 500 $. These object detector models output the road marking detections as bounding boxes on the inverse perspective transformed image. These bounding box detections are transformed to the original image domain as 4-sided polygons using the inverse of the IPT homography matrix ($M$) as the final step.
\subsection{Instance Segmentation Approach}
Figure \ref{roadarch:sf2} depicts the model architecture used for road marking detection under the instance segmentation based approach. The goal of instance segmentation is to predict object instances and their per-pixel segmentation masks. For this task, we employ the widely used Mask R-CNN \cite{mask_rcnn} network architecture with two backbones, Inception-v2 \cite{inception} and ResNet-50 \cite{resnet}. Mask R-CNN \cite{mask_rcnn} extends the Faster R-CNN \cite{FRCNN} architecture by predicting a segmentation mask, in addition to the bounding box, for each region of interest (RoI) identified.
Since instance segmentation networks usually have low inference speeds, we feed the input images directly to the model, after resizing them into a lower resolution of $500 \times 500$, without any additional pre-processing steps. The network outputs both bounding boxes and segmentation masks for the road marking detections. Yet, our interest only lies in the segmentation masks, from which the convex hulls could be obtained for evaluation purposes.
\section{Experiments}
\label{sec:Experiments}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/23.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/23_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/23.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/23.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/23.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/23.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/43c.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/43c_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/43c.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/43c.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/43c.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/43c.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/190.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/190_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/190.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/190.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/190.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/190.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/240a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/240a_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/240a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/240a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/240a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/240a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/895a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/895a_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/895a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/895a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/895a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/895a.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/798.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/798_gt_black.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/798.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/798.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/798.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/798.jpg}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/input/348c.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/gt_polygon/348c_gt_black.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd mbnet/348c.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/ssd inception/348c.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask inception/348c.jpg}
\caption{}
\end{subfigure}%
\begin{subfigure}[b]{0.16\linewidth}
\centering
\includegraphics[width=.98\linewidth]{results/mask resnet/348c.jpg}
\caption{}
\end{subfigure}%
\caption{Visualization of road marking detection results on the CeyMo road marking dataset. The top two rows represent detections under normal conditions and the next five rows represent detections under challenging conditions: crowded, dazzle light, night, rain and shadow. (a) Input image (b) Ground truth (c) SSD-MobileNet-v1 \cite{SSD,mobilenet} (d) SSD-Inception-v2 \cite{SSD,inception} (e) Mask-RCNN-Inception-v2 \cite{mask_rcnn,inception} (f) Mask-RCNN-ResNet50 \cite{mask_rcnn,resnet}}
\label{fi:results}
\vspace{-0.7em}
\end{figure*}
In this section, we describe the experiments carried out, specifically the data augmentation process and the implementation details.
\subsection{Data Augmentation}
As a step towards mitigating the effect of the class imbalance problem, we follow a simple data augmentation technique during training to increase the number of instances of less frequent signs. It can be observed that left arrow and straight-right arrow classes have comparatively fewer instances than their mirrored classes, right arrow and straight-left arrow.
Therefore, we horizontally flip the images, which include arrows to obtain the mirrored signs. However, since flipping instances of cycle lane, bus lane and slow road marking classes would lose their meaning, images with those instances are avoided. Furthermore, we randomly change the brightness, saturation, contrast and hue of the input images while training the detection models.
\subsection{Implementation Details}
For the training and testing of our detection algorithms, we use a computational platform comprising an Intel Core i9-9900K CPU and a Nvidia RTX-2080 Ti GPU. We use TensorFlow Object Detection API \cite{huang2017speed} to train the two Mask R-CNN models \cite{mask_rcnn} under the instance segmentation based approach and the two SSD \cite{SSD} models under the object detection based approach.
For the training of the detection models, the following configurations are used. For both SSD-MobileNet-v1 \cite{SSD,mobilenet} and SSD-Inception \cite{SSD,inception} models, RMSProp \cite{ruder2016overview} optimization is used with an initial learning rate of 0.004 and a momentum of 0.9, and the batch size is set to 24. For Mask-RCNN-Inception \cite{mask_rcnn, inception} model, SGD with momentum \cite{pmlr-v28-sutskever13} optimization is used with an initial learning rate of 0.0001 and a momentum of 0.9, and the batch size is set to 4. For Mask-RCNN-ResNet50 \cite{mask_rcnn, resnet} model, SGD with momentum \cite{pmlr-v28-sutskever13} optimization is used with an initial learning rate of 0.0003 and a momentum of 0.9, and the batch size is set to 2.
\section{Results}
\label{sec:results}
In this section, we present the qualitative and quantitative results we obtained. The performance of the two SSD \cite{SSD} based object detection models and the two Mask R-CNN \cite{mask_rcnn} based instance segmentation networks on our CeyMo road marking benchmark dataset is presented in Table \ref{tab:results_scenario} and Table \ref{tab:results_class}.
Table \ref{tab:results_scenario} shows the $F_{1}$-score values of each model for each of the six categories and the overall $F_{1}$-score for the test set. The inference speed of each model is also listed in frames per second (FPS). It can be observed that Mask R-CNN \cite{mask_rcnn} models under the instance segmentation based approach have been able to achieve better results than the SSD \cite{SSD} models under the object detection based approach. Mask-RCNN-ResNet50 \cite{mask_rcnn,resnet} model has been able to outperform the other models in normal, crowded, dazzle light, rain and shadow categories while Mask-RCNN-Inception-v2 model \cite{mask_rcnn,inception} achieves the highest $F_{1}$-score for the night category. Although the Mask-RCNN-ResNet50 \cite{mask_rcnn,resnet} model has the highest overall $F_{1}$-score of 90.62, its inference speed of 13 FPS is comparatively low which becomes crucial in real-time applications. Mask-RCNN-Inception-v2 \cite{mask_rcnn,inception} model gives a better trade-off between the accuracy and the speed, while SSD-MobileNet-v1 \cite{SSD,mobilenet} and SSD-Inception-v2 \cite{SSD,inception} models along with the inverse perspective transform, result in a moderate accuracy at a higher inference speed. It can be also observed that all models perform better in the normal category and the $F_{1}$-score values are comparatively lower in the five challenging scenarios.
The $F_{1}$-score values for the 11 road marking classes and the Macro $F_{1}$-score of each model are listed in Table \ref{tab:results_class}. Mask-RCNN-ResNet50 \cite{mask_rcnn,resnet} model achieves better results for most of the classes with a Macro $F_{1}$-score of 88.33. It can also be observed that the Macro $F_{1}$-score values of all models are lower than the overall $F_{1}$-score values by around 2 percent or more, which implies that there is a tendency of the models to perform better in certain classes than others. Pedestrian crossings which can be found frequently within the dataset and capture a larger area of the road, are well detected by all models. Classes like slow, bus lane and cycle lane have comparatively lower number of instances in the dataset. Nevertheless, all four models have been able to detect those classes with a good accuracy, due to the distinct shapes and features of those classes. Although we increase the number of arrow signs in the train set through the data augmentation process, the accuracy values of arrow signs are low, when compared with other road markings. This can be mostly due to the similarity of arrow sign classes within themselves, as well as with the lane markings on the road surface.
Qualitative results obtained by our two object detection models and the two instance segmentation networks are visualized in Figure \ref{fi:results}, along with the input images and the ground truth for the six categories in the test set. It can be observed that the Mask R-CNN \cite{mask_rcnn} models under the instance segmentation based approach perform better, especially in challenging scenarios. Furthermore, the segmentation masks used in those models result in more precise localization of road markings than 4-sided polygons used in the object detection based approach.
\section{Conclusion}
\label{sec:conclusion}
In this work, we introduced the CeyMo road marking dataset for road marking detection, addressing the limitations present in the existing datasets. The novel benchmark dataset consists of 2887 images taken under different traffic, lighting and weather conditions, covering 4706 road marking instances belonging to 11 road marking classes. We provide road marking annotations as polygons, bounding boxes and segmentation masks to facilitate a wide range of road marking detection algorithms. The evaluation metrics provided along with the evaluation script will enable direct comparison of future work done on road marking detection. Furthermore, we evaluated the effectiveness of road marking detection firstly, using object detectors on inverse perspective transformed images and secondly, using end-to-end instance segmentation based networks. The speed and accuracy scores for two object detectors and two instance segmentation network architectures are provided as a performance baseline for the benchmark dataset. We believe that the CeyMo road marking dataset can be used to design and evaluate novel road marking detection algorithms stepping towards real-time, accurate road marking detection in challenging environments in the future.
{\small
\bibliographystyle{ieee_fullname}
|
\section{Introduction}
Let $\{\xi_n\}$ be a sequence of independent and identically distributed (i.i.d.) random variables, and $\{a_n\}$ be a sequence of positive real numbers. We consider the random series $\sum_{n=1}^{\infty}a_n\xi_n$. Such random series are basic objects in time series analysis and in regression models (see \cite{Davis-Resnick}), and there have been a lot of research. For example, \cite{Gluskin-Kwapien} and \cite{Latala} studied tail probabilities and moment estimates of the random series when $\{\xi_n\}$ have logarithmically concave tails. Of special interest are the series of positive random variables, or the series of the form $\sum_{n=1}^{\infty}a_n|\xi_n|^p$. Indeed, by Karhunen-Lo\'{e}ve expansion, the $L_2$ norm of a centered continuous Gaussian process $X(t), t\in[0,1],$ can be represented as $\|X\|_{L_2}=\sum_{n=1}^{\infty}\lambda_nZ_n^2$ where $\lambda_n$ are the eigenvalues of the associated covariance
operator, and $Z_n$ are i.i.d. standard Gaussian random variables. It is also known (see \cite{Lifshits-1994}) that the series $\sum_{n=1}^{\infty}a_n|Z_n|^p$ coincides with some bounded Gaussian process $\{Y_t,t\in \mathrm{T}\}$, where $\mathrm{T}$ is a suitable parameter set: $\sum_{n=1}^{\infty}a_n|Z_n|^p=\sup_{\mathrm{T}}Y_t.$
In this paper, we study the the limiting behavior of the upper tail probability of the series
\begin{align}\label{eq:introduction}
\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|^p\geq r\right\}\qquad\text{ as }r\rightarrow\infty.
\end{align}
This probability is also called large deviation probability (see \cite{Arcones}). As remarked in \cite{Gao-Li}, for Gaussian process $\|X\|_{L_2}=\sum_{n=1}^{\infty}\lambda_nZ_n^2,$ the eigenvalues $\lambda_n$ are rarely found exactly. Often, one only knows the asymptotic approximation. Thus, a natural question is to study the relation between
the upper tail probability of the original random series and the one with approximated eigenvalues. Also, it is much easier to analyze the rate function in the large deviation theory when $\{a_n\}$ are explicitly given instead of asymptotic approximation.
Throughout this paper, the following notations will be used. The $l^q$ norm of a real sequence $a=\{a_n\}$ is denoted by $||a||_q=\left(\sum_{n=1}^{\infty} a_n^q\right)^{1/q}.$ In particular, the $l^{\infty}$ norm should be understood as $||a||_{\infty}=\max|a_n|.$
We focus on the following two types of comparisons. The first is at the exact level
\begin{align}\label{comparison-exact}
\frac{\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\|a\|_2\beta+|\alpha|\sum_{n=1}^{\infty} a_n\right\}}{\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq r\|b\|_2\beta+|\alpha|\sum_{n=1}^{\infty} b_n\right\}}\sim1\qquad\text{ as }r\rightarrow\infty
\end{align}
where $\{\xi_n\}$ are i.i.d. Gaussian random variables $N(\alpha,\beta^2);$ see Theorem \ref{exact-standard-normal} and Theorem \ref{exact-general-normal}. This is motivated by \cite{Gao-Hannig-Torcaso} in which the following exact level comparison theorems for small deviations were obtained: as $r\rightarrow0,$
$\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\leq r\right\}\sim c\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\leq r\right\}$ for i.i.d. random variables $\{\xi_n\}$ whose common distribution satisfies several weak assumptions in the vicinity of zero. The proof of the small deviation comparison is based on the equivalence form of $\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\leq r\right\}$ introduced in \cite{Lifshits-1997}. Our proof of upper tail probability comparison (\ref{comparison-exact}) is also based on an equivalent form of $\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\right\}$ in \cite{Lifshits-1994} for Gaussian random variables. The main difficulty is to come up with suitable inequalities which can be used for a specified function $\widehat{\varepsilon}(x,y)$ in Lemma \ref{Lifshits-Theorem2}, and such inequalities are obtained in Lemma \ref{core-lemma-1} and Lemma \ref{core-lemma-2}.
For more general random variables, difficulties arise due to the lack of known equivalent form of $\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\right\}.$ Thus, instead of exact comparison, we consider logarithmic level comparison for upper tail probabilities
\begin{align}\label{comparison-log}
\frac{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\|a\|_q\right\}}{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq r\|b\|_q\right\}}\sim1\qquad\text{ as }r\rightarrow\infty.
\end{align}
It turns out that under suitable conditions on the sequences $\{a_n\}$ and $\{b_n\}$ the comparison (\ref{comparison-log}) holds true for i.i.d. random variables $\{\xi_n\}$ satisfying $$\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{|\xi_1|\geq u\right\}=-c$$ for some finite constants $p\geq1$ and $c>0;$ see Theorem \ref{rough-comparison}. Here we note that logarithmic level comparisons for small deviation probabilities can be found in \cite{Gao-Li}.
From comparisons (\ref{comparison-exact}) and (\ref{comparison-log}), we see that two upper tail probabilities are equivalent as long as suitable scaling is made. We believe that this holds true for more general random variables; see the conjecture at the end of Section \ref{section-exact-comparison} for details.
\section{Exact comparisons for Gaussian random series}\label{section-exact-comparison}
\subsection{The main results}\label{exact-for-Gaussian}
The following two theorems are the main results in this section. The first one is on standard Gaussian random variables.
\begin{theorem}\label{exact-standard-normal}
Let $\{Z_n\}$ be a sequence of i.i.d. standard Gaussian random variables $N(0,1),$ and $\{a_n\},\{b_n\}$ be two non-increasing sequences of positive real numbers such that $\sum_{n=1}^{\infty}a_n<\infty,\sum_{n=1}^{\infty}b_n<\infty,$
\begin{equation}\label{a-n-b-n-converge}
\begin{aligned}
\prod_{n=1}^{\infty}\left(2-\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\right)\text{ and }\prod_{n=1}^{\infty}\left(2-\frac{b_n/\|b\|_2}{a_n/\|a\|_2}\right)\text{ converge}.
\end{aligned}
\end{equation}
Then as $r\rightarrow\infty$
\begin{align*}
\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|Z_n|\geq r\|a\|_2\right\}\sim\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|Z_n|\geq r\|b\|_2\right\}.
\end{align*}
\end{theorem}
For general Gaussian random variables $Z_n,$ it turns out that the condition (\ref{a-n-b-n-converge}) is not convenient to derive the comparison because some more complicated terms appear in the proof. Therefore, an equivalent condition in another form is formulated which forms the following comparison.
\begin{theorem}\label{exact-general-normal}
Let $\{Z_n\}$ be a sequence of i.i.d. Gaussian random variables $N(\alpha,\beta^2),$ and $\{a_n\},\{b_n\}$ be two non-increasing sequences of positive real numbers such that $\sum_{n=1}^{\infty}a_n<\infty,\sum_{n=1}^{\infty}b_n<\infty,$
\begin{equation}\label{a-n-b-n-converge-regular-normal}
\begin{aligned}
\sum_{n=1}^{\infty}\left(1-\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\right)\text{ converges, and }\sum_{n=1}^{\infty}\left(1-\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\right)^2<\infty.
\end{aligned}
\end{equation}
Then as $r\rightarrow\infty$
\begin{align*}
\mathbb{P}&\left\{\sum_{n=1}^{\infty}a_n|Z_n|\geq r\|a\|_2\beta+|\alpha|\sum_{n=1}^{\infty} a_n\right\}\\
&\qquad\qquad\sim\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|Z_n|\geq r\|b\|_2\beta+|\alpha|\sum_{n=1}^{\infty} b_n\right\}.
\end{align*}
\end{theorem}
\subsection{Proofs of Theorem \ref{exact-standard-normal} and Theorem \ref{exact-general-normal}}
The function $\Phi$ stands for the distribution function of a standard Gaussian random variable
$$\Phi(x)=\int_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-u^2/2}du.$$
The first lemma is our starting point.
\begin{lemma}[\cite{Lifshits-1994}]\label{Lifshits-Theorem2} Let $\{\xi_n\}$ be a sequence of i.i.d. Gaussian random variables $N(\alpha,\beta^2),$ and $\{a_n\}$ be a sequence of positive real numbers such that $\sum_{n=1}^{\infty}a_n<\infty.$ Then as $r\rightarrow\infty$
\begin{equation}\label{lemma-Lifshits-Theorem2}
\begin{aligned}
\mathbb{P}&\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\right\}\\
&\sim\prod_{n=1}^{\infty}\widehat{\varepsilon}\left(\frac{a_n(r-|\alpha|\sum_{n=1}^{\infty}a_n)}{||a||_2^2\beta},\frac{\alpha}{\beta}\right)\cdot\left[1-\Phi\left(\frac{r-|\alpha|\sum_{n=1}^{\infty}a_n}{||a||_2\beta}\right)\right]
\end{aligned}
\end{equation}
where $\widehat{\varepsilon}(x,y)=\Phi(x+|y|)+\exp\{-2x|y|\}\Phi(x-|y|).$
\end{lemma}
\begin{lemma}[Lemma 5 in \cite{Gao-Hannig-Torcaso}]\label{lemma5Gao}
Suppose $\{c_n\}$ is a sequence of real numbers such that $\sum_{n=1}^{\infty}c_n$ converges, and $g$ has total variation $D$ on $[0,\infty).$ Then, for any monotonic non-negative sequence $\{d_n\},$
$$\left|\sum_{n\geq N}c_n g(d_n)\right|\leq (D+\sup_x|g(x)|)\sup_{k>N}\left|\sum_{n=N}^k c_n\right|.$$
\end{lemma}
As mentioned in the introduction, the key step of the proofs is to come up with suitable inequalities that can be used for the function $\widehat{\varepsilon}(x,y)$ in Lemma \ref{Lifshits-Theorem2}. For the proof of Theorem \ref{exact-standard-normal}, we need the following
\begin{lemma}\label{core-lemma-1}
For $a\leq0$ and small enough $\delta,$ we have
$$1+a\cdot \delta\leq (1+\delta)^a.$$
\end{lemma}
The proof of this lemma is trivial. The proof of Theorem \ref{exact-general-normal} requires a more complicated inequality as follows.
\begin{lemma}\label{core-lemma-2}
For a fixed $\sigma>0$ and any $\gamma>0,$ there is a constant $\lambda(\sigma)$ only depending on $\sigma$ such that for any $|a|\leq \sigma$ and $|\delta|\leq \lambda,$
$$1+a\cdot \delta+\gamma\leq (1+\delta)^a(1+\delta^2)(1+\gamma)^2.$$
\end{lemma}
The proof of Lemma \ref{core-lemma-2} is elementary (but not trivial) which is given at the end of this section.
\begin{proof}[Proof of Theorem \ref{exact-standard-normal}]
By otherwise considering $\tilde{a}_n=a_n/\|a\|_2$ and $\tilde{b}_n=b_n/\|b\|_2,$ we assume that $\|a\|_2=\|b\|_2=1.$ It follows from Lemma \ref{Lifshits-Theorem2} that
$$\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|Z_n|\geq r\right\}\sim\prod_{n=1}^{\infty}2\Phi\left(ra_n\right)\cdot\left[1-\Phi\left(r\right)\right].$$
Therefore,
\begin{align*}
\frac{\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|Z_n|\geq r\right\}}{\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|Z_n|\geq r\right\}}\sim\prod_{n=1}^{\infty}\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}.
\end{align*}
Now we prove that $\prod_{n=N}^{\infty}\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}$ tends to $1$ as $N\rightarrow\infty$ uniformly in $r.$ Then the limit of $\prod_{n=1}^{\infty}\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}$ as $r\rightarrow\infty$ is equal to $1$ since the limit of each $\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}$ as $r\rightarrow\infty$ is $1.$
By applying Taylor's expansion to $\Phi$ up to the second order, we have
\begin{align*}
\Phi\left(ra_n\right)=&\Phi\left(rb_n\right)+\Phi'\left(rb_n\right)\left(ra_n-rb_n\right)\\
&+\frac{\Phi''(rc_n)}{2}\left(ra_n-rb_n\right)^2
\end{align*}
where $c_n$ is between $a_n$ and $b_n.$ It follows from $\Phi''(rc_n)\leq0$ that
\begin{align*}
\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}
\leq1+\frac{rb_n\Phi'\left(rb_n\right)}{\Phi\left(rb_n\right)}\left(\frac{a_n}{b_n}-1\right).
\end{align*}
Let us introduce a new function $g(x)=-\frac{x\Phi'(x)}{\Phi(x)}.$ Now we apply Lemma \ref{core-lemma-1} with $a=g(rb_n)$ to get
\begin{align*}
\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}
\leq\left(2-\frac{a_n}{b_n}\right)^{g(rb_n)}.
\end{align*}
It then follows from Lemma \ref{lemma5Gao} that
\begin{align*}
\prod_{n\geq N}\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}
&\leq\exp\left\{\sum_{n\geq N}g(rb_n)\log\left(2-\frac{a_n}{b_n}\right)\right\}\\
&\leq\exp\left\{(D+\sup_x|g(x)|)\sup_{k>N}\left|\sum_{n=N}^k\log\left(2-\frac{a_n}{b_n}\right)\right|\right\}\\
\end{align*}
which tends to $1$ uniformly in $r$ from condition (\ref{a-n-b-n-converge}). Thus
\begin{align*}
\limsup_{N\rightarrow\infty}\prod_{n\geq N}\frac{\Phi\left(ra_n\right)}{\Phi\left(rb_n\right)}
\leq 1.
\end{align*}
Similarly,
\begin{align*}
\limsup_{N\rightarrow\infty}\prod_{n\geq N}\frac{\Phi\left(rb_n\right)}{\Phi\left(ra_n\right)}
\leq 1
\end{align*}
which completes the proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{exact-general-normal}]
From Lemma \ref{Lifshits-Theorem2} we get
\begin{align*}
\frac{\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\|a\|_2\beta+|\alpha|\sum_{n=1}^{\infty} a_n\right\}}{\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq r\|b\|_2\beta+|\alpha|\sum_{n=1}^{\infty} b_n\right\}}\sim\prod_{n=1}^{\infty}\frac{h(ra_n/\|a\|_2)}{h(rb_n/\|b\|_2)}
\end{align*}
where $h(x)=\Phi(x+|\alpha/\beta|)+\exp\{-2x|\alpha/\beta|\}\Phi(x-|\alpha/\beta|).$ Without loss of generality, we assume $\|a\|_2=\|b\|_2=1.$ We use the notation $f(x)=\exp\{-2x|\alpha/\beta|\}\Phi(x-|\alpha/\beta|),$ thus
$$h(ra_n)=\Phi(ra_n+|\alpha/\beta|)+f(ra_n).$$
Now we apply Taylor's expansions to $\Phi$ at point $rb_n+|\alpha/\beta|$, and to $f$ at point $rb_n$ both up to the second order, so
\begin{align*}
h(ra_n)=&\Phi(rb_n+|\alpha/\beta|)+rb_n\Phi'(rb_n+|\alpha/\beta|)\left(\frac{a_n}{b_n}-1\right)\\
&\qquad\qquad\qquad+\Phi''(rc_{1,n}+|\alpha/\beta|)\left(ra_n-rb_n\right)^2/2\\
&+f(rb_n)+rb_nf'(rb_n)\left(\frac{a_n}{b_n}-1\right)+\frac{r^2b_n^2f''(rc_{2,n})}{2}\left(\frac{a_n}{b_n}-1\right)^2
\end{align*}
where $c_{1,n}$ and $c_{2,n}$ are between $a_n$ and $b_n.$ Because $\Phi''\leq0,$
\begin{align*}
h(ra_n)\leq&h(rb_n)+rb_n\left[\Phi'(rb_n+|\alpha/\beta|)+f'(rb_n)\right]\left(\frac{a_n}{b_n}-1\right)\\
&\qquad\qquad\qquad+\frac{r^2b_n^2f''(rc_{2,n})}{2}\left(\frac{a_n}{b_n}-1\right)^2.
\end{align*}
Taking into account that $\left|r^2b_n^2f''(rc_{2,n})\right|\leq 2c(|\alpha
|
/\beta|)$ for large $N$ uniformly in $r$ with some positive constant $c$ depending on $|\alpha/\beta|,$ we have
\begin{align*}
\frac{h(ra_n)}{h(rb_n)}\leq1+\frac{rb_n\left[\Phi'(rb_n+|\alpha/\beta|)+f'(rb_n)\right]}{h(rb_n)}\left(\frac{a_n}{b_n}-1\right)+c\left(\frac{a_n}{b_n}-1\right)^2.
\end{align*}
The function $g(x):=x\left[\Phi'(x+|\alpha/\beta|)+f'(x)\right]/h(x)$ is bounded and continuously differentiable on $[0,\infty)$ with a bounded derivative. Therefore it follows from Lemma \ref{core-lemma-2} that
\begin{align*}
\frac{h(ra_n)}{h(rb_n)}\leq\left(\frac{a_n}{b_n}\right)^{g(rb_n)}\left(1+\left(\frac{a_n}{b_n}-1\right)^2\right)\left(1+c\left(\frac{a_n}{b_n}-1\right)^2\right)^2.
\end{align*}
By taking the infinite product, we get
\begin{align*}
\prod_{n=N}^{\infty}\frac{h(ra_n)}{h(rb_n)}\leq\prod_{n=N}^{\infty}\left(\frac{a_n}{b_n}\right)^{g(rb_n)}\prod_{n=N}^{\infty}\left(1+\left(\frac{a_n}{b_n}-1\right)^2\right)\left(1+c\left(\frac{a_n}{b_n}-1\right)^2\right)^2.
\end{align*}
According to Lemma \ref{lemma5Gao}, the first product
\begin{align*}
\prod_{n=N}^{\infty}\left(\frac{a_n}{b_n}\right)^{g(rb_n)}&=\exp\left\{\sum_{n\geq N}g(rb_n)\log\left(\frac{a_n}{b_n}\right)\right\}\\
&\leq\exp\left\{(D+\sup_x|g(x)|)\sup_{k>N}\left|\sum_{n=N}^k \log\left(\frac{a_n}{b_n}\right)\right|\right\}
\end{align*}
which tends to $1$ because the series $\sum_{n=1}^{\infty} \log\left(\frac{a_n}{b_n}\right)$ is convergent (this is from condition (\ref{a-n-b-n-converge-regular-normal}), see Appendix for more details).
For the second product, we use $1+x\leq e^x$ to get
\begin{align*}
&\prod_{n=N}^{\infty}\left(1+\left(\frac{a_n}{b_n}-1\right)^2\right)\left(1+c\left(\frac{a_n}{b_n}-1\right)^2\right)^2\\
&\leq\exp\left\{(1+2c)\sum_{n\geq N}\left(\frac{a_n}{b_n}-1\right)^2\right\}
\end{align*}
and this tends to $1$ because of (\ref{a-n-b-n-converge-regular-normal}). Thus
\begin{align*}
\limsup_{N\rightarrow\infty}\prod_{n=N}^{\infty}\frac{h(ra_n)}{h(rb_n)}\leq 1.
\end{align*}
We can similarly prove $\limsup_{N\rightarrow\infty}\prod_{n=N}^{\infty}\frac{h(rb_n)}{h(ra_n)}\leq 1$ which ends the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{core-lemma-2}] We first show that under the assumptions of Lemma \ref{core-lemma-2}, the following inequality holds
\begin{align}\label{middle}
1+a\cdot \delta\leq (1+\delta)^a(1+\delta^2).
\end{align}
Let us consider the function $p(\delta)$ for $|\delta|<1$ and $|a\delta|<1$ defined as
$$p(\delta)=a\log(1+\delta)+\log(1+\delta^2)-\log(1+a\delta).$$
It is clear that $p(0)=0$ and
\begin{align*}
p'(\delta)=\frac{\delta}{(1+\delta)(1+\delta^2)(1+a\delta)}\left[\delta^2\left(a^2+a\right)+\delta\left(2a+2\right)+\left(a^2-a+2\right)\right]
\end{align*}
which is greater than $3/2$ for sufficiently small $\lambda_1$ depending on $\sigma$ with $|a|\leq \sigma$ and $|\delta|\leq \lambda_1,$ since $a^2-a+2\geq 7/4.$ Inequality (\ref{middle}) is thus proved.
Now we define a new function
$$q(\gamma)=(1+\delta)^a(1+\delta^2)(1+\gamma)^2-(1+a\delta+\gamma).$$
From (\ref{middle}) we have $q(0)\geq0.$ Furthermore,
$$q'(\gamma)=(1+\delta)^a(1+\delta^2)2(1+\gamma)-1$$
which can be made positive for small $\lambda_2$ depending on $\sigma$ with $|\delta|\leq \lambda_2.$ The proof is complete by taking $\lambda=\min\{\lambda_1,\lambda_2\}.$
\end{proof}
\subsection{Appropriate extensions}
By using again an equivalence form for $\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|Z_n|^p\geq r\right\}$ discussed in \cite{Lifshits-1994} with $1\leq p<2,$ we can similarly derive, without much difficulty, exact comparison for the upper tail probabilities of $\sum_{n=1}^{\infty}a_n|Z_n|^p.$ We formulate this as a proposition as follows without a proof.
\begin{prop}\label{exact-general-normal-p}
Let $\{\xi_n\}$ be a sequence of i.i.d. Gaussian random variables $N(\alpha,\beta^2),$ and $\{a_n\},\{b_n\}$ be two sequences of positive real numbers such that $\sum_{n=1}^{\infty}a_n<\infty,\sum_{n=1}^{\infty}b_n<\infty$ and
\begin{equation}\label{abs_sum}
\begin{aligned}
\sum_{n=1}^{\infty}\left|1-\frac{a_n/\sigma_a^p}{b_n/\sigma_b^p}\right|<\infty
\end{aligned}
\end{equation}
for $1\leq p<2,$ $\sigma_a=\left(\sum_{n=1}^{\infty}a_n^{m/p}\right)^{1/m}\beta$ with $m=2p/(2-p).$ Then as $r\rightarrow\infty$
\begin{align*}
\mathbb{P}&\left\{\sum_{n=1}^{\infty}a_n|\xi_n|^p\geq \left(r\sigma_a+|\alpha|\sum_{n=1}^{\infty} a_n^{1/p}\right)^p\right\}\\
&\qquad\qquad\sim\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|^p\geq \left(r\sigma_b+|\alpha|\sum_{n=1}^{\infty} b_n^{1/p}\right)^p\right\}.
\end{align*}
\end{prop}
Based on what we have observed for Gaussian random variables so far, it is reasonable to believe that after suitable scaling, two upper tail probabilities involving $\{a_n\}$ and $\{b_n\}$ separately are equivalent. Namely, we have the following.
\textbf{Conjecture}: Under suitable conditions on $\{a_n\}$ and $\{b_n\},$ for general i.i.d. random variables $\{\xi_n\},$ the following exact comparison holds
\begin{align*}
\mathbb{P}&\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq h\Big(rf^{\xi}(a)+g^{\xi}(a)\Big)\right\}\sim\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq h\Big(rf^{\xi}(b)+g^{\xi}(b)\Big)\right\}
\end{align*}
for some function $h(r)$ satisfying $\lim_{r\rightarrow\infty}h(r)=\infty,$ and for two suitable scaling coefficients $f^{\xi}(a)$ and $g^{\xi}(a)$ whose values at sequence $a=\{a_n\}$ only depend on $a$ and the structure of the distribution of $\xi_1$ (such as the mean, the variance, the tail behaviors, etc).
In the next section, we show that indeed two upper tail probabilities in the logarithmic level are equivalent after some scaling. This adds more evidence of our conjecture.
\section{Logarithmic level comparison}\label{-section-logarithmic-level-comparison}
In this section, we illustrate the logarithmic level comparison for more general random variables $\{\xi_n\}$ other than the Gaussian ones.
\begin{theorem}\label{rough-comparison}
Let $\{\xi_n\}$ be a sequence of i.i.d. random variables whose common distribution satisfies $\mathbb{E}|\xi_1|<\infty$ and
\begin{align}\label{log-condition-c}
\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{|\xi_1|\geq u\right\}=-c
\end{align}for some constants $p\geq1$ and $0<c<\infty.$ Suppose that a sequence of positive real numbers $\{a_n\}$ is such that $\sum_{n=1}^{\infty}a_n^{2\wedge q}<\infty$ with $q$ given by $\frac{1}{p}+\frac{1}{q}=1.$ Then as $r\rightarrow\infty$
\begin{align}\label{end-rough}
\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\right\}\sim -r^p\cdot c\cdot \|a\|_q^{-p}.
\end{align}
\end{theorem}
\begin{remark}
If $\xi_1$ is the standard Gaussian random variable, then $p=2$ and $c=1/2$ in condition (\ref{log-condition-c}). If $\xi_1$ is an exponential random variable with density function $e^{-x}$ on $[0,\infty),$ then $p=c=1.$ One can easily produce more examples. It is straightforward to deduce the following comparison result from (\ref{end-rough}).
\end{remark}
\begin{cor}
Let $\{\xi_n\}$ be a sequence of i.i.d. random variables satisfying the assumptions in Theorem \ref{rough-comparison}. Suppose that two sequences of positive real numbers $\{a_n\}$ and $\{b_n\}$ satisfy $\sum_{n=1}^{\infty}a_n^{2\wedge q}<\infty$ and $\sum_{n=1}^{\infty}b_n^{2\wedge q}<\infty$ with $q$ given by $\frac{1}{p}+\frac{1}{q}=1.$ Then as $r\rightarrow\infty$
$$\frac{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\right\}}{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq r\right\}}\sim\left(\frac{\|b\|_q}{\|a\|_q}\right)^p$$
and
$$\frac{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|\geq r\|a\|_q\right\}}{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|\geq r\|b\|_q\right\}}\sim1.$$
\end{cor}
The proof of Theorem \ref{rough-comparison} is based on the large deviation principle for random series which was derived in \cite{Arcones}. Let us recall a result in \cite{Arcones} (revised a little for our purpose).
\begin{lemma}[\cite{Arcones}]\label{Arones-result}
Let $\{\eta_k\}$ be a sequence of i.i.d. random variables with mean zero satisfying the following condition
\begin{equation}\label{appendix-equation}
\begin{cases}
&\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{\eta_1\leq-u\right\}=-c_1;\\
&\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{\eta_1\geq u\right\}=-c_2,
\end{cases}
\end{equation}
for some $p\geq1$ and $0<c_1,c_2\leq\infty$ with $\min\{c_1,c_2\}<\infty.$ Suppose $\{x_k\}$ is a sequence of real numbers such that $\sum_{k=1}^{\infty}|x_k|^{2\wedge q}<\infty.$ Then the family $\{n^{-1}\sum_{k=1}^{\infty}x_k\eta_k\}$ satisfies the large deviation principle with speed $n^p$ and a rate function
\begin{equation*}
I(z)=\inf\left\{\sum_{j=1}^{\infty}\psi(u_j):\sum_{j=1}^{\infty}u_jx_j=z\right\}, \,\,\,z\in\mathbb{R}
\end{equation*}
where
\begin{align*}
\psi(t)=
\begin{cases}
c_1|t|^p & \text{ if }t<0;\\
0 & \text{ if }t=0;\\
c_2|t|^p & \text{ if }t>0.
\end{cases}
\end{align*}
Namely, for any measurable set $A\subseteq\mathbb{R},$
\begin{align*}
&-\inf\{I(y):y\in\text{interior of }A\}\leq\liminf_{n\rightarrow\infty}n^{-p}\log\mathbb{P}\left\{n^{-1}\sum_{k=1}^{\infty}x_k\eta_k\in A\right\}\\
&\leq\limsup_{n\rightarrow\infty}n^{-p}\log\mathbb{P}\left\{n^{-1}\sum_{k=1}^{\infty}x_k\eta_k\in A\right\}\leq-\inf\{I(y):y\in\text{closure of }A\}.
\end{align*}
\end{lemma}
\begin{proof}[Proof of Theorem \ref{rough-comparison}] We apply Lemma \ref{Arones-result} to the i.i.d. random variables $\eta_k=|\xi_k|-\mathbb{E}|\xi_k|.$ The condition (\ref{log-condition-c}) implies that (\ref{appendix-equation}) is fulfilled. Let us consider a special measurable set $A=[1,\infty).$ By using the Lagrange multiplier, it follows that
\begin{align*}
-\inf\{I(y):y\in\text{interior of }A\}=-\inf\{I(y):y\in\text{closure of }A\}=-c\|x\|_q^{-p}
\end{align*}
(this can be also deduced from Lemma 3.1 of \cite{Arcones}). Then (\ref{end-rough}) follows from the large deviation principle.
\end{proof}
Now let us assume
\begin{align*}
\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{|\xi_1|\geq u\right\}=-c.
\end{align*}
Then it follows easily that
\begin{align*}
\lim_{u\rightarrow\infty}u^{-p/k}\log\mathbb{P}\left\{|\xi_1|^k\geq u\right\}=-c.
\end{align*}
So the logarithmic level comparison for $\xi_n^k$ can be similarly derived as follows.
\begin{prop}\label{rough-comparison-p}
Let $k>0$ be a positive real number, $\{\xi_n\}$ be a sequence of i.i.d. random variables whose common distribution satisfies $\mathbb{E}|\xi_1|^k<\infty$ and
\begin{align*}
\lim_{u\rightarrow\infty}u^{-p}\log\mathbb{P}\left\{|\xi_1|\geq u\right\}=-c
\end{align*}
for some constants $0<c<\infty$ and $p$ such that $p/k\geq1.$ Two sequences of positive real numbers $\{a_n\}$ and $\{b_n\}$ satisfy $\sum_{n=1}^{\infty}a_n^{2\wedge q}<\infty$ and $\sum_{n=1}^{\infty}b_n^{2\wedge q}<\infty$ where $q$ is given by $\frac{1}{p/k}+\frac{1}{q}=1.$ Then as $r\rightarrow\infty$
$$\frac{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|^k\geq r\right\}}{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|^k\geq r\right\}}\sim\left(\frac{\|b\|_q}{\|a\|_q}\right)^{p/k}$$
and
$$\frac{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}a_n|\xi_n|^k\geq r\|a\|_q\right\}}{\log\mathbb{P}\left\{\sum_{n=1}^{\infty}b_n|\xi_n|^k\geq r\|b\|_q\right\}}\sim1.$$
\end{prop}
\section*{Appendix}\label{appendix}
In this section, we make a few remarks on the conditions in Theorem \ref{exact-standard-normal} and Theorem \ref{exact-general-normal}. First, we note that conditions (\ref{a-n-b-n-converge}) and (\ref{a-n-b-n-converge-regular-normal}) are not very restrictive, and examples of sequences satisfying these conditions can be produced. For instance, we can consider two sequences with
$$1-\frac{a_n/\|a\|_2}{b_n/\|b\|_2}=\frac{(-1)^n}{n}.$$
To see the relation between (\ref{a-n-b-n-converge}) and (\ref{a-n-b-n-converge-regular-normal}), let us post part of a useful theorem in \cite{Wermuth-1992} from which many convergence results on infinite products and series can be easily derived.
\begin{lemma}[Part (a) of Theorem 1 in \cite{Wermuth-1992}]
Let $\{x_n\}$ be a sequence of real numbers. If any two of the four expressions
$$\prod_{n=1}^{\infty}(1+x_n),\quad \prod_{n=1}^{\infty}(1-x_n),\quad \sum_{n=1}^{\infty}x_n,\quad \sum_{n=1}^{\infty}x_n^2$$
are convergent, then this holds also for the remaining two.
\end{lemma}
Under condition (\ref{a-n-b-n-converge-regular-normal}) in Theorem \ref{exact-general-normal}, it follows from this result that
$$\prod_{n=1}^{\infty}\left(2-\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\right)\text{ and }\prod_{n=1}^{\infty}\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\text{ converge.}$$
This implies that $\sum_{n=1}^{\infty}\log\left(\frac{a_n/\|a\|_2}{b_n/\|b\|_2}\right)$ is convergent. The facts that
$$\sum_{n=1}^{\infty}\left(1-\frac{b_n/\|b\|_2}{a_n/\|a\|_2}\right)^2<\infty\text{ and }\prod_{n=1}^{\infty}\frac{b_n/\|b\|_2}{a_n/\|a\|_2}\text{ converge}$$
yield
$$\prod_{n=1}^{\infty}\left(2-\frac{b_n/\|b\|_2}{a_n/\|a\|_2}\right)\text{ is convergent.}$$
|
\section{Introduction}
This work discusses the application of Fisher variation principle \cite{II} to such phenomena as financial markets. Those global formations are considered as complex systems, whose modeling should be carried out by using tools and methodologies of statistical mechanics and theoretical physics.
A complete statistical characterization of different markets (stock, commodities, foreign exchange etc.) includes such important aspect as PDF evaluation. Attempts are ongoing to develop the most satisfactory stochastic model describing the main features encountered in empirical analysis \cite{MA}; the non-Gaussian shape of price returns PDF is one of common themes there.
A second area concerns the development of a theoretical model that is able to encompass the essential features of real financial system which is characterized by such PDF. Financial economics borrows results in statistical physics \cite{VO}, and in addition to statistical-mechanic description, a quantum mechanic representation has also emerged \cite{BA}--\cite{CH}. Unfortunately, the constructing of quantum theory for finances often reduces to direct postulation of Schr\"{o}dinger equation and its subsequent solution under some entry conditions. The purpose of present letter is to show that the quantum mechanical framework can be naturally derived from Fisher information thus providing a comprehensive basis for financial economics.
A very common Fisher-technique replaces the entropy by Fisher information measure in many applications (see e.g. \cite{FR}--\cite{SI}, and list of references therein). But, in contrast to the maximum entropy problem with its solution in a fixed exponential form, the minimizing (extremizing) Fisher information function leads to a second order differential equation of Schr\"{o}dinger type, whose solutions exhibit a variety of mathematical forms.
Shaping the potential function in Schr\"{o}dinger equation we give some 'physics' to rise. Then the specific choice for the solutions has to grasp main features of the empirical statistical distributions.
\section{Extremum Fisher information approach}
It is known that prices on financial markets change randomly remaining near the same in average. Taking a sequence of price changes $x$ (logarithmic returns commonly used in financial analysis) obtained through a fixed time interval and estimating their PDF $p(x)$ we can define the dispersion $\sigma^2=\langle( x-\langle x\rangle)^2\rangle$ of observations, or market \emph{volatility}. Fisher's information measure \cite{FR}
\begin{equation} \label{eq.1}
I=\int{dx p(x)\left(\frac{d\ln p(x)}{d x}\right)^2}=4\int{dx \left(\frac{d \psi(x)}{d x}\right)^2}
\end{equation}
($\psi(x)\equiv\sqrt{p(x)}$) arises as a lower bound for the dispersion according to the Cramer-Rao inequality \cite{HA}
$$\sigma^2\cdot I\geq 1,$$
serving thereby as a quality metric of a price decrements predictability.
Suppose, the financial system state can additionally be characterized by mean values \begin{equation}\label{ax} F_k = \int{dx f_k (x)p(x)}, k = 1, ..., M,
\end{equation}
of some $M$ functions $f_k(x)$. The set of measurable values (\ref{ax}) constitutes an available empirical information about system. The statistical moments of very PDF could be taken as $F_k$, if there is no information of other origin.
In this context the relevant PDF extremizes $I$ subject to the prior conditions (\ref{ax}) \cite{II}, \cite{FR}. Normalization entails $\int{dxp(x)}=1$, and the Fisher-based extremization problem adopts the appearance
\begin{equation}\label{var}
\delta_p \left(I[p]-\epsilon\int{dx p(x)}-\sum_{k=1}^M\lambda_k \int{dxf_k(x)p(x)}\right) =0
\end{equation}
with the $(M+1)$ Lagrange multipliers $\epsilon$, and $\lambda_k$ which possible financial meaning is clarified under specific formulation of problem (through a particular choice of set of functions $f_k(x)$).
Variation (\ref{var}) yields now Schr\"{o}dinger equation for the amplitude $\psi(x)$ \cite{II}:
\begin{equation}\label{sch}
-\frac 1 2 \psi''-\frac 1 8 \sum^M_k \lambda_k f_k(x)\psi(x)= \frac{\epsilon}{8}\psi(x),
\end{equation}
where the multiplier $E=\epsilon/8$ plays evidently a role of energy eigenvalue.
Obviously that a choice of the set of functions $f_k(x)$, whose means can be detected, entirely defines the mathematical form of Eq. (\ref{sch}) solutions, and PDFs respectively, and hence the physical model for financial system.
\section{Financial quantum oscillator}
If our prior knowledge about system under investigation is limited to power moments
\begin{equation}\label{*}
F_k=\left< f_k\right>=\left< x^k\right>,
\end{equation}
we can introduce into Schr\"{o}dinger equation (\ref{sch}) the potential function in the manner
\begin{equation}\label{5}
U(x)=-\frac 1 8 \sum_k^M\lambda_k x^k
\end{equation}
It is possible because $U(x)$ belongs to $\mathcal{L}_2$ and thus admits a series of expansion in $x,x^2,x^3$, etc. \cite{FB}.
The number $M$ of terms that should be kept in the potential (\ref{5}), and accordingly, the moments to be fitted, sufficient to approach a goal distribution, depends, in general, on particular distributions. For distribution functions centered at zero the first moment drops out. If one retains in expansion (\ref{5}) the second term only he gets the equation for quantum harmonic oscillator
\begin{equation}\label{*}
-\frac 1 2 \psi''+\frac {\omega^2 x^2} 2 \psi= E\psi
\end{equation}
with the frequency $\omega=\sqrt{|\lambda_2|}/2$ and energy spectrum $$E_n=\omega\left( n+\frac 1 2\right).$$
The empirical price return distributions are typically unimodal. So, among the multiple oscillator's eigenfunctions \cite{FL} (with $H_n(x)$ as Hermite's polynomials)
\begin{equation}\label{*}
\psi_n^0(x)=\sqrt{\frac 1 {2^nn!}\sqrt\frac{\omega}{\pi}}H_n(\sqrt{\omega}x)e^{-1/2\omega x^2},
\end{equation}
one has to choose the ground state solution, $n=0$, which has no nodes. This yields
\begin{equation}\label{*}
p(x)=(\psi^0_0)^2=\sqrt{\frac{\omega}{\pi}}e^{-\omega x^2}.
\end{equation}
It is a Gaussian distribution with dispersion $\sigma^2\equiv\left<x^2\right>=(2\omega)^{-1}=(4E_0)^{-1}$.
The normal PDF is considered by theory of finance often as a first approximation of what is observed in empirical data. In order
|
to take into account the small deviation from normality we could retain more terms in expansion (\ref{5}). With the third and forth moments we come to the problem of unharmonic oscillator:
\begin{equation}\label{*}
-\frac 1 2 \psi''+\left(\frac {\omega^2 x^2} 2+\varepsilon_1(\sqrt{\omega}x)^3+\varepsilon_2(\sqrt{\omega}x)^4\right)\psi= E\psi.
\end{equation}
(Here the re-designations $\lambda_3=-8\varepsilon_1\omega^{3/2}$ and $\lambda_4=-8\varepsilon_2\omega^2$ are performed).
In the frame of quantum mechanical perturbation theory (when the coefficients $\varepsilon_1$ and $\varepsilon_2$ are small enough) the eigenfunction with $n=0$ we are interested in can be written as \cite{LL}
\begin{equation}\label{ex}
\psi_0(x)=\psi_0^0(x)+
\sum_{m\neq 0}\frac{\left<m\mid\varepsilon_1(\sqrt{\omega}x)^3+\varepsilon_2(\sqrt{\omega}x)^4\mid 0\right>}{E_m-E_0}\psi_m^0(x)+...
\end{equation}
As result of standard calculations we obtain in a first approximation the wave function for perturbed ground state
\begin{equation}\label{psi}
\psi_0 (x)=\left(\frac{\omega}{\pi}\right)^{\frac 1 4 }e^{-\frac{\omega x^2} 2}\left[1-\frac{15}{16}\frac{\varepsilon_2}{\omega}-
\frac{2\varepsilon_1}{\sqrt\omega}x+\frac 9 4 \varepsilon_2 x^2+\frac {\sqrt\omega\varepsilon_1} 3 x^3-\frac{\omega\varepsilon_2} 4 x^4\right].
\end{equation}
Correspondingly, the approximated distribution function takes the form
\begin{equation}\label{px}
p(x)=\left(\psi_0 (x)\right)^2=\sqrt\frac{\omega}{\pi}e^{-\omega x^2}C_8(x;\omega,\varepsilon_1,\varepsilon_2)
\end{equation}
where the polynomial of degree eight, $C_8$ is a square of bracketed expression in (\ref{psi}).
The unknown parameters $\omega,\varepsilon_1,\varepsilon_2$ is to be evaluated by fitting (\ref{px}) under empirical curve.
\section{Delta potential and high leptokurtosis}
Although there is no complete agreement in literature with respect to the actual shape of financial returns distributions, some models successfully exploit the idea of power-law tailed PDF (see Refs. in \cite{MA} and \cite{SH}). The fact is that empirical return distributions, while unimodal and approximately symmetric, are typically found to exhibit considerable leptokurtosis, i.e., they are more peaked in the center and have fatter tails than the Gaussian with the same variance. Trying to fit such PDF curve with the function (\ref{px}) does not lead to satisfactory result. So, the information potential $U(x)$ requires modification.
The square potential well
\begin{equation}\label{17}
U(x) =\left\{
\begin{array}{rcl}
-\lambda, & \mid x\mid\leq a\\
0, & \mid x\mid>a
\end{array}
\right.
\end{equation}
seems to be a better choice instead of parabolic one. Among solutions of the Schr\"{o}dinger equation with potential (\ref{17}) the even ones have the view \cite{GK}:
\begin{equation}\label{psiwell}
\psi(x)=\left\{\begin{array}{cc}
A\cos{\sqrt{2(\lambda-|E|)}x}, & |x|\leq a \\
Be^{-\sqrt{2|E|}|x|}, & |x|>a
\end{array}
\right.
\end{equation}
where the eigenvalues $E$ depend on width $a$ and depth $\lambda$ of the well.
The exponential decaying of wave function (\ref{psiwell}) outside the well corresponds to fat tailed PDF, and if the peak of PDF is sharp, the potential well is fine. Upon fineness condition $a^2\lambda\ll 1$, the specific form of wave function middle part ($|x|<a$) loses significance and the only state $$
\psi(x)\approx(2|E|)^{1/4}e^{-\sqrt{2|E|}|x|}
$$
with energy $|E|\approx2\lambda^2a^2$ takes place. In this state the standard deviation $\sigma=(4|E|)^{-1/2}\gg a$, meaning PDF tails fatter than that for Gaussian.
Without going into details of the PDF peak structure one can put $a=0, \lambda\rightarrow\infty$ and transit from (\ref{17}) to delta potential
$$U(x)=-\lambda\delta(x).
$$
This transition means that, according to (\ref{ax}), our prior knowledge about the system regards only to the height of PDF peak:
$$ F=\int dx p(x)\delta(x)=p(0).
$$
Now the unique solution of Schr\"{o}dinger equation is
\begin{equation}\label{*}
\psi(x)=\sqrt\lambda e^{-\lambda \mid x\mid}.
\end{equation}
Corresponding one-parametric PDF,
\begin{equation}\label{la}
p(x)=\lambda e^{-2\lambda \mid x\mid},
\end{equation}
is the Laplace distribution.
Since $x$ denotes log return, Eq. (\ref{la}) implies that the returns $y=e^x$ of asset price has a power law distribution
$$
p(y)=\frac{\lambda}{\mid y\mid^{2\lambda}}
$$
widely discussed in \cite{SH}.
\section{Summary}
The quantum mechanical structure of financial markets has been expressed using Fisher information. In that context the state of financial system characterized by some PDF is correlated with a coupled state of quantum particle in the space of price decrements (velocities). The potential function of related Schr\"{o}dinger equation originates in prior empirical data which one learns from PDF.
It is shown that deep parabolic potential well corresponds to important model of normally distributed random walk. This model is known else as \emph{Efficient market} in the theory of finance. In opposite the non-Gaussian models of price behavior, describing the PDF's tails decay as exponential, are corresponded by a fine potential well, or even delta-function. It is obvious for further research that PDF specific features could be modeled by variation of potential function form.
The nice feature about our approach is that it results in analytically solvable equations, which invites empirical investigations using widely available financial data.
\section*{References}
|
\section{Introduction}
\label{sec_intro}
Quark gluon plasma, which is believed to be the free state of quarks and gluons is one of the most interesting topic of present day hadronic physics. Full understanding of this state can reveal the hidden myth behind the origin of our present day universe and transformation of QGP phase to the hadronic phase. On-going heavy ion collision experiments e.g, Large Hadron Collider (LHC), Relativistic Heavy Ion Collider (RHIC) and the future Compressed Baryonic Matter (CBM) experiment are intended to explore the QCD phase diagram at different values of temperature and density. The LHC and RHIC experiments which work on the condition of high temperature and low dense medium and is compliment to future experiment, CBM, have interpreted the production of QGP \cite{exp,meth}. At such extreme conditions, hadrons melt and quarks and gluons can roam freely in the medium. Direct investigation of this state is difficult due to its existence for short interval of time \cite{ramona}.
Many proposals have been developed to observe the creation of QGP in heavy ion collision experiments \cite{ramona}. Famous signals of QGP are strangeness enhancements in heavy ion collisions as suggested by the Capella and Soff \cite{capella, soff}, jet quenching in the parton energy loss formulated by the Bjorken \cite{bjorken}, enhancements in dileptons spectra in nucleus-nucleus collisions observed in HELIOS and DLS \cite{helios, DLS}, and in heavy ion collisions \cite{srivastava}. Beauty of the dileptons is the least interaction with the medium and these dileptons can give us an undistorted information about QGP medium.
Apart from these, the phenomenon $J/ \psi$ suppression observed
in heavy-ion collision experiments \cite{NA,rhic1,rhic2,alice1,alice2} can also be a signature of the formation of QGP as was proposed first by Matsui and Satz \cite{satz}.
However, not all of the observed $J/ \psi$ suppression in nucleus-nucleus collisions is due to QGP formation \cite{bramb}. As mentioned in \cite{lei}, $J/ \psi$ suppression may be because of co-movers interactions or by inelastic scattering of $J/ \psi$ with surrounding nucleons \cite{gers}. Nuclear dependence of $D$ mesons can also alter the $J/ \psi$ suppression observed by NA50 collaboration \cite{NA, gore, zhang, cassing1, cassing2}. In a similar way, nuclear dependence of $B$ mesons can also interfere the $\Upsilon$ suppression in heavy ion collision experiments.
In addition to $J/ \psi$ suppression, in-medium properties of $D$ mesons can also reveal about the existence of $D-$mesic nuclei \cite{tsu2}. Another peculiar thing about the $D$ meson is about its mass which is 1869 MeV in vacuum and it is much more than the sum of the masses of its constituents which depicts that the Higgs mechanism of mass generation in the standard model is of little importance in an explanation of the mass of the matter around us \cite{hilgert}.
The excess mass of hadrons are considered as a result of interaction of quarks and gluons with the ground state of QCD which is populated with quark and gluon condensates, and to spontaneous breaking of chiral symmetry.
To understand the medium modification of $D$ mesons it is important to understand their production and its collective flow in the medium. It became possible due to the experimental facility at Jefferson Lab, USA \cite{jlab}, in which CEBAF accelerator is used to produce the continous beam of electrons and these are scattered off to produce charm hadrons. At the upcoming FAIR (Facility for Antiproton and Ion Research) project, in PANDA experiment, with annihilation of antiprotons on nuclei, whereas in CBM, with the use of Au nuclei, intentions are to explore the properties of open charm mesons.
Theoretically, many methodologies have been developed to study the in-medium properties of $D$ and $B$ mesons. In \cite{tsu2} authors used quark meson coupling model (QMC) and observed a negative mass shift of the $D$ meson in the nuclear medium. In QMC model, $D$ meson is considered as bound state of one light quark and one charm quark. The interaction of $D$ meson with the nucleons occur through the exchange of the scalar and vector mesons. In \cite{tolo2} using coupled channel $G$ matrix approach authors observed increase in the mass of $D$ meson in nuclear medium.
The chiral hadronic SU(3) model
is generalized to SU(4) and SU(5) sector,
for investigating the in-medium properties of pseudoscalar
$D$ and $B$ mesons \cite{arvind3,amdp1,amdp2}. The QCD sum rules were used in \cite{haya,hilger,azizi,wang2,scalarD,qcd2} to
evaluate the masses and decay constants of $D$ and $B$ mesons at finite density of nuclear matter. Within QCD sum rules, the in-medium properties of heavy mesons are expressed in terms of
quark and gluon condensates and in-medium modification of these condensates lead to medium modification of the properties of heavy meson.
In \cite{arv1,rahul}, using QCD sum rules and chiral SU(3) model \cite{papa}, we investigated the properties of scalar, vector and axial-vector $D$ and $B$ mesons at finite density and temperature of the hadronic matter. The medium modified values of quark and gluon condensates were calculated using chiral effective SU(3) model and these were further used as input in the QCD sum rules to evaluate the properties of $D$ and $B$ mesons. This strategy help us to calculate the properties of heavy mesons at finite density and temperature of medium, for different values of isospin asymmetry and strangeness fraction.
In the present paper, our objective is to calculate the in-medium masses and decay constants of pseudoscalar $D$ and $B$ mesons by using again chiral SU(3) model and QCD sum rules \cite{wang2}. Further, we will use the in-medium masses of pseudoscalar $D$ mesons to calculate the in-medium partial decay width of higher charmonium states $\chi(3556)$, $\psi (3686)$, $\psi (3770)$ to $D \bar{D}$ pairs by using $^3P_0$ model \cite{friman} and shall discuss its possible implications on the $J/\psi$ suppression.
Higher charmonium and bottomonium states are considered as main source of the $J/ \psi$ and $\Upsilon$ states, respectively. If the mass of $D$ ($B$) mesons decrease in the medium then higher charmonium and bottomonium states may decay to $D\bar{D}$ and $B\bar{B}$ pairs, respectively, instead of decaying to ground state charmonium (bottomonium) states. Therefore, the drop of mass of $D$ and $B$ mesons may cause $J/\psi$ and $\Upsilon$ suppression in heavy ion collision experiments. Thus, the decay of higher charmonium and bottomonium states play an important role for the better understanding of the non-perturbative regime of QCD. Many models have been developed in the past to study the decay widths of hadrons e.g., hadrodynamic model in which hadrons are described as elementary point-like objects \cite{R1}, elementary emission models in which mesons are treated as extended objects \cite{R2,R3} but the decays occur via elementary meson emission, and elementary meson emission model in which emitted meson is considered as an elementary particle coupled to the quark \cite{ferre, bonnaz}.
The study of the decay widths of higher charmonium states is also important as a fact that it can be directly measured through the spectrum of dileptons of $p \bar{A}$ and $A A$ reactions in different heavy ion collision experiments \cite{friman}, and this can be taken as a helpful tool to validate the phenomenological methods. To achieve this task many pair creation models were proposed like $^3S_1 $ model (quark-antiquark pair is created from the gluon emitted by a quark of the original meson) \cite{furui}, flux tube model \cite{kokoski} and $^3 P_0$ model in which hadron decay proceed through the $q \bar{q}$ pair with vacuum quantum numbers i.e., $J^{PC}$ = $0^{++}$ \cite{micu}. As far as production of quark and anti-quark is concerned in Ref. \cite{ei}
time-like part of the vector Lorentz confining interaction was considered as main reason for the $q \bar{q}$ production whereas, in Ref. \cite{ackleh,deng} the $q \bar{q}$ production was suggested through the gluon exchange and scalar confining interactions. Further, the spin orbit splitting observed in the heavy quarkonium, predicted a scalar confinement potential \cite{dobb, lees}, while in the study of decay widths of $P-$wave $D$ mesons, the mixture of scalar and vector potential was used \cite{adachi,vij}.
The outline of the paper is as follows: In \cref{sec_chiral_model}, we discuss the chiral SU(3) model to calculate the in-medium scalar fields and use it to compute in-medium quark and gluon condensates.
In \cref{sec_qcdsumrules}, we describe the QCD sum rules to solve the in-medium masses and decay constants of $D$ and $B$ mesons. In \cref{sec_3p0_model}, we briefly discuss $^3P_0$ model to calculate the medium effects on the decay of higher charmonium states $\psi(3686)$, $\psi(3770)$ and $\chi(3556)$ to the $D \bar{D}$ pairs, by considering the medium effects of the $D$ meson mass. In \cref{sec_results_discussions}, we present the results and discussions of the present work and finally in \cref{sec_summary} we shall give a brief summary.
\section{Chiral SU(3) model}
\label{sec_chiral_model}
Chiral SU(3) model is an effective theory applied in non-perturbative regime of QCD \cite{papa}. It is based on the nonlinear realization and broken scale invariance properties of chiral symmetry \cite{w}. Within model we have a general Lagrangian density which consists of a kinetic energy term, baryon meson interaction term which produce baryon mass, self-interaction of vector mesons which generates the dynamical mass of vector mesons, scalar mesons interactions which induce the spontaneous breaking of chiral symmetry, and the explicit breaking term of chiral symmetry. This Lagrangian density can be solved by using mean field approximation under which only the scalar and vector fields contribute to the baryon meson interactions and for all other mesons the expectation values become zero \cite{papa,su3}. From the Lagrangian density, using Euler Lagrange equation of motion, we obtain coupled equations of motion for the scalar fields $\sigma$, $\zeta$, $\delta$ and scalar dilaton field $\chi$ \cite{rahul}. We solve these coupled equations of motion using mean field approximation for the different value of strangness fractions $f_s$, isospin asymmetric parameter $I$, temperature $T$, and, density $\rho_B$ of the medium.
The strangeness fraction is defined as $f_s$ = $\frac{\Sigma_i |s_i|\rho_i}{\rho_B}$, here $s_i$ is the number of strange quarks, $\rho_i$ is number density of $i^{th}$ baryon and isospin asymmetric parameter is defined as $I$ = $\frac{\rho_n - \rho_p}{2\rho_B}$, here $\rho_n$ and $\rho_p$ denote the number density of neutrons and protons, respectively, and, $\rho_B$ is the total baryonic density \cite{arvind2}. To calculate the in-medium masses and decay constants of pseudo-scalar $D$ and $B$ mesons using QCD sum rules, we shall need to calculate the light quark condensates $\langle\bar{u}u\rangle$ and $\langle\bar{d}d\rangle$, and scalar gluon condensates $\langle \frac{\alpha_s}{\pi}G_{\mu\nu}^{a} G^{\mu\nu a}\rangle$. Using chiral effective model, we can express the scalar quark condensates in terms of scalar fields through the explicit symmetry breaking term. We have following expressions for the scalar condensates:
\begin{align}
\left\langle \bar{u}u\right\rangle
= \frac{1}{m_{u}}\left( \frac {\chi}{\chi_{0}}\right)^{2}
\left[ \frac{1}{2} m_{\pi}^{2}
f_{\pi} \left( \sigma + \delta \right) \right],
\label{qu}
\end{align}
and
\begin{align}
\left\langle \bar{d}d\right\rangle
= \frac{1}{m_{d}}\left( \frac {\chi}{\chi_{0}}\right)^{2}
\left[ \frac{1}{2} m_{\pi}^{2}
f_{\pi} \left( \sigma - \delta \right) \right],
\label{qd}
\end{align}
where $m_u$ and $m_d$ are the masses of $u$ and $d$ quarks, having values as $5$ and $7$ MeV, respectively.
Further, from the broken scale invariance property of QCD, we know that the trace of energy momentum
tensor is nonzero and is equal to the scalar gluon condensates $\langle \frac{\alpha_s}{\pi}G_{\mu\nu}^{a} G^{\mu\nu a}\rangle$. We can caricature the trace anomaly in the effective chiral model through the scale breaking term of the effective Lagrangian density which can we further used to
evaluate the trace of energy momentum tensor. Comparing the trace of energy momentum tensor evaluated within chiral model with that of QCD, we can express the gluon condensates, $\langle \frac{\alpha_s}{\pi}G_{\mu\nu}^{a} G^{\mu\nu a}\rangle$ in terms of the scalar fields $\sigma$, $\zeta$, $\delta$, and $\chi$ as
\begin{align}
\left\langle \frac{\alpha_{s}}{\pi} {G^a}_{\mu\nu} {G^a}^{\mu\nu}
\right\rangle = \frac{8}{9} \Bigg [(1 - d) \chi^{4}
+\left( \frac {\chi}{\chi_{0}}\right)^{2}
\left( m_{\pi}^{2} f_{\pi} \sigma
+ \big( \sqrt {2} m_{k}^{2}f_{k} - \frac {1}{\sqrt {2}}
m_{\pi}^{2} f_{\pi} \big) \zeta \right) \Bigg ],
\label{glu}
\end{align}
where ($m_\pi$, $f_\pi$) and ($m_K$, $f_K$) represents the (mass, decay constant) of $\pi$ and $K$ meson, respectively.
The dilaton field $\chi$ is introduced to break the scalar invariance property of QCD. In \cref{glu} $\chi_0$ denotes its vacuum value, parameter $d$ is a constant having value $2/11$, determined through the QCD beta function at one loop level for three colors $N_c$ and three flavors $N_f$ \cite{papa}.
\section{QCD Sum rules for D and B mesons}
\label{sec_qcdsumrules}
The present section is devoted to give an understanding of QCD sum rules used to investigate the masses and decay constants of pseudoscalar $D$ and $B$ mesons in isospin asymmetric strange hadronic matter at finite temperature and density \cite{wang2}. As we will see below the mass shift and decay shift of heavy pseudoscalar mesons will be expressed in terms of quark and gluon condensates.
In QCD sum rule, we start with two point correlation function $\Pi(q)$, which is
the Fourier transformation of the expectation value of the time-ordered product of isospin averaged current $J_5$, i.e., we write \cite{wang2}
\begin{align}
\Pi (q) = i\int d^{4}x\ e^{iq_{\mu} x^{\mu}} \langle T\left\{J_5(x
|
)J_5^{\dag}(0)\right\} \rangle_{\rho_B, T}
\label{tw}
\end{align}
where $ x^\mu = (x^0,\textbf{x})$ is the four coordinate and $q^\mu = (q^0,\textbf{q})$ is four momentum. The spin-isospin averaged current $J_5$ is defined by
\begin{align}
J_5(x) &= J_5^\dag(x) =\frac{\bar{c}(x)i\gamma_5 q(x)+\bar{q}(x)i\gamma_5 c(x)}{2},
\label{psc}
\end{align}
where $q$ is for light quark $u$ or $d$ and $c$ is for charm quark. Note that we are averaging the current of particles and antiparticles i.e., $D$ and $\bar{D}$ mesons. This can be understood as follows: The $D$ meson isospin doublet consist of $D^{+}$ and $D^{0}$ mesons whereas in
$\bar{D}$ meson doublet we have $D^{-}$ and $\bar{D^{0}}$ mesons.
The mesons $D^{+}$, $D^{-}$, $D^{0}$
and $\bar{D^{0}}$ have the quark compositions,
$c\bar{d}$, $d\bar{c}$, $c\bar{u}$ and $u\bar{c}$,
respectively. We see that $D^{+}$ and $D^{-}$ are particle-antiparticles and similarly, $D^{0}$
and $\bar{D^{0}}$ mesons. Thus, when we use $q = d$ in \cref{psc}, we will find average current of $D^{+}$ and $D^{-}$ mesons and when $q = u$, this will be for $D^{0}$
and $\bar{D^{0}}$. Thus, in the present work we shall find the average mass shift and decay shift of $D$ and $\bar{D}$ mesons under centroid approximation \cite{haya,wang2,scalarD}. As we will see later, this will enable us to find the mass-splitting between the isospin doublets due to isospin asymmetry of the medium at finite density and temperature.
The even and odd part of QCD sum rules were used in Ref. \cite{hilger} to calculate the mass splitting between $D$ and $\bar{D}$ mesons.
For $B$ meson, charm quark $c$ will be replaced by the bottom $b$ quark.
The two point correlation function given by \cref{tw} can be divided into the vacuum part, a static one-nucleon part and pion bath contribution, i.e.,
\cite{wang2,zsc,kwon}.
\begin{eqnarray}
\Pi (q) &=&\Pi_{0} (q)+ \frac{\rho_B}{2m_N}T_{N} (q) + \Pi_{P.B.}(q)\,,
\label{pibn}
\end{eqnarray}
where $T_N (q)$ is the forward scattering amplitude \cite{wang2}.
The purpose of third term i.e., pion bath term was to consider the contribution of the finite temperature of the medium \cite{zsc,kwon}. In our present investigation we take the contribution of finite temperature of the medium through the temperature dependence of the scalar fields $\sigma$, $\zeta$, $\delta$ and dilaton field $\chi$ calculated through the chiral SU(3) model.
We have successfully used above idea to calculate the temperature dependence of the masses and decay constants of vector and axial-vector $D$ and $B$ mesons \cite{rahul}. The temperature dependence of the optical potentials of kaons, $D$ and $B$ mesons have been calculated
using the scalar fields $\sigma$, $\zeta$, $\delta$ and dilaton field $\chi$ in \cite{isoamss,arvind2}.
In the limit ${\bf q} \rightarrow 0$, the forward scattering amplitude $T_N(\omega,{\bf q})$ can be related to forward $D-N$ scattering $T$-matrix\cite{koi}
\begin{equation}
{{\cal T}_{D N}}(m_{D},0)= 8\pi(m_N + m_{D})a_{D}.
\end{equation}
where $a_{D}$ is the $DN$ scattering length.
The phenomenological spectral density $\rho(\omega,0)$ can be parametrised into three unknown parameters $a$, $b$ and $c$ as given below \cite{koi},
\begin{align}
\rho(\omega,0) &= -\frac{1}{\pi}
\mbox{Im} \left[\frac{{{\cal T}_{D N}}(\omega,{ 0})}{(\omega^{2}-
m_{D}^2+i\varepsilon)^{2}} \right] \frac{f_D ^2 m_D ^4}{m_c ^2} + \cdots,
\nonumber\\
&= a\,\frac{d}{d\omega^2}\delta(\omega^{2}-m_{D}^2)
+
b\,\delta(\omega^{2} - m_{D}^2) + c\,\theta(\omega^{2}-s_{0})\,.
\label{a1}
\end{align}
The first term in above equation denotes the double-pole term and corresponds to
the on-shell effects of T-matrices,
\begin{equation}
a=-8\pi(m_N + m_{D})
a_{D}f_{D}^2 m_{D}^2\,.
\label{a2}
\end{equation}
The second term in \cref{a1} denotes the single-pole term,
and corresponds to the off-shell (i.e. $\omega^2\neq m_{D}^2$) effects of $T$-matrices. The
third term denotes the continuum term or the remaining effects,
where, $s_{0}$, is the continuum threshold and this define the scale below which
the continuum contribution vanishes \cite{kwon}.
The symbols $m_c$, $m_D$ and $f_D$ denote the mass of charm quark, mass of $D$ meson and decay constant of $D$ meson, respectively.
The shift in mass and decay constant of $D$ and $B$ mesons can be written as \cite{wang2}
\begin{equation}
\delta m_D = 2\pi \frac{m_N + m_D}{m_N m_D} \rho_N a_D,
\label{masshift}
\end{equation}
and
\begin{equation}
\delta f_D = \frac{m_c ^2}{2f_D m^4}\left(\frac{b \rho_N}{2m_N} - \frac{4 f_D^2 m_D^3 \delta m_D}{m_c ^2}\right),
\label{decayshift}
\end{equation}
respectively.
From \cref{a2,masshift,decayshift} we see that to find the mass shift and decay shift, we need to find the unknown parameters $a$ and $b$.
The unknown parameters are eliminated by equating the Borel transformed forward scattering amplitude $T_N(\omega,0)$ on the phenomenological side with the Borel transformed forward scattering amplitude $T_N(\omega,0)$ in operator product expansion side. Finally, we obtain a relation between parameters $a$ and $b$ and the quark and gluon condensates given by \cite{wang2}
\begin{align}
& a \left\{\frac{1}{M^2}\exp\left(-\frac{m_{D}^2}{M^2}\right) - \frac{s_0}{m_{D}^4} \exp\left(-\frac{s_0}{M^2}\right)\right\}
+b \left\{\exp\left(-\frac{m_{D}^2}{M^2}\right) - \frac{s_0}{m_{D}^2} \exp\left(-\frac{s_0}{M^2}\right)\right\}\nonumber\\
-& \frac{2m_N(m_H+m_N)}{(m_H+m_N)^2-m_{D}^2}\left(\frac{f_{D}m_{D}^2g_{DNH}}{m_c}\right)^2\left\{ \left[\frac{1}{M^2}-\frac{1}{m_{D}^2-(m_H+m_N)^2}\right] \exp\left(-\frac{m_{D}^2}{M^2}\right)\right.\nonumber\\
&\left.+\frac{1}{(m_H+m_N)^2-m_{D}^2}\exp\left(-\frac{(m_H+m_N)^2}{M^2}\right)\right\}\nonumber\\
=&-\frac{m_c\langle\bar{q}q\rangle_N}{2}\left\{1+\frac{\alpha_s}{\pi} \left[ 6-\frac{4m_c^2}{3M^2} \right.\right.
\left.\left.-\frac{2}{3}\left( 1-\frac{m_c^2}{M^2}\right)\log\frac{m_c^2}{\mu^2}-2\Gamma\left(0,\frac{m_c^2}{M^2}\right)\exp\left( \frac{m_c^2}{M^2}\right) \right]\right\} \nonumber\\
&\exp\left(- \frac{m_c^2}{M^2}\right) \nonumber\\
&+\frac{1}{2}\left\{-2\left(1-\frac{m_c^2}{M^2}\right)\langle q^\dag i D_0q\rangle_N +\frac{4m_c
}{M^2}\left(1-\frac{m_c^2}{2M^2}\right)\langle \bar{q} i D_0 i D_0q\rangle_N+\frac{1}{12}\langle\frac{\alpha_sGG}{\pi}\rangle_N\right\} \nonumber\\
&\exp\left(- \frac{m_c^2}{M^2}\right)\, .
\label{qcdsumdst}
\end{align}
The nucleon expectation values of quark and gluon condensates appearing in OPE side of above equation can be calculated by using
\begin{eqnarray}
\mathcal{O}_{\rho_{B}} &=&\mathcal{O}_{vacuum} +
4\int\frac{d^{3}p}{(2\pi)^{3} 2 E_{p}}n_{F}\left\langle N(p)\vert \mathcal{O}\vert N(p) \right\rangle+
3\int\frac{d^{3}k}{(2\pi)^{3} 2 E_{k}}n_{B}\left\langle \pi(k)\vert \mathcal{O}\vert\pi(k) \right\rangle \nonumber\\
& =& \mathcal{O}_{vacuum} + \frac{\rho_B}{2 m_N}\mathcal{O}_{N} +
\mathcal{O}_{P.B.},
\label{operator1}
\end{eqnarray}
where $\mathcal{O}_{\rho_{B}}$ gives the expectation value of the operator at finite baryonic density.
The term $\mathcal{O}_{vacuum}$ stands for the vacuum expectation value of the
operator, $\mathcal{O}_{N}$ give us the nucleon expectation value
of the operator and $\mathcal{O}_{P.B.}$ denotes
the contribution from the pion bath at finite temperature.
Also, $n_B$ and $n_F$ are the thermal Boson and Fermion distribution functions and are given by$\left[ e^{E_{k}/T} - 1 \right]^{-1}$ and
$\left[ e^{\left( E_{p}-\mu_{N}\right) /T} - 1 \right]^{-1}$,
respectively.
As discussed earlier, the finite temperature effects in the present investigation will be evaluated through the scalar fields and therefore
contribution of third term will not be considered \cite{arv1}.
Thus, within chiral SU(3) model, we can find the values of
$\mathcal{O}_{\rho_{B}}$ at finite density of the nuclear medium and hence can find
$\mathcal{O}_{N}$ using
\begin{equation}
\mathcal{O}_{N} = \left[ \mathcal{O}_{\rho_{B}} - \mathcal{O}_{vacuum}\right] \frac{2m_N}{\rho_B}.
\label{condexp}
\end{equation}
The values of light quark condensate $<q \bar{q}>_{\rho_{B}}$ and the gluon condensate $\left\langle \frac{\alpha_{s}}{\pi} {G^a}_{\mu\nu} {G^a}^{\mu\nu}\right\rangle_{\rho_{B}}$ are calculated using \cref{qu,qd,glu}. Also, $<\bar{q g_s \sigma G q}>_{\rho_{B}}$ and $<\bar{q} i D_0 i D_0 q>_{\rho_{B}}$ can be approximated in terms of $<q \bar{q}>_{\rho_{B}}$ \cite{arv1,rahul}.
We differentiate \cref{qcdsumdst} w.r.t $\frac{1}{M^2}$ to make one more equation, so that we can solve two coupled equations to eliminate the unknown parameters $a$ and $b$.
Shift in mass and decay constant of $B$ meson can also be calculated through same equations by simply replacing the mass of $D$ meson (charm quark) with mass of $B$ meson (bottom quark).
\section{The $^3 P_0$ model}
\label{sec_3p0_model}
Now we outline the $^3P_0$ model used in the present work to calculate the in-medium strong decay with of higher charmonium states into $D \bar{D}$ pairs \cite{micu,yo,yaouanc}.
The $^3P_0$ model is a quark-antiquark pair creation model in which various transitions
can be studied using the pair creation strength parameter $\gamma$ and oscillator parameter $\beta$
which are fitted to experimental values. As was said earlier, in the $^3P_0$ model $q\bar{q}$ pair is created in the vacuum which combine with the $q\bar{q}$ of parent meson $A$ at rest decaying to $B$ and $C$ mesons.
The invariant matrix element for the particular type of decay $A$ $\rightarrow$ $B$ + $C$ is expressed as \cite{bonnaz,friman}
\begin{eqnarray}
M_{A \rightarrow BC}& \propto &
\int d^3k_q \phi_A(2k_q-2k_B) \phi_B(2k_q-k_B) \phi_C(2k_q-k_B)
\nonumber \\
&& ~~~~~~ \times [\bar{u}_{k_q,s} v_{-k_q,s}]^{{}^3P_0},
\label{overlap}
\end{eqnarray}
where, $k_q - k_B$ and $k_B -k_q$ represent the momentum of heavy quark and anti-quark of $A$ meson
such that total momentum of meson $A$ is zero. The meson $C$ take a anti-quark of momentum $k_q - k_B$ of $A$ meson and one quark of momentum $k_q$ from $^3P_0$ pair such that its total momentum is $-k_B$. Similarly, the momentum of quark and anti-quark composing $B$ meson will be $k_q$ and $k_B - k_q$, respectively so that total momentum of $B$ meson is $k_B$. The term in the square bracket represents the wave function of $q\bar{q}$ pair produced in vacuum.
We take the harmonic oscillator potential for the bound state of wave function \cite{bonnaz,friman}. Considering the nodal structure of the wave function of the mesons $\phi$, we use the method of change of variable i.e., momentum in the following way,
\begin{eqnarray}
k_q^\prime = k_q -{ 1+ r^2 \over 1 +2 r^2} k_B.
\label{change}
\end{eqnarray}
While doing so we allow the parent and daughter mesons to have a different
|
}{u^k}) \QNK k{b_{n-1}, b_{n}}}
\\ &
= 1- \left( \frac{1}{u^k} -1\right) \frac{ \QNK k{b_{n-1}, b_{n}}}{\QNK k{b_{n-1}}+
(\frac{1}{u^k}) \QNK k{b_{n-1}, b_{n}}}
\\ &
\leq 1- \left( \frac{1}{u^k}-1 \right) \frac{ \QNK k{b_{n-1}, b_{n}}}{ (\frac{2}{u^k}) \QNK k{b_{n-1}, b_{n}}}
= 1- \left( \frac{1}{u^k} -1\right) \frac{u^k}{2}
\\ &
= 1- \frac{1}{2} (1-u^k)
\end{split}
\end{equation*}
\end{proof}
First assume that $Q$ is $k$-divergent for all $k$.
Using Lemma~\ref{lem:t2} we then inductively pick the $b_n$ satisfying the following:
\begin{enumerate}
\item \label{lp1}
For all $m_1,m_2 \leq n+1$ and all $i>b_{n-1}$, let $P=P(m_1,m_2,i)$
be defined by: $P\res [0,b_{n-1})= P_{m_1} \res [0,b_{n-1})$ and $P\res [b_{n-1},i)= P_{m_2}
\res [b_{n-1},i)$. Let $w\res [0,b_{n-1})=w_{m_1}$ and $w \res [b_{n-1},i)=w_{m_2}\res [b_{n-1},i)$.
Then for any $B$ with $\|B\|\leq n+1$
we have $|N^P_i(B, w)-P_i(B)|<\frac{1}{2^n} P_i(B)$.
\item \label{lp2}
$(P_m)_{b_{n-1},b_n}(0_{2n}) > 2^n b_{n-1}$ for all $m \leq n$.
\end{enumerate}
Given $x \in \ww$, we define $\varphi_1(x)$ as follows. Suppose $\varphi_1(x)\res b_{n-1}$ has been defined.
Let $y \res [b_{n-1},b_{n})= w_{x(2n+1)}\res [b_{n-1},b_{n})$.
Then we perform the operation $\Theta_{x'(2n),x'(2n)}$ of Section~\ref{section:4cases}
on $y \res [b_{n-1},b_{n})$ to produce
$\varphi_1(x) \res [b_{n-1},b_{n})$.
This defines $\varphi_1(x)\res [b_{n-1},b_{n})$. Doing this for all blocks $[b_{n-1},b_{n})$ produces $\varphi_1(x)$.
If $x \notin C$, that is $x(2n)$ does not tend to $\infty$, then $\varphi(x) \notin \RNQ$.
This is because if $k=\liminf x(2n)$, then there will be infinitely many $n$ for which
$0_k$ occurs in $[0,b_{n})$ at most
\begin{equation}
\begin{split}
N^{P_{x'(2n+1)}}_{b_n}(0_k,w_{x'(2n+1)})(1-\frac{1}{k})+b_{n-1} &
\leq (P_{x'(2n+1)})^{(k)}_{b_{n}} (1-\frac{1}{k})(1+\frac{1}{2^n})+b_{n-1}
\\ &
\leq (P_{x'(2n+1)})^{(k)}_{b_{n}} (1-\frac{1}{k}+\frac{1}{2^{n-1}})
\\ &
\leq (P_{x'(2n+1)})^{(k)}_{b_{n}} (1-\frac{1}{2k})
\end{split}
\end{equation}
many times
while $1_k$ occurs at least
\begin{equation}
\begin{split}
N^{P_{x'(2n+1)}}_{b_{n}}(1_k,w_{x'(2n+1)}) -b_{n-1}& \geq (P_{x'(2n+1)})^{(k)}_{b_{n}} (1-\frac{1}{2^n})-b_{n-1}
\\ & \geq (P_{x'(2n+1)})^{(k)}_{b_{n}} (1-\frac{1}{2^{n-1}})
\end{split}
\end{equation}
many times.
If $x(2n)\to \infty$ but $x(2n+1)$ does not tend to infinity, then $\varphi(x)\in \RNQ\sm \NQ$.
To see this, first note that the point $y$ as above is in $\RNQ$.
Recall $P_x$ is defined by $P_x \res[b_{n-1},b_{n})=P_{x(2n+1)}\res [b_{n-1},b_{n})$.
We show that $y \in \mathscr{N}(P_x)$, which implies $y \in \RNQ$.
For any $B$ and for large enough $n$ and for any
$b_{n-1} \leq i <b_{n}$ we have from property (\ref{lp1}) of the $b_n$:
\begin{equation} \label{qq1}
| N^{P'}_i(B,y')-P'_i(B)|< \frac{1}{2^n} P'_i(B),
\end{equation}
where $y'$ and $P'$ are defined by:
\begin{equation*}
\begin{split}
y'\res[0,b_{n-1})&=w_{x(2n-1)} \res [0,b_{n-1})
\\
y' \res [b_{n-1},i)&=w_{x(2n+1)}\res i
\\
P' \res [0,b_{n-1})&=P_{x(2n-1)}
\\
P'\res [b_{n-1},i)&= P_{x(2n+1)}\res [b_{n-1},i).
\end{split}
\end{equation*}
Also, from property (\ref{lp2}) of the $b_n$ we have:
\begin{equation} \label{qq2}
|N^{P'}_i(B,y')-N^{P_x}_i(B,y)|\leq b_{n-2}<\frac{1}{2^n}
(P_x)_{b_{n-1}}(B)\leq \frac{1}{2^n} (P_x)_i(B).
\end{equation}
Finally,
\begin{equation} \label{qq3}
|P'_i(B)-(P_x)_i(B)|\leq b_{n-2}< \frac{1}{2^n} (P_x)_i(B).
\end{equation}
From Equations \ref{qq1}, \ref{qq2} and \ref{qq3}
we have $|N^{P_x}_i(B,y)-(P_x)_i(B)|\leq \frac{4}{2^{n}} (P_x)_i(B)$.
This shows that $y \in \RNQ$.
Since $x(2n+1)$ does not tend to infinity, then from Claim~\ref{tc} there is a
subsequence
on which $\frac{Q^{(k)}_{i}}{(P_x)^{(k)}_{i}}$ is bounded away from $1$.
Since $y$ is $P_x$-normal,
we have that $y$ is not $Q$-normal.
The operation
applied to $y$ to produce $\varphi_1(x)$ does not affect normality or ratio normality
if $x(2n) \to \infty$ (this is just as in ~\refs{4cases}).
So, $\varphi(x) \in \RNQ \sm \NQ$.
Finally, if $x(n
|
)\to \infty$, then as above $y \in \mathscr{N}(P_x)$.
As $x(2n+1)\to \infty$, we have from Claim~\ref{tc}
that $\lim_i \frac{\QNK ki}{\PxNK ki} \to 1$ and it follows that
$y \in \NQ$.
Since $x(2n)\to \infty$ as well, from the argument in \refs{4cases}
we also have that $y=\varphi(x)\in \NQ$.
So, in all cases we have that $x \in C\sm D$ iff $\varphi(x) \in \RNQ\sm \NQ$.
Suppose now that there is a largest integer $k_0$ such that $Q$ is $k_0$-divergent.
The proof is essentially identical to that above. We let $P_m$ be as before,
and now let $w_m$ be $\leq k_0$ normal with respect to $P_m$, that is, for all $B$
of length $\leq k_0$
we have $\lim_i \frac{N_i(B,w_m)}{(P_m)_i(B)}=1$. We define $\varphi_1(x)$ by first
defining $y$ exactly as before (using the values $x(2n+1)$).
We then modify $y$ to $\varphi_1(x)$, (using $x(2n)$) but in a slightly different manner.
Namely, we get $\varphi_1(x)\res [b_{n-1},b_{n})$ from $y \res [b_{n-1},b_{n})$ as follows.
Let $A\subseteq [b_{n-1},b_n)$ be the integers $i$ in this interval such that
$y \res [i,i+k_0-1)= 0_{k_0}$. Let $A' \subseteq A$ be the last $\lfloor \frac{|A|}{x(2n)}\rfloor$
many elements of $A$. For each $i \in A'$ we change $y(i)$ from a $0$ to a $1$,
and for all other $i$ in this interval we set $\varphi_1(x)(i)=y(i)$.
If $x(2n)$ does not tend to infinity,
then easily $\varphi(x) \notin \RNQ$ as
$\frac{N^{P_x}_i(1_{k_0},\varphi_1(x))}{N^{P_x}_i(0_{k_0},\varphi_1(x))}$
does not tend to $1$. If $x(2n)$ tends to infinity, then we easily have that $\varphi(x)$ is
in $\NQ$ (or $\RNQ$) iff $y$ is in $\NQ$ (resp.\ $\RNQ$). In this case, as above, we have that if
$x(2n+1)\to \infty$ then $y\in \NQ$, and if $x(2n+1)$ does not tend to infinity
then $y \in \RNQ\sm \NQ$. So, in all cases we have $x \in C\sm D$ iff
$\varphi(x)\in \RNQ\sm \NQ$.
\end{proof}
\subsection{Further Discussion}
Theorem~\ref{4cases} can be extended further. First, the hypothesis that
$Q=(q_i)$ in infinite in limit can be weakened to the following condition
studied by T.\ \v{S}al\'{a}t \cite{Salat}: $\lim_{N\to \infty} \frac{1}{N} \sum_{i=1}^N \frac{1}{q_i}=0$.
This condition is equivalent to saying that there is a set $D\subseteq \N$ of
density $0$ such that $(q_i)_{i \notin D}$ tends to infinity (see Theorem~1.20 of \cite{Walters}).
Since changing a sequence on a set of density $0$ may affect normality and ratio normality,
we must now use the argument of Theorem~\ref{NandRN}. At stage $n$ of the construction of
$\varphi_1(x)\res I_n$, we again use two operations $\Theta'_{x'(2n)}$ and $\Xi'_{x'(2n+1)}$.
The first operation $\Theta'_{x'(2n)}$ is the operation implicitly described in the proof of Theorem~\ref{NandRN}.
That is, we define the block $B_{i_0}$ exactly as in that proof, and define the sets
$A, A',A''\subseteq I_n$ as in that proof. We then eliminate the occurrences of the block $B_{i_0}$
at the points of $A''$ by applying the digit changing function $r$ as in Theorem~\ref{NandRN}.
Let $w$ be the result of applying this first operation to $z$ (so $w$ is the
$\varphi_1(x)$ of Theorem~\ref{NandRN}). The proof of Theorem~\ref{NandRN}
did not require that $Q$ be infinite in limit, and so we have that
$x(2n)\to \infty$ implies $w \in \NQ$ and $x(2n) \nrightarrow \infty$ implies
$w \notin \RNQ$. The function $r$ changes digits by at most $1$, and does not affect
distribution normality using Remark~\ref{rdn} and the fact that $D$ has density $0$
(changing a sequence on a set of density $0$ does not affect distribution normality).
So, $w \in \DNQ$. The second operation $\Xi'_{x'(2n+1)}$ is the operation $\Xi_{x'(2n+1)}$
of Theorem~\ref{4cases} except we only apply the operation to digits not in $D$.
We let $\varphi_1(x)$ be the result of applying these operations to $w$.
The operations $\Xi'_{x'(2n+1)}$ do not affect normality or
distribution normality as $q_i\to \infty$ off of $D$, and so for every block $B$,
$|N^Q_m(B,\varphi_1(x))-N^Q_m(B,w)|$ is bounded with $m$. As in Theorem~\ref{4cases}
we have that $\varphi_1(x)\in \DNQ$ iff $x(2n+1)\to \infty$. So, $\varphi=\varphi_2\circ
\varphi_1$ is a reduction of $C\sm D$ (or $D\sm C$ depending on the case) to
the desired set.
Second, we can prove the version of Theorem~\ref{4cases} with $\NQ$ and $\RNQ$
replaced with $\NkQ$ and $\RNkQ$, provided we assume that $Q$ is $k$-divergent
(and infinite in limit, or more generally
$\lim_{N\to \infty} \frac{1}{N} \sum_{i=1}^N \frac{1}{q_i}=0$).
We proceed as above except in defining the block $B_{i_0}$ used in the first
operation, we only consider the first $\sqrt[6]{x'(2n)}$ many good blocks
$B_1,\dots,B_p$ of length $k$. This makes sense since there is some
block of length $k$, namely $0_k$, which has infinite expectation.
If $w$ again denotes the result of applying the first operation in
all of the $I_n$, then the proof of Theorem~\ref{NandRN} shows that
if $x(2n)\to \infty$ then $|N^Q_m(B,w)-N^Q_m(B,z)|/Q_m(B)\to 0$ for all
blocks $B$ of length $k$. It follows that if $x(2n)\to \infty$
then $w \in \NkQ$ and if $x(2n) \nrightarrow \infty$ then
$w \notin \RNkQ$. Also, $w \in \DNQ$ as above. The second operation
works exactly as in the above argument, so $\varphi_1(x) \in \DNQ$
iff $x(2n+1)\to \infty$. So, $\varphi=\varphi_2\circ \varphi_1$
again gives the desired reduction.
\bibliographystyle{amsplain}
\input{genbases_revised_2.bbl}
\end{document}
|
\section{The First Section}
\section{Introduction}
The study presented here is part of our work within a larger collaborative project with the aim to confront
different methods to derive SFHs from integrated light against each other and against the color magnitude diagram
(CMD) approach.
Test object for this ongoing project is a field in the bar of the Large Magellanic Cloud (LMC), for which both an
integrated-light spectrum (obtained with the 3.6m ESO telescope, LaSilla) and data on its resolved stellar
population (obtained with the \emph{Hubble Space Telescope}) are available. That way, the results of the different
groups analysing the spectrum cannot only be compared with each other but can also be compared with the SFH
obtained by an analysis of the CMD of the same field.
A short description of the project can be found in Alloin et al. (2002); an analysis of the CMD for this field is
presented by Smecker-Hane et al. (2002).
In this contribution, we present parts of our analysis of the integrated-light spectrum of the LMC bar field.
Applying the results of our preparatory work presented in Lilly \& Fritze -- v. Alvensleben (2005, these
proceedings), and using our evolutionary synthesis code GALEV, we have performed a set of simulations of galaxies
with systematically varying SFHs, but constant metallicity (Z=0.008). By confrontation of the evolution of the
colors and spectra resulting from the various simulations we then investigate to which extent different SF
scenarios can be discriminated on the basis of their photometric and spectral properties, respectively, and in how
far the detailed SFH obtained by the CMD approach can be reproduced by integrated properties. To keep the study
focused, we keep other parameters, like the initial mass function or the metallicity, constant.
A short overview of our evolutionary synthesis code GALEV and its input physics can be found in Lilly \&
Fritze -- v. Alvensleben (2005, these proceedings).
\section{The LMC bar field and its spectrum}
\begin{figure}[t]
\plottwo{x.full.dr249.dipl.eps}{x.full.dr249.lowres.filter.dipl.eps}
\caption{Integrated-light spectrum of the LMC bar field (dereddened). {\itshape Left:\/} Original spectrum.
{\itshape Right:\/} Spectrum with lowered resolution to be compared with model spectra, and the three filters used
for the analysis.}
\label{abb.lmcspec}
\end{figure}
The integrated-light spectrum of the LMC bar field (FoV: 2.5'$\times$5') was obtained at the ESO 3.6m telescope on
La Silla in Dezember 2000 by E. Pompei and D. Alloin. The 2000 coordinates of the field are: $\alpha$ = 05:23:17
and $\delta$ = --69:45:42; for observational details cf. Alloin et al. 2002.
Figure \ref{abb.lmcspec} (left) shows the original spectrum (dereddened with A$_V$ = 0.249), Figure
\ref{abb.lmcspec} (right) the same spectrum but with lowered resolution to be compared with our model spectra.
For better orientation, Balmer lines $H_\alpha$ to $H_\eta$ are marked by vertical lines in the left panel.
\begin{table}
\caption{Colors derived from the integrated-light spectrum of the LMC bar field, obtained by folding the
spectrum with the respective filter functions.}
\smallskip
\begin{center}
{\small
\begin{tabular}{c c c}
\tableline
\noalign{\smallskip}
(B--V)$_{HST}$ & (V--R)$_{HST}$ & (B--R)$_{HST}$\\
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
0.44mag & 0.53mag & 0.97mag\\
\noalign{\smallskip}
\tableline
\end{tabular}
}
\end{center}
\label{tab.colors}
\end{table}
\noindent Unfortunately, the model spectra we use have a resolution too low to successfully analyse spectral
features; therefore, our analysis mainly depends on broad band colors obtained from the spectrum, and the shape of
the spectral energy distribution.
The usable wavelength range of the spectrum allows for folding with three filters (filter functions shown in Fig.
\ref{abb.lmcspec}, right panel):
\emph{HST} WFPC2 F439W, F555W, and F675W (in the following referred to as B$_{HST}$, V$_{HST}$, R$_{HST}$);
Table \ref{tab.colors} gives the resulting colors obtained by folding the spectrum with the respective filter
functions.
\section{A simple 3-phase SFH}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.47\linewidth]{x.Dipl22.eps}
\caption{Photometric evolution of a simple 3-phase scenario in terms of (B--V)$_{HST}$, (V--R)$_{HST}$, and
(B--R)$_{HST}$ with the corresponding SFH; the observed LMC bar field colors (cf. Table \ref{tab.colors}) are
marked with black dots at 15 Gyr galaxy age.}
\label{abb.lmc22}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.52\linewidth]{x.spec_lmc22.eps}
\caption{Model spectrum of the 3-phase scenario (cf. Fig. \ref{abb.lmc22}) at a galaxy age of 15 Gyr against
the observed LMC spectrum. Both spectra are normalized at 4810\AA\ (arbitrary value).}
\label{abb.lmc22spec}
\end{center}
\end{figure}
From our preparatory work presented in Lilly \& Fritze -- v. Alvensleben (2005, these proceedings), we have learned
that, using integrated light only, variations in the SFH of a galaxy can be traced for only about 1, at the
utmost 4 Gyrs of lookback time.
This means, the relative distribution of SF \emph{within} early epochs ($\geq$ 4 Gyr ago) of galaxy evolution is
almost irrelevant, and that of the nearer history (4 Gyr $\geq$ lookback time $\geq$ 1 Gyr) only of weak relevance
for the observed colors; on the other hand, the relative distribution of SF \emph{between} these 3 phases -- and
not only the SF within the last Gyr -- \emph{is} important.
Therefore, it should be possible to fit any given set of colors (or, as we have also learned at the study,
even Lick indices) using an evolutionary history of only 3 different phases of SF, if the phases are chosen
appropriately:
Figures \ref{abb.lmc22} and \ref{abb.lmc22spec} show that a very simple ``3-phase scenario'' with phase 1 ranging
from 0 to 11 Gyr galaxy age, phase 2 from 11 to 14 Gyr galaxy age, phase 3 from 14 to 15 Gyr galaxy age, can
indeed fit both the observed colors (they are marked with black dots at the right edge of Fig. \ref{abb.lmc22},
top panel) and the observed spectrum (cf. Fig. \ref{abb.lmc22spec}) of the LMC bar field reasonably well;
differences still visible are most likely due to our disregarding of any chemical enrichment history and the fact
that we assume a constant SFR within phase 3.\\
\begin{figure}[t]
\begin{center}
\includegraphics[height=0.29\linewidth]{x.spec_lmc22_3phase.eps}
\includegraphics[height=0.29\linewidth]{x.spec_lmc22_3phaseNORM2.eps}
\caption{Contribution of the 3 phases of the 3-phase scenario to the total spectrum.
{\itshape Left:\/} Absolute contribution (i.e., summation of the 3 subpopulation spectra gives the total
spectrum). {\itshape Right:\/} Subpopulation spectra and total spectrum normalized at 4810\AA.}
\label{abb.phasecontrib}
\end{center}
\end{figure}
Figure \ref{abb.phasecontrib} illustrates the influence of the 3 different epochs of SF by showing, for the
SFH given above, the spectral contributions of stellar populations originating from each of these phases to the
total spectrum after 15 Gyrs:
The left panel shows, e.g., that phase 1, though its duration is almost three times longer than that of phase 2,
contributes only about half of the light of phase 2 to the total spectrum.
The right panel, on the other hand, shows that the first 14 Gyrs together have roughly the same influence on the
final shape of the spectrum as the most recent 1 Gyr of galaxy evolution.
\section{Some experimentation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.32\linewidth]{x.Dipl23.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl25.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl27.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl24.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl26.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl28.eps}
\caption{Variations of the 3-phase scenario (cf. Fig. \ref{abb.lmc22}): Scenarios with systematic variations of
the SFH within phase 1 ({\itshape a, b}), phase 2 ({\itshape c, d}), and phase 3 ({\itshape e, f});
note that the relative distribution of the total amount of SF between the 3 phases remains unchanged.}
\label{abb.lmc23bis28}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.32\linewidth]{x.Dipl22a1.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22b1.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22c1.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22a2.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22b2.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22c2.eps}
\caption{Variations of the 3-phase scenario (cf. Fig. \ref{abb.lmc22}): Scenarios with twice and half the SF,
respectively, within phase 1 ({\itshape g, h}), phase 2 ({\itshape i, j}), and phase 3 ({\itshape k, l});
the relative distribution of the total
|
amount of SF between the 3 phases is changed.}
\label{abb.lmc22a1bis22c2}
\end{center}
\end{figure}
We systematically vary the SFH in each of the 3 phases of our simple 3-phase scenario:\\
In the six scenarios shown in Figure \ref{abb.lmc23bis28}, the constant SF within the 3 phases is, for each phase,
replaced by a burst-like SF at the end and the beginning of the phase, respectively.
Because the relative distribution of SF \emph{between} the 3 phases remains unchanged, this kind of variations has
almost no effect on the final colors ($\Delta$color $<$ 0.05mag even in (B--R)$_{HST}$) if applied to phase 1 or 2;
remarkably, it is not even possible to decide if the galaxy is 15 or 6 Gyrs old in our example (cf. scenario
{\itshape a} and {\itshape b}).
The same kind of variation applied to phase 3 (i.e., to the most recent 1 Gyr of galaxy evolution), however,
results in a significant change of color ($\Delta$(B--R)$_{HST}$ $\approx$ 0.2mag).
In the scenarios shown in Figure \ref{abb.lmc22a1bis22c2}, the SF within the 3 phases remains constant, but,
compared to the original scenario, with half and twice the star formation rate (SFR), respectively. As before, the
variation is applied to each of the phases.
Since this kind of variation changes the relative distribution of SF between the 3 phases, the effect on the final
colors after 15 Gyrs is much larger than before, if applied to phases 1 and 2 ($\Delta$color $\approx$ 0.1mag in
(B--R)$_{HST}$). Variation of phase 3 (scenarios {\itshape k, l}) has a similar effect on colors
($\Delta$(B--R)$_{HST}$ $\approx$ 0.2mag) as the kind of variation applied in Fig. \ref{abb.lmc23bis28} (scenarios
{\itshape e, f}).\\
The experiments sketched above confirm the expectations from our previous work (cf. Section 3).
In phase 1 and 2, changes of the \emph{number of stars produced} within the phase have a stronger effect on colors
than changes of the \emph{mere distribution} of SF within the phase.
A similar behaviour for the effects of SFR changes in phases 1-3 is found in the spectral energy distribution (not
shown here).
\section{The CMD based SFH}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\linewidth]{x.cmdLMCx.200dpi.eps}
\caption{{\itshape Left:} CMD and SFH as presented by Smecker-Hane et al. (2002). {\itshape Right:} Model CMD at
a simulated galaxy age of 15 Gyr, using Smecker-Hane et al.'s SFH. Stellar populations originating from 4
different phases of star formation are coded in different colors (cf. the electronic version of this paper).}
\label{abb.CMDs}
\end{center}
\end{figure}
The resolved stellar population of the LMC bar field was observed with \emph{HST} WFPC2 in 1997 (PI:
Smecker-Hane).\\
Smecker-Hane et al. (2002) presented a SFH derived from their analysis of the high-quality CMD obtained from these
observations; Figure \ref{abb.CMDs} (left) shows this CMD together with the published SFH\footnote{Note that the
time axis is inverted to match the standard used in this paper (evolution of the galaxy from left to right).}. The
right panel of the same figure shows a model CMD at a simulated galaxy age of 15 Gyr, using the SFH from
Smecker-Hane et al. (2002), with 4 color-coded epochs of star formation (cf. the electronic version of this
contribution on astro-ph).
Note that the CMD is computed using a relatively simple approach (cf. Lilly 2003):
The number of stars at each point on the isochrones is determined by the IMF and the relative weight of the
isochrone; the stars are spread around their theoretical position by applying observational errors in I and
V-I, small for bright stars and larger for fainter stars.
For any given SFH (and chemical enrichment history) our code is able to calculate the \emph{time evolution} of the
distribution of stars in the HR diagram and any desired CMD.
However, we do not interpolate between isochrones; therefore, we had to increase the assumed observational errors
in order to reduce the ``gaps'' between isochrones on the CMD (cp. observed with modell errors as shown in Fig.
\ref{abb.CMDs}). Also, features like binary stars are not regarded.
Hence, our model CMDs are not intended to be directly compared with observations but, so far, for principle
investigations only (model-model comparisions).\\
Figures \ref{abb.smeckerhane} and \ref{abb.smeckerhaneCMD}, left panels, show the spectrophotometric evolution
and the model CMD at 15 Gyr galaxy age resulting from Smecker-Hane et al.'s SFH.
The middle panels of the same figures show the same for a scenario using a ``smoothed'' version of this SFH.
``Smoothed'' means that within each of the four epochs of SF (color coded in Fig. \ref{abb.CMDs}), the SFR is
put to a constant value, conserving the number of stars produced in this epoch (i.e., without changing the
relative amounts of SFR \emph{between} the 4 epochs). Note, however, that the latter is not the case when
compared to the 3-phase SFH (right panels in Figs. \ref{abb.smeckerhane} and \ref{abb.smeckerhaneCMD}).
In the next section, both scenarios will be confronted against each other and against the simple 3-phase scenario.
\section{CMDs and integrated light: Comparison and Conclusions}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.32\linewidth]{x.spec_lmcSH004a.eps}
\includegraphics[width=0.32\linewidth]{x.spec_lmcSH004glattB.eps}
\includegraphics[width=0.32\linewidth]{x.spec_lmc22.eps}
\includegraphics[width=0.32\linewidth]{x.DiplSH004a.eps}
\includegraphics[width=0.32\linewidth]{x.DiplSH004glattB.eps}
\includegraphics[width=0.32\linewidth]{x.Dipl22.eps}
\caption{Confrontation of scenarios using Smecker-Hane at al.'s (2002) original SFH (left panels), a
``smoothed'' Smecker-Hane SFH (central panels), and using a simple 3-phase SFH (right panels); see text.\newline
{\itshape Top:\/} Model spectra of the scenarios after 15 Gyr (black) against observed spectrum (grey).
{\itshape Bottom:\/} Photometric evolution of the scenarios in terms of (B-V)$_{HST}$, (V-R)$_{HST}$, and
(B-R)$_{HST}$ with the corresponding SFHs; the observed colors (obtained from the observed spectrum) are marked
with black dots at 15 Gyr.}
\label{abb.smeckerhane}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.32\linewidth]{x.SH004.eps}
\includegraphics[width=0.32\linewidth]{x.SH004glattB.eps}
\includegraphics[width=0.32\linewidth]{x.lmc22.eps}
\includegraphics[width=0.32\linewidth]{x.sfh_SH004a.eps}
\includegraphics[width=0.32\linewidth]{x.sfh_SH004glattB.eps}
\includegraphics[width=0.32\linewidth]{x.sfh_lmc22.eps}
\caption{Confrontation of scenarios using Smecker-Hane et al.'s (2002) original SFH (left panels), a
``smoothed'' Smecker-Hane SFH (central panels), and using a simple 3-phase SFH (right panels); see text.\newline
{\itshape Top:\/} Model CMDs at a simulated galaxy age of 15 Gyr. {\itshape Bottom:\/} Corresponding SFHs.}
\label{abb.smeckerhaneCMD}
\end{center}
\end{figure}
Figures \ref{abb.smeckerhane} and \ref{abb.smeckerhaneCMD} confront the spectrophotometric evolution and the CMDs,
respectively, of three scenarios using Smecker-Hane et al.'s (2002) SFH (left panels), using the ``smoothed''
Smecker-Hane SFH (central panels), and using the ``simple 3-phase'' SFH (right panels).\\
In terms of colors and spectra, the scenario using Smecker-Hane et al.'s SFH is, at a galaxy age of 15 Gyr,
practically identical to the 3-phase scenario; both differ slightly ($\Delta$(B--R)$_{HST}$ $<$ 0.1mag) from the
``smoothed'' Smecker-Hane scenario.
This shows that two scenarios with very different SFHs can result in very similar, observationally
indistinguishable integrated-light properties; on the other hand, scenarios with very similar ``global'' SFH (as
e.g. the ``original'' and ``smoothed'' Smecker-Hane et al. SFH) can differ in their final colors.
The results show not only the ambiguity of SFHs obtained from integrated light but also emphasize again the
importance of the \emph{most recent} (i.e., lookback time $\leq$ 1 Gyr) SFR for the observed colors.\\
In terms of CMDs, the ``original'' and ``smoothed'' Smecker-Hane et al. scenarios seems to be very similar; the
CMD computed using the 3-phase scenario differs from both.
If this result can be validated by a solid numerical comparision between the CMDs (not done yet), this means that
the \emph{global} distribution of SF is crucial for the appearance of CMDs - as opposed to integrated light.
However, note that not even CMDs are free of ambiguity. The ambiguity in the SFH derived from a CMD increases with
increasing lookback time, and SFHs like that presented by Smecker-Hane et al. (2002) are most likely not as exact
as their complicated shape suggests; this was already shown by Lilly (2003) and most recently confirmed by the
``double-blind Cozumel experiment'' (Holtzman 2005, these proceedings).\\
\acknowledgements
TL gratefully acknowledges partial travel support from the organizers;
his work is partially funded by DFG grant Fr 916/11-1-2-3.\\
|
\section{Introduction}
When Niels Bohr and Erwin Schr\"odinger asked decades ago whether new physical principles are needed to explain living systems, the answer seemed ``no'' \cite{bohr33,schroedinger44}.
More recently, however, the field of stochastic thermodynamics with its temporal violations of macroscopic thermodynamic laws at the microscopic scale have provided a new physical perspective on life.
Most remarkable corner stones of far-from-equilibrium thermodynamics are the fluctuation theorems and Seifert's thermodynamic uncertainty relation, stressing the important role of entropy production \cite{searles99,evans02,seifert12,barato15}.
At equilibrium, detailed balance prohibits any entropy production on average, but far from equilibrium such entropy production is a characteristic feature \cite{battle16} and reflects the flow of time \cite{roldan15}.
The fluctuation theorem by Evans and Searles allows the exact calculation of the entropy production along a trajectory from the time-forward and time-reversed path (corresponding to a movie played backwards), where paths can be calculated from e.g.\ Gillespie simulations of the underlying chemical master equation \cite{evans02}.
However, due to its intrinsic connection with the time-reversed path, the theorem cannot be used to calculate the probability of a path simply from its entropy production.
The situation is different when using the least-action principle, which allows the prediction of the most likely path between two points in a stochastic system from minimising the action (integral over the Lagrangian) \cite{assaf17,arnold00}.
This is often done with a Langevin approximation of the master equation, such as using stochastic differential equations incorporating noise terms \cite{perez16,cruz18}.
However, now the link to thermodynamics is less clear as the role of the entropy production is obscured by the action functional.
In this paper, we combine the fluctuation theorem and least-action principle to address the stability of steady states in non-equilibrium systems.
In particular, we will elucidate the roles of steady-state entropy and fluctuations, as well as steady-state and path entropy production in state switching.
For this purpose, we use two different low-dimensional minimal models shown in Fig.~\ref{fig:fig1}, the Schl\"ogl \cite{schloegl72,vellela09,endres15} and the toggle switch \cite{cherry00} models, with fixed external species concentrations to ensure non-equilibrium behaviour.
\section{Merging two approaches for state switching}
To investigate the thermodynamics of state switching we shall study non-equilibrium bistable systems with macrostates denoted $A$ and $B$, where both macrostates correspond to sets of microstates in the discrete space of molecule numbers \textbf{X}, which is a vector for multiple chemical species.
The assumption is made that no significant amount of time is spent outside these macrostates.
In the large volume limit, the process of switching between states can be assumed to be a Poisson process (with exponentially distributed waiting times, see Fig.~\ref{fig:fig1}B inset).
Thus, $\langle\tau_A\rangle=k_{A{\rightarrow}B}\int_0^\infty t\exp(-k_{A{\rightarrow}B}t) dt=k_{A{\rightarrow}B}^{-1}$ where $k_{A{\rightarrow}B}$ is the switching rate from $A$ to $B$, and similarly for the $B$ state.
These switching rates are coarse-grained and will in general have contributions from a number of microscopic paths.
The occupation probability of the $A$ state is then given by $p_A=\langle\tau_A\rangle/(\langle\tau_A\rangle+\langle\tau_B\rangle) = 1/(1+k_{A{\rightarrow}B}/k_{B{\rightarrow}A})$.
Hence, such a two-state system is completely described by the ratio of the switching rates.
How do we calculate these for actual molecular systems?
While switching in equilibrium systems is often determined by Kramer's formula, where the height of the energy barrier matters, the treatment of non-equilibrium systems requires entire paths\cite{wang15,feng14}.
\begin{figure*}[!tbhp]
\includegraphics[width=0.75\textwidth]{fig1.png}
\caption{{\bf Overview of models.} (a) Example simulation of Schl\"ogl model, displaying switching between high and low $X$ (copy number) states with $A$ and $B$ reservoir species held constant.
(b) Example switching paths between $A$ and $B$ states for toggle switch model. (Inset) Waiting time distribution for A to B switching, with a clear exponential distribution beyond a small initial time.
(c) Symbolic chemical equations describing the Schl\"ogl model.
(d) Schematic illustration of 4 species toggle switch model, consisting of two mutually repressing chemical species ($A$,$B$) in addition to substrate (S) and waste (W) species. As the mutual repression between the two chemical species occurs via dimer binding the two rates relevant to this ($r$ and $f$) are not included in the schematic.
The parameter values used to generate the example paths can be found in the supplementary material.}
\label{fig:fig1}
\end{figure*}
\subsection{Microscopic fluctuation theorems}
Using a microscopic description for non-equilibrium systems, the dynamics can be described by a path $\Gamma$, e.g.\ ${\bf X}_0$, ${\bf X}_1$, ${\bf X}_2$,{\dots}, ${\bf X}_N$, obtainable from simulations of the chemical master equation \cite{gaspard04}.
The master equation describes the probability evolution of the microscopic stochastic process and can be formulated as
\begin{equation}
\begin{split}
\frac{d}{dt}P(\mathbf{X};t) = \sum_{i=1}^{N_\rho}&[W_{\rho_i}(\mathbf{X}-\boldsymbol{\nu}_{i}|\mathbf{X})P(\mathbf{X}-\boldsymbol{\nu}_{i};t)\\
&- W_{\overline{\rho}_i}(\mathbf{X}|\mathbf{X}-\boldsymbol{\nu}_{i})P(\mathbf{X};t)],
\end{split}
\end{equation}
where $P(\mathbf{X};t)$ is the probability of being in microstate $\mathbf{X}$ at time $t$ and $N_\rho$ is the total number of reactions (including reverse reactions) with $\overline{\rho}_i$ the corresponding reverse reaction to reaction $\rho_i$.
In particular, $W_{\rho_c}(\mathbf{X}_a|\mathbf{X}_b)$ is the transition rate from state $a$ to state $b$ by reaction $\rho_c$, and $\boldsymbol{\nu}_{i}$ is the change in molecular copy numbers caused by reaction $\rho_i$.
For every time-forward path, there also exists a time-reversed path $\overline{\Gamma}$, ${\bf X}_N$, ${\bf X}_{N-1}$,{\dots}, ${\bf X}_0$.
The probability of observing a particular path, e.g.\ the above time-forward path, is given by the path probability $P_{\Gamma}=\mathcal{N}P({\bf X}_0)\prod^N_{i=1}P(\tau_{i-1,i})W_{\rho_{i}}({\bf X}_{i-1}|{\bf X}_i)$ assuming a continuous-time memory-less Markov process with transition rates $W_{\rho_{i+1}}({\bf X}_i|{\bf X}_{i+1})$.
Specifically, $\rho_i$ is the reaction that causes step $i$, $P(\tau_{i,i+1})$ is the probability that there is no reaction in the time interval between $\tau_{i}$ and $\tau_{i+1}$, and $\mathcal{N}$ is a normalisation factor, ensuring $\sum_{\Gamma}P_{\Gamma} = 1$.
As we are considering a non-equilibrium steady state (NESS) probability distribution, the total entropy production along the path is given by the steady-state fluctuation theorem (FT) \cite{seifert05,malek17} as
\begin{equation}
\Delta S_\Gamma = \ln\left(\frac{P_{\Gamma}}{P_{\overline{\Gamma}}}\right) = \ln\left(\frac{P({\bf X}_0)}{P({\bf X}_N)}\right)+\ln\left(\frac{W_{\Gamma}}{W_{\overline{\Gamma}}}\right),
\label{eq:FT}
\end{equation}
with $W_{\Gamma} = W_{\rho_{1}}({\bf X}_0|{\bf X}_1)\dots W_{\rho_{N}}({\bf X}_{N-1}|{\bf X}_N)$ and $W_{\overline{\Gamma}} = W_{\overline{\rho}_{N}}({\bf X}_N|{\bf X}_{N-1})\dots W_{\overline{\rho}_{1}}({\bf X}_{1}|{\bf X}_0)$ (for details see supplementary material).
The entropy considered here is defined as the Shannon entropy of the probability distribution of states $\mathbf{X}$, i.e.\
\begin{equation}
S(t) = -\sum_{\mathbf{X}} P(\mathbf{X};t)\ln P(\mathbf{X};t)
\end{equation}
in units where Boltzmann's constant $k_B = 1$.
For rare switching, valid for large volume $\Omega$, the steady-state value of $S$ has contributions from $S_A$ with $\mathbf{X}{\in}A$ and $S_B$ with $\mathbf{X}{\in}B$.
Restricting our consideration to paths that start within macrostate $A$ and end within $B$, we can calculate the ensemble-averaged total entropy production $\Delta S_{A{\rightarrow}B}=\sum_{\Gamma|A{\rightarrow}B} P_\Gamma\Delta S_\Gamma=\langle\ln(P({\bf X}_0)/P({\bf X}_N))\rangle_{A{\rightarrow}B}+\langle\ln(W_{\Gamma}/W_{\overline{\Gamma}})\rangle_{A{\rightarrow}B}$, where the first term on the right-hand side (RHS) corresponds to the average change in entropy of the system between start ${\bf X}_0{\in}A$ and end ${\bf X}_N{\in}B$ (note $N$, ${\bf X}_0$ and ${\bf X}_N$ can vary for different paths).
Further, the second term on the RHS corresponds to the flow of entropy from system to medium (here chemical reservoirs), which is termed entropy production in the medium \cite{seifert12}.
In the limit of small fluctuations (large $\Omega$, or small noise approximation), the first term becomes negligibly small and the entropy produced is given by the second term only, e.g.\ $\Delta S_{A{\rightarrow}B} \approx \langle \ln{\left(W_{\Gamma}/W_{\overline{\Gamma}}\right)}\rangle_{A{\rightarrow}B} \approx \ln\left(k_{A{\rightarrow}B}/k_{\overline{A{\rightarrow}B}}\right)$.
This expression is of limited use in determining state switching rates as $k_{A{\rightarrow}B}$ is intrinsically linked to $k_{\overline{A{\rightarrow}B}}$, i.e.\ the switching rate associated with time-reversal of the paths from $A$ to $B$.
Also note that in general due to the dissipative dynamics the ensemble of time-reversed paths does not correspond to the ensemble of reverse-switching paths, and hence $k_{\overline{A{\rightarrow}B}} \neq k_{B{\rightarrow}A}$ \cite{feng14}.
In order to uniquely determine switching rates we need to utilize a macroscopic formalism.
\subsection{Macroscopic Langevin coarse-graining}
In the macroscopic limit, we can make a continuum approximation of the master equation with the chemical Fokker-Planck equation (FPE), i.e.\ $X_i = x_i\Omega$ for $i=1,...,K$ a $K$-dimensional chemical system, with concentrations $x_i$ and volume $\Omega$.
The chemical Fokker-Planck equation can be derived following Gillespie \cite{gillespie00} as
\begin{equation}
\begin{split}
\frac{\partial}{\partial t}P(\mathbf{x};t) = &-\sum^{N}_{i=1}\frac{\partial}{\partial{x_i}}\left[f_i(\mathbf{x})P(\mathbf{x};t)\right]\\
&+ \frac{1}{2} \sum^{N}_{i,j=1}\frac{\partial^2}{\partial{x_i}\partial{x_j}}\left[D_{ij}(\mathbf{x})P(\mathbf{x};t)\right],
\end{split}
\label{eq:CFPE}
\end{equation}
where the deterministic force $\boldsymbol{f}(\mathbf{x})$ is given by
\begin{equation}
f_i(\mathbf{x}) = \sum^{N_\rho}_{j=1} \nu_{ji} w_{\rho_j}(\mathbf{x})
\end{equation}
and the diffusion matrix $\boldsymbol{D}(\mathbf{x})$ is defined as
\begin{equation}
D_{ij}(\mathbf{x}) = \sum^{N_\rho}_{k=1}\nu_{ki}\nu_{kj}w_{\rho_k}(\mathbf{x}).
\label{eq:Diff}
\end{equation}
In the above definitions, $\nu_{ji}$ is the change in molecular copy number of species $i$ that a single occurrence of reaction $\rho_j$ causes, and $w_{\rho_i}(\mathbf{x})$ is the rate (in concentration units) of reaction $\rho_i$ for concentration $\mathbf{x}$, which is the continuum limit of $W_{\rho_i}(\mathbf{X}-\boldsymbol{\nu}_i|\mathbf{X})/\Omega$.
This Fokker-Planck equation is generally a reasonable approximation for large (but finite) values of $\Omega\,\,$ \cite{gillespie00}, particularly near thermodynamic equilibrium \cite{grima11,hanggi84}.
However, while the accurate prediction of switching rates is difficult, the characterization of relative stability is easier.
The chemical Langevin equation corresponds to this Fokker-Planck equation and can be expressed as $\dot x_i=f_i({\bf x})+g_{ij}({\bf x})\xi_j(t)$ with $\xi_j(t)$ uncorrelated white Gaussian noises of zero mean and autocorrelation $\langle\xi_i(t)\xi_j(t')\rangle = \Omega^{-1}\delta_{ij}\delta(t-t')$.
The deterministic force in direction $i$ is given by $f_i$, and $g_{ij}$ determines the propagation of noise from direction $j$ to $i$.
Here, and throughout the paper we adopt the convention that repeated indices (in this case $j$) are summed over.
For the models considered in this paper $g_{ij}$ is always diagonal, representing multiplicative noise.
As Eq.~\ref{eq:CFPE} is an It\^o equation, we treat $g_{ij}$ via It\^o integration \cite{mannella12}.
It is worth noting that our definition of entropy is essentially unchanged as we move to the continuum limit, though it is now the Shannon entropy of a continuous probability distribution.
Paths from this Langevin equation can be treated via path integral methods \cite{chernyak06}, which can be done more simply in our case.
When the probability of escape from a macrostate is sufficiently low, the stochastic transition will be expected to concentrate along a single path $\bf x^*$, with paths significantly diverging having probabilities so low (for large $\Omega$) as to have negligible impact on overall escape probability \cite{touchette09}.
The Wentzel–-Kramers–-Brillouin (WKB) approximation, the classical analogue of the quantum mechanical WKB \cite{assaf17}, can then be used to obtain the conditional probability of this path as $P_{A{\rightarrow}B}\sim\exp(-\Omega \mathcal{A}[{\bf x^*}])$, where $P_{A{\rightarrow}B}$ is the conditional probability of reaching macrostate $B$ from the initial macrostate $A$, and $\mathcal{A}$ is the action, as derived in \cite{ventsel70}.
The path $\bf x^*$ will thus minimize the action $\mathcal{A}[{\bf x^*}] = \min \mathcal{A}[{\bf x}] = \mathcal{A}_{A{\rightarrow}B}$.
This path has a fixed start and end point at the minima of macrostates $A$ and $B$, respectively.
(These points can be reached from any point in their respective macrostates by a zero action path.
Thus, the minimum action path $\mathbf{x}^*$ can represent the general path between macrostates.)
The relevant action for a path with duration $\tau$ is the Freidlin--Wentzell (FW) action \cite{ventsel70}
\begin{equation}
\mathcal{A}[{\bf x}]=\frac{1}{2}\int_0^\tau(\dot{x}-f)_iD_{ij}^{-1}(\dot{x}-f)_j\,dt,\label{eq:LDT}
\end{equation}
with the diffusion matrix given by $D_{ij}=g_{ik}g_{kj}$.
Any choice of components $g_{ij}$ is acceptable provided that they result in a diffusion matrix that matches Eq.~\ref{eq:Diff}.
As we are considering It\^o integration this action possesses one additional term, which can be neglected in the large $\Omega$ limit \cite{touchette09}.
(If we instead use Stratonovich calculus, three additional terms would appear, which also disappear in the large $\Omega$ limit \cite{arnold00}).
Away from this limit, switching paths no longer pass through the same saddle point as their reverse switching paths \cite{feng14}.
All of our results will pertain to the large $\Omega$ limit.
In this limit, the mean-first passage time (MFPT) is given by $T_{A{\rightarrow}B} \sim P^{-1}_{A{\rightarrow}B}=Q_{A{\rightarrow}B}\exp(\Omega \mathcal{A}_{A{\rightarrow}B})$ or $\ln(T_{A{\rightarrow}B}) = \ln(Q_{A{\rightarrow}B}) + \Omega \mathcal{A}_{A{\rightarrow}B}$, so that as $\Omega$ grows the contribution from the prefactor $Q_{A{\rightarrow}B}$ becomes less important and the second term on the RHS describes the MFPT to logarithmic precision \cite{cruz18,bouchet16}.
This approach is often more accurate than treatment based on the FPE obtained by van-Kampen expansion of the master equation \cite{assaf17}.
While our expression for the MFPT applies in the large deviations limit, alternative forms can be obtained outside of it \cite{fiasconaro05}.
An expression for the entropy production based on the time-reversal of Langevin paths can be obtained by noting that the probability of the most probable switching path $A{\rightarrow}B$ is given by $P_{A{\rightarrow}B} = \exp(-\Omega\mathcal{A}_{A{\rightarrow}B})/Q_{A{\rightarrow}B}$.
The probability of the corresponding time-reversed path $\overline{A{\rightarrow}B}$ is then found to be $P_{\overline{A{\rightarrow}B}} = \exp(-\Omega\mathcal{A}_{\overline{A{\rightarrow}B}})/Q_{A{\rightarrow}B}$, where $\mathcal{A}_{\overline{A{\rightarrow}B}}$ is the action of the time-reversed path.
As factor $Q_{A{\rightarrow}B}$ has not changed, this probability is not in general equal to $P_{B{\rightarrow}A}$, which contains factor $Q_{B{\rightarrow}A}$.
Combining the above two expressions as in Eq.~\ref{eq:FT} generates an expression for the entropy production for Langevin paths as $\Delta S^{L}_{A{\rightarrow}B} = \Omega\left(\mathcal{A}_{\overline{A{\rightarrow}B}} - \mathcal{A}_{A{\rightarrow}B}\right)$ \cite{endres17}.
Substituting Eq.~\ref{eq:LDT} into this relation leads to
\begin{equation}
\Delta S^{L}_{A{\rightarrow}B} = 2\Omega\int_0^\tau \dot x_i D_{ij}^{-1} f_j\,dt.
\label{eq:Lent}
\end{equation}
Due to our use of a Langevin equation to describe the paths, the above expression is a coarse-grained (apparent) entropy production \cite{mehl12}, which disappears along with the action at steady steady ($\dot x_i=f_i({\bf x})=0$ in the large $\Omega$ limit).
This suggests that only entropy production along the path matters, and that the Langevin formalism within the steady state is equivalent to a quasi-equilibrium.
In the supplementary material we decompose this coarse-grained entropy production into approximate entropy production (EP) and flow (EF) terms, and find that our EP term does generally correlate with the
|
entropy production as calculated from the master equation.
Hence, despite the obvious difference between the entropy productions from the master equation (Eq.~\ref{eq:FT}) and the Langevin paths (Eq.~\ref{eq:Lent}), the latter has predictive power.
We now want to investigate the special case of one-dimensional systems.
To do so we shall use a relation that is valid at every point along an action minimising path \cite{neu18}
\begin{equation}
\sum_{ij}\dot{x}_iD^{-1}_{ij}\dot{x}_j = \sum_{ij}f_iD^{-1}_{ij}f_j,
\label{eq:PK}
\end{equation}
which can be interpreted as every point along the minimising path having equal kinetic and potential energy (defined below).
In the one-dimensional case this relation simplifies further to
\begin{equation}
\begin{split}
\dot{x}_1D^{-1}_{11}\dot{x}_1&=f_1D^{-1}_{11}f_1\\
|\dot{x}_1|&=|f_1|,
\end{split}
\end{equation}
where $||$ indicates that the magnitude has been taken.
This means that both paths have the same speeds along their lengths as they necessarily pass through the same points (in 1D).
Thus, for one dimensional systems the path $B{\rightarrow}A$ is identical to the time-reversed path $\overline{A{\rightarrow}B}$.
It then follows that for this case $k_{\overline{A{\rightarrow}B}} = k_{B{\rightarrow}A}$, and so the ratio of switching rates are determined solely by the entropy production as
\begin{equation}
k_{A{\rightarrow}B}/k_{B{\rightarrow}A}=\exp(\Delta S^L_{A{\rightarrow}B}).
\label{eq:schFT}
\end{equation}
When considering the broader class of multi-dimensional systems such simple relations no longer apply and we instead consider numerical approaches to explicit minimal models.
This will allow us to investigate how the entropy production varies along the path, and whether diffusion strength and steady-state entropies matter.
\section{Schl\"ogl and toggle switch models}
\begin{figure*}[!tbhp]
\centering
\includegraphics[width=\textwidth]{fig2.png}
\caption{{\bf Dependence of actions on path entropy productions.} (a) Minimum action paths for Schl\"ogl model; aside direction, the only difference between paths is the amount of time spent at the fixed points where there is no contribution to the action. (b) Minimum action paths for toggle switch model, now showing clear differences.
(c) Action ($\mathcal{A}$), entropy production ($\Delta S^L$), kinetic (KE) and potential energy (PE) of Schl\"ogl paths with KE and PE defined in Eq.~\ref{eq:consv}, clearly showing that the entropy production of one path is the opposite of the other. In this and the remaining panels a solid line corresponds to the path from $A{\rightarrow}B$ and a dashed line to $B{\rightarrow}A$. Magnifications are meant to clarify line styles.
(d) Action, differences in action and entropy production, KE and PE along the toggle switch paths. The difference in action is proportional to the difference in entropy produced along the path (this linear relationship is further discussed below).
(e) Entropy production (EP) and entropy flow (EF) terms along the Schl\"ogl paths. The difference (EP-EF) is equal to the entropy production of time-reversed of Langevin paths. (Inset) Plot showing how $f$ and $\dot{x}$ vary along an exemplar minimum action path. (f) Equivalent plot for the toggle switch paths. The parameter values used to generate the paths can be found in the supplementary material.}
\label{fig:fig2}
\end{figure*}
The two models considered are the Schl\"ogl and toggle switch models (Fig.~\ref{fig:fig1}).
For both models concentration-constraints are used to make the models non-equilibrium.
The Schl\"ogl model is a simple one-dimensional model that exhibits bistability \cite{schloegl72,vellela09}.
It involves three chemical species $A$, $B$ and $X$, which obey the chemical dynamics shown in Fig.~\ref{fig:fig1}C.
Holding the concentration of external species fixed (i.e. constant $a$ and $b$), this model can be reexpressed in terms of the dynamics of concentration $x$ as
\begin{equation}
\frac{dx}{dt} = k_{-2}bx^2 - k_{+2}x^3 - k_{-1}x + k_{+1}a + g_{x}(x)\xi(t),
\end{equation}
where
\begin{equation}
g_x(x) = \sqrt{k_{-2}bx^2 + k_{+2}x^3 + k_{-1}x + k_{+1}a}
\end{equation}
and $(k_{\pm 1},k_{\pm 2})$ are rate constants.
See the supplementary material for further details of this model.
The toggle switch model is a two-dimensional model that describes the dynamics of a simple bistable genetic switch \cite{cherry00} (see Fig.~\ref{fig:fig1}D).
We consider the adiabatic limit where the fraction of genes active is completely determined by the concentration of the corresponding repressing protein (see supplementary material for details).
This means that the model can be expressed in terms of change of protein concentrations $a$ and $b$ as
\begin{align}
\frac{da}{dt} &= \frac{kr}{r + fb^2} - k_{-}a - Ka + K_{-} + g_a(a,b)\xi_a(t)\\
\frac{db}{dt} &= \frac{qr}{r + fa^2} - q_{-}b - Qb + Q_{-} + g_b(a,b)\xi_b(t),
\end{align}
where
\begin{equation}
g_a(a,b) = \sqrt{\frac{kr}{r + fb^2} + k_{-}a + Ka + K_{-}},
\end{equation}
and
\begin{equation}
g_b(a,b) = \sqrt{\frac{qr}{r + fa^2} + q_{-}b + Qb + Q_{-}}.
\end{equation}
Further, $r$ is the gene activation rate and $f$ is rate constant for gene repression by protein dimers, the other eight rate constants describe the rates of protein production or degradation.
Here, the constant external concentrations $s$ and $w$ have been absorbed into the relevant rate constants.
In order to study the switching between states for these models, we need to find paths that minimise the action (Eq.~\ref{eq:LDT}).
These paths are usually determined by use of a quasi-Newton method (e.g. the L-BFGS algorithm) to find the minimising path for a particular duration $\tau$, with a gradient descent method used to find the value of $\tau$ that results in the lowest minimum \cite{perez16}.
However, in order to save computational time we made use of the faster (but more complicated) geometric minimum action method \cite{heymann08,heymann08-2,neu18}.
These paths were then be used in Eq.~\ref{eq:LDT} and Eq.~\ref{eq:Lent} in order to generate the corresponding minimum actions and path entropy productions.
\begin{figure*}[!tbhp]
\centering
\includegraphics[width=0.75\textwidth]{fig3.png}
\caption{{\bf Comparison of states and switching paths.} In all panels, red and blue dots denote specific parameterizations of Schl\"ogl and toggle switch models, respectively. In each plot the lines and shaded regions indicate best fits and $95\%$ confidence intervals for the particular data sets, respectively. (a) Comparison of log ratios of occupation probabilities vs difference in entropy productions at steady states from Gillespie simulations. The Schl\"ogl data has a Pearson correlation of $0.5172$ and the toggle switch has $0.3476$.
(b) Log ratio of occupation probabilities obtained from Gillespie simulation vs the difference in minimum action. The dashed line indicates a perfect correspondence. The toggle switch and Schl\"ogl data have correlations of $0.9515$ and $0.9738$, respectively.
The results shown are coarsely discretized due to the low $\Omega$ used to save computational time.
The discretization will effect the Schl\"ogl $B$ state disproportionately as it is formed of significantly fewer microstates than $A$.
This represents a potential explanation for the downwards shift of the Schl\"ogl data.
(c) Comparison of difference in entropy of steady states vs difference in action. Entropies were found by Gillespie simulation with $\Omega=1$. Both sets of data show weak correlations of $0.2892$ and $0.4622$ for the Schl\"ogl and toggle switch models, respectively.
(d) Difference in action vs difference in entropy produced along paths. Both models display a strong linear relationship, with correlations of $0.9445$ and $1.0000$ for the toggle switch and Schl\"ogl models, respectively.}
\label{fig:fig3}
\end{figure*}
Figs.~\ref{fig:fig2}A,B show exemplar minimum action paths for switching in the Schl\"ogl and the toggle switch model, respectively.
As the Schl\"ogl model is one-dimensional its time-reversed switching paths correspond to the switching paths for the contrary direction.
Fig.~\ref{fig:fig2}C shows how this leads to equal and opposite entropy productions (orange lines) along the paths.
For the more complicated toggle switch model this simple relation between paths is lost, but Fig.~\ref{fig:fig2}D shows a linear relation between the difference in minimum action (purple line) and the difference in path entropy productions (gold line).
Despite the systems' quasi-equilibrium behavior, non-equilibrium processes still occur.
In order to illustrate this fact we derived approximate entropy production and flow terms (details in supplementary material).
Figs.~\ref{fig:fig2}E,F show plots of the derived entropy production (EP, blue lines) and flow (EF, red lines), demonstrating non-zero contributions at the steady states.
In order to investigate links between occupation probabilities, entropy, and entropy production, 100 random parameters sets were created for each model.
Specifically, the procedure for the construction of the toggle switch parameter sets was as follows.
First, $k$ was randomly drawn from the (continuous) range $[1,100]$ and $K$ from the range $[0.1,10]$.
Then random numbers were drawn from the range $[0.001,0.1]$ to obtain the ratios $q/k$, $k_{-}/k$, $q_{-}/k$, $Q/K$, $K_{-}/K$, and $Q_{-}/Q$.
Finally, $r$ and $f$ were both drawn from the range $[0,10^4]$.
For the Schl\"ogl parameter sets, all four parameters ($k_{-1}$, $k_{+1}a$, $k_{+2}$, $k_{-2}b$) were drawn from the range $[0.1,10]$.
In both cases the constructed sets were accepted if they resulted in multiple non-zero steady states.
State entropies and occupation probabilities were obtained by direct Gillespie simulation of the chemical master equation \cite{gillespie77}.
Each of these simulations was run for the largest computationally feasible value of $\Omega$ in order to minimise the effect of discretization.
Steady-state entropy productions were calculated using the Schnakenberg method (product of flux and reaction affinity) \cite{schnakenberg76}.
Fig.~\ref{fig:fig3}A shows a weak correlation between state occupation probability and steady-state entropy production, which provides some evidence for the maximum entropy production principle (MaxEPP) \cite{endres17}.
This extremal principle proposes that states with higher entropy production are more dynamically stable (subject to other dynamical constraints) \cite{dewar09,lorenz01}.
We then approximate the log ratio of state occupation probabilities via the Freidlin--Wentzell theorem as, $\ln{(p_A/p_B)} \approx \ln{(P_{B{\rightarrow}A}/P_{A{\rightarrow}B})} = \ln{(Q_{B{\rightarrow}A}/Q_{A{\rightarrow}B})} + \Omega\left(\mathcal{A}_{A{\rightarrow}B} - \mathcal{A}_{B{\rightarrow}A}\right)$.
For large $\Omega$ only the second term would be expected to contribute but this limit is difficult to simulate.
Simulated occupation probabilities match well with this approximation (see Fig.~\ref{fig:fig3}B), demonstrating the validity of our use of the FW action.
Fig.~\ref{fig:fig3}C shows a weak correlation between difference in action and difference in state entropy, as expected from equilibrium theory where higher entropy states are more stable.
However, state entropy increases sublinearly with $\Omega$ so for large $\Omega$ it has no effect on the stability.
Fig.~\ref{fig:fig3}D shows a comparison of the difference in action and the difference in path entropy production, showing that the linear relation observed in Fig.~\ref{fig:fig2}C,D holds generally across parameter sets surveyed.
The effect of diffusion strength was found to be minimal (plots of this are therefore provided in the supplementary material).
Our results suggest a limited form of MaxEPP, which applies to the rate of switching between macrostates.
We shall proceed with our derivation by noting that the action can be split into two parts as $\Omega\mathcal{A}_{A{\rightarrow}B} = \mathcal{C}_{A{\rightarrow}B} - \frac{1}{2}\Delta S^{L}_{A{\rightarrow}B}$, where $\mathcal{C}_{A{\rightarrow}B}$ is the conservative action along the path $A{\rightarrow}B$ and $\Delta S^{L}_{A{\rightarrow}B}$ is the Langevin path entropy production (Eq.~\ref{eq:Lent}) \cite{endres17}.
The conservative action can be expressed in a similar form to Eq. \ref{eq:Lent} as
\begin{equation}
\mathcal{C}_{A{\rightarrow}B} = \frac{\Omega}{2}\int^{\tau}_{0}\left(\dot{x}_iD_{ij}^{-1}\dot{x}_j + f_i\,D_{ij}^{-1} f_j\right)\,dt,
\label{eq:consv}
\end{equation}
where the two terms resemble kinetic (KE) and potential energy (PE) contributions, respectively.
By substituting the expanded form of the action into the expression for switching path probability, a reduced form of MaxEPP can be obtained
\begin{equation}
P_{A{\rightarrow}B} = \frac{\exp{\left(\frac{1}{2}\Delta S^{L}_{A{\rightarrow}B} - \mathcal{C}_{A{\rightarrow}B}\right)}}{Q_{A{\rightarrow}B}},
\label{eq:MaxEPP}
\end{equation}
where $P_{A{\rightarrow}B}$ is the probability of the (most probable) switching path along $A{\rightarrow}B$, and $Q_{A{\rightarrow}B}$ is a constant.
This equation shows that there is a trade-off between minimization of the conservative action (i.e.\ fulfilling the equation of motion) and maximization of the path entropy production (i.e.\ being as dissipative as possible).
If the switching path and its contrary path pass through similar regions of space, then they will have similar kinetic energy contributions along their lengths provided that the deterministic force $\mathbf{f}$ does not vary too rapidly.
From Eq.~\ref{eq:PK} it can been seen that at every point on a minimizing path the potential energy contribution is equal to the kinetic energy contribution.
The approximation that $C_{A{\rightarrow}B}\approx C_{B{\rightarrow}A}$ can thus reasonably be made.
This approximation is exact for the 1D Schl\"ogl model as $\overline{A{\rightarrow}B} = B{\rightarrow}A$, and holds for 90\% of the toggle switch parameterizations used.
In contrast, the dissipative (path entropy production) component Eq.~\ref{eq:Lent} depends on cross terms of velocity $\mathbf{\dot{x}}$ and deterministic force $\mathbf{f}$, and as such are not be expected to cancel, leading to
\begin{equation}
\frac{1}{2}\left(\Delta S^{L}_{B{\rightarrow}A} - \Delta S^{L}_{A{\rightarrow}B}\right) \approx \Omega\left(\mathcal{A}_{A{\rightarrow}B} - \mathcal{A}_{B{\rightarrow}A}\right)
\label{eq:semian}
\end{equation}
in line with the relationship seen in Fig.~\ref{fig:fig3}D.
Significant divergence from the relation was generally observed in cases where the saddle point occurred at a low copy number compared to the steady states, due to the substantially faster variation of the force.
Further discussion of the above derivation can be found in the supplementary material.
\section{Discussion}
Our primary conclusion is that a MaxEPP for switching paths can be obtained within the Langevin approximation (Eq. \ref{eq:MaxEPP}), extending the rule that ``exergonic reactions occur spontaneously'' to switching in multistable systems.
In a system with a large number of potential macrostates our relation predicts that for sufficiently large volumes switches that produce more entropy will be favoured.
If regions of state space with greater entropy productions also require greater path entropy productions to reach, then this could form a basis for a more extensive maximum entropy production principle.
Our secondary conclusion is that there exists a relationship between the difference in action of minimum action paths and difference in entropy produced along these paths (Eq.~\ref{eq:semian}), valid for all paths that do not pass through regions of rapidly varying force.
We additionally found that steady-state entropies had very little effect on the stability of steady states (see Fig.~\ref{fig:fig3}C).
Finally, we found some evidence to support a broader maximum entropy production principle (see Fig.~\ref{fig:fig3}A).
There has been significant interest in the thermodynamics of the transition between different steady-state probability distributions when controlled by an external parameter \cite{bagci13,hatano01}.
This is fundamentally different to our work, which is about the thermodynamics of switching in a bistable system.
In our two-state systems, we would naively expect a net zero entropy production through switching as the entropy produced by a switch in one direction would be cancelled by the eventual switch back.
Only in cases where the switching path differs from the converse switching path are there net fluxes of probability through the system and thus entropy production.
Consistently, for the Schl\"ogl model we find no net entropy production (i.e.\ $\Delta S^L_{A\rightarrow B} = -\Delta S^L_{B\rightarrow A}$).
In other recent work\cite{ruelle16}, bounds on the ratio of transition rates between states in a bistable system based on relative entropy $\Delta H$ and path entropy production are found.
This ratio is determined as $\pi(B\rightarrow A)/\pi(A\rightarrow B) \geq \exp{[-\Delta S_{A\rightarrow B}]}$, where $\pi(A\rightarrow B)$ is the sum of the rates of switching $A$ to $B$ over all possible switching channels.
With the expectation that a single most probable path will dominate, consideration of the two contrary switching paths will be sufficient.
Combining the main result of their paper with our analytic relation (Eq. \ref{eq:semian}) leads to a bound on the path entropy produced as $\Delta S^{L}_{A\rightarrow B}+\Delta S^{L}_{B\rightarrow A}\geq 0$, which becomes an equality in the limit of time-reversed switching paths (e.g.\ Schl\"ogl).
Every parameter set used in Fig.~\ref{fig:fig3} was found to satisfy this condition.
Beyond the chemical physics literature, related frameworks to ours have been used in evolutionary science \cite{vladar11,mustonen10} where the cumulative fitness flux is maximized (like entropy production) subject to the trade-off that speed of allele change and magnitude of selective forces are minimized (like the conservative action).
Our results therefore suggest, that states in evolutionary systems that require greater cumulative fitness fluxes to reach should be expected to be more stable.
Exploring the application of our theory in ecology, and evolutionary science with systems of multiple stable states will be an interesting way forward \cite{scheffer09}.
\section*{Supplementary material}
See supplementary material for further details of the models used, details of how entropy production was calculated from Gillespie simulations, extended derivations of our relations, details of algorithms used to minimize actions, extended discussion of our coarse-grained entropy production, a plot showing the limited effect of diffusion strength, and tables of parameters for Figs.~\ref{fig:fig1} and \ref{fig:fig2}.
\begin{acknowledgments}
J.C. and R.G.E. thankfully acknowledge helpful discussions with Shamil Chandaria, Henrik Jensen, Chiu Fan Lee, Gunnar Pruessner, David Schnoerr, Philipp Thomas, and financial support from the NERC CDT in Quantitative and Modelling Skills in Ecology and Evolution (grant No./ NE/P012345/1), and BBSRC grant BB/N00065X/1.
\end{acknowledgments}
|
\section{Introduction}
Kleene lattices were introduced by J.~A.~Kalman (\cite K) (under a different name) as a special kind of De Morgan lattices which serve as an algebraic axiomatization of a certain propositional logic satisfying the double negation law but not necessary the excluded middle law. If the underlying lattice is not distributive such lattices are called pseudo-Kleene (see e.g.\ \cite{Ch}). It is a question if certain binary operations can be introduced in a Kleene or pseudo-Kleene lattice such that they form an adjoint pair. To solve this problem, we apply an approach using the full twist-product construction and another construction extending a distributive lattice into a Kleene one.
Having a residuated lattice $(L,\vee,\wedge,\cdot,\rightarrow,1)$, M.~Busaniche and R.~Cignoli (\cite{BC}) as well as C.~Tsinakis and A.~M.~Wille (\cite{TW}) introduced binary operations $\odot$ and $\Rightarrow$ on the full twist-product $(L^2,\sqcup,\sqcap)$ to be converted into a residuated lattice $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(1,1))$. It is known that if $\mathbf L=(L,\vee,\wedge,{}')$ is a distributive lattice with an antitone involution, $a\in L$ and $P_a(\mathbf L):=\{(x,y)\in L^2\mid x\wedge y\leq a\leq x\vee y\}$ then $(P_a(\mathbf L),\vee,\wedge,{}')$ is a Kleene lattice. If $\mathbf L$ is not distributive then the situation is different.
Our aim is to combine both of these approaches and hence ask for several questions as follows:
\begin{itemize}
\item When is $(P_a(\mathbf L),\sqcup,\sqcap)$ a sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$, also in the case of a non-distributive lattice $\mathbf L$?
\item When is $P_a(\mathbf L)$ closed under operations $\odot$ and $\Rightarrow$ mentioned above?
\item When can $P_a(\mathbf L)$ be equipped with these operations forming an adjoint pair?
\item Can we define the operations $\odot$ and $\Rightarrow$ in a way different from that of \cite{BC} or \cite{TW} to obtain an integral residuated lattice on the full twist-product $(L^2,\sqcup,\sqcap)$?
\end{itemize}
We answer these question in our paper by giving sufficient and, in some cases, also necessary conditions under which we get a positive solution. Moreover, we present examples showing how our constructions work.
\section{Preliminaries}
We recall several concepts which will be used throughout the paper. Moreover, we recall some results already published on which our present study is based.
Let $\mathbf P=(P,\leq)$ be a poset. An {\em antitone involution} on $\mathbf P$ is a unary operation $'$ on $P$ satisfying
\begin{enumerate}[(i)]
\item $x\leq y$ implies $y'\leq x'$,
\item $x''\approx x$
\end{enumerate}
for all $x,y\in P$. A distributive lattice having an antitone involution is called a {\em De Morgan lattice} or a {\em De Morgan algebra}.
\begin{definition}\label{def1}
A {\em commutative residuated lattice} is an algebra $(L,\vee,\wedge,\cdot,\rightarrow,1)$ of type $(2,2,2,2,0)$ such that
\begin{enumerate}[{\rm(i)}]
\item $(L,\vee,\wedge)$ is a lattice,
\item $(L,\cdot,1)$ is a commutative monoid,
\item for all $x,y,z\in L$, $x\cdot y\leq z$ is equivalent to $x\leq y\rightarrow z$ {\rm(}{\em adjointness property}{\rm)}.
\end{enumerate}
$(L,\vee,\wedge,\cdot,\rightarrow,1)$ is called {\em integral} if $1$ is the top element of the lattice $(L,\vee,\wedge)$. A {\em commutative residuated lattice with $0$} is an algebra $(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ of type $(2,2,2,2,0,0)$ such that $(L,\vee,\wedge,\cdot,\rightarrow,1)$ is a commutative residuated lattice and $0$ is the bottom element of $(L,\vee,\wedge)$. Let $(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ be a commutative residuated lattice with $0$. Define $x':=x\rightarrow0$ for all $x\in L$. $(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ is
\begin{itemize}
\item called a {\em bounded commutative residuated lattice} if $1$ is the top element of $(L,\vee,\wedge)$,
\item said to satisfy the {\em double negation law} if it satisfies the identity $x''\approx x$, i.e. \\
$(x\rightarrow0)\rightarrow0\approx x$.
\end{itemize}
\end{definition}
We say that the operations $\cdot$ and $\rightarrow$ form an {\em adjoint pair} if they satisfy the adjointness (iii) of Definition~\ref{def1}.
The following properties of integral commutative residuated lattices are well-known (cf.\ e.g.\ \cite B).
\begin{proposition}\label{prop1}
Let $(L,\vee,\wedge,\cdot,\rightarrow,1)$ be an integral commutative residuated lattice. \\
Then the following hold for all $x,y,z\in L$:
\begin{enumerate}[{\rm(i)}]
\item $x\leq y$ implies $x\cdot z\leq y\cdot z$,
\item $x\cdot y\leq x,y$,
\item $1\rightarrow x\approx x$,
\item $x\leq y\rightarrow x$,
\item $x\rightarrow y=1$ if and only if $x\leq y$,
\item $x\leq y$ implies $y\rightarrow z\leq x\rightarrow z$,
\item $x\leq y$ implies $z\rightarrow x\leq z\rightarrow y$,
\item $x\rightarrow(y\wedge z)\approx(x\rightarrow y)\wedge(x\rightarrow z)$,
\item $(x\cdot y)\rightarrow z\approx x\rightarrow(y\rightarrow z)$.
\end{enumerate}
\end{proposition}
Let $\mathbf L=(L,\vee,\wedge)$ be a lattice. By the {\em full twist-product} of $\mathbf L$ is meant the lattice $(L^2,\sqcup,\sqcap)$ where $\sqcup$ and $\sqcap$ are defined as follows:
\begin{align*}
(x,y)\sqcup(z,v) & :=(x\vee z,y\wedge v), \\
(x,y)\sqcap(z,v) & :=(x\wedge z,y\vee v)
\end{align*}
for all $(x,y),(z,v)\in L^2$. Hence $(x,y)\leq(z,v)$ if and only if both $x\leq z$ and $v\leq y$. Assume now that $(L,\vee,\wedge,\cdot,\rightarrow,1)$ is an integral commutative residuated lattice. In Theorem~3.1 in \cite{BC} which is a particular case of Corollary~3.6 in \cite{TW}, Busaniche and Cignoli introduced two additional binary operations $\odot$ and $\Rightarrow$ on its full twist-product $(L^2,\sqcup,\sqcap)$ as follows:
\begin{eqnarray}
& & (x,y)\odot(z,v):=(x\cdot z,(x\rightarrow v)\wedge(z\rightarrow y)),\label{equ1} \\
& & (x,y)\Rightarrow(z,v):=((x\rightarrow z)\wedge(v\rightarrow y),x\cdot v)\label{equ2}
\end{eqnarray}
for all $(x,y),(z,v)\in L^2$. They showed that $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(1,1))$ is again a commutative residuated lattice, i.e.\ $\odot$ and $\Rightarrow$ form an adjoint pair. For the convenience of the reader we provide a proof since it is not explicitly contained in \cite{BC} and \cite{TW}.
\begin{theorem}\label{th5}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,1)$ be an integral commutative residuated lattice and $\odot$ and $\Rightarrow$ defined by {\rm(\ref{equ1})} and {\rm(\ref{equ2})}, respectively. Then $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(1,1))$ is a commutative residuated lattice.
\end{theorem}
\begin{proof}
Let $a,b,c,d,e,f\in L$.
\begin{enumerate}[(i)]
\item It is easy to see that $(L^2,\sqcup,\sqcap)$ is a lattice.
\item We prove that $(L^2,\odot,(1,1))$ is a commutative monoid. Because of (iii), (v), (viii) and (ix) of Proposition~\ref{prop1} we have
\begin{align*}
(x,y)\odot(z,v) & \approx(x\cdot z,(x\rightarrow v)\wedge(z\rightarrow y))\approx(z\cdot x,(z\rightarrow y)\wedge(x\rightarrow v))\approx \\
& \approx(z,v)\odot(x,y), \\
((x,y)\odot(z,v))\odot(t,w) & \approx(x\cdot z,(x\rightarrow v)\wedge(z\rightarrow y))\odot(t,w)\approx \\
& \approx((x\cdot z)\cdot t,((x\cdot z)\rightarrow w)\wedge(t\rightarrow((x\rightarrow v)\wedge(z\rightarrow y))))\approx \\
& \approx(x\cdot(z\cdot t),((x\cdot z)\rightarrow w)\wedge(t\rightarrow(x\rightarrow v))\wedge(t\rightarrow(z\rightarrow y)))\approx \\
& \approx(x\cdot(z\cdot t),(x\rightarrow(z\rightarrow w))\wedge(x\rightarrow(t\rightarrow v))\wedge((z\cdot t)\rightarrow y))\approx \\
& \approx(x\cdot(z\cdot t),(a\rightarrow((z\rightarrow w)\wedge(t\rightarrow v)))\wedge((z\cdot t)\rightarrow y))\approx \\
& \approx(x,y)\odot(z\cdot t,(z\rightarrow w)\wedge(t\rightarrow v)\approx \\
& \approx(x,y)\odot((z,v)\odot(t,w)), \\
(x,y)\odot(1,1) & \approx(x\cdot1,(x\rightarrow1)\wedge(1\rightarrow y))\approx(x,1\wedge y)\approx(x,y).
\end{align*}
\item Now we prove the adjointness property. The following are equivalent:
\begin{align*}
& (a,b)\odot(c,d)\leq(e,f), \\
& (a\cdot c,(a\rightarrow d)\wedge(c\rightarrow b))\leq(e,f), \\
& a\cdot c\leq e\text{ and }f\leq(a\rightarrow d)\wedge(c\rightarrow b), \\
& a\cdot c\leq e, f\leq a\rightarrow d\text{ and }f\leq c\rightarrow b, \\
& a\leq c\rightarrow e, a\leq f\rightarrow d\text{ and }c\cdot f\leq b, \\
& a\leq(c\rightarrow e)\wedge(f\rightarrow d)\text{ and }c\cdot f\leq b, \\
& (a,b)\leq((c\rightarrow e)\wedge(f\rightarrow d),c\cdot f), \\
& (a,b)\leq(c,d)\Rightarrow(e,f).
\end{align*}
\end{enumerate}
\end{proof}
It is worth noticing that the operations $\odot$ and $\Rightarrow$ defined above are not independent. Namely one can be expressed by the other by using the antitone involution $'$ defined by $(x,y)':=(y,x)$. Namely,
\begin{align*}
(x,y)\odot(z,v) & \approx(x\cdot z,(x\rightarrow v)\wedge(z\rightarrow y)\approx((x\rightarrow v)\wedge(z\rightarrow y),x\cdot z)'\approx \\
& \approx((x,y)\Rightarrow(v,z))'\approx((x,y)\Rightarrow(z,v)')', \\
(x,y)\Rightarrow (z,v) & \approx((x,y)\Rightarrow(z,v)'')''\approx((x,y)\odot(z,v)')'.
\end{align*}
Moreover, note that the residuated lattice $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(1,1))$ as defined above is not integral since the top element $(1,0)$ of the full twist-product is different from the neutral element $(1,1)$ of the monoid $(L^2,\odot,(1,1))$.
The following concept was introduced in \cite{Ch} .
A {\em pseudo-Kleene lattice} is an algebra $(L,\vee,\wedge,{}')$ of type $(2,2,1)$ such that the following hold for all $x,y\in L$:
\begin{enumerate}[(i)]
\item $\mathbf L=(L,\vee,\wedge)$ is a lattice,
\item $'$ is an antitone involution on $(L,\leq)$,
\item $x\wedge x'\leq y\vee y'$.
\end{enumerate}
(Here and in the rest of the paper $\leq$ denotes the induced order of the lattice $\mathbf L$.) If, moreover, $\mathbf L$ is distributive then $(L,\vee,\wedge,{}')$ is called a {\em Kleene lattice}.
\section{A construction of pseudo-Kleene lattices in the full twist-product}
Let $\mathbf L=(L,\vee,\wedge)$ be a lattice and $(L^2,\sqcup,\sqcap)$ its full twist-product. It is easy to see that $(L^2,\sqcup,\sqcap)$ is distributive if and only if so is $\mathbf L$. The following construction was introduced for distributive lattices in \cite{Ci} and generalized for posets by the authors in \cite{CL}: Let $a\in L$ and consider the following subset of $L^2$:
\[
P_a(\mathbf L):=\{(x,y)\in L^2\mid x\wedge y\leq a\leq x\vee y\}.
\]
Since our paper \cite{CL} is devoted to posets and not to lattices, we are going to show that if $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$ then $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ where the unary operation $'$ on $P_a(\mathbf L)$ is defined by $(x,y)':=(y,x)$ for all $(x,y)\in P_a(\mathbf L)$ is a pseudo-Kleene lattice.
\begin{theorem}\label{th2}
Let $\mathbf L=(L,\vee,\wedge)$ be a lattice and $a\in L$, assume that $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$ and put $(x,y)':=(y,x)$ for all $(x,y)\in L^2$. Then
\begin{enumerate}[{\rm(i)}]
\item $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ is a pseudo-Kleene lattice,
\item the mapping $x\mapsto(x,a)$ is an embedding of $\mathbf L$ into $(P_a(\mathbf L),\sqcup,\sqcap)$,
\item $(P_a(\mathbf L),\sqcup,\sqcap)$ is distributive if and only if so is $\mathbf L$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $(b,c),(d,e)\in P_a(\mathbf L)$ and $f,g\in L$.
\begin{enumerate}[(i)]
\item The following are equivalent:
\begin{align*}
& (b,c)\leq(d,e), \\
& b\leq d\text{ and }e\leq c, \\
& e\leq c\text{ and }b\leq d, \\
& (e,d)\leq(c,b), \\
& (d,e)'\leq(b,c)'.
\end{align*}
Further, we have $(b,c)''=(c,b)'=(b,c)$. Thus $'$ is an antitone involution on $(P_a(\mathbf L),\sqcup,\sqcap)$. Moreover,
\begin{align*}
(b,c)\sqcap(b,c)' & =(b,c)\sqcap(c,b)=(b\wedge c,c\vee b)\leq(a,a)\leq(d\vee e,e\wedge d)= \\
& =(d,e)\sqcup(e,d)=(d,e)\sqcup(d,e)'
\end{align*}
proving that $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ is a pseudo-Kleene lattice.
\item Since we have $(f,a)\leq(g,a)$ if and only if $f\leq g$, it is evident.
\item This can be easily checked.
\end{enumerate}
\end{proof}
In general, $(P_a(\mathbf L),\sqcup,\sqcap)$ need not be a sublattice of $(L^2,\sqcup,\sqcap)$.
\begin{example}
Consider the lattice $\mathbf N_5=(N_5,\vee,\wedge)$ depicted in Figure~1:
\vspace*{-4mm}
\begin{center}
\setlength{\unitlength}{7mm}
\begin{picture}(4,8)
\put(2,1){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(3,4){\circle*{.3}}
\put(1,5){\circle*{.3}}
\put(2,7){\circle*{.3}}
\put(2,1){\line(-1,2)1}
\put(2,1){\line(1,3)1}
\put(1,3){\line(0,1)2}
\put(2,7){\line(-1,-2)1}
\put(2,7){\line(1,-3)1}
\put(1.85,.25){$0$}
\put(.3,2.85){$a$}
\put(.3,4.85){$b$}
\put(3.4,3.85){$c$}
\put(1.85,7.4){$1$}
\put(1.2,-.75){{\rm Fig.\ 1}}
\end{picture}
\end{center}
\vspace*{4mm}
Then $(a,1),(c,b)\in P_a(\mathbf N_5)$, but $(a,1)\sqcup(c,b)=(a\vee c,1\wedge b)=(1,b)\notin P_a(\mathbf N_5)$ since $1\wedge b=b\not\leq a$. This shows that $P_a(\mathbf N_5)$ is not a sublattice of the full twist-product $(N_5^2,\sqcup,\sqcap)$ of $\mathbf N_5$.
\end{example}
We can give a necessary and sufficient condition for $(P_a(\mathbf L),\sqcup,\sqcap)$ being a sublattice of $(L^2,\sqcup,\sqcap)$.
\begin{theorem}\label{th3}
Let $\mathbf L=(L,\vee,\wedge)$ be a lattice and $a\in L$. Then $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$ if and only if the following condition holds for all $x,y,z,v\in L$:
\begin{align*}
& (x\wedge y)\vee(z\wedge v)\leq a\leq(x\vee y)\wedge(z\vee v)\text{ implies} \\
& ((x\vee z)\wedge y\wedge v)\vee(x\wedge z\wedge(y\vee v))\leq a\leq (x\vee z\vee(y\wedge v))\wedge((x\wedge z)\vee y\vee v).
\end{align*}
\end{theorem}
\begin{proof}
Let $b,c,d,e\in L$.
\item The following are equivalent:
\begin{align*}
& (b,c),(d,e)\in P_a(\mathbf L), \\
& b\wedge c\leq a\leq b\vee c\text{ and }d\wedge e\leq a\leq d\vee e, \\
& (b\wedge c)\vee(d\wedge e)\leq a\leq(b\vee c)\wedge(d\vee e).
\end{align*}
Moreover, the following are equivalent:
\begin{align*}
& (b,c)\sqcup(d,e)\in P_a(\mathbf L), \\
& (b\vee d,c\wedge e)\in P_a(\mathbf L), \\
& (b\vee d)\wedge c\wedge e\leq a\leq b\vee d\vee(c\wedge e).
\end{align*}
Finally, the following are equivalent:
\begin{align*}
& (b,c)\sqcap(d,e)\in P_a(\mathbf L), \\
& (b\wedge d,c\vee e)\in P_a(\mathbf L), \\
& b\wedge d\wedge(c\vee e)\leq a\leq(b\wedge d)\vee c\vee e.
\end{align*}
\end{proof}
\begin{corollary}\label{cor1}
Let $\mathbf L=(L,\vee,\wedge)$ be a distributive lattice and $a\in L$. Then $(P_a(\mathbf L),\sqcup,$ $\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$ and $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ where the antitone involution is given by $(x,y)':=(y,x)$ for all $(x,y)\in P_a(\mathbf L)$ is a Kleene lattice.
\end{corollary}
\begin{proof}
If $b,c,d,e\in L$ and
\[
(b\wedge c)\vee(d\wedge e)\leq a\leq(b\vee c)\wedge(d\vee e)
\]
then
\begin{align*}
((b\vee d)\wedge c\wedge e)\vee(b\wedge d\wedge(c\vee e)) & =(b\wedge c\wedge e)\vee(d\wedge c\wedge e)\vee(b\wedge d\wedge c)\vee(b\wedge d\wedge e)\leq \\
& \leq(b\wedge c)\vee(d\wedge e)\vee(b\wedge c)\vee(d\wedge e)\leq a\leq \\
& \leq(b\vee c)\wedge(d\vee e)\wedge(b\vee c)\wedge(d\vee e)\leq \\
& \leq(b\vee d\vee c)\wedge(b\vee d\vee e)\wedge(b\vee c\vee e)\wedge(d\vee c\vee e)\leq \\
& \leq(b\vee d\vee(c\wedge e))\wedge((b\wedge d)\vee c\vee e).
\end{align*}
The rest of proof follows by Theorem~\ref{th2}.
\end{proof}
The following example shows a distributive lattice $\mathbf L$ having an element $a$ such that $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$.
\begin{example}\label{ex1}
Consider the lattice $\mathbf L=(L,\vee,\wedge)$ visualized in Figure~2:
\vspace*{-2mm}
\begin{center}
\setlength{\unitlength}{7mm}
\begin{picture}(1,8)
\put(0,1){\circle*{.3}}
\put(0,3){\circle*{.3}}
\put(0,5){\circle*{.3}}
\put(0,7){\circle*{.3}}
\put(0,1){\line(0,1)6}
\put(-.15,.25){$0$}
\put(.4,2.85){$a$}
\put(.4,4.85){$b$}
\put(-.15,7.4){$1$}
\put(-.8,-.75){{\rm Fig.\ 2}}
\end{picture}
\end{center}
\vspace*{4mm}
If one defines binary operations $\cdot$ and $\rightarrow$ on $L$ by
\[
x\cdot y:=x\wedge y\text{ and }x\rightarrow y:=\left\{
\begin{array}{ll}
1 & \text{if }x\leq y, \\
y & \text{otherwise},
\end{array}
\right.
\]
then $(L,\vee,\wedge,\cdot,\rightarrow,1)$ is an distributive integral commutative residuated lattice. With respect to the binary operations $\odot$ and $\Rightarrow$ defined by {\rm(\ref{equ1})} and {\rm(\ref{equ2})}, respectively, $(P_a(\mathbf L),\sqcup,\sqcap,$ $\odot,\Rightarrow,(0,1),$ $(1,0))$ is a bounded commutative residuated lattice. According to Corollary~\ref{cor1}, $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice if $(L^2,\sqcup,\sqcap)$. The Hasse diagram of $(P_a(\mathbf L),\sqcup,\sqcap)$ is depicted in Figure~3.
\vspace*{-2mm}
\begin{center}
\setlength{\unitlength}{7mm}
\begin{picture}(6,14)
\put(5,1){\circle*{.3}}
\put(3,3){\circle*{.3}}
\put(7,3){\circle*{.3}}
\put(1,5){\circle*{.3}}
\put(5,5){\circle*{.3}}
\put(3,7){\circle*{.3}}
\put(1,9){\circle*{.3}}
\put(5,9){\circle*{.3}}
\put(-1,11){\circle*{.3}}
\put(3,11){\circle*{.3}}
\put(1,13){\circle*{.3}}
\put(5,1){\line(-1,1)2}
\put(5,1){\line(1,1)2}
\put(3,3){\line(-1,1)2}
\put(3,3){\line(1,1)2}
\put(7,3){\line(-1,1)2}
\put(1,5){\line(1,1)4}
\put(5,5){\line(-1,1)4}
\put(1,9){\line(-1,1)2}
\put(1,9){\line(1,1)2}
\put(5,9){\line(-1,1)2}
\put(-1,11){\line(1,1)2}
\put(3,11){\line(-1,1)2}
\put(4.35,.25){$(0,1)$}
\put(1.4,2.85){$(0,b)$}
\put(7.3,2.85){$(a,1)$}
\put(-.6,4.85){$(0,a)$}
\put(5.3,4.85){$(a,b)$}
\put(3.3,6.85){$(a,a)$}
\put(-.6,8.85){$(b,a)$}
\put(5.3,8.85){$(a,0)$}
\put(-2.6,10.85){$(1,a)$}
\put(3.3,10.85){$(b,0)$}
\put(.35,13.45){$(1,0)$}
\put(2.2,-.75){{\rm Fig.\ 3}}
\end{picture}
\end{center}
\end{example}
\vspace{4mm}
In the following, a special role will play the lattices $P_a(\mathbf L)$ all elements of which are comparable with $(a,a)$. We can characterize them as follows.
\begin{theorem}\label{th1}
Let $\mathbf L=(L,\vee,\wedge)$ be a lattice and $a\in L$. Then the following are equivalent:
\begin{enumerate}[{\rm(i)}]
\item $P_a(\mathbf L)\subseteq\{(x,y)\in L^2\mid(x,y)\text{ is comparable with }(a,a)\}$
\item $P_a(\mathbf L)=\{(x,y)\in L^2\mid(x,y)\text{ is comparable with }(a,a)\}$
\item Every element of $L$ is comparable with $a$ and $a$ is $\vee$-irreducible and $\wedge$-irreducible.
\end{enumerate}
If this is the case then $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$.
\end{theorem}
\begin{proof}
Let $b,c\in L$. \\
(i) and (ii) are equivalent since $\{(x,y)\in L^2\mid(x,y)\text{ is comparable with }(a,a)\}\subseteq P_a(\mathbf L)$. \\
(i) $\Rightarrow$ (iii): \\
Since $(a,a)\leq(a,b)$ or $(a,b)\leq(a,a)$ we have $b\leq a$ or $a\leq b$. If $a=b\vee c$ then $b,c<a$ would imply $(b,c)\in P_a(\mathbf L)$ and $(b,c)\parallel(a,a)$, a contradiction. Hence $a$ is $\vee$-irreducible. If $a=b\wedge c$ then $a<b,c$ would imply $(b,c)\in P_a(\mathbf L)$ and $(b,c)\parallel(a,a)$, a contradiction. Hence $a$ is $\wedge$-irreducible. \\
(iii) $\Rightarrow$ (i): \\
Let $(b,c)\in P_a(\mathbf L)$. Then $b\wedge c\leq a\leq b\vee c$. \\
If $b=a$ then $(b,c)=(a,c)$ is comparable with $(a,a)$. \\
If $c=a$ then $(b,c)=(b,a)$ is comparable with $(a,a)$. \\
$b,c<a$ is impossible because of $a\leq b\vee c$. \\
$a<b,c$ is impossible because of $b\wedge c\leq a$. \\
If $b<a<c$ then $(b,c)\leq(a,a)$. \\
If $c<a<b$ then $(a,a)\leq(b,c)$. \\
Now assume that (ii) holds and let $(b,c),(d,e)\in P_a(\mathbf L)$. \\
If $(b,c),(d,e)\leq(a,a)$ then
\begin{align*}
(b,c)\sqcup(d,e) & =(b\vee d,c\wedge e)\leq(a,a), \\
(b,c)\sqcap(d,e) & =(b\wedge d,c\vee e)\leq(a,a).
\end{align*}
If $(b,c)\leq(a,a)\leq(d,e)$ then
\begin{align*}
(b,c)\sqcup(d,e) & =(b\vee d,c\wedge e)\geq(a,a), \\
(b,c)\sqcap(d,e) & =(b\wedge d,c\vee e)\leq(a,a),
\end{align*}
If $(d,e)\leq(a,a)\leq(b,c)$ then
\begin{align*}
(b,c)\sqcup(d,e) & =(b\vee d,c\wedge e)\geq(a,a), \\
(b,c)\sqcap(d,e) & =(b\wedge d,c\vee e)\leq(a,a).
\end{align*}
If $(a,a)\leq(b,c),(d,e)$ then
\begin{align*}
(b,c)\sqcup(d,e) & =(b\vee d,c\wedge e)\geq(a,a), \\
(b,c)\sqcap(d,e) & =(b\wedge d,c\vee e)\geq(a,a).
\end{align*}
Hence $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$.
\end{proof}
\begin{example}
We can see that the lattice $\mathbf L$ and its element $a$ from Example~\ref{ex1} satisfy the conditions of Theorem~\ref{th1} {\rm(iii)}, hence all elements of $P_a(\mathbf L)$ are comparable with the element $(a,a)$, see Figure~3.
\end{example}
\section{Adjoint pairs in $P_a(\mathbf L)$}
Since the element $(1,1)$ of $L^2$ does not belong to $P_a(\mathbf L)$ unless $a=1$, we cannot expect that $(P_a(\mathbf L),\sqcup,\sqcap)$ will be a residuated lattice with respect to operations $\odot$ and $\Rightarrow$ defined by (\ref{equ1}) and (\ref{equ2}), respectively. On the other hand, it would be important to know when $P_a(\mathbf L)$ is closed with respect to $\odot$ and $\Rightarrow$ because then they form an adjoint pair. Hence, if the pseudo-Kleene lattice $P_a(\mathbf L)$ represents a certain logic where $\odot$ is conjunction and $\Rightarrow$ is implication then from the trivial inequality
\[
x\Rightarrow y\leq x\Rightarrow y
\]
we infer by adjointness
\[
(x\Rightarrow y)\odot x\leq y,
\]
in other words, the propositional value of $y$ is at least as high as the propositional values of the conjunction of $x\Rightarrow y$ and $x$. This means that this logic satisfies {\em Modus Ponens} in the fuzzy modification and hence this pseudo-Kleene logic enables deduction.
Now we are ready to state and prove one of our main results.
\begin{theorem}\label{th4}
Let $(L,\vee,\wedge,\cdot,\rightarrow,1)$ be an integral commutative residuated lattice and $a$ an idempotent {\rm(}with respect to $\cdot${\rm)} $\vee$-irreducible and $\wedge$-irreducible element of $L$ which is comparable with every element of $L$ and put $\mathbf L:=(L,\vee,\wedge)$. Then $(P_a(\mathbf L),\sqcup,\sqcap,\odot,\Rightarrow)$ is a subalgebra of $(L^2,\sqcup,\sqcap,\odot,\Rightarrow)$ and hence $\odot$ and $\Rightarrow$ form an adjoint pair if and only if the following two conditions hold for all $x,y\in L$:
\begin{eqnarray}
& & a\cdot x<a\text{ implies }a\cdot x=0,\label{equ3} \\
& & a<x\cdot y\text{ implies }(x\rightarrow a)\wedge(y\rightarrow a)=a.\label{equ4}
\end{eqnarray}
\end{theorem}
\begin{proof}
Let $(b,c),(d,e)\in P_a(\mathbf L)$. According to Theorem~\ref{th1},
\[
P_a(\mathbf L)=\{(x,y)\in L^2\mid (x,y)\text{ is comparable with }(a,a)\}
\]
and we have that $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$. Since $P_a(\mathbf L)$ is closed with respect to $'$, it is closed with respect to $\Rightarrow$ if it is closed with respect to $\odot$. Hence, we need only to check when $P_a(\mathbf L)$ is closed with respect to $\odot$.
\begin{enumerate}[(i)]
\item Assume $(b,c),(d,e)\leq(a,a)$. \\
Because of (ii) and (iv) of Proposition~\ref{prop1} we have
\[
(b,c)\odot(d,e)=(b\cdot d,(b\rightarrow e)\wedge(d\rightarrow c))\leq(a,a).
\]
\item Assume $(b,c)\leq(a,a)\leq(d,e)$. \\
Because of (ii) of Proposition~\ref{prop1} we have $b\cdot d\leq a$. \\
If $b\cdot d=a$ then $(b,c)\odot(d,e)=(b\cdot d,(b\rightarrow e)\wedge(d\rightarrow c))$ is comparable with $(a,a)$. \\
If $b\cdot d<a$ then $(b,c)\odot(d,e)=(b\cdot d,(b\rightarrow e)\wedge(d\rightarrow c))$ is comparable with $(a,a)$ \\
\hspace*{1cm} if and only if $a\leq b\rightarrow e$.
\item Assume $(d,e)\leq(a,a)\leq(b,c)$. \\
Because of the commutativity of $\odot$ this case reduces to the previous one.
\item Assume $(a,a)\leq(b,c),(d,e)$. \\
Because of (i) of Proposition~\ref{prop1} we have $a\leq b\cdot d$. \\
If $a=b\cdot d$ then $(b,c)\odot(d,e)=(b\cdot d,(b\rightarrow e)\wedge(d\rightarrow c))$ is comparable with $(a,a)$. \\
If $a<b\cdot d$ then $(b,c)\odot(d,e)=(b\cdot d,(b\rightarrow e)\wedge(d\rightarrow c))$ is comparable with $(a,a)$ \\
\hspace*{1cm} if and only if $(b\rightarrow e)\wedge(d\rightarrow c)\leq a$.
\end{enumerate}
Hence $(P_a(\mathbf L),\sqcup,\sqcap,\odot,\Rightarrow)$ is a subalgebra of $(L^2,\sqcup,\sqcap,\odot,\Rightarrow)$ if and only if the following statements hold:
\begin{enumerate}[(a)]
\item $b,e\leq a\leq c,d$ and $b\cdot d<a$ imply $a\leq b\rightarrow e$.
\item $c,e\leq a\leq b,d$ and $a<b\cdot d$ imply $(b\rightarrow e)\wedge(d\rightarrow c)\leq a$.
\end{enumerate}
Because of (i) and (vii
|
) of Proposition~\ref{prop1}, (a) is equivalent to the following statements:
\begin{align*}
& b\cdot a<a\text{ implies }a\leq b\rightarrow0, \\
& a\cdot b<a\text{ implies }a\cdot b\leq0, \\
& a\cdot b<a\text{ implies }a\cdot b=0, \\
& (\ref{equ3}).
\end{align*}
Moreover, because of (iv) and (vii) of Proposition~\ref{prop1}, (b) is equivalent to the following statements:
\begin{align*}
& a<b\cdot d\text{ implies }(b\rightarrow a)\wedge(d\rightarrow a)\leq a, \\
& a<b\cdot d\text{ implies }(b\rightarrow a)\wedge(d\rightarrow a)=a, \\
& (\ref{equ4}).
\end{align*}
\end{proof}
\begin{corollary}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,1)$ be an integral commutative distributive residuated lattice and $a\in L$ with $a\cdot a=a$ and assume that every element of $P_a(\mathbf L)$ is comparable with $(a,a)$. Then $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ where $(x,y)':=(y,x)$ for all $(x,y)\in P_a(\mathbf L)$ is a Kleene lattice and $\odot$ and $\Rightarrow$ form an adjoint pair if and only if {\rm(\ref{equ3})} and {\rm(\ref{equ4})} hold.
\end{corollary}
\begin{example}
Consider the lattice $\mathbf L=(L,\vee,\wedge)$ with element $a$ from Example~\ref{ex1}. One can easily check that $\mathbf L$ satisfies the conditions of Theorem~\ref{th4} and hence $(P_a(\mathbf L),\sqcup,\sqcap,\odot,$ $\Rightarrow)$ is a subalgebra of $(L^2,\sqcup,\sqcap,\odot,\Rightarrow)$.
\end{example}
\begin{lemma}\label{lem1}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,1)$ be a distributive commutative residuated lattice and $a\in L$. Then $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$ closed with respect to $\odot$ {\rm(}and hence also with respect to $\Rightarrow${\rm)} if and only if for all $(b,c),(d,e)\in P_a(\mathbf L)$
\[
(b\cdot d)\wedge(b\rightarrow e)\wedge(d\rightarrow c)\leq a\leq(b\cdot d)\vee(b\rightarrow e)\text{ and }a\leq(b\cdot d)\vee(d\rightarrow c).
\]
\end{lemma}
\begin{proof}
According to Theorem~\ref{th2} and Corollary~\ref{cor1}, $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of $(L^2,\sqcup,\sqcap)$. Let $(b,c),(d,e)\in P_a(\mathbf L)$ and put $f:=b\cdot d$, $g:=b\rightarrow e$, $h:=d\rightarrow c$ and $i:=g\wedge h$. Then the following are equivalent:
\begin{align*}
(b,c)\odot(d,e) & \in P_a(\mathbf L), \\
(f,i) & \in P_a(\mathbf L), \\
f\wedge i & \leq a\leq f\vee i, \\
f\wedge i & \leq a\leq f\vee(g\wedge h), \\
f\wedge i & \leq a\leq(f\vee g)\wedge(f\vee h), \\
f\wedge i & \leq a\leq f\vee g\text{ and }a\leq f\vee h.
\end{align*}
\end{proof}
\begin{corollary}\label{cor2}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ be a distributive bounded commutative residuated lattice and $a$ an atom of $\mathbf L$. Then $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$ closed with respect to $\odot$ {\rm(}and hence also with respect to $\Rightarrow${\rm)} if and only if for all $(b,c),(d,e)\in P_a(\mathbf L)$ either {\rm(i)} or {\rm(ii)} hold:
\begin{enumerate}[{\rm(i)}]
\item $(b\cdot d)\wedge(b\rightarrow e)\wedge(d\rightarrow c)=a$,
\item $(b\cdot d)\wedge(b\rightarrow e)\wedge(d\rightarrow c)=0$ and $(a\leq b\cdot d$ or $a\leq(b\rightarrow e)\wedge(d\rightarrow c))$.
\end{enumerate}
\end{corollary}
\begin{proof}
Let $(b,c),(d,e)\in P_a(\mathbf L)$ and put $f:=b\cdot d$, $g:=b\rightarrow e$, $h:=d\rightarrow c$ and $i:=g\wedge h$. According to Lemma~\ref{lem1}, $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$ and $(b,c)\odot(d,e)\in P_a(\mathbf L)$ is equivalent to $(f\wedge i\leq a\leq f\vee g$ and $a\leq f\vee h)$. Now $f\wedge i=a$ implies $a\leq f\vee i$. Using the fact that $a$ is an atom of $\mathbf L$ we see that the following are equivalent:
\begin{align*}
a & \leq f\vee g, \\
a\wedge(f\vee g) & =a, \\
(a\wedge f)\vee(a\wedge g) & =a, \\
a\wedge f=a & \text{ or }a\wedge g=a, \\
a\leq f & \text{ or }a\leq g.
\end{align*}
Analogously, $a\leq f\vee h$ is equivalent to $(a\leq f$ or $a\leq h)$. Finally, the following are equivalent:
\begin{align*}
(b,c)\odot(d,e) & \in P_a(\mathbf L), \\
f\wedge i=a & \text{ or }(f\wedge i=0\text{ and }(a\leq f\text{ or }a\leq g)\text{ and }(a\leq f\text{ or }a\leq h)), \\
f\wedge i=a & \text{ or }(f\wedge i=0\text{ and }(a\leq f\text{ or }(a\leq g)\text{ and }a\leq h))), \\
f\wedge i=a & \text{ or }(f\wedge i=0\text{ and }(a\leq f\text{ or }a\leq g\wedge h)).
\end{align*}
\end{proof}
Analogously as in Corollary~\ref{cor2}, we can consider the operation $\Rightarrow$ instead of $\odot$ and prove a similar result.
\begin{lemma}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ be a distributive bounded commutative residuated lattice and $a$ an atom of $\mathbf L$. Then $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$ closed with respect to $\Rightarrow$ {\rm(}and hence also with respect to $\odot${\rm)} if and only if for all $(b,c),(d,e)\in P_a(\mathbf L)$ either {\rm(i)} or {\rm(ii)} hold:
\begin{enumerate}[{\rm(i)}]
\item $(b\rightarrow d)\wedge(e\rightarrow c)\wedge(b\cdot e)=a$,
\item $(b\rightarrow d)\wedge(e\rightarrow c)\wedge(b\cdot e)=0$ and $(a\leq(b\rightarrow d)\wedge(e\rightarrow c)$ or $a\leq b\cdot e)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $(b,c),(d,e)\in P_a(\mathbf L)$ and put $f:=b\rightarrow d$, $g:=e\rightarrow c$, $h:=b\cdot e$ and $i:=f\wedge g$. According to Theorem~\ref{th2} and Corollary~\ref{cor1}, $(P_a(\mathbf L),\sqcup,\sqcap)$ is a distributive sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$. Now the following are equivalent:
\begin{align*}
(b,c)\odot(d,e) & \in P_a(\mathbf L), \\
(i,h) & \in P_a(\mathbf L), \\
i\wedge h & \leq a\leq i\vee h, \\
i\wedge h & =a\text{ or }(i\wedge h=0\text{ and }a\leq i\vee h), \\
i\wedge h & =a\text{ or }(i\wedge h=0\text{ and }(a\leq i\text{ or }a\leq h)).
\end{align*}
(that $a\leq i\vee h$ is equivalent to $(a\leq i$ or $a\leq h)$ follows like in the proof of Corollary~\ref{cor2}).
\end{proof}
\begin{example}\label{ex2}
Consider the lattice $\mathbf L=(L,\vee,\wedge)$ shown in Figure~4:
\vspace*{-2mm}
\begin{center}
\setlength{\unitlength}{7mm}
\begin{picture}(8,8)
\put(3,1){\circle*{.3}}
\put(1,3){\circle*{.3}}
\put(5,3){\circle*{.3}}
\put(3,5){\circle*{.3}}
\put(7,5){\circle*{.3}}
\put(5,7){\circle*{.3}}
\put(3,1){\line(-1,1)2}
\put(3,1){\line(1,1)4}
\put(1,3){\line(1,1)4}
\put(5,3){\line(-1,1)2}
\put(7,5){\line(-1,1)2}
\put(2.875,.25){$0$}
\put(.3,2.85){$b$}
\put(5.4,2.85){$a$}
\put(2.3,4.85){$c$}
\put(7.4,4.85){$d$}
\put(4.875,7.4){$1$}
\put(3.2,-.75){{\rm Fig.\ 4}}
\end{picture}
\end{center}
\vspace*{4mm}
According to Corollary~\ref{cor1}, $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of the full twist-product $(L^2,\sqcup,\sqcap)$. The Hasse diagram of $(P_a(\mathbf L),\sqcup,\sqcap)$ is depicted in Figure~5.
\vspace*{-2mm}
\begin{center}
\setlength{\unitlength}{7mm}
\begin{picture}(10,14)
\put(5,1){\circle*{.3}}
\put(3,3){\circle*{.3}}
\put(5,3){\circle*{.3}}
\put(7,3){\circle*{.3}}
\put(1,5){\circle*{.3}}
\put(3,5){\circle*{.3}}
\put(5,5){\circle*{.3}}
\put(7,5){\circle*{.3}}
\put(1,7){\circle*{.3}}
\put(3,7){\circle*{.3}}
\put(5,7){\circle*{.3}}
\put(7,7){\circle*{.3}}
\put(9,7){\circle*{.3}}
\put(3,9){\circle*{.3}}
\put(5,9){\circle*{.3}}
\put(7,9){\circle*{.3}}
\put(9,9){\circle*{.3}}
\put(3,11){\circle*{.3}}
\put(5,11){\circle*{.3}}
\put(7,11){\circle*{.3}}
\put(5,13){\circle*{.3}}
\put(5,1){\line(-1,1)4}
\put(5,1){\line(0,1)2}
\put(5,1){\line(1,1)2}
\put(3,3){\line(0,1)2}
\put(3,3){\line(1,1)2}
\put(5,3){\line(-1,1)4}
\put(5,3){\line(1,1)4}
\put(7,3){\line(-1,1)4}
\put(7,3){\line(0,1)4}
\put(1,5){\line(0,1)2}
\put(1,5){\line(1,1)2}
\put(3,5){\line(1,1)4}
\put(5,5){\line(0,1)4}
\put(7,5){\line(-1,1)4}
\put(1,7){\line(1,1)4}
\put(3,7){\line(0,1)4}
\put(7,7){\line(-1,1)4}
\put(7,7){\line(1,1)2}
\put(9,7){\line(-1,1)4}
\put(9,7){\line(0,1)2}
\put(5,9){\line(1,1)2}
\put(7,9){\line(0,1)2}
\put(9,9){\line(-1,1)4}
\put(3,11){\line(1,1)2}
\put(5,11){\line(0,1)2}
\put(4.35,.25){$(0,1)$}
\put(1.4,2.85){$(0,d)$}
\put(5.3,2.85){$(a,1)$}
\put(7.3,2.85){$(0,c)$}
\put(-.6,4.85){$(b,d)$}
\put(1.4,4.85){$(a,d)$}
\put(5.3,4.85){$(0,a)$}
\put(7.3,4.85){$(a,c)$}
\put(-.6,6.85){$(c,d)$}
\put(1.4,6.85){$(b,a)$}
\put(5.3,6.85){$(a,a)$}
\put(7.3,6.85){$(a,b)$}
\put(9.3,6.85){$(d,c)$}
\put(1.4,8.85){$(c,a)$}
\put(5.3,8.85){$(a,0)$}
\put(7.3,8.85){$(d,a)$}
\put(9.3,8.85){$(d,b)$}
\put(1.4,10.85){$(c,0)$}
\put(5.3,10.85){$(1,a)$}
\put(7.3,10.85){$(d,0)$}
\put(4.35,13.45){$(1,0)$}
\put(4.2,-.75){{\rm Fig.\ 5}}
\end{picture}
\end{center}
\vspace*{4mm}
Define an antitone involution $'$ on $(L,\leq)$ and binary operations $\cdot$ and $\rightarrow$ on $L$ by
\[
\begin{array}{c|cccccc}
x & 0 & a & b & c & d & 1 \\
\hline
x' & 1 & c & d & a & b & 0
\end{array}
\]
and
\[
x\cdot y=\left\{
\begin{array}{ll}
x\wedge y\wedge b & \text{if }x,y\in\{a,c\}, \\
x\wedge y & \text{otherwise}
\end{array}
\right.\quad x\rightarrow y=\left\{
\begin{array}{ll}
x'\vee y\vee d & \text{if }x,y\in\{a,c\}, \\
x'\vee y & \text{otherwise}
\end{array}
\right.
\]
for all $x,y\in L$. Then $(L,\vee,\wedge,\cdot,\rightarrow,1)$ is an integral commutative residuated lattice and $x'=x\rightarrow0$ for all $x\in L$. Hence there holds the double negation law. Since $a$ is neither idempotent with respect to $\cdot$ nor meet-irreducible nor comparable with all elements of $L$, we cannot apply Theorem~\ref{th4}. However, since $\mathbf L$ is distributive, $a$ is atom of $\mathbf L$ and conditions {\rm(i)} and {\rm(ii)} of Corollary~\ref{cor2} are satisfied, $P_a(\mathbf L)$ is closed with respect to $\odot$ and hence also with respect to $\Rightarrow$.
\end{example}
If $\mathbf L$ denotes the lattice from Example~\ref{ex2} then $(c,d),(d,b)\in P_d(\mathbf L)$, but
\begin{align*}
(c,d)\odot(d,b) & =(c\cdot d,(c\rightarrow b)\wedge(d\rightarrow d))=(c\wedge d,(c'\vee b)\wedge(d'\vee d))= \\
& =(a,(a\vee b)\wedge(b\vee d))=(a,c\wedge1)=(a,c)\notin P_d(\mathbf L)
\end{align*}
since $d\not\leq c=a\vee c$. This shows that $P_d(\mathbf L)$ is not closed with respect to $\odot$ (and hence also not with respect to $\Rightarrow$).
If $\mathbf L$ satisfies the double negation law then, because of (vi) of Proposition~\ref{prop1}, $'$ is an antitone involution on $(L,\leq)$. Two elements $a$ and $b$ of $L$ are said to be {\em orthogonal to each other} (shortly, $a\perp b$) if $a\leq b'$. If $\mathbf L$ satisfies the double negation law then this is equivalent to $b\leq a'$. $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(0,1),(1,1))$ is said to satisfy the {\em double negation law for orthogonal elements} if $(x,y)''=(x,y)$ for all $(x,y)\in L^2$ with $x\perp y$ where $(x,y)':=(x,y)\Rightarrow(0,1)$ for all $(x,y)\in L^2$.
\begin{theorem}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ be a bounded commutative residuated lattice satisfying the double negation law and $a\in L$. Then the full twist-product $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,$ $ (0,1),(1,1))$ is a commutative residuated lattice with zero-element $(0,1)$ satisfying the double negation law for orthogonal elements.
\end{theorem}
\begin{proof}
If $a,b\in L$ and $a\perp b$ then $b\leq a'$ and hence
\begin{align*}
(a,b)' & =(a,b)\Rightarrow(0,1)=((a\rightarrow0)\wedge(1\rightarrow b),a\cdot1)=(a'\wedge b,a)=(b,a), \\
(a,b)'' & =(b,a)'=(a,b).
\end{align*}
\end{proof}
\section{An alternative construction of adjoint operations}
In this section we show that the operations $\odot$ and $\Rightarrow$ on the full twist-product $(L^2,\sqcup,\sqcap)$ can be defined also in a way different from (\ref{equ1}) and (\ref{equ2}) such that $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(0,1),$ $(1,0))$ becomes a bounded commutative residuated lattice. We formulate it as follows.
\begin{theorem}\label{th6}
Let $(L,\vee,\wedge,\cdot,\rightarrow,0,1)$ be a bounded commutative residuated lattice satisfying the double negation law and define $x':=x\rightarrow0$ for all $x\in L$ and
\begin{eqnarray}
& & (x,y)\odot(z,v):=(x\cdot z,(y'\cdot v')'),\label{equ5} \\
& & (x,y)\Rightarrow(z,v):=(x\rightarrow z,(y'\rightarrow v')')\label{equ6}
\end{eqnarray}
for all $(x,y),(z,v)\in L^2$. Then $(L^2,\sqcup,\sqcap,\odot,\Rightarrow,(0,1),(1,0))$ is a bounded commutative residuated lattice satisfying the double negation law.
\end{theorem}
\begin{proof}
Let $a,b,c,d,e,f\in L$. Obviously, $(L^2,\sqcup,\sqcap,(0,1),(1,0))$ is a bounded lattice. We have
\begin{align*}
(x,y)\odot(z,v) & \approx(x\cdot z,(y'\cdot v')')\approx(z\cdot x,(v'\cdot y')')\approx(z,v)\odot(x,y), \\
((x,y)\odot(z,v))\odot(t,w) & \approx(x\cdot z,(y'\cdot v')')\odot(t,w)\approx((x\cdot z)\cdot t,((y'\cdot v')\cdot w')')\approx \\
& \approx(x\cdot(z\cdot t),(y'\cdot(v'\cdot w'))')\approx(x,y)\odot(z\cdot t,(v'\cdot w')')\approx \\
& \approx(x,y)\odot((z,v)\odot(t,w)), \\
(x,y)\odot(1,0) & \approx(x\cdot1,(y'\cdot0')')\approx(x,(y'\cdot1)')\approx(x,y'')\approx(x,y).
\end{align*}
Moreover, the following are equivalent:
\begin{align*}
& (a,b)\odot(c,d)\leq(e,f), \\
& (a\cdot c,(b'\cdot d')')\leq(e,f), \\
& a\cdot c\leq e\text{ and }f\leq(b'\cdot d')', \\
& a\cdot c\leq e\text{ and }b'\cdot d'\leq f', \\
& a\cdot c\leq e\text{ and }b'\leq d'\rightarrow f', \\
& a\leq c\rightarrow e\text{ and }(d'\rightarrow f')'\leq b, \\
& (a,b)\leq(c\rightarrow e,(d'\rightarrow f')'), \\
& (a,b)\leq(c,d)\Rightarrow(e,f).
\end{align*}
Finally, we have
\begin{align*}
(x,y)' & \approx(x,y)\Rightarrow(0,1)\approx(x\rightarrow0,(y'\rightarrow1')')\approx(x',(y'\rightarrow0)')\approx(x',y''')\approx(x',y'), \\
(x,y)'' & \approx(x',y')'\approx(x'',y'')\approx(x,y).
\end{align*}
\end{proof}
\begin{remark}\label{rem1}
Let us note that under the assumptions of Theorem~\ref{th6}, the antitone involution $(x,y)':=(x',y')$ in the full twist-product $L^2$ as well as in $P_a(\mathbf L)$ can be derived in a natural way by $(x,y)'\approx(x,y)\Rightarrow(0,1)$ since
\[
(x,y)\Rightarrow(0,1)\approx(x\rightarrow0,(y'\rightarrow1')')\approx(x',(y'\rightarrow0)')\approx(x',y''')\approx(x',y').
\]
This does not hold if $\odot$ and $\Rightarrow$ are defined by {\rm(\ref{equ1})} and {\rm(\ref{equ2})}, respectively.
\end{remark}
\begin{remark}
It is worth noticing that the case when the operations $\odot$ and $\Rightarrow$ are defined by {\rm(\ref{equ5})} and {\rm(\ref{equ6})}, respectively, has an interpretation e.g.\ in {\rm MV}-algebras. Namely, an {\rm MV}-algebra is an algebra $(M,\oplus,\neg,0)$ of type $(2,1,0)$ where $(M,\oplus,0)$ is a commutative monoid, $\neg$ satisfies the identity $\neg\neg x\approx x$ and $\oplus$ and $\neg$ are related by the \L ukasiewicz axiom
\[
\neg(\neg x\oplus y)\oplus y\approx\neg(\neg y\oplus x)\oplus x.
\]
Then $(M,\vee,\wedge)$ becomes a distributive lattice where
\begin{align*}
x\vee y & :=\neg(\neg x\oplus y)\oplus y, \\
x\wedge y & :=\neg(\neg x\vee\neg y).
\end{align*}
for all $x,y\in M$. {\rm MV}-algebras serve as an algebraic semantics of the many-valued \L ukasiewicz logics, $\oplus$ is interpreted as disjunction and $\rightarrow$ defined by $x\rightarrow y:=\neg x\oplus y$ for all $x,y\in M$ as implication. If we put $x\cdot y:=\neg(\neg x\oplus\neg y)$ for all $x,y\in M$ then $x\rightarrow y\approx\neg(x\cdot\neg y)$
and $(M,\vee,\wedge,\cdot,\rightarrow,0,1)$ forms a bounded residuated lattice satisfying the double negation law. If we now define $\odot$ and $\Rightarrow$ on the full twist-product $M^2$ by {\rm(\ref{equ5})} and {\rm(\ref{equ6})}, respectively, we obtain
\begin{align*}
(x,y)\odot(z,v) & \approx(x\cdot z,y\oplus v), \\
(x,y)\Rightarrow(z,v) & \approx(x\rightarrow z,\neg y\cdot v).
\end{align*}
In fact, the lattice $\mathbf L=(L,\vee,\wedge)$ from Example~\ref{ex2} is an {\rm MV}-algebra where $\neg x:=x\rightarrow0$ and $x\oplus y:=\neg(\neg x\cdot\neg y)$ for all $x,y\in L$.
\end{remark}
It was shown in \cite{Ch} for Kleene lattices and in \cite{CL} for pseudo-Kleene lattices $(L\vee,\wedge,{}')$ that there exists at most one element $a$ of $L$ satisfying $a'=a$. If such an element exists in a lattice with an antitone involution, we can prove the following result.
\begin{theorem}
Let $\mathbf L=(L,\vee,\wedge,{}')$ be a lattice with an antitone involution and $a\in L$ with $a'=a$, assume that $(P_a(\mathbf L),\sqcup,\sqcap)$ is a sublattice of $(L^2,\sqcup,\sqcap)$ and $(x,y)'=(x',y')$ for all $(x,y)\in L^2$. Then $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ is a pseudo-Kleene lattice if and only if $\mathbf L$ has this property.
\end{theorem}
\begin{proof}
Let $b,c\in L$ and $(d,e),(f,g)\in P_a(\mathbf L)$. We have $(x,y)'\approx(x,y)\Rightarrow(0,1)\approx(x',y')$ as explained in Remark~\ref{rem1}. If $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ is a pseudo-Kleene lattice then $(b,a),(c,a)\in P_a(\mathbf L)$ and hence
\[
(b\wedge b',a\vee a')=(b,a)\sqcap(b',a')=(b,a)\sqcap(b,a)'\leq(c,a)\sqcup(c,a)'=(c,a)\sqcup(c',a')=(c\vee c',a\wedge a'),
\]
i.e.\ $b\wedge b'\leq c\vee c'$ showing that $\mathbf L$ is a pseudo-Kleene lattice. Conversely, assume $\mathbf L$ to be a pseudo-Kleene lattice. Then $d\wedge e\leq a\leq d\vee e$ whence $d'\wedge e'\leq a'\leq d'\vee e'$, i.e.\ $d'\wedge e'\leq a\leq d'\vee e'$ which shows $(d,e)'=(d',e')\in P_a(\mathbf L)$. Hence $P_a(\mathbf L)$ is closed with respect to $'$. Finally, we have
\begin{align*}
(d,e)\sqcap(d,e)' & =(d,e)\sqcap(d',e')=(d\wedge d',e\vee e')\leq(f\vee f',g\wedge g')=(f,g)\sqcup(f',g')= \\
& =(f,g)\sqcup(f,g)'
\end{align*}
showing that $(P_a(\mathbf L),\sqcup,\sqcap,{}')$ is a pseudo-Kleene lattice.
\end{proof}
Our next aim is to show when $P_a(\mathbf L)$ is closed under the operation $\odot$ defined by (\ref{equ5}). We prove the following.
\begin{theorem}
Let $\mathbf L=(L,\vee,\wedge,\cdot,\rightarrow,{}',1)$ be a commutative residuated lattice with an antitone involution, let $a\in L$ be idempotent with respect to $\cdot$, $\vee$-irreducible and $\wedge$-irreducible, assume $a'\cdot a'=a'$ and define $\odot$ by {\rm(\ref{equ5})}. Then $P_a(\mathbf L)$ is closed with respect to $\odot$.
\end{theorem}
\begin{proof}
Let $(b,c),(d,e)\in P_a(\mathbf L)$. We have
\[
(x,y)\odot(z,v)\approx(x\cdot z,(y'\cdot v')')\approx(z\cdot x,(v'\cdot y')')\approx(z,v)\odot(x,y).
\]
According to Theorem~\ref{th1},
\[
P_a(\mathbf L)=\{(x,y)\in L^2\mid(x,y)\text{ is comparable with }(a,a)\}.
\]
In the following we often use (i) and (ii) of Proposition~\ref{prop1}.
\begin{enumerate}[(i)]
\item Assume $(b,c)\leq(a,a)$. \\
We have $b\cdot d\leq b\leq a$ and every one of the following statements implies the next one:
\begin{align*}
a & \leq c, \\
c' & \leq a', \\
c'\cdot e' & \leq a', \\
a & \leq(c'\cdot e')'.
\end{align*}
This shows $(b,c)\odot(d,e)=(b\cdot d,(c'\cdot e')')\leq(a,a)$.
\item Assume $(d,e)\leq(a,a)$. \\
Then $(b,c)\odot(d,e)=(d,e)\odot(b,c)\leq(a,a)$.
\item Assume $(a,a)\leq(b,c),(d,e)$. \\
Then $c,e\leq a$ and hence $a'\leq c',e'$ whence $a'=a'\cdot a'\leq c'\cdot a'\leq c'\cdot e'$ from which we conclude $(c'\cdot e')'\leq a$. Because of $a\leq b,d$ we have $a=a\cdot a\leq b\cdot a\leq b\cdot d$. Together we obtain $(a,a)\leq(b\cdot d,(c'\cdot e')')=(b,c)\odot(d,e)$.
\end{enumerate}
\end{proof}
Unfortunately, $P_a(\mathbf L)$ is not closed under $\Rightarrow$ defined by (C2) provided $L$ in non-trivial, i.e.\ if it has more than one element.
\begin{theorem}
Let $(L,\vee,\wedge,\cdot,\rightarrow,{}',0,1)$ be a bounded commutative residuated lattice with an antitone involution and $a\in L$ and put $(x,y)\Rightarrow(z,v):=(x\rightarrow z,(y'\rightarrow v')')$ for all $(x,y),(z,v)\in L^2$. Then $P_a(\mathbf L)$ is closed with respect to $\Rightarrow$ if and only if $|L|=1$.
\end{theorem}
\begin{proof}
Assume $P_a(\mathbf L)$ to be closed with respect to $\Rightarrow$. Since $(0,a),(a,0),(a,1),(1,a)\in P_a(\mathbf L)$ we have
\begin{align*}
(0,0) & =(0,1')=(1\rightarrow0,(a'\rightarrow a')')=(1,a)\Rightarrow(0,a)\in P_a(\mathbf L), \\
(1,1) & =(1,0')=(1,(1\rightarrow0)')=(a\rightarrow a,(0'\rightarrow1')')=(a,0)\Rightarrow(a,1)\in P_a(\mathbf L)
\end{align*}
whence $a\leq0\vee0=0\leq1=1\wedge1\leq a$ and therefore $0=1$, i.e.\ $|L|=1$.
\end{proof}
|
\section{Introduction}
\subsection{Introduction}
The hypothesis that critical two-dimensional lattice models should have conformally invariant scaling limits was formulated in the physics community in the 1980s. Starting with \cite{BPZ} conformal field theory (CFT) was developed to exploit conformal invariance and subsequently applied to many lattice models, producing predictions of, e.g., critical exponents and correlation functions.
The Schramm-Loewner evolution (SLE) processes \cite{Schramm_LERW} provide a precise mathematical approach by describing scaling limits of random cluster interfaces and self-avoiding walks in the lattice models. To date, convergence in a sense described below, and in particular conformal invariance, has been established in several cases: loop-erased random walk (LERW), the uniform spanning tree, critical percolation, the Ising model, and the discrete Gaussian free field \cite{LSW04, smirnov-perc, smirnov-ising, SS}. The uniform measure on self-avoiding walks is strongly believed to belong to this collection of models, but whether it actually does remains one of the most interesting and apparently difficult open problems in probability. Once a convergence result is established one can use SLE computations to rigorously derive properties such as critical exponents or dimensions of the discrete interfaces, see, e.g., \cite{smirnov_werner, lsw_arm_exponents, Masson}. Some field theoretic statements may also be interpreted and given probabilistic and geometric meaning, see, e.g., \cite{friedrich-werner, kang-makarov} and the references in the latter.
SLE curves are constructed using Loewner's differential equation. It gives the dynamics of a family of Riemann maps from a reference domain onto a continuously decreasing family of simply connected domains. Under favorable circumstances, as in the case of SLE, there is a non-crossing, continuous curve such that one gets the decreasing domains by taking the complements of the growing curve. This \emph{Loewner curve} comes equipped with a particular parametrization by capacity which it inherits from the Loewner equation. Studying SLE in this parametrization is practical for many problems and we have information about, e.g., sharp Holder exponents, continuity properties, and finer multifractal relations \cite{RS, lind-holder, JVL, JVL2, JVRW}. Before the present paper all SLE convergence results we know of consider a discrete curve reparametrized by capacity, and proves convergence in that parametrization. This is sufficient to study many properties of discrete models converging to SLE.
However, information is lost when reparametrizing the discrete curve. A more detailed analysis (see \cite{benoist-dumaz-werner} for an example) is possible by considering the discrete process parametrized by length, in what is sometimes called its \emph{natural parametrization}. By this we mean that the curve traverses each lattice edge in the same amount of time. Since the limiting trace is fractal, one needs to rescale so that whole curve in a smooth
bounded domain is traversed in time of order $1$. It then seems reasonable to expect that the discrete curve in its natural parametrization converges to SLE equipped with a different parmatrization than capacity. Indeed, this is widely believed to be true in all the cases where convergence to SLE is known.
The SLE curve with parameter $\kappa \in (0,8)$ is a random fractal of almost sure dimension $d=1+\kappa/8$. With the length rescaling of the discrete curve in mind, we are looking for a parametrization $\gamma(t)$ such that $r\gamma[0,t]$ equals $\gamma[0,r^dt]$ in distribution. That is, one in which it takes about time $O(r^d)$ for the curve to travel distance $r$. (Compare this with the discrete interpretation of dimension.) It would be natural to try to parametrize by $d$-dimensional Hausdorff content, but it turns out that this does not work: the Hausdorff content is $0$ almost surely \cite{rezaei-hausdorff}. What does work is to parametrize by $d$-dimensional Minkowski content, so that
\[
\lim_{\epsilon \to 0+} \epsilon^{2-d} \text{Area}\left(\left\{z: \, \operatorname{dist}(z, \gamma[0,t]) \le \epsilon \right\} \right) = t
\]
holds for all $t \ge 0$. To make sense of this requires work, see \cite{LR}. The resulting parametrization is also called the natural parametrization of SLE$_\kappa$. The first construction \cite{lawler_sheffield} did not use Minkowski content, but went via the Doob-Meyer decomposition of a supermartingale obtained by integrating the SLE Green's function. The ``natural time'' was defined as the increasing part in this decomposition. Both approaches are important for this paper.
Our main theorem is that LERW parametrized by renormalized length converges to SLE$_2$ parametrized by Minkowski content. Let us give a rough statement. Fix an analytic simply connected domain $D$ with distinct boundary points $a,b$ and for $N=1,2,\ldots,$ a lattice spacing $N^{-1}$. We take $D_{N}$ to be an appropriate simply connected component of $(N^{-1} \mathbb{Z}^2) \cap D$ with boundary edges $a_N, b_N$ approximating $a,b$. We will measure distance between curves using a metric on parametrized curves defined as follows: If $\gamma^1:[s_1,t_1] \rightarrow {\mathbb C}$ and $\gamma^2:[s_2,t_2]
\rightarrow {\mathbb C}$ are continuous curves, then
\begin{equation} \label{metric} \rho(\gamma^1,\gamma^2) = \inf\left[\sup_{s_1 \leq t \leq t_1}
\left|\alpha(t) - t \right| + \sup_{s_1 \leq t \leq t_1}
\left|\gamma^2\left(\alpha(t) \right) -\gamma^1( t) \right|\right] ,\end{equation}
where the infimum is over all increasing homeomorphisms
$\alpha:[s_1,t_1] \rightarrow [s_2,t_2]$.
\begin{thm}\label{main-thm-rough}
There is a universal (but presumed lattice dependent) constant $\check{c}$ and an explicit sequence $\epsilon_N \to 0+$ as $N \to \infty$ such that the following holds. For each $N$, let $\eta(t), t\in [0, T_\eta],$ be LERW in $D_N$ from $a_N$ to $b_N$ viewed as a continuous curve parametrized so that each edge is traversed in time $\check{c}N^{-5/4}$. Let $\gamma(t), t \in [0, T_\gamma],$ be chordal SLE$_2$ in $D$ from $a$ to $b$ parametrized by $5/4$-dimensional Minkowski content. There is a coupling of $\eta$ and $\gamma$ such that
\[
\mathbf{P}\left\{\rho \left(\eta,\gamma \right) > \epsilon_N \right\} < \epsilon_N.
\]
In particular, $\eta$ converges to $\gamma$ weakly with respect to the metric $\rho$.
\end{thm}
See Section~\ref{sect:main_complete} and Theorem~\ref{thm:main_complete} in particular for a complete statement, but we mention here that we obtain an estimate on the convergence rate and one may take $\epsilon_N = c \, \left(\log N \right)^{-1/60}$ in the theorem, where $c$ is a constant depending on the domain configuration. The recent paper \cite{benoist-dumaz-werner} gives an already worked out application of Theorem~\ref{main-thm-rough}. See also \cite{AMK, BCK, kennedy_montecarlo} for additional discussions of discrete models and their relations to SLE in the natural parametrization. There is a version of Theorem~\ref{main-thm-rough} for radial LERW. The proof requires some extra work which is done in \cite{LV_lerw_radial_note}.
The starting point of the proof is the main result of \cite{BLV}: the renormalized probability that LERW uses a fixed interior edge converges towards the SLE$_2$ Green's function. (See \cite{BLV} and the references therein for a discussion of the LERW growth exponent and related work.) It is important that this result holds for general domains and that we have estimates on the convergence rate. With these facts in hand, the next step is to revisit the convergence in the capacity parametrization \cite{LSW04}. We need to work with a slightly different coupling than the ones previously constructed and we need to be careful about certain measurability properties. We carry out the needed work in a separate paper \cite{LV_lerw_chordal_note}. There we give proofs using the Green's function as martingale observable and derive quantitative bounds on error terms. It is convenient to work with a discrete difference version of Loewner's equation and we discuss this and develop the required estimates in \cite{LV_lerw_chordal_note}.
Given these results we thus have a coupling of LERW with SLE$_2$ in which with large probability the Loewner chains and paths are uniformly close when parametrized by capacity. The main goal of this paper is to show that in this coupling, uniformly as the capacity of the paths is varied, the renormalized length of the LERW is nearly the same as the Minkowski content of the SLE, except for an event of small probability. In order to do this we consider martingales given by taking conditional expectations of the total number of steps and the total content of the LERW and SLE, respectively, given the growing coupled curves sampled at mesoscopic capacity increments. The idea is to look at the Doob-Meyer decompositions of the martingales and use the fact that the Green's functions are very close in order to show that the supermartingale parts are close. From this it is possible to deduce that the increasing parts, that is, the naural times, must also be close. A significant complication is to control the contribution of regions in the complement of the curves where the result of \cite{BLV} gives only trivial information. We handle
this by discretizing and at each step restricting attention to ``open'' squares for which certain geometric estimates hold that allow us to estimate using \cite{BLV}. The contribution of ``closed'' squares is shown to be negligible.
Although many of our estimates are specific to LERW, our general method of proof is not. We do not see any obstructions for it to work for other models as well, if (and this is a big if!) the analogs of the Green's function convergence and second moment estimates for the discrete model
are available.
\subsection{Overview of the proof of Theorem~\ref{main-thm-rough}} \label{sect:overview}
Let us now be more precise about what is needed to carry out this
idea and where in the paper it is done. We will give more detailed definitions in Section~\ref{sect:preliminaries}.
\begin{itemize}
\item{We fix an analytic simply connected domain $D$ with distinct boundary points $a',b'$.}
\item{For any lattice spacing $N^{-1}$ we approximate $(D,a',b')$ by a triple $(A,a,b)$ where $A = A(D,N) \subset \Z^2=\Z + i\Z$ is a
simply connected lattice set with boundary edges $a,b$ near $Na',Nb'$. We often identify edges with their midpoints.}
\item{
We identify each $\zeta \in \Z^2 $ with the closed square
$\Square_\zeta$ of side length $1$ centered at $\zeta$. We let $D_A$ be the simply connected complex domain generated by $A$ by taking the (interior of the) union of the squares corresponding the points of $A$. Note that $N^{-1}D_A$ approximates the domain $D$. We will sometimes slightly abuse language and refer to $D_A$ as a ``union of squares domain''.
}
\item{We write $\check a = N^{-1}a, \check b = N^{-1}b$, and $\check D = \check{D}_A=N^{-1}D_A$ for the quantities scaled by
$N^{-1}$. As $N \to \infty$, $\check D$ converges to $D$ in the Carath\'eodory sense, and it is not hard to estimate the convergence rate. Indeed (see Lemma
\ref{mar5.lemma1}) there exists a conformal transformation $\psi: \check D\rightarrow D$
with $\psi(0) = 0, \psi'(0) > 0, $
\[ |\psi(z) - z| \leq \frac{c \, \log N}{N} , \qquad z \in \check D, \]
\[ |\psi'(z) - 1| \leq \frac{c }{N \, \operatorname{dist}(z, \partial \check D)},
\qquad z \in \check D, \qquad \operatorname{dist}(z, \partial D) \geq \frac c N.\]}
\item{We fix a conformal transformation
\[ F:D_A \rightarrow {\mathbb H}, \quad F(a) = 0, \quad F(b) = \infty.\]
Note that this map is defined only up to a final scaling. We will consider the paths
only up to the time that their half plane capacity reaches $1$. This half plane capacity is defined in terms of the image under $F$ and so depends on the scaling. We will be able
to consider the entire path in $D$ by varying the initial $F$. }
\item{
We choose a mesoscopic scale $h = N^{-2u/3}$ where $u>0$ is the exponent (also denoted by $u$) in the error term
in the main estimate from \cite{BLV}, see \eqref{nov3.1}. We choose the scale this coarse so that
this error does not contribute significantly in estimates. Let $n_{0}= \lfloor h^{-1} \rfloor$. }
\item{ We
grow a LERW in $A$ from $a$ to $b$ which we denote by $\eta$. We write $\mathbf{P}_{A,a,b}$ for the associated probability measure. We stop the path each
time its capacity has increased by $h$, and write $\eta^n$ for path stopped after $n$ mesoscopic increments. By removing the vertices of $\eta^n$ from $A$ (taking an appropriate connected component if needed), we have a
sequence of configurations $(A_0,a_0,b), (A_1,a_1,b), (A_2,a_2,b), \ldots$ with
$A_0 \supset A_1 \supset \cdots$ and we let
$D_n = D_{A_n}$. By mesoscopic capacity increment, we mean the
half-plane capacity of ${\mathbb H} \setminus F(D_n)$
so that $\operatorname{hcap}\left[{\mathbb H} \setminus F(D_n)\right]
\approx h \, n$. }
\item{
We let $g_n: {\mathbb H} \setminus F(D_n) \rightarrow
{\mathbb H}$ be the conformal transformation with $g_n(z) = z + o(1),
z \rightarrow \infty$; let $F_n = g_n \circ F$ for
$g_n = g^n \circ \cdots \circ g^1$
where $g^n$ is the corresponding transformation $g^n: F_{n-1}(D_{n-1}
\setminus D_n) \rightarrow {\mathbb H}$ normalized at infinity. Let \[U_n
= F_n(a_n), \quad \xi_n = U_n- U_{n-1}\] so that $U_n$ is a discrete ``driving term'' for the LERW.}
\item{ Let $\mathbf{P}_n = \mathbf{P}_{A_n,a_n,b}$. In an accompanying paper
\cite{LV_lerw_chordal_note} we use the LERW Green's function and Loewner
difference estimates to couple the LERW with an SLE$_2$.
To be more precise, we
find a standard Brownian motion $W_t$ and a sequence of stopping times
$0 = \tau_0 < \tau_1 < \tau_2 < \cdots$ such that except for an event
of small probability,
\[ \max_{n \le n_0}|U_n - W_{\tau_n}| \leq c \, h^{1/5}. \] }
\item{
Given the Brownian motion, there is a corresponding SLE$_2$ path
in ${\mathbb H}$, that is, there is a simple curve $ \gamma:[0,\infty)
\rightarrow {\mathbb H}$ and conformal maps $g_t^\text{\tiny SLE}: {\mathbb H} \setminus \gamma_t
\rightarrow {\mathbb H}$ satisfying
\[ \partial_t g_t^\text{\tiny SLE}(z) = \frac{1}{g_t^\text{\tiny SLE}(z) - W_t}. \]
Here we write $\gamma_t = \gamma[0,t]$ for the trace in ${\mathbb H}$ and we have parametrized
the curve so that $\operatorname{hcap}[\gamma_t] = t$.
We obtain the SLE in $\check D$ by $\check \gamma(t) =
N^{-1}F^{-1} \left[\gamma(t) \right]$; here, we have retained the capacity
parametrization.
}
\item{We let \[\phi_n^{\text{\tiny LERW}}(z) = \left(g_n \circ F \right)(Nz)-U_n, \quad \phi_{\tau_n}^{\text{\tiny SLE}}(z) = \left(g_{\tau_n}^\text{\tiny SLE} \circ F \right) (Nz) - W_{\tau_n}\] and let $\G_n$ be the $\sigma$-algebra of the coupling, that is, the $\sigma$-algebra
generated by the discrete LERW domains $A_k, k \le n,$ and the Brownian motion $W_s, 0 \leq s \leq \tau_n$. We are careful in
our construction to make sure that $\{W_t- W_{\tau_n}: t \geq \tau_n\}$
is independent of $\G_n$. If $\Im \phi^{\text{\tiny SLE}}_{\tau_{n}}(z) \geq h^{1/20}$, then with large probability the two uniformizing maps are close:
\[ \max_{n \le n_{0}}\left|\phi_n^\text{\tiny LERW}(z) - \phi_{\tau_n}^\text{\tiny SLE}(z) \right| \leq ch^{1/30}.\]}
\item{Let $T$ and $T_n$ be the number of steps of $\eta$ and $\eta^n,$ respectively, and let $\check{T} = c_{*}^{-1} N^{-5/4}T$ and $\check{T}_n = c_*^{-1} N^{-5/4}T_n$ be the scaled quantities. Here $c_{*}$ is the constant appearing in \eqref{nov3.1} below. Similarly let $\check{\Theta}$ be the $5/4$-dimensional Minkowski content of $\check{\gamma}_\infty$ and let $\check{\Theta}_t$ times the $5/4$-dimensional Minkowski content of $\check{\gamma}_t$. }
\item{We consider two discrete time $\mathcal{G}_n$-martingales:
\[
M^{\text{\tiny LERW}}_n = \mathbf{E}\left[\check{T} \mid \mathcal{G}_n \right] = \check{T}_n + c_*^{-1} N^{-5/4}\sum_{z \in A_n}\mathbf{P}_n\left\{z \in \eta\right\}.
\]
and
\[
M^{\text{\tiny SLE}}_n = \mathbf{E}\left[\check{\Theta} \mid \mathcal{G}_n \right] = \check{\Theta}_{\tau_n} + \int_{\check D \setminus \check \gamma_{\tau_n
}}
G_{\check D \setminus \check \gamma_{\tau_n
}}(z;\check \gamma(\tau_n),
\check b) \, dA(z).
\]
Here \[ G_{\check D \setminus \check \gamma_{\tau_n
}}\left(z;\check \gamma(\tau_n),
\check b \right) = \tilde{c}\, r_n(z)^{-3/4} \sin^3\left[\arg \phi_{\tau_n}^\text{\tiny SLE}(z) \right]\] is the Euclidean
Green's function for SLE$_2$ in $\check{D}\setminus \check{\gamma}_{\tau_n}$ from $\check\gamma(\tau_n)$ to $b$ and we are writing $r_n(z)$ for the conformal radius of $\check{D}\setminus \check{\gamma}_{\tau_n}$ seen from $z$. The value of the constant $\tilde{c} \in (0,\infty)$ is unknown.}
\item{
We form the difference of the two $\G_n$-martingales:
\begin{equation} \label{nov4.1}
M_n = M_n^\text{\tiny SLE} -
M_n^{\text{\tiny LERW}}.
\end{equation}
We can then write $M_n = B_n + Y_n$,
where
\[
B_n = \Check{\Theta}_{\tau_n} - \check{T}_n,
\]
and
\[ Y_n = \int_{\check D \setminus \check \gamma_{\tau_n
}}
G_{\check D \setminus \check \gamma_{\tau_n
}}(z;\check \gamma(\tau_n),
\check b) \, dA(z)
- c_*^{-1} \, N^{-5/4} \, \sum_{\zeta \in A_n} \mathbf{P}_{n}\{\zeta \in \eta\}.\] Notice that $B_n$
is a difference of two increasing processes so it is a process
of bounded variation, and
$ Y_n$ is a difference of two supermartingales.}
\item{The main result of \cite{BLV} tells us that there are constants $c_* \in (0, \infty)$ and $u > 0$ such that
\begin{equation} \label{nov3.1}
\mathbf{P}_n\{\zeta \in \eta\} = c_* \, G_{D_n}(\zeta; a_n, b)\left(1 + O\left(N^{- u} \right) \right),
\end{equation}
at least if the interior point $\zeta$ is not too close to $\partial
A_n$. So after rescaling and integrating this relation, taking regularity properties into account, we expect $Y_n$ to be uniformly small.
}
\item{In Section~\ref{sect:main-proof} we will (roughly speaking) use estimates for the coupling and \eqref{nov3.1} to find a $\delta>0$ so that if $\epsilon_N = \left(\log N \right)^{-\delta}$ then there is a ``large'' stopping time $\tau$ such that \begin{itemize}
\item{$|Y_{n}| \le \epsilon_N$ for all $n < \tau$,}
\item{ $\mathbf{E} \left[Y_\tau^2 \right] \le \epsilon_N$,}
\item{ and $\left|B_n' -B_{n-1}'\right| \le \epsilon_N$ for all $n \le \tau$, where $B_n'$ is a predictable version of $B_n$.}
\end{itemize}}
\item{Given this, an argument using the $L^2$-maximum principle shows that $\max_{n \le \tau}|B_{n}'|$ is bounded terms of $\epsilon_N$, with large probability. In Section~\ref{bvsec} we use this to bound $\max_{n \le \tau}|B_n|$, and this is the estimate we want.}
\end{itemize}
A substantial complication in this approach is that the Loewner difference
equation only shows that for suitable $\epsilon > 0$ the uniformizing LERW and SLE maps $\phi_n^\text{\tiny LERW}$ and $\phi_{\tau_n}^\text{\tiny SLE}$
are uniformly close for $z \in D$ with $\Im \left[ \phi_{\tau_n}^\text{\tiny SLE}(\zeta) \right] \geq h^\epsilon$.
We need to also control the contribution of points
for which $\Im\left[\phi_{\tau_n}^\text{\tiny SLE}(\zeta)\right]$ is small. Moreover, the error in the precise version of \eqref{nov3.1} depends on the geometry of the domain seen from $\zeta$. In fact, the curves may \emph{a priori} both create large regions of ``bad'' points, but we will show that the proportion of bad points that are subsequently visited goes to zero and so do not actually contribute. We will achieve this by showing that, roughly speaking, all
such points satisfy at least one of the following conditions for each $n$, and estimate the contribution differently depending on which. Here we summarize the definition for SLE, see Section~\ref{sect:open} the slightly different definition for LERW and further discussion. We set $\lambda = h^{1/100}$ and describe the two conditions. \begin{enumerate}
\item[I.]{We have $\Im\left[ \phi_{\tau_n}^\text{\tiny SLE}(\zeta) \right] \ge \lambda$ and there exists $j \leq n$, such that
\[S_j(\zeta) \leq \left(\log N \right)^{-2/5}, \quad \text{ where } S_j(\zeta)
= \sin \left[\arg \phi_{\tau_n}^\text{\tiny SLE}(\zeta) \right].\] Roughly speaking, this means the the path ``screens'' $\zeta$, e.g., by almost closing a bubble around it, but the distance between the curve and $\zeta$ may still be large. }
\item[II.]{We have $\Im \left[\phi_{\tau_n}^\text{\tiny SLE}(\zeta) \right] < \lambda$ and the distance at time $\tau_n$
from $\zeta$ to the curve is less than $\left(\log N \right)^{-5}$
but the tip of the curve is at least distance
$\left(\log N \right)^{-1}$ from $\zeta$, so that the curve ``got close to $\zeta$ and then away''.
}
\end{enumerate}
A square $\Square_\zeta$ becomes ``closed'' at time $n$ (and stays closed forever) if either of the conditions I or II hold for $\zeta$ at time $n$. A square is ``open'' at a given time if it is not closed. The idea is to do the argument as sketched above but instead redefining the processes $M_n^{\text{\tiny SLE}},
M_n^{\text{\tiny LERW}},M_n,B_n,Y_n$ to be the corresponding
quantities referring to the amount of
natural time spent in open squares, that is, time spent before the
square has become closed. For this to work we have to show that it is enough to consider the open squares and this part of the argument is given in Section~\ref{badsec}. The proof of Theorem~\ref{main-thm-rough} is completed in Section~\ref{sect:main-proof}, assuming some statements that are proved in later sections.
The proof of Theorem~\ref{main-thm-rough} requires sharp one and two-point
estimates for both SLE and LERW. For SLE they have been developed
in several recent papers, and the sharp one-point estimate for
LERW is \eqref{nov3.1}.
In Section~\ref{LERWsec} we have collected the needed estimates about LERW.
We have separated them from the main argument because they have independent interest and because this section can be read independently.
The two-point estimates for LERW
need both the sharp one-point estimate and an
appropriate separation lemma that states that that
two-sided LERW conditioned to reach a ball about the origin have
a good chance of having the endpoints at the first visits from
the two directions ``separated''. We leave the exact
statements for Section \ref{LERWsec}. This section, which comprises
almost half of this paper, does not use any facts about SLE.
\subsection{Acknowledgments}
Lawler was supported by National Science Foundation grant DMS-1513036. Viklund was supported by the Knut and Alice Wallenberg Foundation, the Swedish Research Council, the Gustafsson Foundation, and National Science Foundation grant DMS-1308476. We also wish to thank the Isaac Newton Institute for Mathematical Sciences where part of this work was carried out.
\section{Preliminaries}\label{sect:preliminaries}
\subsection{Discrete set-up and loop-erased random walk}\label{sect:set-up}
Here we will give precise definitions of our discrete quantities.
\begin{itemize}
\item If $A$ is a finite subset of $\Z^2$, we let $\partial_e A$ denote
the edge boundary of $A$, that is, the set of edges of
$\Z^2$ with exactly one endpoint in $A$. We will specify
elements of $\partial_e A$ by $a$, the midpoint of the edge. Note
that
$a$ specifies the edge uniquely up to the orientation. We will write\
$a_-,a_+$ for the endpoints of the edge in $\Z^2\setminus A$ and $A$, respectively.
Note that
\[ a_-, b_- \in \partial A:= \{z \in \Z^2 \setminus A: \operatorname{dist}(z, A) = 1\}, \]
\[ a_+, b_+ \in \partial_iA := \{z \in A: \operatorname{dist}(z, \partial A) = 1\}. \]
We also write the edge as
$e_a = [a_-,a_+], e_b = [b_-,b_+]$ for the edges oriented from the outside to the
inside.
\item Let $\whoknows$
denote the set of triples $(A, a, b)$ where
$A$ is a finite, simply connected subset of $\Z^2$
containing the origin, and $ a, b$ are
elements
of $\partial_e A$ with $a_- \neq b_-$.
We allow $a_+ = b_+$.
\item let $\Square= \{x+iy \in {\mathbb C}: |x|,|y| \leq 1/2\}$
be the closed square of side length one centered at the origin and
$\Square_z = z + \Square$. If $(A,a,b) \in \whoknows$, let
$D_A$ be the corresponding simply connected domain defined
as the interior of
\[ \bigcup_{z \in A} \Square_z . \]
This is a simply connected Jordan domain whose boundary is a subset of the edge set of the dual graph of $ \Z^2$. Note that $a,b \in \partial D_A$. We refer to $D_A$ as a ``union of squares'' domain.
\item{Let
$F = F_{A,a,b}$ denote a conformal map from
$D_A$ onto ${\mathbb H}$ with $F( a ) = 0, F( b ) = \infty$. This
map is defined only up to a dilation; later we will fix
a particular choice of $F$.
Note that $F$ and $F^{-1}$
extend continuously to the boundary of the domain (with the appropriate
definition of continuity at infinity).
}
\item For $z \in D_A$, we define
\[ \theta_A(z;a,b) = \arg F(z) , \;\;\; \Sine_{A,a,b}(z) = \sin \theta_A(z;a,b),
\]
which are
independent of the choice of $F$, since $F$ is unique up to scaling.
Also for $z \in {\mathbb H}$, we write
\[ \Sine(z) = \sin[\arg(z)].\]
\item We write $r_A(z)=r_{D_A}(z)$ for the conformal radius of $D_A$
with respect to $z$. It can be computed from $F$ by
\[ r_A(z) = 2\, \frac{\Im F(z) }{|F'(z)|}, \]
which is independent of the choice of $F$.
\item{A walk $\omega = [ \omega_0, \ldots, \omega_n]$ is a sequence of nearest neighbors in $\Z^2$. The length $|\omega| = n$ is by definition the number of traversed edges.}
\item{If $z,w \in A$, we write
$\paths_A(z,w)$ for the
set of walks $\omega$ starting at $z$, ending
at $w$, and otherwise staying in $A$. }
\item{
The simple random walk measure $p$ assigns to each walk measure $p(\omega) = 4^{-|\omega|}$. The total measure
of $\paths_A(z,w)$ equals $G_A(z,w)$, the simple random
walk Green's function.}
\item{
If $a,b \in \partial_e A$, there is an obvious bijection between
$\paths_A(a_+,b_+)$ and $\paths_A(a,b)$, the set of
walks starting with edge $e_a$, ending with
$e_b^R$ and otherwise staying in $A$.
(Here and
throughout this section we write $\omega^R$ for the
reversal of the path $\omega$, that is, if
$\omega = [\omega_0,\omega_1,\ldots,\omega_k]$, then
$\omega^R = [\omega_k,\omega_{k-1},\ldots,\omega_0]$.)
We sometimes write $\omega: a \rightarrow b$ for walks in $\paths_A(a,b)$ with the condition to stay in $A$ implicit. }
\item{We
write $H_{\partial A}(a,b)$ for the total random walk measure of $\paths_A(a,b)$.
It is easy to see that
$ H_{\partial A}(a,b) = G_A(a_+,b_+)/16$,
The factor of $1/16 = (1/4)^2$ comes from the $p$-measure of the edges $e_a,e_b$. $H_{\partial A}(a,b)$ is called the boundary Poisson kernel.}
\item{A self-avoiding walk (SAW) is a walk visiting each point at most once. We write $\mathcal{W}_A(z,w) \subset \mathcal{K}_A(z,w)$ for the set of SAWs from $z$ to $w$ staying in $A$.
We will write $\omega$ for
general nearest neighbor paths and reserve $\eta$ for SAWs. We write $\mathcal{W}_A(a,b)$ similarly when $a,b$ are boundary edges. }
\item{The loop-erasing procedure takes a walk and outputs a SAW, the \emph{loop-erasure} of $\omega$. Suppose a walk $\omega = [\omega_0, \ldots, \omega_n]$ is given.
\begin{itemize}
\item{If $\omega$ is self-avoiding, set $\text{LE}[\omega] = \omega$.}
\item{Otherwise, define $s_0 = \max\{ j \le n: \omega_j = \omega_0\}$ and let $\text{LE}[\omega]_0 =\omega_{s_0}$. }
\item{For $i \ge 0$, if $s_i < n$, define $s_{i+1} = \max\{ j \le n: \, \omega_j = \omega_{s_i}\}$ and set $\text{LE}[\omega]_{i+1} = \omega_{s_i+1}$.}
\end{itemize}
Note that if $e_a \oplus \omega
\oplus e_b^R \in \paths_A(a,b)$, then
LE$[e_a \oplus \omega
\oplus e_b^R] = e_a \oplus \text{LE}[\omega]
\oplus e_b^R $.
}
\item The loop-erasing procedure induces a natural measure on SAWs as follows. We define $\hat P_{A,a,b}$, the ``loop-erased'' measure, on $\mathcal{W}_A(a,b)$ by
\[ \hat P_{A,a,b}(\eta) = \sum_{\omega \in \paths_A(a,b): \; \text{LE}(
\omega) = \eta}
p(\omega). \]
Note that $\hat P_{A,a,b}[\mathcal{W}_A(z,w)] = H_{\partial A}
(a,b)$. Let \[\mathbf{P}_{A,a,b} = \frac{\hat P_{A,a,b}}{
H_{\partial A}
(a,b)}\] denote the probability measure obtained by normalization. This is the probability law of loop-erased random walk (LERW) in $A$ from $a$ to $b$.
\end{itemize}
We state the main result from \cite{BLV}.
\begin{lemma} There exists $\hat c > 0$ and $u >0$ such that the following holds. Suppose $(A,a,b) \in \whoknows$ and that $\zeta \in A$ is such that $S_{A,a,b}(\zeta) \ge r_A(\zeta)^{-u}$ , then
\begin{equation}
\label{BLV1} \mathbf{P}_{A,a,b}\{ \zeta \in \eta\} = \hat c \, \, r_A(\zeta)^{-3/4}\Sine_{A,a,b}^3(\zeta)
\, \left[1 +O\left(r_A(\zeta)^{-u}\Sine_{A,a,b}^{-1}(\zeta) \right)\right].
\end{equation}
\end{lemma}
We do not have an explicit bound on $u$ except $u>0$. We will fix a value of $u$ such that \eqref{BLV1}
holds for the remainder of the paper.
For our purpose it is more useful to write \eqref{BLV1} in terms of the
Euclidean Green's function of SLE$_2$, see Section~\ref{SLE:defs} for the definition. For now we recall that in this case
\[ G_{D_A}(\zeta;a,b) = \tilde c \, r_A(\zeta)^{-3/4}
\, S_{A,a,b}^3(\zeta), \]
for some universal (but unknown)
$\tilde c >0$. Therefore, we may rewrite \eqref{BLV1}
as
\begin{equation} \label{BLV2}
\mathbf{P}_{A,a,b}\{ \zeta \in \eta\} = c_* \,
G_{D_A} (\zeta;a,b) \,\left[1
+ O\left(r_A(\zeta)^{-u} \right) \; \Sine_{A,a,b}^{-1}(\zeta)
\right],
\end{equation}
where $c_* = \hat c/\tilde c$.
\subsection{SLE and Minkowski content}\label{SLE:defs}
Chordal SLE$_\kappa$ in ${\mathbb H}$ is defined by first solving the Loewner equation
\[
\partial_t g_t(z) = \frac{2/\kappa}{g_t(z) - B_t}, \quad g_0(z)=z,
\]
with $B_t$ a standard Brownian motion. For each $t \ge 0$, $g_t(z)$ is a conformal map from a simply connected domain $H_t$ onto ${\mathbb H}$ normalized so that $g_t(z) = z + (2/\kappa)
t/z + O(1/|z|^2)$ as $z \to \infty$. The family $(g_t(z))$ is called the SLE$_\kappa$ Loewner chain. The SLE$_\kappa$ path is the continuous curve defined by \[\gamma(t)=\lim_{y \to 0+} g^{-1}_t(iy+B_t).\] The curve generates the Loewner chain in the sense that $H_t$ is the unbounded component of ${\mathbb H} \setminus \gamma_t$, where $\gamma_t = \gamma[0,t]$. As $t \to \infty$, this curve connects $0$ with $\infty$ in ${\mathbb H}$. The compact set which is disconnected from $\infty$ by $\gamma_t$ is called the SLE$_\kappa$ hull (in general, a hull is a compact set such that ${\mathbb H} \setminus K$ is unbounded and simply connected) and is denoted $K_t$. If $\kappa \le 4$, then $\gamma$ is simple so that $K_t = \gamma_t$.
Given a hull $K$ there is a Riemann map $g : {\mathbb H} \setminus K \to {\mathbb H}$ such that $g(z) = z + o(1)$ as $z \to \infty$. We define the half-plane capacity of $K$ by
\[
\operatorname{hcap}[K] = \lim_{|z| \to \infty} z \left( g(z)-z \right).
\]
If $\gamma$ is parametrized so that $\operatorname{hcap}[K_t] = (2/\kappa) t $, we say that $\gamma$ is parametrized by capacity.
Given a simply connected domain $D$ with marked boundary points (prime ends in general) $a,b$, we define SLE$_\kappa$ in $D$ from $a$ to $b$ by conformal invariance. That is, we choose a conformal map $\phi:D \to {\mathbb H}$ such that $\phi(a) = 0, \, \phi(b) = \infty$ and consider the image of $\gamma$ under $\phi^{-1}$. The map $\phi$ is only unique up to scaling, but allowing for a linear reparametrization the law of $\gamma$ is scale invariant.
The Green's function for SLE$_\kappa$ in ${\mathbb H}$ is the function defined by \[G_{{\mathbb H}}(z; 0,\infty)=\lim_{\epsilon \to 0+} \epsilon^{d-2} \mathbf{P} \left\{ \operatorname{dist}(z,\gamma_\infty) \le \epsilon \right\}, \quad d=1+\frac{\kappa}{8}.\]
(We suppress the $\kappa$-dependence in writing $G_{D}$.) We have the formula
\[
G_{{\mathbb H}}(z; 0,\infty)=\tilde{c} \, r_{{\mathbb H}}(z)^{d-2}\sin^{\beta}\left( \arg z \right), \quad \beta=\frac 8 \kappa-1,
\]
where $\tilde{c} \in (0, \infty)$ is a $\kappa$-dependent but unknown constant. Note that $r_{{\mathbb H}}(z) =2 \Im z$. (Replacing Euclidean distance by conformal radius on the left-hand side in the definition results in the same formula with a computable constant.)
Using conformal covariance we can see that
\[
G_D(z; a,b) = \tilde{c}\, r_D(z)^{d-2} \sin^{\beta}\left( \arg \phi(z) \right),
\]
where $\phi : D \to {\mathbb H}$ is as in the previous paragraph.
Besides the capacity parametrization, the \emph{natural parametrization} of SLE is important for this paper. Let us review a few facts about it, see \cite{LR} for proofs and further discussion.
The simplest definition to state is in terms of $d$-dimensional Minkowski content: given $\gamma_t$, we can define
\[
\Theta_t = \Cont_d\left( \gamma_t\right) = \lim_{\epsilon \to 0+}\epsilon^{d-2} \text{Area}\left\{ z: \, \operatorname{dist}(z, \gamma_t) \le \epsilon \right\}.
\]
Then almost surely this limit exists for all $t$ and $t \mapsto \Theta_t$ is Holder continuous. Setting \[s(t) = \inf \left\{s \ge 0 : \Theta_s = t \right\},\] we may reparametrize $\gamma$ by Minkowski content, that is, consider $t \mapsto \gamma \circ s(t)$, which can be seen to be almost surely Holder continuous. This is SLE$_\kappa$ in the natural parametrization (or SLE$_\kappa$ parametrized by natural time). When we do not specify the dimension $d$, e.g., by simply writing $\Cont(\cdot)$ we are assuming $\kappa=2$ and $d=5/4$.
Suppose $D$ is a bounded simply connected domain with (say) analytic boundary. An important property of the Minkowski content is that if $\gamma$ is SLE$_\kappa$ in $D$ from $a$ to $b$, and $V \subset D$, then
\[
\mathbf{E} \left[ \Cont_d(\gamma_\infty \cap V) \mid \gamma_t \right] = \Cont_d(\gamma_t \cap V) + \int_{V \setminus \gamma_t} G_{D\setminus \gamma_t}(z; \gamma(t), b)\, dA(z).
\]
In particular, $\mathbf{E}[\Theta_\infty] = \int_D G_D(z; a,b) \, dA(z) <\infty$ and
\[
\mathbf{E} \left[\Theta_\infty \mid \gamma_t \right] = \Theta_t + \int_{D \setminus \gamma_t} G_{D \setminus \gamma_t}(z; \gamma(t), b) \, dA(z)
\]
is a martingale, and the two terms on the right hand side form its Doob-Meyer decomposition into an increasing process and a supermartingale, respectively.
In several places, sometimes without explicit reference, we will use the one-point estimate for SLE. We state one version here, see Section~2.2 of \cite{LR} for this and other versions.
\begin{lemma}\label{lem:SLE-one-point}
Suppose $0 < \kappa < 8$. There exist positive constants $c_*, \alpha$ such that the following holds. Let $\gamma$ be SLE$_\kappa$ in $D$ from $a$ to $b$, where $D$ is a simply connected domain with distinct boundary points (prime ends) $a,b$. Then for all $z \in D$ with $\operatorname{dist}(z,\partial D) \ge 2\epsilon$,
\[
\mathbf{P}\left\{\gamma \cap \mathcal{B}(z,\epsilon) \neq \emptyset \right\} = c_*\epsilon^{2-d}G_D(z;a,b)\left[1+O(\epsilon^\alpha)\right],
\]
where $G_D(z;a,b)$ is the Green's function for SLE$_\kappa$ from $a$ to $b$ in $D$.
\end{lemma}
\subsection{Complete statement of main result}\label{sect:main_complete}
We will now give a complete statement of our main result. In order to do so, we will have to scale
the lattice path.
\begin{itemize}
\item Given $\eta \in \mathcal{W}_A(a,b)$, of the form
\[ \eta = [\eta_0 = a_-, \eta_1 = a_+,
\ldots ,\eta_{n} = b_+, \eta_{n+1} = b_-] , \]
we write $\eta(t)$ for the curve obtained by going from $a$ to $b$
along $\eta$ at speed one. More precisely, $\eta(t), 0 \leq t \leq n$,
is defined by
$\eta(0) = a; \eta(n) = b$;
\[ \eta\left(j- \frac 12\right) = \eta_j, \;\;\;\;j=1,\ldots, n;\]
and $\eta(t)$ is defined for other $t$ by
linear interpolation.
\item If $\eta(t), 0 \leq t \leq n$ is a curve as above
and $N > 0$, we let $\eta^N(t)$ denote the scaled map
\[ \eta^N(t) = N^{-1} \, \eta(t \,c_* \, N^{5/4}) ,
\;\;\;\;\; 0 \leq t \leq \frac{n}{c_* \, N^{5/4}}.\]
Here $c_*$ is the constant from \eqref{BLV2}.
\item We write $\mathbf{P}_{A,a,b}^N$ for the probability measure
obtained from $\mathbf{P}_{A,a,b}$ by considering the curves scaled
as above.
\item If $\eta = [\eta_0,\ldots,\eta_k] \in \mathcal{W}_A(a,b)$,
let $\eta^j = [\eta_0,\ldots,\eta_j]$, $A_j
= A \setminus \eta^j$ and $a_j = [\eta_{j} + \eta_{j+1}]/2$
so that $a_j \in \partial_e A_j$.
The tuples $(A_j, a_j, b), \, j=0,1,\ldots$ form a sequence of decreasing discrete domains with two marked boundary edges. We write $D_j = D_{A_j}$.
Note that $D_j$ is obtained from $D_A$ by removing the
$j$ squares associated to first $j$ steps of $\eta$ plus
any squares that have become disconnected from $0$.
\item{We will assume that we have a bounded analytic
simply connected domain $D$ containing the origin with
analytic boundary and two distinct boundary points
$a',b'$. We will consider lattice approximations of $D$.
The lattice scaling will be $N^{-1}$. We will define some
scaled quantities, but the dependence on $N$ will be implicit.}
\item{The assumption that $D$ is analytic is of course not necessary -- we will make it for convenience, but remark that by an approximation argument our main result can be extended to more general domains, assuming local analyticity at $a',b'$.}
\item{ If $N > 0$, let $A = A_{N,D}$ be the connected
component containing the origin of the set of $\zeta \in \Z^2$
with $\Square_\zeta \subset ND$. Let $D_A$ be the corresponding
domain obtained by taking the interior of the union of the $\Square_\zeta$. Let $\check D_A = N^{-1} \, D_A$. If $a \in \partial_e
A$, we write $\check a $ for the midpoint of the edge $a/N$. We sometimes identify an edge with its midpoint.}
\end{itemize}
\begin{itemize}
\item We consider the metric \eqref{metric} on continuous curves and write $\wp_\rho$ for the corresponding Prokhorov metric
on probability measures on curves.
\item If $D$ is a domain as above and $a,b$
are distinct boundary points, then $\mu_D(a,b)$ denotes
the probability measure given by SLE$_2$ with the natural
parametrization. (In other papers of the first author,
the notation measure $\mu_D(a,b)$
refers to SLE with total mass of the partition function
and the probability measure and the corresponding probability
measure
denoted by $\mu_D^\#(a,b)$. However, since we only
need to use the probability measure in this paper, we
choose the simpler notation.)
\end{itemize}
\begin{thm}\label{thm:main_complete} Let $D$ be a bounded analytic domain containing
the origin with distinct
boundary points $a',b'$. For each $N$, let $A_N = A_{N,D}$
and let $a_N,b_N \in \partial_e A_N$ with
\[ \check a := a_N/N \stackrel{N \to \infty}{\longrightarrow} a', \;\;\;\;
\check b:= b_N/N \stackrel{N \to \infty}{\longrightarrow} b'.\]
Then,
\[ \lim_{N \to \infty} \mathbf{P}^N_{A_N,a_N,b_N} =
\mu_D(a',b'), \]
where the convergence is with respect to the Prokhorov
metric as above.
\end{thm}
We start by making some reductions. It is not
difficult (see Corollary \ref{mar5.cor3}) to show that
\[ \lim_{N \rightarrow \infty}
\wp\left[\mu_{\check D}(\check a,\check b) ,
\mu_D(a',b')\right] = 0.\]
Hence, it suffices to show that
$ \lim_{N \to \infty} \wp\left[\nu_N ,\tilde \nu_N
\right]=0,$
where \[\nu_N^{\text{\tiny LERW}} = \mathbf{P}^N_{A_N,a_N,b_N}, \qquad
\nu_N^{\text{\tiny SLE}} = \mu_{\check D}(\check a, \check b).\]
In order to compare
$\nu_N^{\text{\tiny LERW}}, \nu_N^{\text{\tiny SLE}}$ we consider the paths parametrized
by half-plane capacity. This capacity is defined by
first taking $F:D_A \rightarrow {\mathbb H}$ with
$F_N(a_N) = 0, F_N(b_N) = \infty$ and measuring
capacities of the images under $F$. The map $F$
is unique up to a final dilation.
For every $k < \infty$, we can consider the measures
$\nu_N^{\text{\tiny LERW}}, \nu_N^{\text{\tiny SLE}}$ on paths stopped when the capacity
reaches $k$. Since the total capacity of the curves
is infinite, this truncation is well defined and does not
give the entire curve. In order to get convergence
in the Prokhorov metric we need two facts. The first:
\begin{itemize}
\item The curves parametrized by capacity are very close
in supremum norm except for small probability.
\end{itemize}
We prove this in the separate paper \cite{LV_lerw_chordal_note}, but we will give the needed statements below.
This result has been proved previously for convergence
in capacity parametrization using a slightly different coupling, see \cite{LSW04, zhan}. The second is the one that we focus on:
\begin{itemize}
\item If we reparametrize by length (using normalized
number of steps for the LERW and Minkowski content for
the SLE), the reparametrizations are very close in
supremum norm except for small probability.
\end{itemize}
Let $\nu_{N,k}^{\text{\tiny LERW}}, \nu_{N,k}^{\text{\tiny SLE}}$ denote the corresponding
measures on curves
parametrized by length but {\em truncated when their
capacity reaches $k$}. We will show that
$\wp \left[\nu_{N,k}^{\text{\tiny LERW}}, \nu_{N,k}^{\text{\tiny SLE}} \right] $ is small.
We also need to show for LERW and for SLE that as
$k \rightarrow \infty$, both
$\wp \left[\nu_{N,k}^{\text{\tiny LERW}},\nu_N^{\text{\tiny LERW}} \right]$ and $\wp\left[ \nu_{N,k}^{\text{\tiny SLE}},
\tilde \nu_k^{\text{\tiny SLE}} \right]$ are bounded by $\epsilon _k$ for some $\epsilon_k
\rightarrow 0$ (independent of $N$).
This is discussed in Section \ref{metricsec}.
Finally, rather than take a particular $F$ and showing
the estimates for paths truncated at capacity $k$, we
will start with any $F$ and truncate at capacity $1$.
Note that paths truncated at capacity $k$ for a given
$F$ are the same as those truncated at capacity $1$
for the map $z \mapsto k^{-1/2} \, F(z).$
This is the main theorem of this paper and
precise statements can be
found in \eqref{feb15.1}
and \eqref{feb15.2}.
We prove a theorem on about the chordal version of LERW connecting two boundary
points but one can derive from this a corresponding result
about LERW from a boundary point to an interior point.
The details can be found in \cite{LV_lerw_radial_note}, but
we state the result here.
\begin{thm}\label{main-thm-rough-radial}
There is a universal constant $\check{c}$ and an explicit sequence $\epsilon_N \to 0+$ as $N \to \infty$ such that the following holds. For each $N$, let $\eta(t), t\in [0, T_\eta],$ be LERW in $A_N$ from $a_N$ to the origin viewed as a continuous curve parametrized so that each edge is traversed in time $\check{c}N^{-5/4}$. Let $\gamma(t), t \in [0, T_\gamma],$ be radial SLE$_2$ in $D$ from $a$ to $0$ parametrized by $5/4$-dimensional Minkowski content. There is a coupling of $\eta$ and $\gamma$ such that
\[
\mathbf{P}\left\{\rho \left(\eta,\gamma \right) > \epsilon_N \right\} < \epsilon_N.
\]
In particular, $\eta$ converges to $\gamma$ weakly with respect to the metric $\rho$.
\end{thm}
There are two approaches to this last theorem. One would be to
redo the work in this paper in the radial setting. In fact, the
proof of convergence of LERW with respect to capacity
parametrization in \cite{LSW04} was for the radial case,
and a version for the chordal case was first done in \cite{zhan} by following
the same outline. However, for our result this
would take a fair amount of work; in particular, the radial analogue
of the estimate \eqref{BLV2} would need to be proved. Fortunately,
we now know that one can go between the chordal and radial
results using Radon-Nikodym derivatives and this is the appraoch
we use in \cite{LV_lerw_radial_note}.
\section{Deterministic estimates and coupling}\label{sect:deterministic}
The first step to the proof is to construct the coupling between SLE$_2$
and LERW. Our argument is similar to that in \cite{LSW04} although there
are some differences. First, we use a difference equation form of the Loewner theory.
This is useful because our
domains $D_n=D_{A_n}$ are derived from $D_A$ by cutting out
squares rather than a curve.
We could work with slit domains and translate results, but we feel the difference equation approach produces cleaner arguments in the present setting.
The other change is that we use the LERW Green's
function rather than the discrete Poisson kernel as our observable.
It is important that these results are
redone in our context, but because the proofs are similar to those
in previous coupling, we will only state the important results here. Complete proofs and additional discussion
can be found in
\cite{LV_lerw_chordal_note}.
\subsection{Loewner difference equation}
Suppose
$\gamma:(0,\infty) \rightarrow {\mathbb H}$ is a simple
curve with $\gamma(0+) =0$ parametrized so that
$\operatorname{hcap}[\gamma_t] = t$, where $\gamma_t = \gamma[0,t]$.
Let $g_t:{\mathbb H} \setminus \gamma_t
\rightarrow {\mathbb H}$ be the conformal transformation with
$g_t(z) = z +o(1)$ as $z \rightarrow \infty$. Then we have the chordal Loewner differential equation,
\[ \partial_t g_t(z) = \frac{1 }{g_t(z) - U_t} , \quad g_0(z)=z, \]
where $U_t = g_t(\gamma(t))$. The proof of this correspondence, at
least as given in \cite{Lbook}, starts by proving a
``difference estimate'' to show that for small
$t$,
\begin{equation} \label{oct9.1}
g_t(z) - z = \frac{t}{z}
+ O\left(\frac{tr}{|z|^2} \right),
\end{equation}
where $r$ denotes the radius of $\gamma_t$. This
estimate does not require $\gamma_t$ to be the image of a curve
and in fact holds with an error term uniformly
bounded over all hulls (see below) of half-plane capacity
$t$ and radius $r$.
We will say that
$K \subset {\mathbb H}$ is a {\em (compact ${\mathbb H}$-)hull},
if $K$ is bounded and
$D_K := {\mathbb H} \setminus
K$ is a simply connected domain. Define
\[ r_{K} = {\rm rad}(K) = \max\{|z|: z \in K\},\;\;\;\;
h_K = \operatorname{hcap}(K),\]
and recall that $h_K \leq r_K^2$. If $r_K$ is small, $K$ is located near $0$.
Let $g_{K}$ be the unique conformal transformation
of $D_K$ onto ${\mathbb H}$ whose expansion at infinity is
\[ g_{K}(z) = z + \frac{h_K}{z} + O(|z|^{-2}).\]
Suppose now we have a sequence of hulls of small capacity
$K_1,K_2,\ldots$ and ``locations'' $U_1,U_2,\ldots \in {\mathbb{R}}$, so that, roughly speaking $K_j + U_j$ is near $U_j$.
Let \[r_j = r_{K_j}, \quad h_j = h_{K_j}= h_{K_j + U_j}, \quad g^j
= g_{K_j + U_j}\] and let
\[ g_j = {g^j \circ \cdots
\circ g^1}.\]
Since the right-hand side of \eqref{oct9.1}
depends only on $r$ and not on the exact shape
of $K$, it follows that if we have two sequences
for which the capacity increments and ``driving terms'', $h_j$ and $U_j$, are close, then the functions $g_n$
are close. We give a precise formulation of this
in the next two proposition.
\begin{prop}\label{prop:loewner-comparison} There exists $1 < c < \infty$ such
that the following holds. Suppose $(K_1,U_1),
$ $ (K_2,U_2)\ldots$ and $(\tilde K_1,\tilde U_1),
(\tilde K_2,\tilde U_2),$ $\ldots$ are two sequences as above
with corresponding $r_j, h_j, g^j, g_j$ and
$\tilde r_j, \tilde h_j, \tilde g^j, \tilde g_j$.
Let \[0 < h <
r^2 < \epsilon^2 < \delta^8 < 1/c,\]
and $n \leq 1/h$ and suppose that
for all $j=1,\ldots,n$,
\[ |h_j - h| \leq hr/\delta , \;\;\;\;
|\tilde h_j - h| \leq hr/ \delta , \]
\[ r_j, \tilde r_j \leq r , \]
\[ |U_j -\tilde U_j| \leq \epsilon. \]
Suppose $z = x+iy \in {\mathbb H}$ and let
$z_n = x_n +iy_n = g_n(z), \tilde z_n =
\tilde x_n + i \tilde y_n = \tilde g_n(z).$
Then, if $y_n,\tilde y_n \geq \delta$,
\begin{equation} \label{halloween.1}
|g_n(z) - \tilde g_n(z)| \leq c \, (\epsilon/\delta) \,
(y \wedge 1).
\end{equation}
\end{prop}
\begin{prop}
\label{prop:deriv-lb} There exists $1 < c < \infty$ such
that the following holds. Suppose $(K_1,U_1),
$ $ (K_2,U_2)\ldots$ is a sequence as above
with corresponding $r_j, h_j, g^j, g_j$.
Let \[0 < h <
r^2 < \delta^8 < 1/c,\]
and $n \leq 1/h$ and suppose that
for all $j=1,\ldots,n$,
\[ |h_j - h| \leq hr/\delta, \quad r_j \le r. \]
Suppose $z = x+iy \in {\mathbb H}$ and let
$z_n = x_n +iy_n = g_n(z)$.
Then if $y_n \geq \delta$,
\begin{equation} \label{halloween.2}
|g_n'(z)| = \exp\left\{-\sum_{j=0}^{n-1} \Re \frac{h}{(z_{j} - U_{j})^2}\right\} \left(1+O(\delta) \right).
\end{equation}
In particular, there is a constant $c$ such that if
\begin{equation}\label{jan26.1}
\nu=\min_{0\le j \le n} \left\{\sin\left[ \arg\left(g_j(z) - U_j \right) \right] \right\},
\end{equation}
then,
\begin{equation}\label{jan26.2}
|g'_n(z)| \ge c \left(\frac{y_n}{y}\right)^{1-2\nu^2}.
\end{equation}
\end{prop}
\subsection{Coupling}\label{sect:coupling}
We will consider 4-tuples $(A,a,b,F)$ where $(A,a,b)
\in \whoknows$. and $F:D_A \rightarrow {\mathbb H}$ is a conformal transformation
with $F(a) = 0, F(b) = \infty$. As we have noted before, there is a one-parameter
family of such transformations $F$, so we will fix one of them. We define \[N = N(A,a,b,F) = |(F^{-1})'(10i)|\] and note that $N$ is half the conformal radius of $D_A$ seen from $F^{-1}(10i)$. All of our results will hold only for
$N$ sufficiently large, and we will not always be explicit about this.
We fix a mesoscopic scale $h$, defined by
\[
h = N^{-2u/3},
\]
where $u$ is the exponent from \eqref{BLV1}. This is a somewhat arbitrary
choice, but we will use that $N^{-u} = O(h^{6/5})$.
Let $(A_0,a_0,b) = (A,a,b), D_0 = D_A, F_0^\text{\tiny LERW} = F$. We will define a sequence $(A_n,a_n,b)$ with corresponding
simply connected domains $D_n$ and functions $F_n^\text{\tiny LERW}$ recursively
by saying that the conditional distribution of $(A_n,a_n,b)$
given $(A_{n-1},a_{n-1},b)$ is that of the LERW
probability measure $\mathbf{P}_{n-1}:= \mathbf{P}_{A_{n-1},a_{n-1},b}$
where the walk (taking microscopic lattice steps) is stopped at the first time $m = m_n$ such that
\[ \operatorname{diam}[K^m] \geq h^{2/5} \;\;\; \mbox{ or } \;\;\;
\operatorname{hcap}[K^m] \geq h , \]
where
\begin{equation} \label{feb16.2}
K^j = F_{{n-1}}(D_{A_{n-1}} \setminus D_{A_{n-1} \setminus \eta^j}) \subset {\mathbb H}
\end{equation}
and $\eta$ is LERW in $A_{n-1}$ from $a_{n-1}$ to $b$.
We set
$D_n = D_{A_n}$ and
\begin{equation} \label{feb16.1}
F_n^\text{\tiny LERW} = g_n \circ F_0 - U_n, \;\;\;\; U_n:=
g_n \circ F_0 (a_n) .
\end{equation}
where $ g_n: F_0(D_0 \setminus D_n)
\rightarrow {\mathbb H}$ is the conformal transformation
normalized so that $ g_n(\infty) = \infty$,
$ g_n'(\infty) = 1$. Note that the transformation $F_n^\text{\tiny LERW}$ is
translated so that $F_n^\text{\tiny LERW}(a_n) = 0, F_n^\text{\tiny LERW}(b) = \infty$.
Let $\xi_n = U_n - U_{n-1}$.
Let us be more precise. We write $D_0 = D_A, D_j = D_{A_j}$. Set $m_0=0, m_1=m$, where
\[
m=\min\left\{j \ge 0: \, \operatorname{hcap}\left[K_{j} \right]
\geq h \text{ or } \operatorname{diam}\left[K_j \right] \geq h^{2/5} \right\},\]
where $K_j= F(D \setminus D_{A_j})$,
and for $n=0,1, \dots$, and $j =0, 1, \ldots$,
\[
K_j = F(D \setminus D_j), \quad K^n_j = F_{m_n}(D_{m_n} \setminus D_{m_n + j}),
\]
and
\[
\Delta_n = \min\left\{ j \ge 0: \, \operatorname{hcap}[K_j^n] \ge h \, \text{ or } \, \operatorname{diam}[K_j^n] \ge h^{2/5} \right\},
\]
\[
m_{n+1} = m_n + \Delta_n.
\]
Write
\[
K^n = K^n_{\Delta_n}.
\]
Then
\[t_{m_{n+1}} =t_{m_n} + \operatorname{hcap}\left[K^n \right].\]
and we set
\[
r_{m_{n+1}} = \operatorname{diam}\left[K^n \right].
\]
Let $g^{n+1}:{\mathbb H} \setminus K^n \rightarrow {\mathbb H}$
be the conformal transformation with $g^{n+1}(z)
- z = o(1)$ and set $F_{m_{n+1}} = g^{n+1} \circ F_{m_n}$ and \[ g_{n+1} = g^{n+1} \circ g^{n} \circ \cdots
\circ g^1 . \]
We also define the ``driving process increment'',
\[ \xi_{n+1} = g^{n+1} \circ F_{m_n}(a_{m_n})-\xi_{n}, \]
giving a ``driving process''
\begin{equation} \label{jan12.2} U_{n+1} = \xi_1 + \cdots + \xi_{n+1}.\end{equation}
Write also
\[
H_n = F(D_{m_n}) \subset \mathbb{H}.
\]
We continue this process until $n_0$, the first time $n$ such
that
\[ \operatorname{diam} \left[F(K_{m_n})
\right] \geq 2 \;\;\;\;\;\; \mbox{ or }
\;\;\;\;\;\; \operatorname{hcap} \left[ F(K_{m_n})
\right]\geq 2 . \]
Note that $n_0 -1 \leq 1/h$ and that for $n < n_0$,
\[ \operatorname{hcap} \left[F(D_0 \setminus D_{n})
\right] \leq 2, \;\;\;\;|X_n| \leq 2, \]
\[ |(F_{m_n}^{-1})'(10i)| \asymp |(F^{-1})'(10i)|
= N,\]
Using the Beurling estimate, we can see that
for $n < n_0$, the mesoscopic increments satisfy
\[ t_{m_n} -t_{m_{n-1}} \leq h + O(N^{-1}),\;\;\;\;
r_{m_n}-r_{m_{n-1}} \leq h^{2/5} + O(N^{-1/2}).\]
Let ${\cal F}_n$ denote the $\sigma$-algebra
generated by $(A_0,a_0,b),\cdots,
(A_{m_n},a_{m_n},b)$.
\begin{lemma}\label{lem:coupling-pt2}
There is a coupling of the LERW $\eta$ and a standard Brownian motion $(W_t, \mathcal{\tilde{F}}_t)$ and
a sequence of strictly increasing stopping times $\{\tau_n\}$ for $(W_t, \mathcal{\tilde{F}}_t)$
such that the following holds. Let $n_*$ be the minimum of $n_0 + 1$ and the
smallest $n'$ such that one of the following does not hold:
\[
\max_{n \leq n'}|\tau_n-nh| \leq h^{1/5} ,
\]
\[
\max_{n \le n'}|W_{\tau_n}-U_n| \leq h^{1/10} ,
\]
\[ \max_{n \le n'} \max_{\tau_{n-1} \leq t \leq \tau_n}
|W_{t} - W_{\tau_{n-1}}| \leq h^{2/5} ,
\]
\[ \max_{t \leq \tau_{n'}}\;\;
\max_{t - h^{1/5} \leq s \leq t}\;\;
|W_t - W_s| \leq h^{1/12} .
\]
Then $\mathbf{P}\{n_* \leq n_0\} = O(h^{1/10})$.
Moreover, if $\G_{n}$ denotes the $\sigma$-algebra generated by $\hat {\cal F}_{n}$
and ${\cal F}_{\tau_{n}}$, then $t \mapsto W_{t + \tau_n}-
W_{\tau_n}$ is independent of $\G_{n}$ and
the distribution of the LERW given
$\G_n$ is the same as the distribution given
$\hat {\cal F}_n$.
\end{lemma}
Given the Brownian motion $W_t$, there is a corresponding SLE$_2$ Loewner chain $(g_t^\text{\tiny SLE})$ obtained by solving the Loewner differential equation with $W_t$ as driving term. The Loewner chain is generated by an SLE$_2$ path in
${\mathbb H}$ that we denote by $\gamma(t)$. Let $\hat \gamma(t)
= \hat F^{-1}[\gamma(t)]$ which is an SLE$_2$ path
from $\hat a$ to $\hat b$ in $\hat D_A$
parametrized by capacity in ${\mathbb H}$ (this parametrization depends
on $F$ but we have fixed $F$.)
We write
\[
F^{\text{\tiny SLE}}_n(z) = (g_{\tau_n}^\text{\tiny SLE} \circ F)(z)-W_{\tau_n}
\]
and
\[
F^{\text{\tiny LERW}}_n(z) = (g_n \circ F)(z) - U_n.
\]
Combining the coupling with the deterministic
estimates we get the following.
\begin{lemma}\label{lem:coupling-of-maps}
If $n < n_*$, we have uniformly in $\zeta \in A$ such that $\Im F_n^\text{\tiny SLE}(\zeta) \ge h^{1/80}$,
\[
\left| F_n^{\text{\tiny LERW}}(\zeta) - F_n^{\text{\tiny SLE}}(\zeta)\right| \le c h^{1/15}.
\]
\end{lemma}
\section{Core argument}
\subsection{Setup}
At this point we will quickly review our setup.
\begin{itemize}
\item{We start with an analytic domain $D$ containing
the origin and with distinct boundary points $a',b'$. For each integer $N>0$ we define $( A ,a ,b )$ (and the associated union of squares domain $D_{A }$) as
the discrete approximations of $(ND,Na',Nb')$ with a choice of conformal transformation
$ F: D_{A} \rightarrow {\mathbb H} $
with $ F(a) = 0, F(b ) = \infty, \Im[F(0)] = 1 + o(1)$. All constants, implicit
or explicit, may depend on $D,a',b', F$, but are otherwise
universal.}
\item Let \[h=h_N=N^{-2u/3}, \quad n_0 = n_{0,N} = \lfloor h^{-1}\rfloor,\]
be the mesoscopic scale, where $u$ is the exponent
in \eqref{BLV1}.
\item Let $T_n = c_*^{-1} \left(m_1 + \cdots + m_n \right)$ denote
the number of steps of the LERW taken after $n$
mesoscopic steps, see Section~\ref{sect:deterministic}, rescaled by the constant in \eqref{nov3.1}.
\item The scaled LERW $\eta(t), 0 \leq t \leq 1$,
in $\check D$
parametrized by capacity is given by
\[ \check \eta(nh) = N^{-1} \eta_{c_{*}T_n}.\]
We choose the parametrization
to linearly interpolate between times $(n-1)h$ and $nh$.
\item As in Section~\ref{sect:coupling} we couple the LERW with an SLE$_2$ path
from $a'$ to $b'$
in $ D_A$, denoted $\hat{\gamma}$, parameterized so
that
\[ \operatorname{hcap} \left( F \circ \hat{\gamma}[0,t]\right) = t .\]
We let $ \check \gamma(t) = N^{-1} \, \hat{\gamma}(t)$ be the
corresponding SLE$_2$ in $\check D = \check{D}_A$.
\item Let
\[ \check T_{nh} = N^{-5/4} \, T_n \]
be the rescaled number of steps in the walk with $\check{T}_{t}$ defined by linear interpolation between times $(n-1)h$ and $nh$. Let
\[ \check \Theta_t = \Cont \left(\check \gamma[0,t] \right) \]
be the $5/4$-dimensional Minkowski content of $\check \gamma[0,t]$.
\end{itemize}
We can now state the main result.
\begin{thm}\label{thm:main-thm-core-sec} There exists $\epsilon_N \rightarrow 0$
such that
except for an event of probability
at most $ \epsilon_N$,
\begin{equation} \label{feb15.1}
\max_{0 \leq t \leq 1} \left|\check \eta(t) - \check \gamma(t) \right|
\leq \epsilon_N ,
\end{equation}
\begin{equation} \label{feb15.2}
\max_{0 \leq t \leq 1} \left|\check T_t - \check{\Theta}_t \right|
\leq \epsilon_N.
\end{equation}
\end{thm}
We will prove the theorem with
\[\epsilon_N = c\, \left(\log N \right)^{-1/60}\] where the constant $c$ depends
on $D,a',b',F$.
The estimate \eqref{feb15.1} with an unspecified sequence $\epsilon_N$ and for a slightly different coupling was done in \cite{LSW04}.
The convergence rate in the coupling of \cite{LSW04} was estimated in \cite{BJK, JV}. In \cite{LV_lerw_chordal_note} we obtain a polynomial convergence rate for \eqref{feb15.1} in the case of the coupling used in this paper. In this paper, we will only worry about proving the second estimate
\eqref{feb15.2}.
We encourage the reader to recall the general idea of the proof as outlined in Section~\ref{sect:overview}.
For the remainder, we fix $N$ and a coupling as above.
Where we use $n$, we will assume that $n < n_*$ where
$n_*$ is as in
Lemma \ref{lem:coupling-pt2}.
\subsection{Maximal estimate}
We will need to know that neither the Minkowski content nor the scaled
number of steps visited by the loop-erased random walk can get large
on a small region. To make this precise, let $\mathcal{B}(z,\epsilon)$
denote the closed disk of radius $\epsilon$ about $z$ and
define
\begin{eqnarray*}
\maxsle& = & N^{-5/4} \, \sup_{z \in {\mathbb C}} \Cont\left[
\hat \gamma \cap \mathcal{B}(z,
N/\log N)\right] \\
& = & \sup_{z \in {\mathbb C}} \Cont\left[ {\check \gamma} \cap \mathcal{B}(z,1/\log N)\right].
\end{eqnarray*}
The LERW analogue is
\[ \maxlerw = N^{-5/4} \, \sup_{z \in {\mathbb C}}
\sum_{\zeta \in A \cap \mathcal{B}(z,N/\log N)}
1\{\zeta \in \eta\}.\]
\begin{prop} \label{maxprop}
There exists $c < \infty$ such that
\[ \mathbf{E}\left[\maxsletwo\right] + \mathbf{E}\left[\maxlerwtwo\right]
\leq c \, \left(\log N \right)^{-5/4}.\]
\end{prop}
\begin{proof}
The estimate for SLE was done in
\cite{LR} where a similar maximal estimate is a key step
for establishing H\"older continuity of the Minkowski content
with respect to capacity parametrization. In
Proposition \ref{prop:maximal-estimate} we
use a similar
argument for LERW after establishing
a bound
on the second moment for the number of steps of the walk.
\end{proof}
\subsection{Open and closed squares: definitions}\label{sect:open}
\begin{figure}[t]
\centering
\def0.8\columnwidth{0.95\columnwidth}
\input{Badpoints3.pdf_tex}
\caption{Left: The square is open. Middle: Closed square of Type I. The distance from the real line after uniformizing is still larger than $\lambda$, but the sine of the argument along the path has gotten small. Right: Closed square of Type II. The curve got close to the square and then escaped while the distance from the real line after uniformizing dropped below $\lambda$. } \label{fig:badpoints}
\end{figure}
Throughout this section we set
\[ \lambda = \lambda_N = h^{ 1/100}.\]
In the following definition recall that $ \Im[ F_n^\text{\tiny SLE}(\zeta)] $
is a decreasing function of $n$. See Figures~\ref{fig:badpoints} and \ref{fig:badpoints2}.
\begin{definition} \label{sleopendef}
We will say that the square $\Square_\zeta, \zeta \in A,$ is {\bf closed for
SLE} at step $n$ if either of $I$ or $II$ holds at
$t= \tau_n$, where:
\begin{itemize}
\item[\bf{I}:] We have \[
\lambda \leq \Im[ F_{ n}^\text{\tiny SLE}(\zeta)] \leq 10 ,\]
and
\[ \Sine_{{\mathbb H} \setminus \gamma_{\tau_n}}(F(\zeta);\gamma({\tau_n}),\infty)
\leq \frac{1}{\left(\log N \right)^{2/5}}.\]
\end{itemize}
\begin{itemize}
\item[\bf{II}:] We have
\[ \Im[F_n^\text{\tiny SLE}(\zeta)] < \lambda , \]
\[\operatorname{dist}( \zeta, \partial ( D_A \setminus \hat {\gamma}_t)) \leq \frac{N}{\left(\log N \right)^5}, \]
and
\[
| \zeta - \hat {\gamma}(t)| \geq \frac{N }{\log N}.\]
\end{itemize}
\end{definition}
\begin{definition}\label{lerwopendef}
We will say that the square $\Square_\zeta, \, \zeta \in A,$ is {\bf closed for
LERW} at step $n$ if either
of $I$ or $II$ holds at
$t= \tau_n$, where:
\begin{itemize}
\item[\bf{I}:] We have
\[
\lambda \leq \Im[ F_{{ n}}^{\text{\tiny SLE}}(\zeta)] \leq 10 ,\]
and $\Square_\zeta$ is closed for SLE.
\end{itemize}
\begin{itemize}
\item[\bf{II}:] We have
\[ \Im[ F_{ n}^{\text{\tiny SLE}}(\zeta)] < \lambda , \]
\[\operatorname{dist}( \zeta,\partial {D}_n) \leq \frac{N}{\left(\log N \right)^5}, \]
and \[ | \zeta - a_k|
\geq \frac{N}{\log N}.\]
\end{itemize}
\end{definition}
\begin{figure}[t]
\centering
\def0.8\columnwidth{0.65\columnwidth}
\input{goodbadinH22.pdf_tex}
\caption{The image of an open square near the boundary. If $z=F(\zeta)$ and $S(z)=\Sine_{{\mathbb H} \setminus \gamma_{\tau_n}}(z;\gamma({\tau_n}),\infty) \ge \left(\log N \right)^{-2/5}$ we have derivative estimates. The conclusion is that the distance to the boundary is $o(N/\left(\log N \right)^5)$ and so if the square is open it is at distance $O(\left(\log N \right)/N)$ of the tip.} \label{fig:badpoints2}
\end{figure}
In both cases, once a square is closed it stays closed forever.
A square is said to be open for SLE (for LERW) at step $n$ if it is not closed for SLE (for LERW) at step $n$.
We will write $\opensle_{n,\zeta}$ and $\openlerw_{n,\zeta}$ for the indicator functions
of the event that $\Square_\zeta$ is open for SLE and LERW, respectively.
Then we have the following properties:
\begin{itemize}
\item If $\Im[F_n^\text{\tiny SLE}(\zeta) ] \geq \lambda$, then
$\opensle_{n,\zeta} = \openlerw_{n,\zeta}$.
\item If $n \leq m$, and $\opensle_{n,\zeta} = 0$, then
$\opensle_{m,\zeta} = 0. $ If $\openlerw_{n,\zeta}
= 0$ then $\openlerw_{m,\zeta} = 0$.
\end{itemize}
The next observation is that a square $\Square_\z
|
eta$
cannot still be open if it both
far from the tip and the conformal map has a
small imaginary part.
The essential idea is that in order for the imaginary part to be small
but for the curve to not get close to the point, there must be a time
when the sine of the angle was small.
\begin{prop} \label{prop.feb13}
$\;$
\begin{itemize}
\item Suppose $\Square_ \zeta$ is open for SLE at step $n$. Then either $\Im[F_{n}^\text{\tiny SLE}(\zeta)]
\geq \lambda $ or $|\zeta-\hat\gamma(t)| \leq N/\left(\log N \right)$.
\item Suppose $\Square_ \zeta$ is open for LERW at step $n$. Then either $\Im[F_{n}^\text{\tiny SLE}(\zeta)]
\geq \lambda $ or $|\zeta - a_n| \leq N/\left(\log N \right)$.
\end{itemize}
\end{prop}
\begin{proof} Let $z = F(\zeta)$ and
suppose $\Im(z) \leq 20$ and let $\rho=\tau_k$ where $k$ is the first $n$
with $\Im[g_{\tau_n}(z)] \leq \lambda $. Then using Koebe's theorem, \[\operatorname{dist}\left(\zeta, \partial(D_A \setminus \hat\gamma_\rho) \right) \asymp N\frac{\lambda}{|g'_\rho(z)|}.\]If $\Square_\zeta$ is still open,
then \eqref{jan26.2} implies that
\[
\frac{\lambda}{|g'_\rho(z)|} \le c \lambda^{2 \nu^2}, \quad \nu = \left(\log N \right)^{-2/5}.
\]
Combining these estimates gives $\operatorname{dist}(\zeta,\partial ( D_A \setminus
\hat{\gamma}_\rho) )= o(N/\left(\log N \right)^5)$. A similar argument (using Lemma~\ref{lem:coupling-of-maps})
shows the same for the LERW.
\end{proof}
We can restate this as follows. Suppose $\zeta \in A$ with $\Im[F(\zeta)]
\geq \lambda$,
\begin{itemize}
\item The square $\Square_\zeta$ stays open until either the sine of the argument
gets too small or the imaginary part drops below $\lambda.$ We measure
the argument using the SLE path but by the coupling, since the imaginary part is at least $\lambda$, it is almost the same
as measuring using the LERW.
\item If the sine gets too small, $\Square_\zeta$ closes.
\item If the imaginary part of $F_n^\text{\tiny SLE}(\zeta)$
drops below $\lambda$ and $\Square_\zeta$ has
not closed, we know that $\zeta$ is within distance $N/\left(\log N \right)^{5}$
of the boundary.
\item The square now closes when the tip of the path
gets distance $N/\log N$ away
from $\zeta$. (This is defined separately for ``closed for SLE'' and
``closed for LERW''.) It is possible that the square
$\Square_\zeta$ will be visited before it is closed; indeed, this is the
``typical'' behavior if the path will visit $\Square_\zeta$.
\end{itemize}
We will work with contents restricted to open squares. Define
\[ I_\zeta = c_*^{-1}\,1\{\zeta \in \eta\},\]
and
\[ I_\zeta^\circ = c_*^{-1}\, 1\{\exists k \mbox{ such that } \eta_k = \zeta
\mbox{ and } \Square_\zeta \mbox{ is open for LERW at step } k-1\}. \]
Let
\[ T = \sum_{\zeta \in \eta } I_\zeta, \;\;\;\;
T_n = \sum_{\zeta \in \eta^n } I_\zeta, \]
\[ T^\circ = \sum_{\zeta \in \eta } I_\zeta^\circ , \quad T_n^\circ = \sum_{\zeta \in \eta^n } I_\zeta^\circ \]
denote the number of points and
number of open points visited by $\eta$ and $\eta^n$, respectively (both scaled by $c_*$).
Now we define the corresponding SLE quantities. For each $\zeta \in A$, let
\[j(\zeta) = \min\{n: \Square_\zeta {\text{ is closed for SLE at step }} n \} \] be the step at which $\Square_\zeta$ closes for SLE and let $\Theta^\circ_\zeta$
denote the $5/4$-dimensional Minkowski content of the path in $\Square_\zeta$ before closing,
\[ \Theta_\zeta =
\, \Cont\left[ \hat\gamma \cap
\Square_\zeta \right], \;\;\;\; \Theta^\circ_\zeta =
\, \Cont\left[ \hat\gamma[0,\tau_{j(\zeta)}] \cap
\Square_\zeta \right].\]
Then we set
\[ \Theta = \Cont[\hat\gamma] = \sum_{\zeta \in A} \Theta_\zeta, \;\;\;\;
\Theta_n = \Cont\left[\hat\gamma[0,\tau_n]\right], \]
\[ \Theta^\circ = \sum_{\zeta \in A} \Theta^\circ_\zeta, \quad \Theta^\circ_n =
\sum_{\zeta \in A} \Cont\left[\hat \gamma[0,\tau_{j(\zeta)} \wedge
\tau_n ] \cap
\Square_\zeta \right].\]
(There is some ambiguity in this notation. We write $\Theta_\zeta$ and
$\Theta_n$ and they mean different things whether or not the subscript
is a point in $\Z^2$ ($\zeta$) or a nonnegative integer ($j,k,m,n$).
We hope this will not cause confusion.)
\subsection{Proof of Theorem~\ref{thm:main-thm-core-sec}}\label{sect:main-proof}
The goal of this section is to prove the main result but we will leave proofs of some
facts for later sections. We will achieve this by proving the following statement.
\begin{prop} \label{main-prop}
There exists $c$ such that for $N$ sufficiently large,
\begin{equation} \label{feb18.5}
\mathbf{P}\left\{\max_{0 \leq n \leq n_*} N^{-5/4} \, |T_n - \Theta_n|
\geq c\, \left(\log N \right)^{-1/60} \right\}
\leq c\, \left(\log N \right)^{-1/30}. \end{equation}
\end{prop}
We will argue that we can replace $T_n$ and $\Theta_n$ by $T_n^\circ$ and $\Theta_n^\circ$ as defined in the previous section.
In this section stopping times and martingales will be discrete time
with respect to the filtration $\{\G_n\}$ of the coupling.
Note that
\[ \mathbf{E}\left[T^\circ \mid \G_n\right]
= T_n^\circ + R_n^\circ, \;\;\;\;\;
\mbox{ where } R_n^\circ=
\sum_{\zeta \in A_n} \openlerw_{n,\zeta}\,
\mathbf{E}_n \left[I_\zeta^\circ\right], \]
where we write
\[ \mathbf{E}_n \left[I_\zeta^\circ\right] =
\mathbf{E}_{A_n,a_n,b} \left[I_\zeta^\circ \right].\]
In particular, $T_n^\circ + R_n^\circ$ is a martingale.
The corresponding SLE martingale is
\begin{equation} \label{feb18.1}
\mathbf{E}\left[\Theta^\circ \mid \mathcal{G}_n \right] = \Theta^\circ_n + \sum_{\zeta \in A}
\opensle_{n,\zeta} \,\mathbf{E}_n\left[\Theta^\circ_\zeta\right],
\end{equation}
where
$ \mathbf{E}_n\left[\Theta^\circ_\zeta\right]$
is the expected value
of $\Theta^\circ_\zeta$ with respect to SLE$_2$ from $\hat{\gamma}(\tau_n)$ to $b$ in
$D_A \setminus \hat\gamma_{\tau_n}$.
We consider the difference, which is also a martingale:
\[ N^{-5/4}\mathbf{E}\left[ \Theta^\circ - T^\circ
\mid \G_n\right] = Y_n^\circ + \tilde{B}_n^\circ , \]
where
\[ Y_n^\circ = N^{-5/4}\sum_{\zeta \in A}\left(\mathbf{E}_n \left[I_\zeta^\circ \right] - \mathbf{E}_n\left[\Theta^\circ_\zeta\right] \right)
, \;\;\;\;\;\;
\tilde{B}_n^\circ = N^{-5/4} \left[\Theta_n^\circ - \, T_n^\circ \right].\]
It turns out to be convenient to modify this and replace $\tilde{B}_n^\circ$ by a predictable (i.e., $\mathcal{G}_{n-1}$-measurable) version. For this we set
\[
B_n^\circ = \sum_{j=1}^n\mathbf{E} \left[\tilde{B}_j^\circ-\tilde{B}_{j-1}^\circ \mid \mathcal{G}_{j-1} \right].
\]
and define the martingale
\[
M_n^\circ = Y_n^\circ + B_n^\circ.
\]
The next lemma whose proof we delay shows that it suffices to
prove \eqref{feb18.5} with $B_n^\circ$ in place of $N^{-5/4} \, (\Theta_n -
T_n)$.
\begin{lemma} \label{bvlemma}
There exists $c < \infty$ such that
\[ \mathbf{P}\left\{ \max_{n \leq n_0} \left|B_n^\circ - N^{-5/4}\, (\Theta_n -
T_n) \right| \geq c \, \left(\log N \right)^{-5/128}
\right\} \leq c \, \left(\log N \right)^{-5/32}.\]
\end{lemma}
\begin{proof} See Section~\ref{bvsec}, and in particular Proposition \ref{bvprop}.
\end{proof}
Given this, the strategy is to apply the following general lemma to the martingale $M_n^\circ = Y_k^\circ + B_k^\circ$ with $\epsilon, \delta$ being chosen as suitable negative exponents of $\log N$.
\begin{lemma}\label{lem:l2-lemma} Suppose $B_k, {M}_k$ are discrete time processes
with $M_k$ a square-integrable
martingale with respect to a filtration $\{{\cal F}_k\}$
with $M_0 = 0$. Assume
that $B_k = X_k - Z_k$ where $X_k,Z_k$ are positive increasing
predictable (that is, $X_k,Z_k$ are ${\cal F}_{k-1}$-measurable)
processes with $X_0 = Z_0 = 0$. Let $Y_k = M_k - B_k$.
Suppose that $\tau$ is a stopping time such that
\[ \mathbf{E}[X_\tau + Z_\tau ] \leq c_1, \]
and
\[ |Y_j| \leq \epsilon, \quad |B_{j+1} - B_{j}| \le \epsilon, \quad \quad j < \tau.\]
Then for every $y > 0$,
\[ \mathbf{P}\left\{\max_{0 \leq j \leq k \wedge \tau}
|B_j| \geq y +2\epsilon \right\}
\leq
y^{-2} \, \left(\mathbf{E} \left[ Y_{k \wedge \tau}^2 \right]
+ 3 \, \epsilon \, c_1\right). \]
\end{lemma}
\begin{proof}See the end of the section.
\end{proof}
With this lemma in mind we see that we need to find a stopping time $\tau$ for which it holds that $\max_{n < \tau}|Y_n^\circ|$, $\max_{n < \tau}|B_{n}^\circ-B_{n-1}^\circ|$, and $\mathbf{E}\left[|Y_\tau^\circ|^2 \right]$ are all small. We will define the stopping time in terms of an estimate of $|Y_n^\circ|$, which we will now derive. For a fixed $n$, let
\[
S_n(\zeta) = \sin\left[\arg F_n^{\text{\tiny SLE}}(\zeta) \right]
\]
and then
\begin{align*} A_n' & = \left\{\zeta \in A:\, \Im\left[F_n^\text{\tiny SLE}(\zeta)\right]
\geq \lambda\, ; \,S_n(\zeta) \geq \left(\log N \right)^{-3/8} \right\},\\
A_n'' &= \left\{\zeta \in A:\, \Im\left[F_n^\text{\tiny SLE}(\zeta)\right]
\geq \lambda\, ; \,S_n(\zeta) < \left(\log N \right)^{-3/8} \right\}. \end{align*}
The choice of $3/8$ is somewhat arbitrary and we have not optimized it.
We will use the fact that
$\frac 13 <\frac 38
<\frac 25$.
We write
\[ \mathbf{E}\left[T^o \mid \G_n \right] = T_n^\circ+ \sum_{\zeta \in A_n'
\cup A_n''} \opensle_{n,\zeta} \, \mathbf{E}_n\left[I_\zeta^o \right]
+ \sum_{\zeta \in A_n \setminus( A_n' \cup A_n'')}
\openlerw_{n,\zeta} \, \mathbf{E}_n\left[I_\zeta^o \right],\]
\[ \mathbf{E} \left[\Theta^\circ\mid \G_n \right] = \Theta_n^\circ +
\sum_{\zeta \in A_n'
\cup A_n'' } \opensle_{n,\zeta} \,\mathbf{E}_n\left[\Theta_\zeta^\circ \right]
+ \sum_{\zeta \in A_n \setminus ( A_n' \cup A_n'')}
\opensle_{n,\zeta} \, \mathbf{E}_n \left[\Theta_\zeta^\circ \right]. \]
Here we are using the fact that $\opensle_{n,\zeta}
= \openlerw_{n,\zeta}$ in $A' \cup A''$.
Since
\[Y_n^\circ = N^{-5/4}\sum_{\zeta \in A}\left(\mathbf{E}_n \left[I_\zeta^\circ \right] - \mathbf{E}_n\left[\Theta^\circ_\zeta\right] \right) \]
we can estimate
\begin{equation} \label{feb23.18}
|Y_n^\circ| \leq | Y_n'|
+ Q_n + \tilde Q_n,
\end{equation}
where
\[ Y_n' = N^{-5/4} \sum_{\zeta \in A_n'} \left(
\mathbf{E}_n[I_\zeta^\circ] - \mathbf{E}_n[\Theta_\zeta^\circ]\right), \]
\[ Q_{n} = N^{-5/4} \sum_{\zeta \in A_n''} \opensle_{n,\zeta} \, \left(
\mathbf{E}_n[I_\zeta^\circ] + \mathbf{E}_n[\Theta_\zeta^\circ]\right), \]
\[ \tilde Q_n = N^{-5/4} \sum_{\zeta \in A \setminus (A_n' \cup A_n'')}
\left( \openlerw_{n,\zeta}\,
\mathbf{E}_n[I_\zeta^\circ] +
\opensle_{n,\zeta} \, \mathbf{E}_n[\Theta_\zeta^\circ]\right). \]
We can then describe the stopping time as follows. Let $n_1$ be the minimum
of $n_*$ and the first $n$ such that either
\[ Q_n \geq \frac 12 \, \left(\log N \right)^{-1/30}\]
or
\[ \mathbf{E}\left[\maxsle + \maxlerw \mid \G_n \right]
\geq \left(\log N \right)^{-1/2}. \]
\begin{lemma}\label{lem:n_1}
We have
\begin{equation}\label{feb26.1}
\mathbf{P}\left\{n_1 < n_* \right\} = o\left(\left(\log N \right)^{-1/30}\right),
\end{equation}
\begin{equation}\label{feb23.192}
Q_{n} \le \left(\log N \right)^{-1/30},\quad n \le n_1,
\end{equation}
\begin{equation}\label{feb23.212}
\tilde{Q}_n \le \left(\log N \right)^{-1/2},\quad n< n_1,\end{equation}
and \begin{equation}\label{feb23.213}
\mathbf{E}\left[\tilde{Q}^2_{n_1} \right] = O\left(\left(\log N \right)^{-5/4}\right).
\end{equation}
\end{lemma}
\begin{proof}Write $S_n(\zeta) = \sin\left[ \arg F_n^{\text{\tiny SLE}}(\zeta)\right]$.
Note that if $n \leq n_1$, and $\zeta \in A_n''$, then deterministically
(for $N$ sufficiently large) \[S_{n-1}(\zeta) < 2\,\left(\log N \right)^{-3/8}, \quad \Im[F_{n-1}^\text{\tiny SLE}(\zeta)] \geq \lambda.\]
We shall prove in Proposition~\ref{feb23.prop1} that this gives \eqref{feb23.192}.
On the other hand, as we will see, Proposition \ref{feb23.prop1} also shows that for any stopping time $\tau$ we have the estimate
$\mathbf{E}[Q_\tau] \leq O \left(\left(\log N \right)^{-1/8} \right)$, and hence
\begin{equation} \label{feb23.10}
\mathbf{P}\{Q_{n_1} \geq \left(\log N \right)^{-1/16} \}\leq c \, \left(\log N \right)^{-1/16}.
\end{equation}
Using Proposition \ref{prop.feb13}, we see that for any
stopping time $n \leq n_0$,
\[ \tilde Q_n \leq \mathbf{E} \left[\maxsle \mid \G_n \right] + \mathbf{E}\left[\maxlerw \mid \G_n \right]\]
so we get \eqref{feb23.212}. Using Proposition \ref{maxprop} we see that
\[ \mathbf{E}\left[\mathbf{E}(\maxsle \mid \G_n)^2 \right]
\leq \mathbf{E}\left[\mathbf{E}(\maxsletwo \mid \G_n) \right]
\leq \mathbf{E}(\maxsletwo) \leq c\, {\left(\log N \right)^{-5/4}}, \]
and similarly for $\mathbf{E}\left[\mathbf{E}(\maxlerw \mid \G_n)^2 \right]$.
Hence \eqref{feb23.213} follows.
Also, using Chebyshev's inequality,
\begin{equation} \label{feb23.11}
\mathbf{P} \left\{\mathbf{E}(\maxsle+
\maxlerw \mid \G_n) \geq \left(\log N \right)^{-1/2} \right\}
\leq c \, \left(\log N \right)^{-1/4}.
\end{equation}
Combining \eqref{feb23.10} and \eqref{feb23.11}, we get \eqref{feb26.1}. \end{proof}
It remains to handle the main term, $Y_n'$.
\begin{lemma}
There is a constant $c< \infty$ such that if $n_1$ is as above, then
\begin{equation} \label{feb23.22}
\mathbf{E}\left[ (Y_{n_1}')^2 \right] \leq c\, \left(\log N \right)^{-1/4}.
\end{equation}
\end{lemma}
\begin{proof}
Suppose $n \le n_1$.
We first use Lemma~\ref{lem:coupling-of-maps}
to see that if $\zeta \in A_n'$, then
\[ F_n^{\text{\tiny LERW}}(\zeta) = F_n^{\text{\tiny SLE}}(\zeta) \, \left[1 + O \left(h^{1/20} \right)
\right]. \]
Moreover, the Beurling estimate shows that (if $N$ is sufficiently large), $S_n(\zeta) \ge r_A(\zeta)^{-u}$ for all $\zeta \in A_n'$. Hence, from \eqref{BLV1}, integrating the Green's function over $\Square_\zeta$,
\[ \mathbf{E}_n \left[ I_\zeta \right] = \mathbf{E}_n\left[\Theta_\zeta\right]
\, \left[1 + O \left(h^{1/30} \right)
\right].\]
Note that all closed squares in $A_n'$ are of Type I. Therefore using Proposition~\ref{feb23.prop1}, we see that
\[ \mathbf{E}_n \left[ I_\zeta^\circ \right] = \mathbf{E}_n \left[ I_\zeta\right]
\, \left[1+O\left(\left(\log N \right)^{-1/8}\right)\right], \]\[
\mathbf{E}_n\left[ \Theta_\zeta^\circ \right]
= \mathbf{E}_n \left[\Theta_\zeta \right]
\, \left[1+O\left(\left(\log N \right)^{-1/8} \right)\right], \]
and hence
\[ \left|\mathbf{E}_n \left[ I_\zeta^\circ \right] - \mathbf{E}_n \left[ \Theta_\zeta^\circ \right]\right|
\leq c\, \left(\log N \right)^{-1/8}\, \mathbf{E}_n \left[ \Theta_\zeta^\circ \right]. \]
Since
\[
\sum_{\zeta \in A_n'}\mathbf{E}_n\left[ \Theta_\zeta^\circ\right] \le \left| \mathbf{E}_n \left[ \Theta_\infty \right] -\Theta_n \right|,
\]
after recalling that $Y_n'$ is rescaled, it follows that
\[ \mathbf{E}\left[(Y_n')^2 \right] \leq c\, \left(\log N \right)^{-1/4}\, N^{-5/2} \,
\mathbf{E}\left[ \left|\mathbf{E}_n\left[ \Theta_\infty - \Theta_n \right] \right|^2
\right].\]
However, as shown in \cite{LZhou}, if $\kappa < 8$, and $\check{D}_t = \check{D} \setminus \check{\gamma}_t$, then for
any stopping time $\tau$,
\begin{align*}
\lefteqn{ N^{-5/2}\mathbf{E}\left[\mathbf{E} \left[\Theta_\infty - \Theta_\tau \mid \G_\tau \right]^2\right]}
\hspace{.5in}\\
& = \mathbf{E}\left[\int_{\check D_\tau \times \check D_\tau} G_{\check D_\tau}(z;\check \gamma(\tau),\check b)
\, G_{\check{D}_\tau} (w;\check \gamma(\tau),\check b) \, dA(w)\, dA(z) \right]\\
& \leq
c\, \mathbf{E}\left[\int_{\check D_\tau \times \check D_\tau} G_{\check D_\tau}(z,w;
\check \gamma(\tau),\check b) \, dA(w)\, dA(z) \right]\\
& = c\, \mathbf{E}\left[\int_{\check D_\tau \times \check D_\tau} \mathbf{E}\left[G_{\check D_\tau}(z,
w;\check \gamma(\tau),\check b)\mid \mathcal{G}_{\tau}\right] \, dA(w)\, dA(z) \right]\\
& \leq c\, \int_{\check D \times \check D } G_{\check D}(z,w) \,
dA(w)\, dA(z) < \infty.
\end{align*}
Here $G_{\check D_t}(z,w;\check\gamma(t), \check b)$ denotes the (unordered) two-point
SLE$_\kappa$ Green's function which is a positive supermartingale justifying
the last equality. The first inequality is a general estimate
about the two-point Green's function. The conclusion is that we have proved \eqref{feb23.22}. \end{proof}
\begin{proof}[Proof of Proposition~\ref{main-prop}]
Combining \eqref{feb23.18},
\eqref{feb23.192}, \eqref{feb23.212},
and \eqref{feb23.22}, we see that
\[ \mathbf{E}\left[Y_{n_1}^2\right] \leq c \, \left(\log N \right)^{-1/15}.\]
Proposition~\ref{main-prop} then follows from Lemma~\ref{lem:l2-lemma} using
\[ \epsilon = \left(\log N \right)^{-1/15} , \;\;\;\;
y = \left(\log N \right)^{-1/60},\]
to get
\[ \mathbf{P}\left \{\max_{0 \leq j \leq n_1}
|B_j| \geq 3\, \left(\log N \right)^{-1/60} \right\}
\leq c\, \left(\log N \right)^{-1/30}.\]
\end{proof}
It remains to prove Lemma~\ref{lem:l2-lemma}.
\begin{proof}[Proof of Lemma~\ref{lem:l2-lemma}]
We write $\Delta Y_j = Y_j - Y_{j-1},\,
\Delta B_j = B_{j} - B_{j-1},$ and $\Delta M_j = M_j - M_{j-1}$.
Using the assumptions that $B_k$ is ${\cal F}_{k-1}$-measurable
and $M_k$ is a martingale, we get
\begin{align*}
\lefteqn{\mathbf{E}[Y_{k \wedge \tau}^2 \mid {\cal F}_{k-1}] - Y_{(k-1) \wedge \tau}^2}
\hspace{.4in}\\
& = 1\{\tau > n-1\} \, \left(2 Y_{k-1} \,\mathbf{E}\left[ \Delta Y_k
\mid {\cal F}_{k-1}\right] + \mathbf{E} \left[(\Delta Y_k)^2 \mid
{\cal F}_{k-1} \right] \right)\\
& = 1\{\tau > k-1\} \, \left(2 \,Y_{k-1} \, \Delta B_k
+ (\Delta B_k)^2 + \mathbf{E}\left[(\Delta M_k)^2 \mid {\cal F}_{k-1} \right] \right)
\end{align*}
By taking expectations of both sides and adding we see that
\begin{align*}
\mathbf{E}[Y_{k \wedge \tau}^2]
& = \mathbf{E} \left[M_{k \wedge \tau}^2 \right]\\
& \hspace{6ex}+
2\sum_{j=1}^k \mathbf{E} \left[Y_{j-1} \, \Delta B_j
; \tau > j-1 \right] +\sum_{j=1}^k \mathbf{E} \left[(\Delta B_j)^2
; \tau > j-1 \right]\\
& \geq
\mathbf{E}\left[M_{k \wedge \tau}^2 \right]\\
& \hspace{6ex} -
2\epsilon \sum_{j=1}^n \mathbf{E} \left[ |\Delta B_j|
; \tau > j-1 \right] -\epsilon \sum_{j=1}^n \mathbf{E} \left[|\Delta B_j|
; \tau > j-1 \right]\\
& \geq \mathbf{E}\left[M_{k \wedge \tau}^2 \right] -
3 \epsilon \, \mathbf{E} \left[X_\tau + Z_\tau \right ] .
\end{align*}
Therefore,
\[ \mathbf{E}\left[M_{k \wedge \tau}^2 \right]
\leq \mathbf{E}\left[Y_{k \wedge \tau}^2 \right] + 3 \epsilon \,
\mathbf{E} \left[X_\tau + Z_\tau \right] \leq \mathbf{E}\left[Y_{k \wedge \tau}^2 \right]
+ 3 \, \epsilon \, c_1. \]
Hence by the $L^2$ maximal principle,
\[ \mathbf{P}\left\{\max_{0 \leq j \leq k \wedge \tau}
|M_j| \geq y \right\} \leq y^{-2} \, \left( \mathbf{E}\left[Y_{k \wedge \tau}^2 \right]
+ 3 \, \epsilon \, c_1 \right). \]
Hence, recalling that $|B_j| = |M_j - Y_j|$, and $|Y_j|1_{j < \tau} \le \epsilon 1_{j < \tau}$,
\begin{align*}
\mathbf{P}\left\{\max_{0 \leq j \leq k \wedge \tau}
|B_j| \geq y +2\epsilon \right\}
& \leq \mathbf{P}\left\{\max_{0 \leq j < k \wedge \tau}
|B_j| \geq y + \epsilon \right\}\\
& \leq \mathbf{P}\left\{\max_{0 \leq j <k \wedge \tau}
\left( |M_j| + |Y_j| \right) \geq y + \epsilon \right\} \\
& \leq \mathbf{P}\left\{\max_{0 \leq j <k \wedge \tau}
|M_j| \geq y \right\} \\
& \le
y^{-2} \, \left(\mathbf{E}\left[Y_{k \wedge \tau}^2 \right]
+ 3 c_1 \epsilon \right),
\end{align*}
which is what we wanted to prove.
\end{proof}
\section{Open and closed squares: estimates} \label{badsec}
\subsection{Expected number of visits in closed squares}
In this subsection, we will show that the expected contribution
to the natural time for squares that are closed goes to zero
by proving the following.
\begin{prop} \label{feb21.1}
There exists $c < \infty$ such that
\[ \mathbf{E}[\Theta - \Theta^\circ]
+ \mathbf{E}[T - T^\circ] \leq c \, \left(\log N \right)^{-1/5}
\, N^{5/4}.\]
\end{prop}
Before giving the proof we need several lemmas.
Recall that
\[ \Theta - \Theta^\circ= \sum_{\zeta \in A}
\mathbf{E}[\Theta_\zeta - \Theta^\circ_\zeta], \;\;\;\;
\mathbf{E
|
tt{push} sequence have been compiled, the AST stack will be a collection of type-safe ASTs that can be evaluated as programs. To select the single AST associated with the compiled genome, the top-most AST with a data type matching the target program's return type is selected.
Figure \ref{fig:neg-to-zero-genome} shows one possible genome that compiles to the solution of the negative-to-zero problem shown in Figure \ref{fig:neg-to-zero}. The slice of genome between the first pair of \texttt{OPEN} and \texttt{CLOSE} genes will be saved as chunk that will compile into a function body when the \texttt{ABS[Int]} stack instruction is processed. The first \texttt{LocalVar(1)} gene will resolve to the argument of the anonymous function created by \texttt{ABS[Int]}. The AST created by \texttt{ABS[Int]} will become the definition of the local variable produced by the \texttt{LET} at the end of the genome. The slice of genome between the second pair of \texttt{OPEN} and \texttt{CLOSE} genes will be saved as a chunk that is compiled into the body of the \texttt{LET} expression, and the second \texttt{LocalVar(1)} will resolve to the local variable defined by the \texttt{LET}. Notice that the \texttt{LET} expression is the root node of the AST (and the root symbol of the Clojure code in Figure \ref{fig:neg-to-zero-code}), therefore it is the last gene in the genome.
The genome in Figure \ref{fig:neg-to-zero-genome} is artificially simple for demonstration purposes. During evolution, most genomes are much longer and contain genes that either noop or build additional ASTs which do not get selected as the program because they do not return the correct data type for the problem or are buried deep in the stack at the end of compilation.
\subsection{Evolution}
For this work, a standard generational genetic algorithm was used to evolve programs. An initial population of random genomes was produced using the method described in Section \ref{sec:genomes}. Genomes are compiled into a type-safe ASTs (Section \ref{sec:compilation}) which are executed identically to a native function in the host language. These programs are evaluated based on a set of training cases in the form of input-output pairs.
The program's error on each training case is determined by a user provided error function. The collection of errors across all training cases is referred to as an individual's ``error vector''. If no AST with the problem's target return type is produced after compilation the individual is given a penalty error on every training case. If the compiled AST produces a runtime error, such as ``index out of bounds,'' when called on a training case, it is given a penalty error.
Parents are selected from the population of evaluated individuals on the basis of error vectors using Lexicase Selection~\cite{Helmuth:2015:UncompromisingLexicase, Helmuth:2019:LexicaseSpecialists}. The next generation of genomes is produced through variation of parent genomes.
If a individual is found to have an error of zero on all training cases, or if the maximum number of generations is reached, evolution is stopped and the individual with the lowest total error, given by the sum of its error vector, is returned. If this individual has a total error of zero, it is called a ``solution.''
\subsection{Simplification}
The best individual found during evolution is extracted for simplification. It has been shown that simplification acts as a form of regularization which improves the program's generalizability to unseen data cases~\cite{Helmuth:2017:simplification}. In addition, a simplified program may be easier for a human to understand. The best individual from evolution is simplified using a hill-climbing algorithm, as follows:
\begin{enumerate}
\item Create a new genome using an order-preserving random subset of the best individual's genome.
\item Compile and evaluate the new genome to create a new individual.
\item If the total error of the new individual is equal to, or lower than, the current best individual it replaces the best individual.
\item If iteration limit is reached, return best individual. Otherwise, return to step 1.
\end{enumerate}
The best individual's program after all iterations of simplification is reported as the output of the evolutionary search. For our experiments, this is the program that is tested for generalization on an unseen set of test cases.
\section{Experimental Design}
\label{sec:experiment}
We assess the ability of CBGP to perform automatic program synthesis using a subset of 14 problems from the program synthesis benchmark suite PSB1~\cite{Helmuth:2015:BenchmarkSuite}. The problems in PSB1 originate from introductory computer science textbooks, allowing us to assess how the system performs on the types of programming problems we ask new programmers to solve. We chose 14 problems that represent a wide range of requirements, such as data types and control flow, and difficulties. We purposefully avoided some problems that have typically been most difficult for other GP techniques and we have not yet benchmarked CBGP using the more recent (and difficult) problems of the suite PSB2~\cite{Helmuth:2021:GECCO:PSB2}, since this is CBGP's first benchmarking (and easier problems seem warranted) and because there is more data available for comparing CBGP with other program synthesis systems. However, assessing performance on PSB2 soon would supplement our experiments here.
Each problem is specified by a set of input/output examples defining the desired behavior of the program, in the form of supervised learning. Each run uses 100 training cases composed of hand-coded examples and a subset of a large set of randomly-generated inputs. Additionally, we use a set of 300 additional random examples to test each program that passes the training set for generalization; only those programs that perfectly pass all 300 examples are reported as solutions. The error functions used to measure the differences between program outputs and correct outputs are the same ones described in PSB1~\cite{Helmuth:2015:BenchmarkSuite}.
\begin{table}[t]
\rowcolors{2}{gray!15}{white}
\centering
\begin{tabular}{l l}
\toprule
\textbf{Hyperparameter} & \textbf{Value} \\ \midrule
Population Size & 1000 \\
Max Generations & 300 \\
Parent Selection & Lexicase Selection~\cite{Helmuth:2015:UncompromisingLexicase} \\
Variation & UMAD~\cite{Helmuth:2018:GECCO:UMAD} \\
Mutation Rate & 0.1 \\
Simplification Steps & 2000 \\
Initial Genome Sizes & [50, 250] \\
Number of Training Cases & 100 \\
Number of Unseen Test Cases & 300 \\
\bottomrule
\end{tabular}
\caption{The evolutionary hyperparameters used for all runs of CBGP associated with the results presented in this paper.}
\label{fig:hyperparameters}
\end{table}
We conduct 100 runs of CBGP per problem, and primarily report the success rate for each problem as measured by generalizing solutions.
The generational genetic algorithm was configured with the hyperparameters given in Table \ref{fig:hyperparameters}. This configuration was selected due to its similarity to the configuration of PushGP in published results on the same set of benchmark problems. We leave the optimization of this configuration to future research.
With an aim to enhancing comparability between systems, we created a genetic source (function set) that largely matches that used in the comparison PushGP runs~\cite{Helmuth:2015:BenchmarkSuite, Helmuth:2018:GECCO:UMAD}. However, due to representational differences, there is not a one-to-one match between the genetic sources. The functions handling data operations are largely the same, but control flow is handled quite differently in CBGP with a functional host language compared to Push or the grammar-based programs of G3P and GE. In particular, much of the control flow in this implementation of CBGP is handled by higher-order functions that iterate over lists, such as \texttt{map}, \texttt{filter}, and \texttt{reduce}.
The description of the problems in PSB1 recommends not using every single available function for every problem~\cite{Helmuth:2015:BenchmarkSuite, Helmuth:2020:ALife:source}. For example, including functions that manipulate strings when solving a problem that only relates to lists of integers would expand the search space unnecessarily. As such, we follow these recommendations by creating type-tuned genetic sources for each problem in the fashion recommended by PSB1: for each problem, only include functions that manipulate the data types deemed relevant by PSB1. This ensures that we do not cherry-pick instructions known to be useful for a problem, while not including instructions that have no bearing on it.
\subsection{Comparison Methods}
We compare our CBGP results with those of other GP representations: PushGP, G3P, and GE. We choose comparison results from papers using comparable evolutionary hyperparameters as much as possible.
For PushGP, we use results from the paper introducing Uniform Mutation by Additions and Deletions (UMAD)~\cite{Helmuth:2018:GECCO:UMAD}. Like this paper, our CBGP runs use UMAD as the only genetic operator, making a reasonable comparison.
We use the paper introducing grammar design patterns as the results for G3P~\cite{Forstenlechner:2017:G3P}. The paper uses similarly type-tuned grammars to determine the instructions available to evolving programs.
Our reported GE results are taken from a paper exploring the use of domain knowledge and novelty in program synthesis~\cite{Hemberg:2019:DomainKnowledgeAndNoveltyImproveGE}. Since neither of those ideas are used in our work here, we use the baseline control results reported in the paper.
\section{Results}
\label{sec:results}
\begin{table*}[t]
\centering
\rowcolors{3}{gray!15}{white}
\begin{tabular}{l rrrr r}
\toprule
& \multicolumn{4}{l}{\textbf{Generalized Solution Rate}} & \multicolumn{1}{c}{\textbf{Generalization}} \\
\textbf{Problem} & \multicolumn{1}{l}{\textbf{CBGP}} & \multicolumn{1}{l}{\textbf{PushGP}} & \multicolumn{1}{l}{\textbf{G3P}} & \multicolumn{1}{l}{\textbf{GE}} & \multicolumn{1}{c}{\textbf{Rate (CGBP)}} \\
\midrule
smallest & 100 & 100 & {\ul 89} & 100 & 1.0 \\
mirror-image & 100 & 100 & {\ul 1} & {\ul 25} & 1.0 \\
number-io & 100 & 98 & 96 & 100 & 1.0 \\
vectors-summed & 100 & {\ul 11} & {\ul 85} & {\ul 1} & 1.0 \\
negative-to-zero & 99 & {\ul 80} & 98 & {\ul 32} & 1.0 \\
median & 98 & {\ul 55} & {\ul 65} & 99 & 1.0 \\
vector-average & 88 & 88 & {\ul 0} & {\ul 0} & 0.99 \\
compare-string-lengths & 22 & 32 & {\ul 3} & 30 & 0.79 \\
last-index-of-zero & 10 & \textbf{62} & \textbf{24} & 13 & 0.92 \\
replace-space-with-newline & 0 & \textbf{87} & 0 & \multicolumn{1
|
}{r}{-} & \multicolumn{1}{r}{-} \\
small-or-large & 0 & \textbf{7} & 5 & 0 & \multicolumn{1}{r}{-} \\
count-odds & 0 & \textbf{8} & \textbf{10} & 0 & \multicolumn{1}{r}{-} \\
digits & 0 & \textbf{19} & 0 & \textbf{70} & - \\
for-loop-index & 0 & 2 & \textbf{25} & 0 & - \\
\bottomrule
\end{tabular}
\caption{Percentage of runs that found a generalized solution on each problem. Underlined values indicate the comparison method has a statistically significantly worse solution rate than CBGP according to a chi-squared test with at a p-value of 0.05. Values in bold indicate a statistically significantly better success rate using the same test. The generalization rate column denotes the proportion of runs for which the program which solved all training cases also solved the unseen test data.}
\label{table:solution_rates}
\end{table*}
Table~\ref{table:solution_rates} compares the success rates of CBGP to other GP representations on the 14 benchmark problems. CBGP performs quite well on 7 of the problems, producing success rates near or at 100. On all of these problems except number-io, at least one of the other methods performs significantly worse than CBGP. On the other hand, CBGP performs significantly worse than at least one other method on the last 6 problems.
The last column in Table~\ref{table:solution_rates} gives the proportion of solutions on the training data that perfectly generalize to the unseen test set. Compared to the other three GP representations, which have typically produced low generalization rates on some, but not all, of these problems, the generalization rate of CBGP solutions is quite high across the board. For example, compare-string-lengths, last-index-of-zero, median, and negative-to-zero all produced generalization rates lower than 0.75, while almost no problem exhibited a generalization rate of 1.0, in a study of generalization using PushGP~\cite{Helmuth:2017:simplification}.
\begin{table}[t]
\centering
\caption{Solution sizes for each problem that CBGP solved. Min gives the minimum size of any solution program, while Pre and Post give the mean sizes before and after applying automatic simplification to the solution genomes.}
\label{table:sizes}
\rowcolors{2}{gray!15}{white}
\begin{tabular}{lrrr}
\toprule
\textbf{Problem} & \textbf{Min} & \textbf{Pre} & \textbf{Post} \\
\midrule
smallest & 7 & 7.55 & 7.18 \\
mirror-image & 4 & 4.54 & 4.06 \\
number-io & 4 & 4.92 & 4.03 \\
vectors-summed & 4 & 4.25 & 4.00 \\
negative-to-zero & 7 & 7.90 & 7.02 \\
median & 9 & 10.52 & 10.03 \\
vector-average & 7 & 9.74 & 8.89 \\
compare-string-lengths & 10 & 12.34 & 11.79 \\
last-index-of-zero & 8 & 12.42 & 10.33 \\
\bottomrule
\end{tabular}
\end{table}
Table~\ref{table:sizes} presents the sizes of solution programs found for each problem solved by CBGP. Program size is measured in number of nodes in the Clojure S-expression representation of the program, which is identical to the number of nodes in the AST. We find that three of these problems have been solved by programs containing only 4 nodes, while the remainder have been solved by programs with 10 or fewer nodes. Interestingly, the mean solution sizes pre- and post-simplification tend to be quite close to the minimum sizes. This means that evolved solutions rarely have unnecessary code in the programs themselves. Note that genomes, on the other hand, may have lots of unnecessary genes that either produce unused ASTs or have no effect on AST compilation. It seems that removing such unnecessary genes during simplification does not result in dramatically simpler programs, as the mean post-simplification program size is not much smaller than pre-simplification.
\subsection{Example Solution Programs}
The supplementary materials to this paper include a file containing every solution evolved by CBGP. In Figure \ref{fig:example-solutions} we give some examples of those solution programs and note some of their interesting features below.
The solution to negative-to-zero interestingly maps the subtract function over two copies of the input vector, which produces a vector entirely made of zeros. It then maps the \texttt{max} function over the zeros and the input vector, changing every negative integer into 0 as required.
The vector-average solution behaves as expected. One thing to note is that it converts the length of the input vector to a $Double$, since the \texttt{count} function is typed to return an $Int$. Future work into allowing for subtyping or type classes could allow for all $Int$ expressions to be considered valid $Double$ expressions, but for now, the conversion must happen explicitly.
The smallest problem requires the program to find the minimum of four inputs. Instead of simply applying the \texttt{min} function 3 times, this solution unnecessarily defines a new function that finds the \texttt{min} of \texttt{input4} and its argument, and then applies that function to \texttt{input1}.
The last-index-of-zero problem requires the program to find the last index where 0 appears in the input list. This solution reverses the input, finds the first index of zero, and then subtracts that from the decremented length of the input. This strategy is similar to some solutions to this problem that have been evolved in PushGP.
\begin{figure}
\begin{verbatim}
(defn negative-to-zero
[input1]
(map max (map - input1 input1) input1))
(defn vector-average
[input1]
(safe-div (reduce + input1)
(float (count input1))))
(defn smallest
[input1 input2 input3 input4]
(min (min input3
((fn [a-639347] (min input4 a-639347)) input1))
input2))
(defn last-index-of-zero
[input1]
(- (count (butlast input1))
(index-of (reverse input1) 0)))
\end{verbatim}
\caption{A sample of solution Clojure programs evolved by CBGP. Anonymous function argument symbols were generated using a unique natural number prefixed with an \texttt{a-}. Whitespace was adjusted for readability.}
\label{fig:example-solutions}
\end{figure}
\section{Discussion and Future Work}
CBGP has demonstrated that it can readily find solutions to some problems, but on others the solution rate of CBGP quickly drops to zero. These trends correlate somewhat to the problems found difficult by other GP representations; however, there are some problems that CBGP solves readily that others do not and vice versa.
When initially tested on the PSB1 benchmark problems, the other genetic programming systems saw similar trends, and have since increased performance as the methods mature through continued research. We hope to see a similar rise in the search performance of CBGP in the future.
The large variety in which problems each system finds easier or harder points to the importance of program representation for search performance. This area is not well understood in GP, and we hope that CBGP can help better illuminate important differences in representation. We suggest this as an area of future research such that we can understand what makes problems difficult under a given representation.
One hypothesis regarding different representations producing wildly different results on some problems is the impact of representations on the size of programs needed to form a particular computation. Solutions to problems with high solution rates tend to be smaller than solutions to problems with a low success rate, regardless of representation. The problems that CBGP solves most readily have small solution programs, and similar results have been shown for PushGP on the same problems~\cite{Helmuth:2015:BenchmarkSuite}. We do not know if similarly small solution programs are possible for the problems that CBGP did not solve, and it simply did not find them, or if they require larger programs and therefore are more difficult to find in the search space. Further research into CBGP solutions to these problems could help us understand whether it is simply the size of the solution programs preventing them from being solved, or whether CBGP has issues traversing the search space effectively regardless of solution size for some problems.
The exceptionally high generalization rate of CBGP is not easily explained. When considered in combination with the inability to find solutions on harder problems, this may be an indication that CBGP cannot fall back on memorizing or bloating the program into something that overfits the training data. In CBGP, an increase in genome size does not necessarily cause an increase in program size because additional genes may simply result in more ASTs being left on the stack after compilation, rather than larger ASTs. This hypothesis is further supported by the minimum and average program sizes of solutions found by CBG. The problems with high solution (and generalization) rates are solved by small programs.
When looking through solution programs, we found very few instances of programs that define and use anonymous functions effectively. Defining such functions is an integral part of functional programming for human programmers. Thus one piece of important future work is to try to assess why CBGP is not making use of function definition, and considering ways to encourage this behavior.
One insight from this research that may be helpful to the wider research fields of genetic programming and program synthesis is the value of introducing formalisms, such as type theory, into our systems. The body of work accumulated in fields of theoretical computation provide the program synthesis community with tools to guide synthesis towards programs with desirable properties, such as type safety.
Functional CBGP can, in theory, represent programs using any data type or language construct supported by the type system, and the unification algorithm in particular. A valuable direction of future research is to implement the common extensions to the Hindley-Milner type system which add support for function overloading, sub-types, and variadic functions~\cite{Smith:1992:TypeInferenceOverloadingSubtyping, Pottier:1998:TypeInferenceWithSubtyping, CARDELLI:1994:SystemFSubtyping, Dolan:2017:PolymorphismSubtypingTypeInference}. The primary benefit of these extensions would be the ability to represent programs using all the features of a modern functional programming language and possibly approach any programming task that can be well-specified by types. Another benefit of supporting additional kinds of polymorphism is the ability to use a smaller genetic source with considerably fewer, more general, functions which could dramatically reduce the search space and improve solution rates on complex problems.
\section{Conclusion}
In this paper we present functional Code Building Genetic Programming and show how it leverages type theory to ensure synthesized programs are type safe while also allowing polymorphic functions, anonymous functions, and higher order functions to be expressed. We report on empirical benchmarks that show CBGP can find solution programs more consistently than other contemporary GP methods on some problem, while it struggles to find any solutions on others. Investigations into solution programs show repeated use of polymorphic functions and higher order functions, but little use of anonymous function definitions.
When CBGP does find a solution on training data, we observe an exceedingly high rate of generalization to unseen test data. This phenomenon is in contrast to the comparatively low generalization rates of all other GP systems included in our comparison~\cite{Helmuth:2017:simplification,Forstenlechner:2017:G3P,Sobania:2021:EuroGP}. Furthermore, the solution programs found by CBGP are small, even without the use of typical genome simplification techniques.
Finally, we direct future research towards a deeper utilization of type theory in general program synthesis systems. We also suggest the genetic programming field perform broader studies into the impact of representation on problem difficulty.
|
\section{Introduction} \label{sec1}
Throughout $R$ is a commutative noetherian local ring.
Foxby \cite{f}, Vasconcelos \cite{v} and Golod \cite{g} independently initiated the study of
semidualizing modules.
A finite (i.e. finitely generated) $R$-module $C$ is called \emph{semidualizing} if the natural
homothety map $\chi_C^R: R \longrightarrow \mathrm{Hom}_R(C, C)$ is an isomorphism
and $\mathrm{Ext}^{\geqslant1}_R(C, C)=0$ ( see \cite[Definition 1.1]{hj} ).
Examples of semidualizing $R$-modules include $R$ itself and a dualizing $R$-module when one exists.
The set of all isomorphism classes of semidualizing $R$-modules is denoted by $\mathfrak{G}_0(R)$,
and the isomorphism class of a semidualizing $R$-module $C$ is denoted $[C]$.
The set $\mathfrak{G}_0(R)$ have caught attentions of several authors; see, for example \cite{fs-w2}, \cite{cs-w}, \cite{ns-w} and \cite{s-w1}.
In \cite{cs-w}, Christensen and Sather-Wagstaff show that $\mathfrak{G}_0(R)$ is finite when $R$ is Cohen-Macaulay and equicharacteristic.
Then Nasseh and Sather-Wagstaff, in \cite{ns-w}, settle the general assertion that
$\mathfrak{G}_0(R)$ is finite.
Also, in \cite{s-w1}, Sather-Wagstaff studies the cardinality of $\mathfrak{G}_0(R)$.
Each semidualizing $R$-module $C$ gives rise to a notion of reflexivity for finite $R$-modules.
For instance, each finite projective $R$-module is totally $C$-reflexive.
For semidualizing $R$-modules $C$ and $B$, we write $[C] \trianglelefteq [B]$ whenever
$B$ is totally $C$-reflexive.
In \cite{gerko}, Gerko defines chains in $\mathfrak{G}_0(R)$. A \emph{chain} in
$\mathfrak{G}_0(R)$ is a sequence
$[C_n]\trianglelefteq \cdots \trianglelefteq[C_1] \trianglelefteq [C_0]$,
and such a chain has length $n$ if $[C_i]\neq[C_j]$ whenever $i\neq j$.
In \cite{s-w1}, Sather-Wagstaff uses the length of chains in $\mathfrak{G}_0(R)$
to provide a lower bound for the cardinality of $\mathfrak{G}_0(R)$.
It is well-known that a Cohen-Macaulay ring which is homomorphic
image of a Gorenstein local ring, admits a dualizing module (see \cite[Theorem 3.9]{s} ).
Then Foxby \cite{f} and Reiten \cite{reiten}, independently, prove the converse.
Recently Jorgensen et.\! al. \cite{jls-w}, characterize the Cohen-Macaulay local rings which admit
dualizing modules and non-trivial semidualizing modules ( i.e. neither free nor dualizing ).
In this paper, we are interested in characterization of a Cohen-Macaulay ring $R$ which admits
a dualizing module and a certain chain in $\mathfrak{G}_0(R)$.
We prove that, when a Cohen-macaulay ring $R$ with dualizing module has a
\emph{suitable chain} in $\mathfrak{G}_0(R)$ (see Definition~\ref{suitable}) of length $n$,
then there exist a Gorenstein ring $Q$ and ideals $I_1, \cdots, I_n$ of $Q$ such that
$R\cong Q/(I_1+\cdots+I_n)$ and, for each $\Lambda\subseteq [n]=\{1, \cdots, n\}$,
the ring $Q/(\Sigma_{l\in \Lambda} I_l)$
has certain homological and cohomological properties (see Theorem~\ref{T42}).
Note that, this result gives the result of Jorgensen et.\! al. when $n=2$ and the result of
Foxby and Reiten in the case $n=1$.
We prove a partial converse of Theorem~\ref{T42} in Propositions \ref{P42} and \ref{P43}.
\section{Preliminaries} \label{sec2}
This section contains definitions and background material.
\begin{defn}\label{reflexive} (\cite[Definition 2.7]{hj} and \cite[Theorem 5.2.3 and Definition 6.1.2]{s-w2})
Let $C$ be a semidualizing $R$-module. A finite $R$-module $M$ is \emph{totally} $C$-\emph{reflexive}
when it satisfies the following conditions\,:
\begin{itemize}
\item[(i)] the natural homomorphism
$\delta^C_M: M \longrightarrow \mathrm{Hom}_R(\mathrm{Hom}_R(M, C), C)$
is an isomorphism, and
\item[(ii)]
$\mathrm{Ext}^{\geqslant1}_R(M, C)=0=\mathrm{Ext}^{\geqslant1}_R(\mathrm{Hom}_R(M, C), C)$.
\end{itemize}
A totally $R$-reflexive is referred to as totally reflexive.
The $\mbox{G}_C$-dimension of a finite $R$-module $M$, denoted $\mbox{G}_C$-$\mathrm{dim}_R(M)$,
is defined as
$\hspace{1cm}\mbox{G}_C$-$\mathrm{dim}_R(M)= \mathrm{inf}\left\{ \begin{array}{lll}
n\geqslant 0 & {\Bigg |} & {\small\begin{array}{l}
\mathrm{there\ is\ an\ exact\ sequence\ of\ } R\!-\!\mathrm{modules}\\ 0\rightarrow G_n\rightarrow \cdots\rightarrow G_1\rightarrow G_0\rightarrow M\rightarrow0\\ \mathrm{such\ that\ each\ } G_i\ \mathrm{is\ totally}\ C\!-\!\mathrm{reflexive}\end{array}}
\end{array}\right\}.$
\end{defn}
\begin{rem}\label{R1} \cite[Theorem 6.1]{c2}
Let $S$ be a Cohen-Macaulay local ring equipped with a module-finite local ring
homomorphism $\tau: R \rightarrow S$ such that $R$ is Cohen-Macaulay.
Assume that $C$ is a semidualizing $R$-module.
Then $\mbox{G}_C$-$\mathrm{dim}_R(S)<\infty$ if and
only if there exists an integer $g\geqslant0$ such that $\mathrm{Ext}^i_R(S, C) = 0$ for all $i$,
$i\neq g$, and $\mathrm{Ext}^g_R(S, C)$
is a semidualizing $S$-module; when these conditions hold, one has $g=\mbox{G}_C$-$\mathrm{dim}_R(S)$.
\end{rem}
\begin{f}\label{f2}
Define the order $\trianglelefteq$ on $\mathfrak{G}_0(R)$. For $[B], [C]\in \mathfrak{G}_0(R)$,
write $[C]\trianglelefteq[B]$ when $B$ is totally $C$-reflexive (see, e.g., \cite{s-w1}).
This relation is reflexive and antisymmetric \cite[Lemma 3.2]{fs-w1},
but it is not known whether it is transitive in general.
Also, write $[C]\vartriangleleft[B]$ when $[C]\trianglelefteq [B]$ and $[C]\neq[B]$.
For a semidualizing $C$, set
$$\mathfrak{G}_C(R)= \big\{[B]\in \mathfrak{G}_0(R)\ \big|\ [C]\trianglelefteq [B] \big\}.$$
In the case $D$ is a dualizing $R$-module, one has $[D]\trianglelefteq [B]$ for any semidualizing $R$-module $B$, by \cite[(V.2.1)]{h}, and so $\mathfrak{G}_D(R)= \mathfrak{G}_0(R)$.
If $[C]\trianglelefteq[B]$ then $\mathrm{Hom}_R(B, C)$ is a semidualizing and
$[C]\trianglelefteq[\mathrm{Hom}_R(B, C)]$ (\cite[Theorem 2.11]{c2}).
Moreover, if $A$ is another semidualizing $R$-module with $[C]\trianglelefteq[A]$, then $[B]\trianglelefteq[A]$
if and only if $[\mathrm{Hom}_R(A, C)]\trianglelefteq[\mathrm{Hom}_R(B, C)]$ (\cite[Proposition 3.9]{fs-w1}).
\end{f}
\begin{thm}\label{T01}\cite[Theorem 3.1]{gerko}
Let $B$ and $C$ be two semidualizing $R$-modules such that $[C]\trianglelefteq[B]$.
Assume that $M$ is an $R$-module which is both totally $B$-reflexive and totally $C$-reflexive,
then the composition map
$$\varphi : \mathrm{Hom}_R(M, B)\otimes_R\mathrm{Hom}_R(B, C) \longrightarrow
\mathrm{Hom}_R(M, C)$$
is an isomorphism.
\end{thm}
\begin{cor}\label{C01}\cite[Corollary 3.3]{gerko}
If $[C_n]\trianglelefteq \cdots \trianglelefteq[C_1] \trianglelefteq [C_0]$ is a chain in $\mathfrak{G}_0(R)$, then one gets
$$C_n\cong C_0 \otimes_R \mathrm{Hom}_R(C_0, C_1) \otimes_R \cdots \otimes_R\mathrm{Hom}_R(C_{n-1}, C_n).$$
\end{cor}
Assume that $[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$
is a chain in $\mathfrak{G}_0(R)$.
For each $i\in [n]$ set $B_i=\mathrm{Hom}_R(C_{i-1}, C_i)$.
For each sequence of integers $\mathbf{i} = \{i_1, \cdots , i_j\}$ with $j \geqslant 1$
and $1\leqslant i_1 < \cdots < i_j \leqslant n$, set
$B_{\mathbf{i}}=B_{i_1}\otimes_R\cdots \otimes_R B_{i_j}$.
( $B_{\{i_1\}} =B_{i_1}$ and set $B_\emptyset=C_0$.)
In order to facilitate the discussion, we list some results from \cite{s-w1}. We first recall the following definition.
\begin{defn}
Let $C$ be a semidualizing $R$-module.
The \emph{Auslander class} $\mathcal{A} _C(R)$ with respect to $C$
is the class of all $R$-modules $M$ satisfying the following conditions.
(1) The natural map $ \gamma^C_M: M \longrightarrow \mathrm{Hom}_R(C, C\otimes_R M)$ is an isomorphism.
(2) $\mathrm{Tor}^R_{\geqslant1}(C, M)=0=\mathrm{Ext}^{\geqslant1}_R(C, C\otimes_R M)$.
\end{defn}
\begin{pro}\label{P01}
Assume that $[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$
is a chain in $\mathfrak{G}_0(R)$ such that \\
$\mathfrak{G}_{C_1}(R)\subseteq\mathfrak{G}_{C_2}(R)\subseteq\cdots \subseteq \mathfrak{G}_{C_n}(R)$.
\begin{itemize}
\item[(1)] \cite[Lemma 4.3]{s-w1} For each $i, p$, $1\leqslant i\leqslant i+p\leqslant n$
$$B_{\{i, i+1,\cdots, i+p\}}\cong \mathrm{Hom}_R(C_{i-1}, C_{i+p}).$$
\item[(2)] \cite[Lemma 4.4]{s-w1} If $1\leqslant i< j-1 \leqslant n-1$, then
$$B_{\{i, j\}}\cong \mathrm{Hom}_R(\mathrm{Hom}_R(B_i, C_{j-1}), C_j).$$
\item[(3)] \cite[Lemma 4.5]{s-w1} For each sequence $\mathbf{i}=\{i_1, \cdots , i_j\}\subseteq [n]$,
the $R$-module $B_{\mathbf{i}}$ is a semidualizing.
\item[(4)] \cite[Lemma 4.6]{s-w1} If $\mathbf{i}=\{i_1, \cdots , i_j\}\subseteq [n]$ and $\mathbf{s}=\{s_1, \cdots , s_t\}\subseteq [n]$
are two sequences with $\mathbf{s}\subseteq \mathbf{i}$, then
$[B_{\mathbf{i}}] \trianglelefteq[B_{\mathbf{s}}]$ and
$\mathrm{Hom}_R(B_{\mathbf{s}}, B_{\mathbf{i}})\cong B_{\mathbf{i}\setminus \mathbf{s}}$.
\item[(5)] \cite[Theorem 4.11]{s-w1} If $\mathbf{i}=\{i_1, \cdots , i_j\}\subseteq [n]$ and $\mathbf{s}=\{s_1, \cdots , s_t\}\subseteq [n]$
are two sequences, then the following conditions are equivalent.
\begin{itemize}
\item[(i)] $B_{\mathbf{i}} \in \mathcal{A}_{B_{\mathbf{s}}}(R)$.
\item[(ii)] $B_{\mathbf{s}} \in \mathcal{A}_{B_{\mathbf{i}}}(R)$.
\item[(iii)] The $R$-module $B_{\mathbf{i}}\otimes_R B_{\mathbf{s}} $ is semidualizing.
\item[(iv)] $ \mathbf{i}\cap \mathbf{s}=\emptyset $.
\end{itemize}
\end{itemize}
\end{pro}
At the end of this section we recall the definition of trivial extension ring.
Note that this notion is the main key in
the proof of the converse of Sharp's result \cite{s}, which is given by Foxby \cite{f} and Reiten \cite{reiten}.
\begin{f}\label{f5}
For an $R$-module $M$, the \emph{trivial extension} of $R$ by $M$ is the ring $R \ltimes M$,
described as follows. As an $R$-module, we have $R \ltimes M=R\oplus M$. The multiplication
is defined by $(r, m)(r', m')=(rr', rm'+r'm)$.
Note that the composition $R\rightarrow R \ltimes M\rightarrow R$ of the natural homomorphisms is the identity map of $R$.
Note that, for a semidualizing $R$-module $C$,
the trivial extension ring $R\ltimes C$ is a commutative noetherian local ring.
If $R$ is Cohen-Macaulay then $R\ltimes C$ is Cohen-Macaulay too.
For more information about the trivial extension rings one may see, e.g., \cite[Section 2]{jls-w}.
\end{f}
\section{Results} \label{sec3}
This section is devoted to the main result, Theorem~\ref{T42}, which extends the results
of Jorgensen et.\!~al. \cite[Theorem 3.2]{jls-w} and of Foxby \cite{f} and Reiten \cite{reiten}.
For a semidualizing $R$-module $C$, set ${(-)}^{\dag_C}=\mathrm{Hom}_R(-, C)$.
The following notations are taken from \cite{s-w1}.
\begin{defn}\label{suitable}
Let $[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$ be a chain in $\mathfrak{G}_0(R)$ of length $n$.
For each sequence of integers $\mathbf{i} = \{i_1, \cdots , i_j\}$ such that $j \geqslant 0$ and $1\leqslant i_1 < \cdots < i_j \leqslant n$, set
$C_{\mathbf{i}}= C_0^{\dag_{C_{i_1}}\dag_{C_{i_2}}\cdots\dag_{C_{i_j}}}$.
(When $j = 0$, set $C_{\mathbf{i}} = C_\emptyset= C_0$ ).
We say that the above chain is \emph{suitable} if $C_0=R$ and
$C_{\mathbf{i}}$ is totally $C_t$-reflexive, for all
$\mathbf{i}$ and $t$ with $i_j\leqslant t \leqslant n$.
Note that if $[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [R]$
is a suitable chain, then $C_{\mathbf{i}}$ is a semidualizing $R$-module for each
$\mathbf{i}\subseteq [n]$.
Also, for each sequence of integers $\{ x_1, \cdots, x_m\}$ with
$1\leqslant x_1<\cdots <x_m\leqslant n$, the sequence
$[C_{x_m}]\vartriangleleft \cdots \vartriangleleft[C_{x_1}] \vartriangleleft [R]$
is a suitable chain in $\mathfrak{G}_0(R)$ of length $m$.
\end{defn}
Sather-Wagstaff, in \cite[Theorem 3.3]{s-w1}, proves that if $\mathfrak{G}_0(R)$ admits a chain
$[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$ such that
$\mathfrak{G}_{C_0}(R)\subseteq\mathfrak{G}_{C_1}(R)\subseteq\cdots \subseteq \mathfrak{G}_{C_n}(R)$, then $|\mathfrak{G}_0(R)|\geqslant 2^n$.
Indeed, the classes $[C_{\mathbf{i}}]$, which are parameterized by the
allowable sequences $\mathbf{i}$, are precisely the $2^n$
classes constructed in the proof of \cite[Theorem 3.3]{s-w1}.
\begin{thm}\cite[Theorem 4.7]{s-w1}\label{T03}
Let $\mathfrak{G}_0(R)$ admit a chain
$[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$
such that
$\mathfrak{G}_{C_1}(R)\subseteq\mathfrak{G}_{C_2}(R)\subseteq\cdots \subseteq \mathfrak{G}_{C_n}(R)$. If $C_0=R$,
then the $R$-modules $B_{\mathbf{i}}$ are precisely
the $2^n$ semidualizing modules constructed in \cite[Theorem 3.3]{s-w1}.
\end{thm}
\begin{rem}\label{R40}
In Proposition~\ref{P01} and Theorem~\ref{T03},
if we replace the assumption of existence of a chain
$[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$
in $\mathfrak{G}_0(R)$ such that
$\mathfrak{G}_{C_1}(R)\subseteq\mathfrak{G}_{C_2}(R)\subseteq\cdots \subseteq \mathfrak{G}_{C_n}(R)$
by the existence of a suitable chain, then the assertions hold true as well.
\end{rem}
The next lemma and proposition give us sufficient tools to treat Theorem~\ref{T42}.
\begin{lem}\label{L41}
Assume that $R$ admits a suitable chain
$[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]=[R]$ in $\mathfrak{G}_0(R)$.
Then for any $k\in [n]$, there exists a suitable chain
\begin{equation}\label{e40}
[C_n]\vartriangleleft \cdots \vartriangleleft[C_{k+1}]\vartriangleleft[C_k]\vartriangleleft[C_1^{\dag_{C_k}}]\vartriangleleft \cdots\vartriangleleft[C_{k-2}^{\dag_{C_k}}] \vartriangleleft[C_{k-1}^{\dag_{C_k}}] \vartriangleleft [R]
\end{equation}
in $\mathfrak{G}_0(R)$ of length $n$.
\end{lem}
\begin{proof}
For $i,j$, $0\leqslant j<i\leqslant k$, as $[C_i]\vartriangleleft[C_j]$
one has
$[C_{j}^{\dag_{C_k}}]\vartriangleleft[C_{i}^{\dag_{C_k}}]$.
As $[C_k]\neq[C_{i}^{\dag_{C_k}}]$, one gets
$[C_t]\vartriangleleft[C_{i}^{\dag_{C_k}}]$ for each $t$, $k\leqslant t\leqslant n$.
Thus (\ref{e40}) is a chain in $\mathfrak{G}_0(R)$ of length $n$.
Next, we show that (\ref{e40}) is a suitable chain.
For $r, t \in\{0, 1, \cdots , n\}$ and a sequence $\{x_1,\cdots, x_m\}$ of integers with
$r\leqslant x_1<\cdots < x_m\leqslant t$, repeated use of Theorem~\ref{T01} implies
$$C_r^{\dag_{C_t}}\cong C_r^{\dag_{C_{x_1}}}\otimes_R C_{x_1}^{\dag_{C_{x_2}}}\otimes_R \cdots \otimes_R C_{x_m}^{\dag_{C_t}}.$$
For each $r$, $0<r<k$, set $C'_r=C_r^{\dag_{C_k}}$.
If $\mathbf{i} = \{i_1, \cdots , i_j\}$ and $\mathbf{u} = \{u_1, \cdots , u_s\}$
are sequences of integers such that $j, s\geqslant 0$ and
$1\leqslant i_j < \cdots < i_1 < k\leqslant u_1< \cdots < u_s\leqslant n$, then we set
$$C_{\mathbf{i},\mathbf{u}}=
C_0^{\dag_{C'_{i_1}}\cdots\dag_{C'_{i_j}}\dag_{C_{u_1}}\cdots\dag_{C_{u_s}}}.$$
When $s=0$ (resp., $j=0$ or $j=0=s$) we have $C_{\mathbf{i}, \mathbf{u}}=C_{\mathbf{i}, \emptyset}$
(resp., $C_{\mathbf{i}, \mathbf{u}}=C_{\emptyset, \mathbf{u}}$ or
$C_{\mathbf{i}, \mathbf{u}}=C_{\emptyset, \emptyset} =C_0$).
By Proposition~\ref{P01}(4) and Remark~\ref{R40}, one has
$C_0^{\dag_{C'_{i_1}}\dag_{C'_{i_2}}}\cong\mathrm{Hom}_R(C_{i_1}^{\dag_{C_k}}, C_{i_2}^{\dag_{C_k}})\cong C_{i_2}^{\dag_{C_{i_1}}}$ and so
$C_0^{\dag_{C'_{i_1}}\dag_{C'_{i_2}}\dag_{C'_{i_3}}}\cong
\mathrm{Hom}_R(C_{i_2}^{\dag_{C_{i_1}}}, C_{i_3}^{\dag_{C_k}})\cong C_{i_3}^{\dag_{C_{i_2}}}\otimes_R C_{i_1}^{\dag_{C_k}}$.
By proceeding in this way one obtains the following isomorphism
\begin{equation}\label{e41}
C_0^{\dag_{C'_{i_1}}\cdots\dag_{C'_{i_j}}}\cong\left\{ \begin{array}{ll}
C_{i_j}^{\dag_{C_{i_{j-1}}}}\otimes_R C_{i_{j-2}}^{\dag_{C_{i_{j-3}}}}\otimes_R\cdots\otimes_R C_{i_2}^{\dag_{C_{i_1}}} & \mathrm{if}\ j\ \mathrm{is\ even}, \\
& \\
C_{i_j}^{\dag_{C_{i_{j-1}}}}\otimes_R C_{i_{j-2}}^{\dag_{C_{i_{j-3}}}}\otimes_R\cdots\otimes_R C_{i_1}^{\dag_{C_k}} & \mathrm{if}\ j\ \mathrm{is\ odd}.
\end{array} \right.
\end{equation}
Therefore, by Proposition~\ref{P01}(2) and Remark~\ref{R40},
$$C_0^{\dag_{C'_{i_1}}\cdots\dag_{C'_{i_j}}}\cong\left\{ \begin{array}{ll}
C_0^{\dag_{C_{i_j}}\cdots\dag_{C_{i_1}}} & \mathrm{if}\ j\ \mathrm{is\ even}, \\
& \\
C_0^{\dag_{C_{i_j}}\cdots\dag_{C_{i_1}}\dag_{C_k}} & \mathrm{if}\ j\ \mathrm{is\ odd},
\end{array} \right.$$
and thus
$$C_{\mathbf{i}, \mathbf{u}}\cong\left\{ \begin{array}{ll}
C_0^{\dag_{C_{i_j}}\cdots\dag_{C_{i_1}}\dag_{C_{u_1}}\cdots\dag_{C_{u_s}}} & \mathrm{if}\ j\ \mathrm{is\ even}, \\
& \\
C_0^{\dag_{C_{i_j}}\cdots\dag_{C_{i_1}}\dag_{C_k}\dag_{C_{u_1}}\cdots\dag_{C_{u_s}}} & \mathrm{if}\ j\ \mathrm{is\ odd}.
\end{array} \right.$$
Hence, by assumption, $[C_t]\trianglelefteq[C_{\mathbf{i}, \mathbf{u}}]$ for all $t$, $t\geqslant u_s$.
If $s=0$, then
$C_{\mathbf{i}, \mathbf{u}}=C_{\mathbf{i}, \emptyset}=C_0^{\dag_{C'_{i_1}}\cdots\dag_{C'_{i_j}}}$. \\
On the other hand, for each $l$, $1\leqslant l\leqslant i_j$, we have
$$C_l^{\dag_{C_k}}\cong C_l^{\dag_{C_{i_j}}}\otimes_R C_{i_j}^{\dag_{C_{i_{j-1}}}}\otimes_R \cdots
\otimes_R C_{i_3}^{\dag_{C_{i_2}}}\otimes_R C_{i_2}^{\dag_{C_{i_1}}}\otimes_R C_{i_1}^{\dag_{C_k}}.$$
Thus, by Proposition~\ref{P01}(4) and (\ref{e41}),
$[C_l^{\dag_{C_k}}]\trianglelefteq [C_{\mathbf{i}, \mathbf{u}}]$.
Hence the chain (\ref{e40}) is suitable.
\end{proof}
\begin{rem}\label{R42}
Let $R$ be Cohen-Macaulay and $[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [C_0]$ be a suitable chain in $\mathfrak{G}_0(R)$.
For any $k$, $1\leqslant k\leqslant n$, set $R_k=R\ltimes C_{k-1}^{\dag_{C_k}}$ the trivial extension of $R$ by $C_{k-1}^{\dag_{C_k}}$.
Then $R_k$ is totally $C_l^{\dag_{C_k}}$-reflexive and totally $C_t$-reflexive $R$-module for all $l, t$ with $1\leqslant l< k \leqslant t\leqslant n$.
Set
$$C_l^{(k)}=\left\{ \begin{array}{ll}
\mathrm{Hom}_R(R_k, C_{k-1-l}^{\dag_{C_k}}) & \text{if}\ \ 0\leqslant l< k-1\\
\mathrm{Hom}_R(R_k, C_{l+1}) & \text{if}\ \ k-1\leqslant l\leqslant n-1 \hspace{0.1cm}.
\end{array} \right.$$
Then, by Remark~\ref{R1}, $C_l^{(k)}$ is a semidualizing $R_k$-module for all $l$, $0\leqslant l\leqslant n-1$.
\end{rem}
\begin{pro}\label{P41}
Under the hypotheses of Remark \ref{R42}, for all $k$, $1\leqslant k\leqslant n$,
$$[C_{n-1}^{(k)}] \vartriangleleft \cdots \vartriangleleft [C_1^{(k)}] \vartriangleleft [R_k]$$
is a suitable chain in $\mathfrak{G}_0(R_k)$ of length $n-1$.
\end{pro}
\begin{proof}
Let $k\in[n]$.
For integers $a, b$ with $a\neq b$ and $0\leqslant a, b\leqslant n-1$,
we observe that $[C_a^{(k)}]\neq[C_b^{(k)}]$. Indeed, we consider the three cases
$0\leqslant a, b < k-1$, $0\leqslant a<k-1 \leqslant b \leqslant n-1$, and
$k-1 \leqslant a, b \leqslant n-1$. We only discuss about the first case. The other cases are treated in a similar way. For $0\leqslant a, b < k-1$,
if $[C_a^{(k)}]=[C_b^{(k)}]$, then
$\hspace{0.2cm}\mathrm{Hom}_R(R_k, C_{k-1-a}^{\dag_{C_k}})\cong \mathrm{Hom}_R(R_k, C_{k-1-b}^{\dag_{C_k}})\hspace{0.1cm}$ and so
$\mathrm{Hom}_{R_k}(R, \mathrm{Hom}_R(R_k, C_{k-1-a}^{\dag_{C_k}}))\cong \mathrm{Hom}_{R_k}(R, \mathrm{Hom}_R(R_k, C_{k-1-b}^{\dag_{C_k}}) )$.
Thus, by adjointness, $C_{k-1-a}^{\dag_{C_k}} \cong C_{k-1-b}^{\dag_{C_k}}$ ,
which contradicts with (\ref{e40}) in Lemma~\ref{L41}.
In order to proceed with the proof, for an $R_k$-module $M$, we invent the symbol
$(-)^{\dag^k_M}=\mathrm{Hom}_{R_k}(-, M)$.
Note that, for $R_k$-modules $M_1, \cdots, M_t$, we have
$$(-)^{\dag^k_{M_1} \dag^k_{M_2} \cdots \dag^k_{M_t}}=
\bigg(\Big(\big( (-)^{\dag^k_{M_1}}\big)^{\dag^k_{M_2}}\Big)^{\cdots}\bigg)^{\dag^k_{M_t}}=
\mathrm{Hom}_{R_k}\big((-)^{\dag^k_{M_1} \dag^k_{M_2} \cdots \dag^k_{M_{t-1}}}, M_t\big).$$
For two sequences of integers $\mathbf{p} = \{p_1, \cdots , p_r\}$ and $\mathbf{q} = \{q_1, \cdots , q_s\}$ such that $r, s\geqslant 0$ and
$0< p_1 < \cdots < p_r < k-1\leqslant q_1< \cdots < q_s\leqslant n-1$, set
$$C^{(k)}_{\mathbf{p}, \mathbf{q}}=
R_k^{\dag^k_{C_{p_1}^{(k)}}\cdots\dag^k_{C_{p_r}^{(k)}}\dag^k_{C_{q_1}^{(k)}}\cdots
\dag^k_{C_{q_s}^{(k)}}}.$$
Therefore one gets the following $R$-module isomorphisms
$$\begin{array}{llll}
C^{(k)}_{\mathbf{p}, \mathbf{q}}& = &
\mathrm{Hom}_{R_k}(\cdots \mathrm{Hom}_{R_k}(\mathrm{Hom}_{R_k}(
\cdots \mathrm{Hom}_{R_k}(R_k, C_{p_1}^{(k)})\cdots , C_{p_r}^{(k)}),
C_{q_1}^{(k)}) \cdots , C_{q_s}^{(k)})\\
& \cong &
\mathrm{Hom}_{R}(\cdots \mathrm{Hom}_{R}(\mathrm{Hom}_{R}(
\cdots \mathrm{Hom}_{R}(R_k, C_{k-1-p_1}^{\dag_{C_k}}) \cdots , C_{k-1-p_r}^{\dag_{C_k}}), C_{q_1+1}) \cdots , C_{q_s+1})\\
& \cong &
R^{\dag_{C'_{k-1-p_1}}\cdots\dag_{C'_{k-1-p_r}}\dag_{C_{q_1+1}}\cdots\dag_{C_{q_s+1}}}
\oplus
R^{\dag_{C'_{k-1}}\dag_{C'_{k-1-p_1}}\cdots\dag_{C'_{k-1-p_r}}\dag_{C_{q_1+1}}\cdots\dag_{C_{q_s+1}}}\\
& = & C_{\mathbf{i}, \mathbf{u}}\oplus C_{\mathbf{i'}, \mathbf{u}}\, ,
\end{array}$$
where $\mathbf{i} = \{k-1-p_1, \cdots , k-1-p_r\}$, $\mathbf{i'} = \{k-1, k-1-p_1, \cdots , k-1-p_r\}$,
$\mathbf{u} = \{q_1+1, \cdots , q_s+1\}$, $C'_l=C_l^{\dag_{C_k}}$, for all $0< l< k$,
and $C_{\mathbf{i}, \mathbf{u}}$ and $C_{\mathbf{i'}, \mathbf{u}}$ are as in the
proof of Lemma~\ref{L41}.
As $[C_{t+1}]\trianglelefteq [C_{\mathbf{i}, \mathbf{u}}]$ and
$[C_{t+1}]\trianglelefteq [C_{\mathbf{i'}, \mathbf{u}}]$ in $\mathfrak{G}_0(R)$
for all $t$, $q_s\leqslant t\leqslant n-1$, one gets
$[C^{(k)}_t]\trianglelefteq [C^{(k)}_{\mathbf{p}, \mathbf{q}}]$ in
$\mathfrak{G}_0(R_k)$, by \cite[Theorem 6.5]{c2}.
When $s=0$ we have
$C^{(k)}_{\mathbf{p}, \mathbf{q}}=C^{(k)}_{\mathbf{p}, \emptyset}\cong C_{\mathbf{i}, \emptyset}\oplus C_{\mathbf{i'}, \emptyset}$.
By Lemma~\ref{L41}, for all $m$, $p_r\leqslant m<k-1$, one has
$[C_{k-1-m}^{\dag_{C_k}}]\trianglelefteq [C_{\mathbf{i}, \emptyset}]$ and
$[C_{k-1-m}^{\dag_{C_k}}]\trianglelefteq [C_{\mathbf{i'}, \emptyset}]$ in $\mathfrak{G}_0(R)$.
Thus, by \cite[Theorem 6.5]{c2}, one gets
$[C^{(k)}_m]\trianglelefteq [C^{(k)}_{\mathbf{p}, \emptyset}]$ in $\mathfrak{G}_0(R_k)$.
Hence $[C_{n-1}^{(k)}] \vartriangleleft \cdots \vartriangleleft [C_1^{(k)}] \vartriangleleft [R_k]$
is a suitable chain in $\mathfrak{G}_0(R_k)$ of length $n-1$.
\end{proof}
To state our main result, we recall the definitions of Tate homology and Tate cohomology ( see \cite{am} and \cite{jls-w} for more details ).
\begin{defn}\label{Tate1}
Let $M$ be a finite $R$-module. A \emph{Tate resolution} of $M$ is a diagram
$\mathbf{T}\stackrel{\vartheta}{\longrightarrow}\mathbf{P}\stackrel{\pi}{\longrightarrow}M$, where $\pi$ is an $R$-projective resolution of $M$,
$\mathbf{T}$ is an exact complex of projectives such that $\mathrm{Hom}_R(T, R)$ is exact, $\vartheta$ is a morphism, and $\vartheta_i$ is isomorphism for all $i\gg 0$.
By \cite[Theorem 3.1]{am}, a finite $R$-module has finite $\mbox{G}$-dimension if and only if it admits a Tate resolution.
\end{defn}
\begin{defn}\label{Tate2}
Let $M$ be a finite $R$-module of finite $\mbox{G}$-dimension, and let
$\mathbf{T}\stackrel{\vartheta}{\longrightarrow}\mathbf{P}\stackrel{\pi}{\longrightarrow}M$ be a Tate resolution of $M$.
For each integer $i$ and each $R$-module $N$, the $i$th \emph{Tate homology} and \emph{Tate cohomology} modules are
$$\widehat{\mathrm{Tor}}^R_i(M, N)= \mathrm{H}_i(\mathbf{T}\otimes_R N) \hspace{1.7cm} \widehat{\mathrm{Ext}}_R^i(M, N)=\mathrm{H}_{-i}(\mathrm{Hom}_R(\mathbf{T}, N)).$$
\end{defn}
\begin{thm}\label{T42}
Let $R$ be a Cohen-Macaulay ring with a dualizing module $D$.
Assume that $R$ admits a suitable chain
$[C_n]\vartriangleleft \cdots \vartriangleleft[C_1] \vartriangleleft [R]$ in $\mathfrak{G}_0(R)$
and that $C_n\cong D$. Then there exist a Gorenstein local ring $Q$ and
ideals $I_1, \cdots, I_n$ of $Q$, which satisfy the following conditions.
In this situation, for each $\Lambda\subseteq [n]$, set $R_{_\Lambda}=Q/(\Sigma_{l\in \Lambda} I_l)$, in particular $R_{_\emptyset}=Q$.
\begin{itemize}
\item[(1)] There is a ring isomorphism $R\cong Q/{(I_1+\cdots+I_n)}$.
\item[(2)] For each $\Lambda\subseteq [n]$ with $\Lambda\neq \emptyset$, the ring $R_{_\Lambda}$ is non-Gorenstein Cohen-Macaulay with a dualizing module.
\item[(3)] For each $\Lambda\subseteq [n]$ with $\Lambda\neq \emptyset$, we have $\bigcap_{l\in \Lambda}I_l=\prod_{l\in \Lambda}I_l$.
\item[(4)] For subsets $\Lambda$, $\Gamma$ of $[n]$ with $\Gamma\subsetneq\Lambda$, we have $\mathrm{G}$-$\dim_{R_{_\Gamma}} R_{_{\Lambda}}=0$, and
$\mathrm{Hom}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})$ is a non-free semidualizing $R_{_{\Lambda}}$-module.
\item[(5)] For subsets $\Lambda$, $\Gamma$ of $[n]$ with $\Lambda\neq \Gamma$, the module
$\mathrm{Hom}_{R_{_{\Lambda\cap \Gamma}}} (R_{_\Lambda}, R_{_{\Gamma}})$
is not cyclic and
$$\mathrm{Ext}^{\geqslant1}_{R_{_{\Lambda\cap \Gamma}}} (R_{_\Lambda}, R_{_{\Gamma}})=0=\mathrm{Tor}^{R_{_{\Lambda\cap \Gamma}}}_{\geqslant1}(R_{_\Lambda}, R_{_{\Gamma}}).$$
\item[(6)] For subsets $\Lambda$, $\Gamma$ of $[n]$ with $|\Lambda\setminus \Gamma|=1$, we have
$$\mathrm{\widehat{Ext}}^i_{R_{_{\Lambda\cap \Gamma}}} (R_{_\Lambda}, R_{_{\Gamma}})=0=\mathrm{\widehat{Tor}}^{R_{_{\Lambda\cap \Gamma}}}_i(R_{_\Lambda}, R_{_{\Gamma}})$$
for all $i \in \mathbb{Z}$.
\end{itemize}
\end{thm}
The ring $Q$ is constructed as an iterated trivial extension of $R$. As an $R$-module,
it has the form $Q=\oplus_{\mathbf{i}\subseteq [n]} B_{\mathbf{i}}$.
The details are contained in the following construction.
\begin{cons}\label{cons}
We construct the ring $Q$ by induction on $n$.
We claim that the ring $Q$, as an $R$-module, has the form
$Q=\oplus_{\mathbf{i}\subseteq [n]} B_{\mathbf{i}}$ and the ring structure on it is as follows.\\
For two elements $\big(\alpha_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}$ and $\big(\theta_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}$ of $Q$
$$\big(\alpha_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]} \big(\theta_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}=\big(\sigma_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}\ , \ \text{where}\ \ \ \ \sigma_{\mathbf{i}}=\sum_{\scriptsize{\begin{array}{c}\mathbf{v}\subseteq \mathbf{i}\\
\mathbf{w}= \mathbf{i}\setminus \mathbf{v} \end{array}} }
\alpha_{\mathbf{v}}\cdot\theta_{\mathbf{w}}\, .$$
For $n=1$, set $Q=R\ltimes C_1$ and $I_1=0\oplus C_1$, which is the result of Foxby \cite{f} and
Reiten \cite{reiten}.
The case $n=2$ is proved by Jorgensen et.\!~al. \cite[Theorem 3.2]{jls-w}.
They proved that the extension ring $Q$ has the form
$Q=R\oplus C_1 \oplus C_1^{\dag_{C_2}}\oplus C_2$ as an $R$-module
(i.e. $Q=B_\emptyset\oplus B_1\oplus B_2 \oplus B_{\{1,2\}}$).
Also the ring structure on $Q$ is given by
$(r, c, f, d)(r', c', f', d')=(rr', rc' + r'c, rf'+r'f, f'(c)+f(c')+rd'+r'd)$.
The ideal $I_l$, $l=1,2$, has the form $I_l=0\oplus 0\oplus B_l \oplus B_{\{1,2\}}$.
Let $n>2$. Take an element $k\in [n]$. By Proposition~\ref{P41}, the ring
$R_k=R\ltimes C_{k-1}^{\dag_{C_k}}$
has the suitable chain
$[C_{n-1}^{(k)}] \vartriangleleft \cdots \vartriangleleft [C_1^{(k)}] \vartriangleleft [R_k]$
in $\mathfrak{G}_0(R_k)$ of length $n-1$.
Note that $C_{n-1}^{(k)}=\mathrm{Hom}_R(R_k, C_n)\cong\mathrm{Hom}_R(R_k, D)$
is a dualizing $R_k$-module.
We set $B_i^{(k)}=\mathrm{Hom}_{R_k}(C_{i-1}^{(k)}, C_{i}^{(k)})$, $i=1, \cdots, n-1$.
For two sequences $\mathbf{p}=\{p_1, \cdots, p_r\}$, $\mathbf{q}=\{q_1, \cdots, q_s\}$
such that $r, s\geqslant 1$ and
$1\leqslant p_1 < \cdots < p_r < k-1\leqslant q_1< \cdots < q_s\leqslant n-1$, we set
\begin{equation}\label{e46}
B^{(k)}_{\mathbf{p}, \mathbf{q}}=B_{p_1}^{(k)}\otimes_{R_k}\cdots\otimes_{R_k}B_{p_r}^{(k)}\otimes_{R_k} B_{q_1}^{(k)}\otimes_{R_k}\cdots\otimes_{R_k}B_{q_s}^{(k)},
\end{equation}
$$B^{(k)}_{\mathbf{p}, \emptyset}=B_{p_1}^{(k)}\otimes_{R_k}\cdots\otimes_{R_k}B_{p_r}^{(k)},\ \
B^{(k)}_{\emptyset, \mathbf{q}}= B_{q_1}^{(k)}\otimes_{R_k}\cdots\otimes_{R_k}B_{q_s}^{(k)},\ \ \mathrm{and}\ \
B^{(k)}_{\emptyset, \emptyset}=C_0^{(k)}=R_k.$$
By applying the induction hypothesis on $R_k$ there is an extension ring, say $Q_k$, which is
Gorenstein local and, as an $R_k$-module, has the form
$$Q_k= \bigoplus_{\scriptsize{\begin{array}{c}\mathbf{p}\subseteq \{1, \cdots, k-2\}\\
\mathbf{q}\subseteq \{k-1, \cdots, n-1\}\end{array}}} B^{(k)}_{\mathbf{p}, \mathbf{q}}\ .$$
Moreover, the ring structure on $Q_k$ is as follows.\\
For
$\phi=\big(\phi_{\mathbf{p}, \mathbf{q}}\big)_{\tiny{
\mathbf{p}\subseteq \{1, \cdots, k-2\},\
\mathbf{q}\subseteq \{k-1, \cdots, n-1\}}}$
and
$\varphi=\big(\varphi_{\mathbf{p}, \mathbf{q}}\big)_{\tiny{
\mathbf{p}\subseteq \{1, \cdots, k-2\},\
\mathbf{q}\subseteq \{k-1, \cdots, n-1\}}}$
of $Q_k$
\begin{equation}\label{e45}
\phi\, \varphi=\psi=\big(\psi_{\mathbf{p}, \mathbf{q}}\big)_{\tiny{
\mathbf{p}\subseteq \{1, \cdots, k-2\},\
\mathbf{q}\subseteq \{k-1, \cdots, n-1\}}}\ , \ \text{where} \ \ \ \
\psi_{\mathbf{p}, \mathbf{q}}=\sum_{\scriptsize{\begin{array}{c}
\mathbf{a} \subseteq \mathbf{p}, \mathbf{b}\subseteq\mathbf{q}\\
\mathbf{c}=\mathbf{p}\setminus \mathbf{a} \\
\mathbf{d}=\mathbf{q}\setminus \mathbf{b}
\end{array}}}
\phi_{\mathbf{a}, \mathbf{b}}\cdot\varphi_{\mathbf{c}, \mathbf{d}}\, .
\end{equation}
For each $\mathbf{p}, \mathbf{q}$, Proposition~\ref{P01}(2), Remark~\ref{R40} and (\ref{e46})
imply the following $R$-module isomorphism
\begin{equation}\label{e42}
B^{(k)}_{\mathbf{p}, \mathbf{q}}\cong \left\{ \begin{array}{l}
B_{\{k-p_r,\cdots, k-p_1, q_1+1, \cdots, q_s+1\}}\oplus B_{\{k-p_r, \cdots, k-p_1, k, q_1+1,\cdots, q_s+1\}}, \\
\mathrm{or} \\
B_{\{1, k-p_r, \cdots, k-p_1, q_2+1,\cdots, q_s+1\}}\oplus B_{\{1, k-p_r, \cdots, k-p_1, k, q_2+1,\cdots, q_s+1\}}.
\end{array} \right.
\end{equation}
Therefore one gets an $R$-module isomorphism $Q_k\cong \oplus_{\mathbf{i}\subseteq [n]} B_{\mathbf{i}}$. Set $Q=Q_k$.
Assume that $\mathbf{p}, \mathbf{p}'\subseteq \{1, \cdots, k-2\}$ and $\mathbf{q}, \mathbf{q}'\subseteq \{k-1, \cdots, n-1\}$ such that
$\mathbf{p}\cap\mathbf{p}'=\emptyset$ and $\mathbf{q}\cap\mathbf{q}'=\emptyset$.
By Proposition~\ref{P01}(5) and Remark~\ref{R40}, the $R_k$-module
$B^{(k)}_{\mathbf{p}, \mathbf{q}}\otimes_{R_k}B^{(k)}_{\mathbf{p}', \mathbf{q}'}$ is a semidualizing and so
$B^{(k)}_{\mathbf{p}, \mathbf{q}}\otimes_{R_k}B^{(k)}_{\mathbf{p}', \mathbf{q}'}=
B^{(k)}_{\mathbf{p}\cup\mathbf{p}' , \mathbf{q}\cup \mathbf{q}'}$.
If $\phi_{\mathbf{p}, \mathbf{q}}\in B^{(k)}_{\mathbf{p}, \mathbf{q}}$ and
$\varphi_{\mathbf{p}', \mathbf{q}'}\in B^{(k)}_{\mathbf{p}', \mathbf{q}'}$, then
by the isomorphism (\ref{e42}), one has
$\phi_{\mathbf{p}, \mathbf{q}}=(\beta_{\mathbf{p}, \mathbf{q}}, \gamma_{\mathbf{p}, \mathbf{q}})$ and
$\varphi_{\mathbf{p}', \mathbf{q}'}=(\beta_{\mathbf{p}', \mathbf{q}'}, \gamma_{\mathbf{p}', \mathbf{q}'})$, so that
$$\phi_{\mathbf{p}, \mathbf{q}} \cdot \varphi_{\mathbf{p}', \mathbf{q}'}=(\beta_{\mathbf{p}, \mathbf{q}}\cdot \beta_{\mathbf{p}', \mathbf{q}'},\ \beta_{\mathbf{p}, \mathbf{q}}\cdot \gamma_{\mathbf{p}', \mathbf{q}'} + \beta_{\mathbf{p}', \mathbf{q}'}\cdot\gamma_{\mathbf{p}, \mathbf{q}}) .$$
Thus by means of the ring structure on $Q_k$, (\ref{e45}), one can see that the resulting ring structure on $Q$ is as claimed.
The next step is to introduce the ideals $I_1, \cdots, I_n$. We set
$I_l = (\underbrace{0\oplus\cdots\oplus 0}_{2^{n-1}})\oplus (\oplus_{\mathbf{i}\subseteq [n],\, l\in\mathbf{i}}B_{\mathbf{i}})$, $1\leqslant l\leqslant n$,
which is an ideal of $Q$.
Also we have the following sequence of $R$-isomorphisms which preserve ring isomorphisms:
$$\begin{array}{llll}
Q/{(I_1+\cdots+I_n)}& = & (\oplus_{\mathbf{i}\subseteq [n]} B_{\mathbf{i}})/(\Sigma_{l=1}^n (\underbrace{0\oplus\cdots\oplus 0}_{2^{n-1}})\oplus (\oplus_{\mathbf{i}\subseteq [n],\, l\in\mathbf{i}}B_{\mathbf{i}}) )\\
& \cong & (\oplus_{\mathbf{i}\subseteq [n]} B_{\mathbf{i}})/(\oplus_{\mathbf{i}\subseteq [n], \mathbf{i}\neq \emptyset} B_{\mathbf{i}})\\
& \cong & R\, .
\end{array}$$
Note that each ideal
$I_{k,l}$, $1\leqslant l\leqslant n-1$, of $Q_k$ has the form
$I_{k,l}=(\underbrace{0\oplus\cdots\oplus 0}_{2^{n-2}})\oplus
(\oplus_{l\in\mathbf{p}\cup \mathbf{q}}B^{(k)}_{\mathbf{p}, \mathbf{q}})$.
Then, by (\ref{e42}), one has the following $R$-module isomorphism
$$I_{k,l}\cong \left\{ \begin{array}{ll}
I_{k-l} & \text{if}\ \ 1\leqslant l\leqslant k-1\\
I_{l+1} & \text{if}\ \ k\leqslant l\leqslant n-1.
\end{array} \right.$$
Also, by means of the ring isomorphism $Q_k\rightarrow Q$, we have the natural correspondence
between ideals:
$$I_{k,l}\stackrel{\text{correspond}}{\longleftarrow\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-
\hspace{-0.2cm}\longrightarrow} \left\{ \begin{array}{ll}
I_{k-l} & \text{if}\ \ 1\leqslant l\leqslant k-1\\
I_{l+1} & \text{if}\ \ k\leqslant l\leqslant n-1.
\end{array} \right.$$
Therefore for each $\Lambda\subseteq [n]\setminus\{k\}$,
there is a ring isomorphism
$Q/(\Sigma_{l\in \Lambda} I_l) \cong Q_k/(\Sigma_{l\in\Lambda'} I_{k,l})$,
for some $\Lambda'\subseteq[n-1]$.
\end{cons}
The proof of Theorem~\ref{T42}, which is inspired by the proof of \cite[Theorem 3.2]{jls-w},
is rather technical and needs some preparatory lemmas.
\begin{lem}\label{L43}
Assume that $\Lambda\subseteq [n]$. Under the hypothesis of Theorem \ref{T42}, if $[n]\setminus \Lambda=\{b_1, \cdots, b_t\}$ with $1\leqslant b_1<\cdots<b_t\leqslant n$,
then there is an $R$-isomorphism
$$R_{_\Lambda}\cong \oplus_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}} B_{\mathbf{i}}$$
which induces a ring structure on $R_{_\Lambda}$ as follows.
For elements $\big(\alpha_{\mathbf{i}}\big)_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}$ and $\big(\theta_{\mathbf{i}}\big)_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}$
of $R_{_\Lambda}$
$$\big(\alpha_{\mathbf{i}}\big)_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}
\big(\theta_{\mathbf{i}}\big)_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}=
\big(\sigma_{\mathbf{i}}\big)_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}\ , \ where\ \ \ \ \
\sigma_{\mathbf{i}}=\sum_{\scriptsize{\begin{array}{c}
\mathbf{v}\subseteq \mathbf{i}\\
\mathbf{w}=\mathbf{i}\setminus\mathbf{v} \end{array}} }
\alpha_{\mathbf{v}}\cdot\theta_{\mathbf{w}}\, .$$
\end{lem}
\begin{proof}
We prove by induction on $n$. The case $n=1$ is clear. The case $n=2$ is proved in \cite{jls-w}.
Assume that $n>2$ and the assertion holds true for $n-1$.
If $\Lambda=[n]$, there is nothing to prove. Suppose that $|\Lambda|\leqslant n-1$
then there exists $k\in [n]$ such that $\Lambda\subseteq [n]\setminus\{k\}$. Thus, by Construction~\ref{cons}, there exists a subset $\Lambda'$ of $[n-1]$ such that
$R_{_\Lambda}\cong Q_k/(\Sigma_{l\in \Lambda'} I_{k,l})$ as ring isomorphism.
Note that $|[n-1]\setminus \Lambda'|=t-1$. Set $[n-1]\setminus \Lambda'=\{d_1, \cdots, d_u, d_{u+1}, \cdots, d_{t-1}\}$ such that
$1\leqslant d_1<\cdots<d_u<k-1$ and $k-1\leqslant d_{u+1}<\cdots< d_{t-1}\leqslant n-1$.
Then by induction there exists an $R_k$-isomorphism
$$Q_k/(\Sigma_{l\in \Lambda'} I_{k,l})\cong \bigoplus_{\scriptsize{\begin{array}{c}
\mathbf{p}\subseteq \{d_1, \cdots, d_u\}\\
\mathbf{q}\subseteq \{d_{u+1}, \cdots, d_{t-1}\} \end{array}}}
B_{\mathbf{p}, \mathbf{q}}^{(k)}.$$
Proceeding as Construction~\ref{cons}, there is an $R$-isomorphism
$$\Big(\bigoplus_{\scriptsize{\begin{array}{c}
\mathbf{p}\subseteq \{d_1, \cdots, d_u\}\\
\mathbf{q}\subseteq \{d_{u+1}, \cdots, d_{t-1}\} \end{array}}}
B_{\mathbf{p}, \mathbf{q}}^{(k)}\Big)\
\cong \ \Big(\bigoplus_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}} B_{\mathbf{i}}\Big).$$
Therefore one has an $R$-isomorphism
$R_{_\Lambda}\cong \oplus_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}} B_{\mathbf{i}}$.
Similar to Construction~\ref{cons}, $R_{_\Lambda}$ has the desired ring structure.
\end{proof}
\begin{lem}\label{L44}
Under the hypothesis of Theorem~\ref{T42}, if $\Gamma\subsetneq \Lambda\subseteq [n]$, then
$\mathrm{Ext}^{\geqslant1}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})=0$ and
$\mathrm{Hom}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})$ is a non-free semidualizing $R_{_{\Lambda}}$-module.
\end{lem}
\begin{proof}
The case $n=1$ is clear and the case $n=2$ is proved in
\cite[Lemma 3.8]{jls-w}.
Let $n>2$ and suppose that the assertion is settled for $n-1$.
First assume that $\Lambda=[n]$. Set $[n]\setminus \Gamma=\{a_1, \cdots, a_s\}$ with $1\leqslant a_1<\cdots<a_s\leqslant n$.
By Lemma~\ref{L43}, $R_{_{\Gamma}}\cong \oplus_{\mathbf{i}\subseteq \{a_1, \cdots, a_s\}} B_{\mathbf{i}}$.
By Proposition~\ref{P01}(4) and Remark~\ref{R40},
$[B_{\{a_1, \cdots, a_s\}}]\trianglelefteq [B_{\mathbf{i}}]$ and
$\mathrm{Hom}_R(B_{\mathbf{i}}, B_{\{a_1, \cdots, a_s\}}) \cong B_{\{a_1, \cdots, a_s\}\setminus\mathbf{i}}$,
for all $\mathbf{i}\subseteq\{a_1, \cdots, a_s\}$.
Therefore there are $R$-isomorphisms
$$\mathrm{Hom}_R (R_{_{\Gamma}}, B_{\{a_1, \cdots, a_s\}})\cong \mathrm{Hom}_R (\oplus_{\mathbf{i}\subseteq \{a_1, \cdots, a_s\}} B_{\mathbf{i}}, B_{\{a_1, \cdots, a_s\}})\cong \oplus_{\mathbf{i}\subseteq \{a_1, \cdots, a_s\}} B_{\mathbf{i}}\cong R_{_{\Gamma}}$$
and, for all $i\geqslant 1$,
$$\mathrm{Ext}^i_R(R_{_{\Gamma}}, B_{\{a_1, \cdots, a_s\}})\cong \mathrm{Ext}^i_R(\oplus_{\mathbf{i}\subseteq \{a_1, \cdots, a_s\}} B_{\mathbf{i}}, B_{\{a_1, \cdots, a_s\}})=0.$$
Let $\mathbf{E}$ be an injective resolution of $B_{\{a_1, \cdots, a_s\}}$ as an $R$-module. Thus $\mathrm{Hom}_R (R_{_{\Gamma}}, \mathbf{E})$ is an injective resolution of $R_{_{\Gamma}}$ as an $R_{_{\Gamma}}$-module. Note that the composition of natural homomorphisms \\
$R\rightarrow R_{_{\Gamma}}\rightarrow R$ is the identity $\mathrm{id}_R$.
Therefore
$$\mathrm{Hom}_{R_{_{\Gamma}}}(R, \mathrm{Hom}_R (R_{_{\Gamma}}, \mathbf{E}))\cong \mathrm{Hom}_R(R\otimes_{R_{_{\Gamma}}}R_{_{\Gamma}},\mathbf{E})\cong \mathrm{Hom}_R(R, \mathbf{E})\cong \mathbf{E}\,.$$
Hence
$$\begin{array}{llll}
\mathrm{Ext}^i_{R_{_{\Gamma}}} (R, R_{_{\Gamma}}) & \cong & \mathrm{H}^i(\mathrm{Hom}_{R_{_{\Gamma}}}(R, \mathrm{Hom}_R (R_{_{\Gamma}}, \mathbf{E}))) \\
& \cong & \mathrm{H}^i(\mathbf{E}) \\
& \cong & \left\{ \begin{array}{ll}
0 &\text{if}\ \ i>0\\
B_{\{a_1, \cdots, a_s\}} &\text{if}\ \ i=0\,.
\end{array} \right.
\end{array}$$
As $\{a_1, \cdots, a_s\}\neq \emptyset$, the $R$-module $B_{\{a_1, \cdots, a_s\}}$ is a non-free semidualizing.
Now assume that $|\Lambda|\leqslant n-1$. There exist $k\in [n]$, and subsets $\Gamma'$, $\Lambda'$ of $[n-1]$ such that there are $R$-isomorphisms and ring isomorphisms
$R_{_{\Gamma}}\cong Q_k/(\Sigma_{l\in \Gamma'} I_{k,l})$ and $R_{_{\Lambda}}\cong Q_k/(\Sigma_{l\in \Lambda'} I_{k,l})$,
where $Q_k$ and $I_{k,l}$ are as in Construction~\ref{cons}.
By induction we have
$$\mathrm{Ext}^{i}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})\cong \mathrm{Ext}^{i}_{Q_k/(\Sigma_{l\in \Gamma'} I_{k,l})} (Q_k/(\Sigma_{l\in \Lambda'} I_{k,l}), Q_k/(\Sigma_{l\in \Gamma'} I_{k,l}))=0$$
for all $i\geqslant 1$, and
$$\mathrm{Hom}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})\cong \mathrm{Hom}_{Q_k/(\Sigma_{l\in \Gamma'} I_{k,l})} (Q_k/(\Sigma_{l\in \Lambda'} I_{k,l}), Q_k/(\Sigma_{l\in \Gamma'} I_{k,l}))$$
is a non-free semidualizing $Q_k/(\Sigma_{l\in \Lambda'} I_{k,l})$-module.
Then $\mathrm{Hom}_{R_{_{\Gamma}}} (R_{_{\Lambda}}, R_{_{\Gamma}})$ is a non-free semidualizing $R_{_{\Lambda}}$-module.
\end{proof}
\begin{lem}\label{L45}
Under the hypothesis of Theorem \ref{T42}, if ${\Lambda}$ and ${\Gamma}$
are two subsets of $[n]$, then
$\mathrm{Tor}^{R_{_{{\Lambda}\cup {\Gamma}}}}_{\geqslant 1}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$. Moreover, there is an $R_{_{\Lambda}}$-algebra isomorphism
$R_{_{\Lambda}}\otimes_{R_{_{{\Lambda}\cup {\Gamma}}}} R_{_{\Gamma}}\cong R_{_{{\Lambda}\cap {\Gamma}}}$.
\end{lem}
\begin{proof}
We prove by induction. If $n=1$, there is nothing to prove. The case $n=2$ is proved in
\cite[Lemma 3.9]{jls-w}.
Let $n>2$ and suppose that the assertion holds true for $n-1$.
First assume that ${\Lambda}\cup {\Gamma}=[n]$ and set $[n]\setminus {\Lambda}=\{b_1, \cdots, b_t\}$, $[n]\setminus {\Gamma}=\{a_1, \cdots, a_s\}$.
Then $[n]\setminus ({\Lambda}\cap {\Gamma})=\{b_1, \cdots, b_t, a_1, \cdots, a_s\}$.
By Lemma~\ref{L43},
$R_{_{\Lambda}}\cong \oplus_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}} B_{\mathbf{i}}\, \ \mathrm{and}\ \ R_{_{\Gamma}}\cong \oplus_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}} B_{\mathbf{u}}$.
As $\{b_1, \cdots, b_t\}\cap \{a_1, \cdots, a_s\}=\emptyset$, for each
$\mathbf{i}\subseteq \{b_1, \cdots, b_t\}$ and $\mathbf{u}\subseteq \{a_1, \cdots, a_s\}$,
by Proposition~\ref{P01}(5) and Remark~\ref{R40}, one has
$B_{\mathbf{i}}\in \mathcal{A}_{B_{\mathbf{u}}}(R)$ and so
$\mathrm{Tor}^R_{\geqslant1}(B_{\mathbf{i}}, B_{\mathbf{u}})=0$.
Hence $\mathrm{Tor}^R_{\geqslant1}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$.
By Proposition~\ref{P01}(5) and Remark~\ref{R40}, the $R$-module
$B_{\mathbf{i}}\otimes_R B_{\mathbf{u}}$ is semidualizing and so \\
$B_{\mathbf{i}}\otimes_R B_{\mathbf{u}}= B_{\mathbf{i}\cup\mathbf{u}}$.
Therefore one has the natural $R$-module isomorphism
$$\eta: R_{_{\Lambda}}\otimes_R R_{_{\Gamma}}\longrightarrow
R_{_{{\Lambda}\cap{\Gamma}}}\, , \ \ \ \
\eta\big( (\alpha_{\mathbf{i}})_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}
\otimes(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}\big)
=\big(\alpha_{\mathbf{i}}.\theta_{\mathbf{u}}\big)_{\scriptsize{\begin{array}{l}
\mathbf{i}\subseteq \{b_1, \cdots, b_t\}\\
\mathbf{u}\subseteq \{a_1, \cdots, a_s\}
\end{array}}}$$
It is routine to check that $\eta$ is also a ring isomorphism.
On the other hand the natural maps
$$\zeta:R_{_{\Lambda}}\rightarrow R_{_{\Lambda}}\otimes_R R_{_{\Gamma}}\, , \ \ \ \
\zeta{\big(}(\alpha_{\mathbf{i}})_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}{\big)}=(\alpha_{\mathbf{i}})_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}
\otimes(\stackrel{\centerdot}{\theta}_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}$$
and
$$\varepsilon:R_{_{\Lambda}}\rightarrow R_{_{{\Lambda}\cap {\Gamma}}}\, , \ \ \ \
\varepsilon((\alpha_{\mathbf{i}})_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}{\big)}=(\chi_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1,\cdots, a_s, b_1, \cdots, b_t\}}\, ,
\hspace{1cm}$$
$$\mathrm{where}\ \ \ \ \ \ \ \ \ \ \ \ \
\stackrel{\centerdot}{\theta}_{\mathbf{u}}= \left\{ \begin{array}{ll}
0 &\text{if}\ \ \mathbf{u}\neq \emptyset\\
1 &\text{if}\ \ \mathbf{u}=\emptyset
\end{array} \right. \ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \ \
\chi_{\mathbf{v}}=\left\{ \begin{array}{ll}
\alpha_{\mathbf{v}}& \text{if}\ \ \mathbf{v}\cap \{a_1, \cdots, a_s\}=\emptyset\\
0 & \text{if}\ \ \mathbf{v}\cap \{a_1, \cdots, a_s\}\neq\emptyset\, ,
\end{array} \right. $$
are ring homomorphism.
It is easy to check that $\eta\zeta=\varepsilon$. Hence
$R_{_{\Lambda}}\otimes_R R_{_{\Gamma}}\stackrel{\eta}{\longrightarrow} R_{_{{\Lambda}\cap {\Gamma}}}$
is an $R_{_{\Lambda}}$-algebra isomorphism.
Now let ${\Lambda}\cup {\Gamma}\subsetneq [n]$, then, by Construction~\ref{cons},
there exist $k\in [n]$ and ${\Lambda}', {\Gamma}'\subseteq[n-1]$
such that there are $R$-isomorphisms and ring isomorphisms
$$R_{_{\Lambda}}\cong Q_k/(\Sigma_{l\in {\Lambda}'} I_{k, l})\, , \ \ \ \ \ \ \ \ \
R_{_{\Gamma}}\cong Q_k/(\Sigma_{l\in {\Gamma}'} I_{k, l})$$
$$R_{_{{\Lambda}\cup {\Gamma}}}\cong Q_k/(\Sigma_{l\in {\Lambda}'\cup {\Gamma}'} I_{k, l})\, ,\ \mathrm{and}\ \ \ R_{_{{\Lambda}\cap {\Gamma}}}\cong Q_k/(\Sigma_{l\in {\Lambda}'\cap {\Gamma}'} I_{k, l})\,.$$
Thus, by induction, for all $i\geqslant 1$
$$\mathrm{Tor}^{R_{_{{\Lambda}\cup {\Gamma}}}}_{i}(R_{_{\Lambda}}, R_{_{\Gamma}})\cong \mathrm{Tor}^{Q_k/(\Sigma_{l\in {\Lambda}'\cup {\Gamma}'} I_{k,l})}_{i}(Q_k/(\Sigma_{l\in {\Lambda}'} I_{k,l}), Q_k/(\Sigma_{l\in {\Gamma}'} I_{k,l}))=0$$
and there is $Q_k/(\Sigma_{l\in {\Lambda}'} I_{k, l})$-algebra isomorphism and so $R_{_{\Lambda}}$-algebra isomorphism as follows
$$\begin{array}{llll}
R_{_{\Lambda}}\otimes_{R_{_{{\Lambda}\cup {\Gamma}}}} R_{_{\Gamma}} & \cong & Q_k/(\Sigma_{l\in {\Lambda}'} I_{k,l})\otimes_{Q_k/(\Sigma_{l\in {\Lambda}'\cup {\Gamma}'} I_{k, l})} Q_k/(\Sigma_{l\in {\Gamma}'} I_{k, l}) \\
& \cong & Q_k/(\Sigma_{l\in {\Lambda}'\cap {\Gamma}'} I_{k, l}) \\
& \cong & R_{_{{\Lambda}\cap {\Gamma}}}\,.
\end{array}$$
\end{proof}
\begin{lem}\label{L46}
Under the hypothesis of Theorem \ref{T42}, if ${\Lambda}$ and $\Gamma$ are two subsets of $[n]$, then $\mathrm{Tor}^{R_{_{\Lambda}}}_{\geqslant 1}(R_{_{{\Lambda}\cup {\Gamma}}}, R_{_{{\Lambda}\cap {\Gamma}}})=0$. Moreover, there is an $R_{_{{\Lambda}\cap {\Gamma}}}$-module isomorphism
$R_{_{{\Lambda}\cup {\Gamma}}}\otimes_{R_{_{\Lambda}}} R_{_{{\Lambda}\cap {\Gamma}}}\cong R_{_{\Gamma}}$.
\end{lem}
\begin{proof}
It is proved by induction on $n$. If $n=1$, there is nothing to prove. The case $n=2$ is proved in \cite[Lemma 3.11]{jls-w}.
Let $n>2$ and suppose that the assertion holds true for $n-1$.
First assume that ${\Lambda}\cup {\Gamma}=[n]$.
Let $\mathbf{P}$ be an $R$-projective resolution of $R_{_{\Gamma}}$. Lemma~\ref{L45} implies that
$R_{_{\Lambda}}\otimes_R \mathbf{P}$ is an $R_{_{\Lambda}}$-projective resolution of
$R_{_{\Lambda}}\otimes_R R_{_{\Gamma}}\cong R_{_{{\Lambda}\cap {\Gamma}}}$.
One has the following natural isomorphisms
$$R\otimes_{R_{_{\Lambda}}}(R_{_{\Lambda}}\otimes_R\mathbf{P})\cong (R\otimes_{R_{_{\Lambda}}}R_{_{\Lambda}})\otimes_R\mathbf{P}\cong R\otimes_R \mathbf{P} \cong \mathbf{P}$$
and then, for all $i\geqslant 1$,
$$\hspace{-0.5cm}\mathrm{Tor}^{R_{_{\Lambda}}}_{i}(R, R_{_{{\Lambda}\cap {\Gamma}}}) \cong
\mathrm{H}_i(R\otimes_{R_{_{\Lambda}}}(R_{_{\Lambda}}\otimes_R\mathbf{P})) \cong \mathrm{H}_i(\mathbf{P}) =0\, .$$
Set $[n]\setminus {\Lambda}=\{b_1, \cdots, b_t\}$ and
$[n]\setminus {\Gamma}=\{a_1, \cdots, a_s\}$.
Then
$[n]\setminus ({\Lambda}\cap {\Gamma})=\{b_1, \cdots, b_t, a_1, \cdots, a_s\}$.
Consider the $R$-module isomorphism
$\xi : R_{_{\Gamma}}\stackrel{\cong}{\longrightarrow} R\otimes_{R_{_{\Lambda}}}R_{_{{\Lambda}\cap {\Gamma}}}$
which is the composition
$$R_{_{\Gamma}}\stackrel{\cong}{-\hspace{-0.2cm}\longrightarrow} R\otimes_R R_{_{\Gamma}} \stackrel{\cong}{-\hspace{-0.2cm}\longrightarrow}
R\otimes_{R_{_{\Lambda}}}(R_{_{\Lambda}}\otimes_R \, R_{_{\Gamma}}) \stackrel{\cong}{\underset{R\otimes \eta}{-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}\longrightarrow}} R\otimes_{R_{_{\Lambda}}}R_{_{{\Lambda}\cap {\Gamma}}}$$
given by
$$\begin{array}{llll} (\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}\ \mapsto
\ 1\otimes (\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}&\mapsto& 1\otimes[(\stackrel{\centerdot}{\alpha}_{\mathbf{i}})_{\mathbf{i}\subseteq \{b_1, \cdots, b_t\}}\otimes(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}]\\
&\mapsto& 1\otimes (\lambda_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}\, ,
\end{array}$$
$$\mathrm{where} \hspace{1.2cm} \stackrel{\centerdot}{\alpha}_{\mathbf{i}}= \left\{ \begin{array}{ll}
0 &\text{if}\ \ \mathbf{i}\neq \emptyset\\
1 & \text{if}\ \ \mathbf{i}=\emptyset
\end{array} \right. \ \ \ \ \ \ \ \ \mathrm{and}\ \ \ \ \ \ \ \ \
\lambda_{\mathbf{v}}=\left\{ \begin{array}{ll}
\theta_{\mathbf{v}}&\text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}=\emptyset\\
0 &\text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}\neq\emptyset
\end{array} \right. . \hspace{2.2cm} $$
We claim that $\xi$ is an $R_{_{{\Lambda}\cap {\Gamma}}}$-module isomorphism.
\textit{Proof of the claim:}
The $R_{_{{\Lambda}\cap {\Gamma}}}$-module structure of $R_{_{\Gamma}}$, which is given via the natural surjection
$R_{_{{\Lambda}\cap {\Gamma}}}\rightarrow R_{_{\Gamma}}$, is described as
$$(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}= (\gamma_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}},$$
where $(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}$ is an element of $R_{_{{\Lambda}\cap {\Gamma}}}$.
In the following we check that
$$\xi\big((\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}\big)=
(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}[\xi((\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}})].$$
Note that
$$\begin{array}{llll}
\xi\big((\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}\big) & = & \xi((\gamma_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}) \\
& = & \xi\big( (\sigma_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}\big) \\
& = & 1\otimes (\mu_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}},
\end{array}$$
where
$(\sigma_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}= (\gamma_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}}(\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}} \ \mathrm{and}\ \mu_{\mathbf{v}}= \left\{ \begin{array}{ll}
\sigma_{\mathbf{v}} &\text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}=\emptyset\\
0 &\text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}\neq\emptyset
\end{array} \right.. $
On the other hand
$$\begin{array}{llll}
(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}[\xi((\theta_{\mathbf{u}})_{\mathbf{u}\subseteq \{a_1, \cdots, a_s\}})]&\hspace{-0.15cm} = &\hspace{-0.2cm}(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}[1\otimes (\lambda_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}]\\
&\hspace{-0.15cm} = &\hspace{-0.2cm} 1\otimes [(\gamma_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}(\lambda_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}]\\
&\hspace{-0.15cm} = &\hspace{-0.2cm} 1\otimes (\varrho_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}\\
&\hspace{-0.15cm} = &\hspace{-0.2cm} [1\otimes (\mu_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}] + [1\otimes \delta],
\end{array}$$
where
$\delta=(\delta_{\mathbf{v}})_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}$ with
$\delta_{\mathbf{v}}=\left\{ \begin{array}{ll}
0 & \text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}=\emptyset\\
\varrho_{\mathbf{v}} & \text{if}\ \ \mathbf{v}\cap \{b_1, \cdots, b_t\}\neq\emptyset
\end{array}\right. .$
It is enough to show that $1\otimes \delta=0$.
To this end, we have
$$1\otimes \delta=
\sum_{\scriptsize{\begin{array}{c}
\mathbf{w}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}\\
\mathbf{w}\cap \{b_1, \cdots, b_t\}\neq \emptyset
\end{array}}}
1\otimes \delta({\tiny{\mathbf{w}}}),$$
where
$\delta({\tiny{\mathbf{w}}})=\big(\delta({\tiny{\mathbf{w}}})_{\mathbf{v}} \big)_{\mathbf{v}\subseteq
\{a_1, \cdots, a_s, b_1, \cdots, b_t\}}$
with
$\delta({\tiny{\mathbf{w}}})_{\mathbf{v}}=\left\{ \begin{array}{ll}
0 & \text{if}\ \ \mathbf{v}\neq \mathbf{w}\\
\delta_{\mathbf{w}} & \text{if}\ \ \mathbf{v}= \mathbf{w}
\end{array}\right. .$
For each $\mathbf{w}$ there exist $\mathbf{w}'\subseteq \{ b_1, \cdots, b_t\}$ and $\mathbf{w}''\subseteq\{a_1, \cdots, a_s\}$ with
$\mathbf{w}'\cup \mathbf{w}''=\mathbf{w}$. Thus $B_{\mathbf{w}'}\otimes_R B_{\mathbf{w}''}\stackrel{\rho_{\mathbf{w}}}{\cong} B_{\mathbf{w}}$ and there exist
$\delta'_{\mathbf{w}}\in B_{\mathbf{w}'}$ and $\delta''_{\mathbf{w}}\in B_{\mathbf{w}''}$
such that
$\delta_{\mathbf{w}}=\rho_{\mathbf{w}}(\delta'_{\mathbf{w}}\otimes \delta''_{\mathbf{w}})$.
Set
$\alpha(\tiny{\mathbf{w}})=\big(\alpha(\tiny{\mathbf{w}})_{\mathbf{i}}\big)_{\mathbf{i}
\subseteq \{ b_1, \cdots, b_t\}}$, where
$\alpha({\tiny{\mathbf{w}}})_{\mathbf{i}}=\left\{\begin{array}{ll}
0 & \text{if}\ \ \mathbf{i}\neq \mathbf{w}'\\
\delta'_{\mathbf{w}} & \text{if}\ \ \tiny{\mathbf{i}= \mathbf{w}'}
\end{array} \right.$.
As the $R_{_\Lambda}$-module structure on $R$ is given via the natural surjection $R_{_\Lambda}\longrightarrow R$, and $\alpha(\tiny{\mathbf{w}})$ is an element
of the kernel of this map,
$0\oplus(\oplus_{\mathbf{i}\subseteq \{ b_1, \cdots, b_t\}, \mathbf{i}\neq \emptyset} B_{\mathbf{i}})$,
we have $1\,\alpha(\tiny{\mathbf{w}})=0$.
Set
$\beta(\tiny{\mathbf{w}})=\big( \beta(\tiny{\mathbf{w}})_{\mathbf{v}}\big)_{\mathbf{v}\subseteq \{a_1, \cdots, a_s, b_1, \cdots, b_t\}}$, where
$\beta({\tiny{\mathbf{w}}})_{\mathbf{v}}=\left\{\begin{array}{ll}
0 & \text{if}\ \ \mathbf{v}\neq \mathbf{w}''\\
\delta''_{\mathbf{w}} & \text{if}\ \ \tiny{\mathbf{v}=\mathbf{w}''}
\end{array} \right.\!.$
Note that $\beta(\tiny{\mathbf{w}})$ is an element of $R_{_{{\Lambda}\cap {\Gamma}}}$ and
$\delta({\tiny{\mathbf{w}}})=\alpha(\tiny{\mathbf{w}})\beta(\tiny{\mathbf{w}})$.
Then
$$1\otimes \delta=\sum_{\mathbf{w}} 1\otimes \delta({\tiny{\mathbf{w}}})=
\sum_{\mathbf{w}} 1\otimes[\alpha(\tiny{\mathbf{w}}) \beta(\tiny{\mathbf{w}})]
= \sum_{\mathbf{w}} [1\alpha(\tiny{\mathbf{w}})]\otimes
\beta(\tiny{\mathbf{w}})
=\sum_{\mathbf{w}} 0\otimes\beta(\tiny{\mathbf{w}})=0.$$
Therefore the claim is proved and also the assertion holds in the case
${\Lambda}\cup {\Gamma}= [n]$.
\hspace{-0.3cm}We treat the case ${\Lambda}\cup {\Gamma}\subsetneq [n]$ by induction and its details are similar to the proof of Lemma~\ref{L45}.
\end{proof}
\vspace{0.2cm}
\emph{Proof of Theorem \ref{T42}}. \ (1) is proved in Construction~\ref{cons}.
(2). It is proved by induction on $n$. The case $n=1$ is clear by assumptions.
Let $n>1$ and the claim is settled for $n-1$.
If $\Lambda=[n]$, then $R_{_\Lambda}\cong R$ and is Cohen-Macaulay with the dualizing module $D$ and is not Gorenstein.
Let $\Lambda\subsetneq [n]$. There exists $k\in [n]$ such that
$\Lambda\subseteq [n]\setminus \{k\}$. By Construction~\ref{cons}, there exists
a subset $\Lambda'\neq \emptyset$ of $[n-1]$ such that
$R_{_\Lambda}\cong Q_k/(\Sigma_{l\in \Lambda'} I_{k,l})$ as ring isomorphism. Thus, by induction, $R_{_\Lambda}$ is non-Gorenstein Cohen-Macaulay ring with dualizing module.
(3). It is clear that $\prod_{l\in \Lambda} I_l\subseteq \bigcap_{l\in \Lambda}I_l$\,. Let $\alpha=\big(\alpha_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}$ be an element of $\bigcap_{l\in \Lambda}I_l$\,.
Then, by Construction~\ref{cons}, $\alpha_{\mathbf{i}}=0$ for all $\mathbf{i}\subseteq [n]$ with $\Lambda\nsubseteq \mathbf{i}$\,. We have
$\alpha=\sum_{\Lambda\subseteq \mathbf{v}\subseteq[n]}\alpha(\tiny{\mathbf{v}})$,
where
$\alpha({\tiny{\mathbf{v}}})=\big(\alpha({\tiny{\mathbf{v}}})_{\mathbf{i}} \big)_{\mathbf{i}\subseteq [n]}$
with
$\alpha({\tiny{\mathbf{v}}})_{\mathbf{i}}=\left\{ \begin{array}{ll}
0 &\text{if}\ \ \mathbf{i}\neq \mathbf{v}\\
\alpha_{\mathbf{v}} &\text{if}\ \ \mathbf{i}= \mathbf{v}
\end{array}\right.$. Set $\Lambda=\{a_1, \cdots, a_m\}$.
If $\mathbf{v}\subseteq [n]$ such that
$\Lambda\subseteq \mathbf{v}$, then
$\mathbf{v}=\{a_1\}\cup\{a_2\}\cup\cdots \cup\{a_{m-1}\}\cup(\mathbf{v}\setminus\{a_1, \cdots, a_{m-1}\})$.
Thus
$$B_{\mathbf{v}}\stackrel{\Phi}{\cong }B_{\{a_1\}}\otimes_R \cdots \otimes_R B_{\{a_{m-1}\}}\otimes_R B_{\mathbf{v}\setminus\{a_1, \cdots, a_{m-1}\}}.$$
Therefore there exist $\theta_{\mathbf{v}, m}\in B_{\mathbf{v}\setminus\{a_1, \cdots, a_{m-1}\}}$ and $\theta_{\mathbf{v}, r}\in B_{\{a_r\}}$, $1\leqslant r< m$, such that
$\alpha_{\mathbf{v}}=\Phi(\theta_{\mathbf{v}, 1}\otimes\cdots \otimes\theta_{\mathbf{v}, m-1}\otimes \theta_{\mathbf{v}, m})$. Set
$\varphi({\tiny{\mathbf{v}}}, r)=\big(\varphi({\tiny{\mathbf{v}}}, r)_{\mathbf{i}}\big)_{\mathbf{i}\subseteq [n]}$, $1\leqslant r\leqslant m$, where, for
$1\leqslant r < m$,
\vspace{0.2cm}
\hspace{-0.4cm}$\varphi({\tiny{\mathbf{v}}}, r)_{\mathbf{i}}=\left\{\begin{array}{ll}
0 & \text{if}\ \ \mathbf{i}\neq \{a_r\}\\
\theta_{\mathbf{v}, r} & \text{if}\ \ \mathbf{i}=\{a_r\}
\end{array} \right.$
and \
$\varphi({\tiny{\mathbf{v}}}, m)_{\mathbf{i}}=\left\{\begin{array}{ll}
0 & \text{if}\ \ \mathbf{i}\neq \mathbf{v}\setminus \{a_1, \cdots, a_{m-1}\}\\
\theta_{\mathbf{v}, m} & \text{if}\ \ \mathbf{i}=\mathbf{v}\setminus \{a_1, \cdots, a_{m-1}\}\,.
\end{array} \right.$
\vspace{0.2cm}
Note that $\varphi({\tiny{\mathbf{v}}}, r)\in I_{a_r}$, $1\leqslant r\leqslant m$.
Hence
$\varphi({\tiny{\mathbf{v}}}, 1)\cdots \varphi({\tiny{\mathbf{v}}}, m-1)
\varphi({\tiny{\mathbf{v}}}, m)\in \prod_{l\in \Lambda} I_l$.
On the other hand
$\varphi({\tiny{\mathbf{v}}}, 1)\cdots \varphi({\tiny{\mathbf{v}}}, m-1)
\varphi({\tiny{\mathbf{v}}}, m)= \alpha(\tiny{\mathbf{v}})$. Thus
$\alpha(\tiny{\mathbf{v}})$ is an element of $\prod_{l\in \Lambda} I_l$ and so
$\alpha\in\prod_{l\in \Lambda} I_l$.
(4) is followed by Remark~\ref{R1} and Lemma~\ref{L44}.
(5). Let $\mathbf{P}$ be a projective resolution of $R_{_{\Lambda\cup \Gamma}}$ over $R_{_{\Lambda}}$.
Lemma~\ref{L46} implies that the complex
$\mathbf{P}\otimes_{R_{_{\Lambda}}} R_{_{{\Lambda}\cap {\Gamma}}}$
is a $R_{_{{\Lambda}\cap {\Gamma}}}$-projective resolution of
$R_{_{{\Lambda}\cup {\Gamma}}}\otimes_{R_{_{\Lambda}}} R_{_{{\Lambda}\cap {\Gamma}}}\cong R_{_{\Gamma}}$.
From the isomorphisms
$$(\mathbf{P}\otimes_{R_{_{\Lambda}}} R_{_{{\Lambda}\cap {\Gamma}}})\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}}R_{_{\Lambda}}
\cong \mathbf{P}\otimes_{R_{_{\Lambda}}}R_{_{\Lambda}}\cong \mathbf{P}$$
one gets
$$\mathrm{Tor}^{R_{_{{\Lambda}\cap {\Gamma}}}}_{i}(R_{_{\Gamma}}, R_{_{\Lambda}})
\cong \mathrm{H}_i((\mathbf{P}\otimes_{R_{_{\Lambda}}} R_{_{{\Lambda}\cap {\Gamma}}})\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}}R_{_{\Lambda}}) \cong
\mathrm{H}_i(\mathbf{P}) =0\, .$$
for all $i\geqslant 1$. There is a natural isomorphism
$R_{_{\Lambda}}\otimes_{R_{_
|
{{\Lambda}\cap {\Gamma}}}}R_{_{\Gamma}}\cong R_{_{{\Lambda}\cup {\Gamma}}}$
which is both $R_{_{{\Lambda}\cap {\Gamma}}}$- and $R_{_{\Gamma}}$-isomorphism.
Let $\mathbf{P}'$ be an $R_{_{{\Lambda}\cap {\Gamma}}}$-projective resolution of $R_{_{\Lambda}}$. As seen in the above,
$\mathbf{P}'\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}}R_{_{\Gamma}}$
is a projective resolution of $R_{_{{\Lambda}\cup {\Gamma}}}$ over
$R_{_{\Gamma}}$. Therefore we have
$$\begin{array}{lll}
\mathrm{Ext}^i_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})&\cong &
\mathrm{H}^i(\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(\mathbf{P}', R_{_{\Gamma}})\\
& \cong & \mathrm{H}^i(\mathrm{Hom}_{R_{_{\Gamma}}}(\mathbf{P}'\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}}R_{_{\Gamma}}, R_{_{\Gamma}})\\
& \cong & \mathrm{Ext}^i_{R_{_{\Gamma}}}(R_{_{{\Lambda}\cup {\Gamma}}}, R_{_{\Gamma}})
\end{array}$$
for all $i\geqslant 0$. By (4), $\mbox{G}$-$\mathrm{dim}_{R_{_{\Gamma}}}R_{_{{\Lambda}\cup {\Gamma}}}=0$, and so one gets
$\mathrm{Ext}^{\geqslant 1}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})=0.$
Also, by (4), $\mathrm{Hom}_{R_{_{\Gamma}}}(R_{_{{\Lambda}\cup {\Gamma}}}, R_{_{\Gamma}})$ is a non-free semidualizing
$R_{_{{\Lambda}\cup {\Gamma}}}$-module and thus $\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})$ is not cyclic.
(6). As $R_{_{{\Lambda}\cap {\Gamma}}}=Q/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l) $ and
$R_{_{\Lambda}}=Q/(\Sigma_{l\in {\Lambda}} I_l)\cong R_{_{{\Lambda}\cap {\Gamma}}}/
({\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)})$,
one has the natural isomorphism
$$\kappa: \mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}} , R_{_{{\Lambda}\cap {\Gamma}}})\longrightarrow\big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big), \ \
\kappa(\psi)=\psi(\stackrel{\centerdot}{\alpha}),$$
where
$\stackrel{\centerdot}{\alpha}=(\stackrel{\centerdot}{\alpha}_{\mathbf{i}})_{\mathbf{i}\subseteq [n]\setminus \Lambda}$
with
$\stackrel{\centerdot}{\alpha}_{\mathbf{i}}= \left\{ \begin{array}{ll}
0 &\text{if}\ \ \mathbf{i}\neq \emptyset\\
1 & \text{if}\ \ \mathbf{i}=\emptyset
\end{array} \right.$
is the identity element of $R_{_\Lambda}$.
Next we show that
$\big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big)= {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}$.
Set $\Lambda\setminus\Gamma=\{a\}$.
Let
$\gamma=(\gamma_{\mathbf{i}})_{\mathbf{i}\subseteq [n]\setminus{{\Lambda}\cap {\Gamma}}}$
be an element of
$\big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big)$.
If $\gamma\notin {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}$, then
there exists $\mathbf{v}\subseteq [n]\setminus{{\Lambda}\cap {\Gamma}}$ such that
$a\notin \mathbf{v}$ and $\gamma_{\mathbf{v}}\neq 0$.
Set $M=R\gamma_{\mathbf{v}}$, which is a non-zero submodule of $B_{\mathbf{v}}$.
As $B_a$ is a semidualizing $R$-module and $M\neq 0$, we have $B_a\otimes_R M\neq0$.
Thus there exists an element $e$ of $B_a$ such that $e\otimes\gamma_{\mathbf{v}}\neq0$.
Set $\theta=(\theta_{\mathbf{i}})_{\mathbf{i}\subseteq [n]\setminus{{\Lambda}\cap {\Gamma}}}$,
where \\
$\theta_{\mathbf{i}}= \left\{ \begin{array}{ll}
0 &\text{if}\ \ \mathbf{i}\neq \{a\}\\
e & \text{if}\ \ \mathbf{i}=\{a\}
\end{array} \right.$.
Note that $\theta$ is an element of
${\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}$ and
$\gamma \theta \neq0$, which contradicts with
$\gamma\in \big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big)$.
Therefore
$$\big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big)\subseteq {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}.$$
On the other hand
${\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}\subseteq
\big(0 :_{R_{_{{\Lambda}\cap {\Gamma}}}} {\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)} \big)$. Indeed, if
$\alpha=(\alpha_{\mathbf{i}})_{\mathbf{i}\subseteq [n]\setminus{{\Lambda}\cap {\Gamma}}}$
and
$\alpha'=(\alpha'_{\mathbf{i}})_{\mathbf{i}\subseteq [n]\setminus{{\Lambda}\cap {\Gamma}}}$
are two elements of ${\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}$,
then $\alpha_{\mathbf{i}}=0=\alpha'_{\mathbf{i}}$ for all $\mathbf{i}$ such that $a\notin\mathbf{i}$.
Hence, by Lemma~\ref{L43}, $\alpha\alpha'=0$.
Thus
\begin{equation}\label{e47}
\kappa: \mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}} , R_{_{{\Lambda}\cap {\Gamma}}})\longrightarrow{\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}, \ \
\kappa(\psi)=\psi(\stackrel{\centerdot}{\alpha})
\end{equation}
is an $R_{_{{\Lambda}\cap {\Gamma}}}$-isomorphism.
By (4), $\mbox{G}$-$\mathrm{dim}_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Lambda}}=0$.
Let $\mathbf{F}$ be a minimal free resolution of $R_{_{\Lambda}}$ over $R_{_{{\Lambda}\cap {\Gamma}}}$. Note that ${\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}$ is the first syzygy of $R_{_{\Lambda}}$ in $\mathbf{F}$. By \cite[Construction 3.6]{am} and
(\ref{e47}), we can construct a Tate resolution of $R_{_{\Lambda}}$ as $\mathbf{T}\rightarrow \mathbf{F}\rightarrow R_{_{\Lambda}}$, where $\mathbf{T}$ construct by splicing $\mathbf{F}$ with
$\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(\mathbf{F} , R_{_{{\Lambda}\cap {\Gamma}}})$. Hence
$\mathbf{T}\cong \mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(\mathbf{T} , R_{_{{\Lambda}\cap {\Gamma}}})$.
This explains the first isomorphism in the next sequence
\begin{equation}\label{e43}
\begin{array}{lll}
\widehat{\mathrm{Tor}}^{R_{_{{\Lambda}\cap {\Gamma}}}}_i(R_{_{\Lambda}}, R_{_{\Gamma}})& = & \mathrm{H}_i\big(\mathbf{T}\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Gamma}}\big)\\
& \cong & \mathrm{H}_i\big(\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(\mathbf{T} , R_{_{{\Lambda}\cap {\Gamma}}})\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Gamma}}\big)\\
& \cong & \mathrm{H}_i\big(\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(\mathbf{T} , R_{_{\Gamma}})\big)\\
& = & \widehat{\mathrm{Ext}}^{-i}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})
\end{array}
\end{equation}
for all $i\in \mathbb{Z}$.
As each $R_{_{{\Lambda}\cap {\Gamma}}}$-module $\mathbf{T}_i$ is finite and free,
the second isomorphism follows.
By (4), $\mbox{G}$-$\mathrm{dim}_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Lambda}}=0$ and so,
by \cite[Theorem 5.2]{am}, one has
\begin{equation}\label{e44}
\widehat{\mathrm{Tor}}^{R_{_{{\Lambda}\cap {\Gamma}}}}_i(R_{_{\Lambda}}, R_{_{\Gamma}})
\cong \mathrm{Tor}^{R_{_{{\Lambda}\cap {\Gamma}}}}_i(R_{_{\Lambda}}, R_{_{\Gamma}})
\ \ \ \ \mathrm{and}\ \ \ \ \
\widehat{\mathrm{Ext}}^{i}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}}) \cong \mathrm{Ext}^{i}_{R_{_{{\Lambda}\cap {\Gamma}}}}
(R_{_{\Lambda}}, R_{_{\Gamma}})
\end{equation}
for all $i\geqslant 1$. Thus, by (\ref{e43}), (\ref{e44}) and (5), one gets
$$\widehat{\mathrm{Ext}}^{-i}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})\cong \widehat{\mathrm{Tor}}^{R_{_{{\Lambda}\cap {\Gamma}}}}_i(R_{_{\Lambda}}, R_{_{\Gamma}})\cong \mathrm{Tor}^{R_{_{{\Lambda}\cap {\Gamma}}}}_i(R_{_{\Lambda}}, R_{_{\Gamma}})=0\,,$$
$$\widehat{\mathrm{Tor}}^{R_{_{{\Lambda}\cap {\Gamma}}}}_{-i}(R_{_{\Lambda}}, R_{_{\Gamma}}) \cong \widehat{\mathrm{Ext}}^{i}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})
\cong \mathrm{Ext}^{i}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$$
for all $i\geqslant 1$. Therefore, by (\ref{e43}), to complete the proof it is enough to show that $\widehat{\mathrm{Ext}}^{0}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$.
As $\widehat{\mathrm{Ext}}^{-1}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$ and $R_{_{\Lambda}}$ is totally reflexive as an
$R_{_{{\Lambda}\cap {\Gamma}}}$-module one has, by \cite[Lemma 5.8]{am},
the exact sequence
\begin{equation}\label{e48}
0\rightarrow \mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}} , R_{_{{\Lambda}\cap {\Gamma}}})\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Gamma}}
\stackrel{\nu}{\longrightarrow} \mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})
\longrightarrow \widehat{\mathrm{Ext}}^{0}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})\rightarrow 0\,,
\end{equation}
where the map $\nu$ is given by
$$\nu(\psi\otimes \theta)=\psi_{_\theta}\, , \ \ \ \psi_{_\theta}(\alpha)=\psi(\alpha)\theta\, .$$
In a similar way to (\ref{e47}), one gets the natural
isomorphism
$\tau: \mathrm{Hom}_{R_{_{\Gamma}}}(R_{_{{\Lambda}\cup {\Gamma}}},
R_{_{\Gamma}})\longrightarrow
{\Sigma_{l\in {\Lambda}\cup {\Gamma}} I_l/(\Sigma_{l\in {\Gamma}} I_l)}$
given by $\tau(\psi)=\psi(\stackrel{\centerdot}{\varphi})$,
where $\stackrel{\centerdot}{\varphi}$
is the identity element of $R_{_{{\Lambda}\cup {\Gamma}}}$.
It is straightforward to show that the following diagram commutes:
$$\begin{array}{ccc}
\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}} , R_{_{{\Lambda}\cap {\Gamma}}})\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Gamma}} &
\stackrel{\nu}{-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}
\longrightarrow} &
\mathrm{Hom}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}}) \\
\begin{array}{ll} \kappa\otimes R_{_\Gamma} {\Bigg\downarrow} \cong \end{array} &
& \begin{array}{ll} f {\bigg\downarrow} \cong \end{array}\\
{\Sigma_{l\in {\Lambda}} I_l/(\Sigma_{l\in {\Lambda}\cap {\Gamma}} I_l)}\otimes_{R_{_{{\Lambda}\cap {\Gamma}}}} R_{_{\Gamma}} &
\stackrel{\cong}{\underset{g}{\longrightarrow}}
I_a/(\Sigma_{l\in {\Gamma}} I_aI_l)
\stackrel{\cong}{\underset{h}{\longrightarrow}}
\Sigma_{l\in {\Lambda}\cup {\Gamma}} I_l/(\Sigma_{l\in {\Gamma}} I_l)
\stackrel{\cong}{\underset{\tau}{\longleftarrow}}&
\mathrm{Hom}_{R_{_{\Gamma}}}(R_{_{{\Lambda}\cup {\Gamma}}}, R_{_{\Gamma}})\,,
\end{array}$$
where the maps $f, g$ and $h$ are natural isomorphisms.
Hence $\nu$ is surjective and (\ref{e48}) implies that
$\widehat{\mathrm{Ext}}^{0}_{R_{_{{\Lambda}\cap {\Gamma}}}}(R_{_{\Lambda}}, R_{_{\Gamma}})=0$.
\hspace{10.3cm}$\Box$
The following results give a partial converse to Theorem~\ref{T42}.
Note that Proposition~\ref{P43} is a generalization of the result
of Jorgensen et.\! al. \cite[Theorem 3.1]{jls-w}.
\begin{pro}\label{P42}
Let $R$ be a Cohen-Macaulay ring. Assume that there exist a Gorenstein
local ring $Q$ and ideals $I_1, \cdots, I_n$ of $Q$
satisfying the following conditions.
\begin{itemize}
\item[(1)] There is a ring isomorphism $R\cong Q/(I_1+\cdots + I_n)$.
\item[(2)] The ring $R_k=Q/(I_1+\cdots +I_k)$ is Cohen-Macaulay for all $k\in [n]$.
\item[(3)] $\mathrm{fd}_{R_j}(R_k)<\infty$ for all $k\in [n]$ and all $1\leqslant j \leqslant k$.
\item[(4)] For each $k\in [n]$ and all $0\leqslant j < k$,
$\mathrm{I}_{R_k}^{R_k}(t)\neq t^e \mathrm{I}_{R_{j}}^{R_{j}}(t)$ for any integer $e$. ($R_0=Q$)
\end{itemize}
Then there exist integers $g_0, g_1, \cdots, g_{n-1}$ such that
$$[\mathrm{Ext}_{Q}^{g_0}(R, Q)] \vartriangleleft [\mathrm{Ext}_{R_1}^{g_1}(R, R_1)]\vartriangleleft \cdots \vartriangleleft
[\mathrm{Ext}_{R_{n-1}}^{g_{n-1}}(R, R_{n-1})]\vartriangleleft [R]$$
is a chain in $\mathfrak{G}_0(R)$ of length $n$.
\end{pro}
\begin{proof}
We prove by induction. For $n=1$, it is clear that $\mathrm{Ext}_{Q}^{g_0}(R, Q)$ is a dualizing
$R$-module for some integer $g_0$. It will be shown in following that the condition (4) implies
$[\mathrm{Ext}_{Q}^{g_0}(R, Q)]\vartriangleleft[R]$.
Let $n=2$. As $\mathrm{fd}_{R_1}(R)<\infty$, one has $\mbox{G}$-$\mathrm{dim}_{R_1}(R)<\infty$.
Then, by Remark~\ref{R1}, there exists an integer $g_1$ such that
$\mathrm{Ext}_{R_1}^{i}(R, R_1)=0$ for all $i\neq g_1$ and
$C_1=\mathrm{Ext}_{R_1}^{g_1}(R, R_1)$ is a semidualizing $R$-module.
Therefore there is an isomorphism $C_1\simeq \Sigma^{g_1}\mathbf{R}\mathrm{Hom}_{R_1}(R, R_1)$ in the derived category $\mathrm{D}(R)$.
Thus, by \cite[(1.7.8)]{c2}, $\mathrm{I}_R^{C_1}(t)=t^{-g_1}\mathrm{I}_{R_1}^{R_1}(t)$.
Also there exists an integer $g_0$ such that
$\mathrm{Ext}_{Q}^{i}(R, Q)=0$ for all $i\neq g_0$ and
$D=\mathrm{Ext}_{Q}^{g_0}(R, Q)$ is a dualizing $R$-module and then
$D\simeq \Sigma^{g_0}\mathbf{R}\mathrm{Hom}_Q(R, Q)$ in $\mathrm{D}(R)$.
Assumption (4) implies that $C_1$ is a non-trivial semidualizing $R$-module and so
$[D]\vartriangleleft[C_1]\vartriangleleft[R]$ is a chain in
$\mathfrak{G}_0(R)$ of length $2$.
Let $n>2$ and suppose that the assertion holds true for $n-1$. By induction there exist
integers $h_0, h_1, \cdots, h_{n-2}$ such that
\begin{equation}\label{e49}
[\mathrm{Ext}_Q^{h_0}(R_{n-1}, Q)] \vartriangleleft [\mathrm{Ext}_{R_1}^{h_1}(R_{n-1}, R_1)]\vartriangleleft \cdots \vartriangleleft
[\mathrm{Ext}_{R_{n-2}}^{h_{n-2}}(R_{n-1}, R_{n-2})]\vartriangleleft [R_{n-1}]
\end{equation}
is a chain in $\mathfrak{G}_0(R_{n-1})$ of length $n-1$.
( In fact, there is an isomorphism
$\mathrm{Ext}_{R_i}^{h_i}(R_{n-1}, R_i)\simeq \Sigma^{h_i}\mathbf{R}\mathrm{Hom}_{R_i}(R_{n-1}, R_i)$
in $\mathrm{D}(R_{n-1})$ for all $0\leqslant i\leqslant n-2$.)
As $\mathrm{fd}_{R_k}(R)<\infty$, one has $\mbox{G}$-$\mathrm{dim}_{R_k}(R)<\infty$
for all $k\in [n]$ and so, by Remark~\ref{R1}, there exists an integer $g_k$ such that
$\mathrm{Ext}_{R_k}^{i}(R, R_k)=0$ for all $i\neq g_k$ and
$C_k=\mathrm{Ext}_{R_k}^{g_k}(R, R_k)$ is a semidualizing $R$-module.
We have $C_k\simeq \Sigma^{g_k}\mathbf{R}\mathrm{Hom}_{R_k}(R, R_k)$ in $\mathrm{D}(R)$.
Also there exists an integer $g_0$ such that
$\mathrm{Ext}_{Q}^{i}(R, Q)=0$ for all $i\neq g_0$ and
$D=\mathrm{Ext}_{Q}^{g_0}(R, Q)$ is a dualizing for $R$ and so
$D\simeq \Sigma^{g_0}\mathbf{R}\mathrm{Hom}_Q(R, Q)$ in $\mathrm{D}(R)$.
Note that there is an isomorphism
$\mathbf{R}\mathrm{Hom}_{R_k}(R, R_k)\simeq \mathbf{R}\mathrm{Hom}_{R_{n-1}}(R, \mathbf{R}\mathrm{Hom}_{R_k}(R_{n-1}, R_k))$, $0\leqslant k \leqslant n-1$,
in $\mathrm{D}(R)$, and $R$ is a finite $R_{n-1}$-module with
$\mathrm{fd}_{R_{n-1}}(R)<\infty$.
Thus, by \cite[Theorem 5.7]{fs-w1} and (\ref{e49}), one obtains
$[\mathrm{Ext}_{R_{k-1}}^{g_{k-1}}(R, R_{k-1})]\trianglelefteq[\mathrm{Ext}_{R_k}^{g_k}(R, R_k)]$ for all $1\leqslant k \leqslant n-1$.
By \cite[(1.7.8)]{c2}, $\mathrm{I}_R^{C_k}(t)=t^{-g_k}\mathrm{I}_{R_k}^{R_k}(t)$ for all
$1\leqslant k \leqslant n-1$ and $\mathrm{I}_R^{D}(t)=t^{-g_0}\mathrm{I}_{Q}^{Q}(t)$.
Therefore, by condition (4),
$[\mathrm{Ext}_{R_{k-1}}^{g_{k-1}}(R, R_{k-1})]\vartriangleleft[\mathrm{Ext}_{R_k}^{g_k}(R, R_k)]$ for all $1\leqslant k \leqslant n-1$, and
$[\mathrm{Ext}_{R_{n-1}}^{g_{n-1}}(R, R_{n-1})]\vartriangleleft[R]$.
Hence
$$[\mathrm{Ext}_Q^{g_0}(R, Q)] \vartriangleleft [\mathrm{Ext}_{R_1}^{g_1}(R, R_1)]
\vartriangleleft \cdots \vartriangleleft
[\mathrm{Ext}_{R_{n-1}}^{g_{n-1}}(R, R_{n-1})]\vartriangleleft [R]$$
is a chain in $\mathfrak{G}_0(R)$ of length $n$.
\end{proof}
\begin{pro}\label{P43}
Let $R$ be a Cohen-Macaulay ring. Assume that there exist a Gorenstein
local ring $Q$ and ideals $I_1, \cdots, I_n$ of $Q$
satisfying the following conditions.
\begin{itemize}
\item[(1)] There is a ring isomorphism $R\cong Q/(I_1+\cdots + I_n)$.
\item[(2)] For each $\Lambda\subseteq [n]$,
the ring $R_{_\Lambda}=Q/(\Sigma_{l\in \Lambda} I_l)$ is Cohen-Macaulay.
\item[(3)] For subsets $\Lambda$, $\Gamma$ of $[n]$ with $\Lambda\cap \Gamma=\emptyset$
\begin{itemize}
\item[(i)] $\mathrm{Tor}^Q_{\geqslant1}(R_{_\Lambda}, R_{_\Gamma})=0;$
\item[(ii)] For all $i \in \mathbb{Z}$, $\hspace{0.3cm}\mathrm{\widehat{Ext}}^i_Q (R_{_\Lambda}, R_{_\Gamma})=0=\mathrm{\widehat{Tor}}^Q_i(R_{_\Lambda}, R_{_\Gamma}).$
\end{itemize}
\item[(4)] For two subsets $\Lambda$, $\Gamma$ of $[n]$ with $\Lambda\neq \Gamma$ and
for any integer $e$,
$\mathrm{I}_{R_{_\Lambda}}^{R_{_\Lambda}}(t)\neq t^e \mathrm{I}_{R_{_\Gamma}}^{R_{_\Gamma}}(t)$.
\end{itemize}
Then, for each $\Lambda\subseteq [n]$, there is an integer $g_{_\Lambda}$ such that
$\mathrm{Ext}^{g_{_\Lambda}}_{R_{_\Lambda}}(R, R_{_\Lambda})$
is a semidualizing $R$-module.
As conclusion, $R$ admits $2^n$ non-isomorphic semidualizing modules.
\end{pro}
\begin{proof}
For two subsets $\Lambda$, $\Gamma$ of $[n]$ with $\Gamma\subseteq \Lambda$, we have
$\mbox{G}$-$\mathrm{dim}_{R_{_\Gamma}}(R_{_\Lambda})<\infty$.
Indeed, $\mbox{G}$-$\mathrm{dim}_Q(R_{_{\Lambda\setminus\Gamma}})
<\infty$, since $Q$ is Gorenstein. Thus $R_{_{\Lambda\setminus\Gamma}}$
admits a Tate resolution
$\mathbf{T}\stackrel{\vartheta}{\longrightarrow}\mathbf{P}\stackrel{\pi}{\longrightarrow}
R_{_{\Lambda\setminus\Gamma}}$ over $Q$, where
$\vartheta_i$ is isomorphism for all $i\gg 0$.
We show that the induced diagram
$\mathbf{T}\otimes_Q R_{_\Gamma}\stackrel{\vartheta\otimes_Q R_{_\Gamma}}{-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}\longrightarrow}
\mathbf{P}\otimes_Q R_{_\Gamma}
\stackrel{\pi\otimes_Q R_{_\Gamma}}{-\hspace{-0.2cm}-\hspace{-0.2cm}-\hspace{-0.2cm}\longrightarrow}
R_{_{\Lambda\setminus\Gamma}}\otimes_Q R_{_\Gamma}$
is a Tate resolution of
$R_{_{\Lambda\setminus\Gamma}}\otimes_Q R_{_\Gamma}\cong R_{_\Lambda}$
over $R_{_\Gamma}$.
By condition (3)(i), $\mathbf{P}\otimes_Q R_{_\Gamma}$ is a free resolution of
$R_{_\Lambda}$ over $R_{_\Gamma}$.
Also by assumption,
$\mathrm{\widehat{Tor}}^Q_i(R_{_{\Lambda\setminus\Gamma}}, R_{_\Gamma})=0$ for all
$i \in \mathbb{Z}$ and then $\mathbf{T}\otimes_Q R_{_\Gamma}$ is an exact complex
of finite free $R_{_\Gamma}$-modules.
Of course, the map $\vartheta_i\otimes_Q R_{_\Gamma}$ is an isomorphism for all $i\gg 0$.
In order to show that
$\mathrm{Hom}_{R_{_\Gamma}}(\mathbf{T}\otimes_Q R_{_\Gamma}, R_{_\Gamma})$
is exact we note that the sequence of isomorphisms
$$\mathrm{Hom}_{R_{_\Gamma}}(\mathbf{T}\otimes_Q R_{_\Gamma}, R_{_\Gamma})
\cong \mathrm{Hom}_Q(\mathbf{T}, \mathrm{Hom}_{R_{_\Gamma}}( R_{_\Gamma}, R_{_\Gamma}))\cong \mathrm{Hom}_Q(\mathbf{T}, R_{_\Gamma}),$$
implies that
$$\mathrm{H}_i(\mathrm{Hom}_{R_{_\Gamma}}(\mathbf{T}\otimes_Q R_{_\Gamma}, R_{_\Gamma}))\cong\mathrm{H}_i( \mathrm{Hom}_Q(\mathbf{T}, R_{_\Gamma}))
\cong \mathrm{\widehat{Ext}}^{-i}_Q (R_{_{\Lambda\setminus\Gamma}}, R_{_\Gamma})$$
which is zero, by condition (3)(ii), for all $i \in \mathbb{Z}$.
Hence the complex
$\mathrm{Hom}_{R_{_\Gamma}}(\mathbf{T}\otimes_Q R_{_\Gamma}, R_{_\Gamma})$
is exact and so
$R_{_\Lambda}$ admits a Tate resolution over $R_{_\Gamma}$. Therefore
$\mbox{G}$-$\mathrm{dim}_{R_{_\Gamma}}(R_{_\Lambda})<\infty$.
In particular, $\mbox{G}$-$\mathrm{dim}_{R_{_\Lambda}}(R)<\infty$ for all $\Lambda\subseteq [n]$.
Hence, by Remark~\ref{R1}, $\mathrm{Ext}^i_{R_{_\Lambda}}(R, R_{_\Lambda})=0$ for all $i\neq g_{_\Lambda}$,
where $g_{_\Lambda}:=\mbox{G}$-$\mathrm{dim}_{R_{_\Lambda}}(R)$, and
$C_{_\Lambda}:=\mathrm{Ext}^{g_{_\Lambda}}_{R_{_\Lambda}}(R, R_{_\Lambda})$
is a semidualizing $R$-module.
Note that there is an isomorphism
$C_{_\Lambda}\simeq \Sigma^{g_{_\Lambda}}\mathbf{R}\mathrm{Hom}_{R_{_\Lambda}}(R, R_{_\Lambda})$ in the derived category $\mathrm{D}(R)$. Therefore, by \cite[(1.7.8)]{c2},
$$\mathrm{I}_R^{C_{_\Lambda}}(t)=\mathrm{I}_R^{\Sigma^{^{g_{_\Lambda}}}\mathbf{R}\mathrm{Hom}_{R_{_\Lambda}}(R, R_{_\Lambda})}(t)=t^{-g_{_\Lambda}}\mathrm{I}_{R_{_\Lambda}}^{R_{_\Lambda}}(t).$$
Now the condition (4) implies that the $2^n$ semidualizing $C_{_\Lambda}$ are
pairwise non-isomorphic.
\end{proof}
|
\section*{Acknowledgment}
L.F. and B.V. acknowledges J\"org Fink for useful discussions. B.V. acknowledges Roser
Valenti for discussion and for sharing her ab-initio calculations of FeSe with
pressure. L.B. acknowledges financial support by Italian MIUR under project
PRIN-R
|
IDEIRON-2012X3YFZ2 and by MAECI under the Italian-India collaborative
project SUPERTOP-PGR04879. B.V. acknowledges funding from MINECO (Spain) via Grants No.FIS2014-53219-P and
Fundaci\'on Ram\'on Areces.
|
\section{Introduction}
\vspace{2.0mm}
All graphs considered in this work are simple and undirected.
Let $ G= (V(G),E(G))$ be an undirected graph where $V(G)$ and $E(G)$ are the sets of its vertices and
edges, respectively. If
$A \subset V(G)$, we denote by $\langle A\rangle$ the induced subgraph
generated by $A$. For any vertex $x$ of a graph G,
the neighborhood of $x$ is the set $N(x) = \{y \in V(G) | xy \in E(G) \}$. The degree of a vertex $x$ is the cardinality of $N(x)$ and it is denoted by $d(x)$. We denote by $\Delta(G)$ the maximum degree of $G$. If $x,y \in V(G)$, the distance between $x$ and $y$ (that is, the length of the shortest $x-y$-path) is represented by $d(x,y)$ and $N^2(x)= \{y \in V(G) | d(x,y)=2\}$.
The {\it girth} of $G$ is the length of its shortest cycle, and is
denoted by $g(G)$.
Let $G$ be a graph with a proper vertex coloring. Let us denote by $C_i$ the set of vertices of color $i$, herein called the class of color $i$. Let $x_i$ denote a vertex $x$ of color $i$; $x_i$ is said a \textit{color-dominating vertex} (or,
\textit{b-dominating vertex}) if $x_i$ is adjacent to at least one vertex in each of the other classes. A color $i$ is a \textit{dominating color} if there is at least one vertex $x_i$ that is color-dominating.
If $y_i$ is a vertex which is not color-dominating, at least one color $j \neq i$
does not appear in $N(y_i)$. The color $j$ is said a \textit{missing color} in
$N(y_i)$ or simply a missing color of $y_i$.
A {\it b-coloring} is a proper coloring of its vertices
such that each color class contains a color-dominating vertex.
The b-chromatic number $b(G)$ is the largest integer $k$ such that $G$ admits a b-coloring with $k$ colors.
Since this parameter has been introduced by R. W. Irving
and D. F. Manlove \cite{Irving}, it aroused the interest of many researchers
as we can see in \cite{Hoang}, \cite{Bonomo}
and \cite{Ana} and, more recently, in \cite{Jak}.
For a vertex $x$ of $G$, let $G-\{x\}$ be the vertex-deleted subgraph of $G$
obtained by deleting $x$ and all edges incident to $x$.
It is known that the chromatic number of $G-\{x\}$ can have a maximum
variation of one unit compared to the chromatic number of $G$.
However, this is not true for the b-chromatic number - the difference between
$b(G)$ and $b(G-\{x\})$ can be arbitrarily large.
This fact motivates the search for bounds to the b-chromatic number
(see \cite{Bala} and \cite{Kouider}).
For general graphs S.F.Raj and R.Balakrishnan proved that:
\begin{theorem}{\cite{Bala}}
For any connected graph of order $n\geq 5$, and for any vertex $x \in V(G)$,
$b(G)-(\lceil n/2 \rceil -2) \leq b(G-\lbrace x\rbrace) \leq b(G)+(\lfloor n/2 \rfloor -2)$
\end{theorem}
The bounds are sharp.
\vspace{2.0mm}
In \cite{KouiderM}, some upper bounds of $b(G -\lbrace x\rbrace)$ have been established in
some classes of graphs as quasi-line graphs, graphs of large girth and chordal graphs. A {\it chordal} graph is a graph such that every cycle of length at least $4$ has a chord.
A graph is a {\it quasi-line} graph if the neighborhood of each vertex is covered by at most two cliques. In particular, claw-free graphs (i.e. graphs without induced $K_{1,3}$) are quasi-line graphs.
The following results have been shown.
\begin{theorem}\cite{KouiderM} \label{quasi}
Let $G=(V,E)$ a graph.
1) If $G$ is a quasi-line-graph, then for each vertex $x$, $$b(G-\lbrace x\rbrace) \leq b(G)+2$$
2) If $G$ is any graph of girth at least $5$, then for each vertex $x$, $$b(G-\lbrace x\rbrace) \leq b(G)+1$$
\end{theorem}
\vspace{4mm}
\begin{theorem} \cite{KouiderM} \label{coch}
Let $G=(V,E)$ be a chordal graph of clique-number $\omega$
and b-chromatic number $b(G)$.\
Then, for each vertex $x$,
$$b(G-\lbrace x\rbrace) \leq b(G)+1 + \sqrt{d(x)-1}$$
$$b(G-\lbrace x\rbrace) \leq b(G)+1 + \sqrt{\omega -1}$$
\end{theorem}
In this work we present lower bounds for $b(G-\{x\})$ in terms of $b(G)$,
particularly regarding two important classes of graphs, quasi-line and chordal
graphs. We also obtain a lower bound for $b(G-\{x\})$ for graphs of large girth.
Besides this introduction we have three more sections. In the second
one we present a lower bound for $b(G-\{x\})$, when $G$ is a general graph
and another bound for quasi-line graphs.
The third section is devoted to the study of chordal graphs, obtaining also
a lower bound for $b(G-\{x\})$ in this class. Finally, in the last section,
we analyse graphs with girth at least $5$, presenting also here a lower bound
for $b(G-\{x\})$.
\section{General bound and quasi-line graphs}
We begin this section with a lower bound for the b-chromatic number of a
vertex deleted subgraph of any graph.
\begin{proposition}
For every vertex $x \in V(G)$, $b(G-\{x\})\geq b(G)-d(x).$
\end{proposition}
\proof
Let $x \in V(G)$ be a fixed vertex. For a b-coloring of $G$, let $i$ be the color of $x$. We consider
two cases. First suppose
that each color has at least one color-dominating vertex in $G-\{x\}$; then the b-coloring of $G$ is also
a b-coloring of $G-\{x\}$, so $b(G-\{x\})\geq b(G)$.
Now, let us consider that there is a color with no color-dominating vertices
in $G-\{x\}$. We have then two possibilities:
\begin{itemize}
\item There is no color-dominating vertex of color $i$ in $G-\{x\}$, that is, $C_i$ has no color-dominating vertex then; for each vertex $z$ in $C_i$, there is at least one color missing in $N(z)$. We can change
the color of each vertex $z$ in $C_i$ by a missing color in $N(z)$, eliminating the color $i$.As $C_i$ is a stable set, the new coloring is proper.
For this case we have $b(G-\{x\})\geq b(G)-1 \geq b(G)-d$.
\item There is a vertex $y \in N(x)$ such that $y$ was the color-dominating vertex of color $s, s \neq i$ in $G$ and there is no more color-dominating vertex
of color $s$ in $G-\{x\}$. As $C_s$ has no color-dominating vertices, we then change the color $s$ of $y$ by $i$ and, for each other vertex $z$ in $C_s$, we change the color $s$ for a missing color of $z$, eliminating the color $s$. As $C_s$ is a stable set, the new coloring is proper. We repeat this process for
all vertices in $N(x)$ in the same conditions as $y$.
We do this for at most $d(x)$ vertices.
In this case we obtain $b(G-\{x\})\geq b(G)-d(x)$.
\end{itemize}
\begin{flushright}
$\blacksquare$
\end{flushright}
If $\Delta(G) < \left\lceil\frac{n}{2}\right\rceil -2 $, this bound is better than the lower bound in \cite{Bala}.
Note that there exist chordal (resp. quasi-line) graphs $G$ such that
$b(G-\{x\})$ is stricly less than $b(G)$.
For example, let $G_0$ be a chordal graph obtained from a chordal graph $H$ and a new vertex $x$ joined to every vertex of $H$. Then
$b(G_0~-x)= b(G_0)-1$.\\
\begin{theorem}
If $G$ is a quasi-line graph then, for every vertex $x \in V(G)$,
$b(G-\{x\})\geq b(G)-2$.
\end{theorem}
\proof
Let $x \in V(G)$ be a fixed vertex.
$N(x)$ is covered by at most two cliques $K_1$ and $K_2$. Let $i$ be the color of $x$.
Considering a b-coloring of $G$, there is at most two vertices
$u_k, u'_k \in N(x)$ with the same color $k$. Again, by the
fact that $G$ is quasi-line, $N^{2}(x) \cap N(u_k)$ is a clique as it is
independent from the neighbour $x$ of $u_k$. Analogously
$N^2(x) \cap N(u'_k)$ is a clique.\\
We delete the vertex $x$.If in $G-\{x\}$ each color is dominating, then
$b(G-\{x\}) \geq b(G).$
Let $i$ be the color of $x$in $G$ . We may suppose that $G-\{x\}$ has a color-dominating vertex of color $i$ otherwise we color each
vertex of color $i$ by a missing color and we get a b-coloring of $G-\{x\}$ by
$b(G)-1$ colors .\\
We choose a color $s$ that is no more dominating, which means that there is no more color-dominating vertices of color $s$. Each vertex of color $s$ has a
missing color.
If the color $s$ had more than one color-dominating vertex in $G$, then it had
exactly two color-dominating vertices $w_s \in K_1$ and $w'_s \in K_2$. We recolor both of them by $i$. We then recolor each other vertex of color $s$ by a missing color. In this way we obtain a b-coloration of $G-\{x\}$, eliminating one color.
If in $G$,there was only one color-dominating vertex $w_s$ of color $s$ in $N(x)$, say
in $K_1$, we recolor $w_s$ by $i$. We eliminate the color $s$ by coloring each vertex of $C_s$ by a missing color.
If there is a color $t$ no more dominating we choose one, then
the color-dominating vertex $w_t$ was necessarily in $K_2$.
We color $w_t$ by $i$. We color any other vertex of $C_t$ by a missing color.
Necessarily all the remaining colors are dominating.
We conclude that $b(G-\{x\})\geq b(G)-2$.
\begin{flushright}
$\blacksquare$
\end{flushright}
Note that there exists a quasi-line graph $G$ such that $b(G-x)=b(G)-1$
for at least a vertex $x$.We give an example.
Let $\omega \geq 3$ be an integer, and let $p=2 \omega-1$.
Let $P= \{x_0,x_1,...,x_p,x_{p+1} ~\}$ be a path. We consider the graph $G_1$
obtained by replacing each edge $[x_i,x_{i+1}]$ by a clique $K_i$ of order
$\omega$. The graph $G_1$ is a claw-free graph; we have $b(G_1)=p$ and
$b(G_1-\{x\})=b(G_1)-1$.
\section{Chordal graphs}
We want to show the following result.
\begin{theorem}
Let $G$ be a chordal graph and $x$ be a fixed vertex of $G$. Then
$b(G-\{x\})\geq b(G)-\omega_G $
where $\omega_G $ is the clique number of $G$.
\end{theorem}
\vspace{4mm}
We will need first the next lemma, about the adjacencies in chordal graphs.
\begin{lemma}
Let $G$ be a chordal graph, and $a,x,b,$ be three consecutive vertices of a cycle $\Gamma$ of $G$. Suppose that the vertex $x$ of $G$ has no neighbours in $\Gamma- \{a,b \}$. Then $a$ and $b $
are adjacent in $G$.
\end{lemma}
\proof
The proof is by contradiction. We suppose $a$ and $b$ independent.
Suppose $\Gamma$ is a shortest cycle containing the path $axb$.
If the length of $\Gamma$ is at least $4$, then as $G$ is chordal,
and by minimality of $\Gamma$,
it contains a chord incident with $x$ whose second endvertex is distinct from $a$ and $b$, a contradiction.
\begin{flushright}
$\blacksquare$
\end{flushright}
In what follows, consider a $b$-coloring of $G$, with $b=b(G)$ colors. Let
$x\in V(G)$ be a fixed vertex and let $i$ be the color of $x$.
Let $I_a$ be the set of colors without color-dominating vertices in $G-\{x\}$ and
let $J_a$ be their set of color-dominating vertices in $G$. We remark that
$J_a \subset N(x)$ and no vertex in $J_a$ is neighbour of a vertex of color $i$
in $G-\{x\}$. \\
Before proving our main result, we introduce a necessary definition.
\begin{definition}
Let $x_i'$ be a fixed color-dominating vertex of color $i$, different from $x$.
Let $W=\{w\mid w\notin C_i, w ~\mbox{color-dominating vertex in} ~G\}$.
Let $k\leq b$ be a fixed integer.
We denote by $W_k$ the set of color-dominating vertices of color $k$.
A path $P$ of $G-\{x\}$ is said a {\it pseudo-alternating path} of $G-\{x\}$, and denoted by $P_k[x'_{i},z]$, if it is
a path of endvertices $x'_{i}$ and $z$, such that:
\begin{itemize}
\item $V(P_k)\subset C_i\cup C_k\cup W \cup \{z\}$
\item each $w\in W \cap V(P_k)$ , $w\neq x'_{i}$ and $w\neq z$, is preceded
by a vertex of color $C_k$
(resp. of $C_i$) and succeded by a vertex of $C_i$ (resp. of $C_k$).
\item $V(P_k - \{z\})\cap N(x)= \varnothing$.
\end{itemize}
\end{definition}
A pseudo-alternating path $P[x_{i}',z]$ is an {\it alternating path} if $z$ is neighbour of $x$ in $G$.
We remark that if $z\notin C_i\cup C_k$, $z$ is necessarily preceded by a vertex of $C_i\cup C_k$ and, if\
$P$ is maximal, $z$ has neighbours in $C_i$ and $C_k$, belonging to $P$ or
$z\in I_a.$
{\bf Proof of the Theorem 5}
We consider a $b$-coloring of $G$, with $b(G)$ colors. Suppose $x \in C_i$.
We may assume that:\\
There is no $b$-coloring of $G-\{x\}$ by $b(G)-1$ colors \hspace{4mm} \bf(a)\\
\rm
otherwise we have the inequality of the theorem.
\rm
If $C_i-\{x\}$ contains no color-dominating vertex, we recolor each vertex of
that set by a missing color in its neighborhood. We get a $b$-coloring of
$G-\{x\}$ by $b(G)-1$ colors, a contradiction with assumption (a).
We may suppose from now that $C_i-\{x\}$ has color-dominating vertices.
$x_{i}',x_{i}", \ldots, x_{i}^{(r)}$.
Let us take $k$ in $I_a$. We recolor each vertex of $C_k$ by a missing color.
We eliminate color $k$. In this new
coloring, if there is a color $k'$ with no color-dominating vertex, then $k' \in I_a$.
We recolor $C_{k'}$ and we
eliminate color $k'$. Repeating this process, we get finally a $b$-coloring by
at least $b(G)-|I_a|$.
In view to establish the bound of the theorem, we want to bound $|I_a|$.
The bound will be established by three claims.
\vspace{2.0mm}
We denote by $Q[y,v]$ any path of $G-\{x\}$ such that $V(Q)\cap N(x)=\{v\}$.
Let $v \bar{~}$ be the neighbour of $v$ in that path.
Let $x_i$ be
|
a color-dominating vertex.
Let $F(x_{i})$ be the set of neighbours $z$ of $x$ such that $z$ is extremity
of an alternating path $P_k[x_i, z]$ and $k$ is the color of $z$ i.e.,
$F(x_{i})~= F_1(x_{i}) \cup F_2(x_{i})$, where $F_1(x_{i})$
and $F_2(x_{i})$ are defined below:\\
\noindent
$F_1(x_{i})=\{z \in N(x)| z = z_k ~\mbox{ and in some} P_k[x_{i},z_k],
{z_k}\bar{~}~\mbox{in} C_i ~ \}$ \\
and\\
$F_2(x_{i})=\{z \in N(x)| z = z_k ~\mbox{and in some} P_k[x_{i}',z_k],
~{z_k}\bar{~} ~\mbox{in~ W} \} \diagdown F_1(x_{i}')$.\\
Let $\cal{G}$ be a component of $G-(\{x\}\cup N(x))$
Consider $X_i$ the set of color-dominating vertices of color $i$ contained in
$\cal{G}$.\\
Let $F'(x_i)= \{v \in N(x)|, \mbox {there exists~} Q[x_{i},v] \}$.\\
Let $F'= \{v \in N(x)|, \mbox {there exists~} x_i~\rm {in}~X_i ~\rm{and}~ Q[x_{i},v]\}$.
Note that for any $x_i$, we have $F'(x_i)= F'= N(x) \cap N(\cal{G} ).$
\vspace{2mm}
Let $F_1= \cup \{F_1(x_{i}^{r}), x_{i}^{r} \in \cal{G} \}$.It is a subset of
$F'.$
\vspace{2mm}
By assumption (a) there exists a component $\cal{G}$
for which there is no recoloring of $\cal{G}$ by $\{1,...,b_G\}-\{i\}$ such
that all the colors of $W \cap({\cal{G}}~\cup N(x))$ have a color-dominating
vertex in $G-x.$ From now we use such a component.
We get the following assertion as corollary of Lemma 1.\\
\noindent \bf{Claim 1:}
\rm
For vertex $x_{i}$ of $X_i$,
$F'(x_{i}')$ is a clique containing $F_1$.\\
\noindent
Proof of claim 1:
It is sufficient to note that $F'(x_i)$ is not empty, otherwise there is no
alternating path, we choose $k\neq i$ and we exchange colors $k$ and $i$ in
the pseudo-alternating paths.
No color-dominating vertex loses a color.
$X_i=\{x_{i}',\ldots, x_{i}^{r}\}$ is recolored by $k$.
A contradiction with the definition of $\cal G$.\\
$F'(x_i)$ is a clique by Lemma 1 $\bullet$
\\
At this moment we need to introduce another definition\\
Let $P_s[x_{i}',z_1]$ be an alternating path for $s\neq i$, $s\neq 1$, with
$z_1\in F_1(x_{i}')$. An {\it extension} of $P_s$, denoted by $R_s[x_{i}',z']$,
is a path of the form $P_s[x_{i}',z_1] \cup [z_1,y_i ] \cup L(y_i,z']$, where
\begin{itemize}
\item $V(L)\subset C_i\cup C_s\cup W $; $(V(L)-\{z'\})\cap N(x)\subset W$.
\item For each $z\in V(L)-\{z'\}$ ,
\item if $z=z_k \in N(x)$, then $z \in W $ a color-dominating vertex of color $k$ and
$W_k \subset N(x)$; $z_k$ is preceded in $R_s$ by a vertex of $C_s$,
followed by a vertex $y_{i}(k)$.
If $z\in F_{1}(x_{i}^{l})$, then $y_{i}(k)\in P_{k}(x_{i}^{l},z_k)$
\item if $z\in W - N(x)$ then $z$ is preceded by a vertex of $C_r$, followed
by a vertex of $C_r'$, where $\{r,r'\}=\{i,s\}$
\item If $z'\in W$, $z'$ is preceded by a vertex of $C_{i} \cup C_{s}$.
\\
\end{itemize}
And now, we have two claims:\\
Let $\cal K ~(A)$ be the set of colors which appear in the subset $A$ of $V$.
\noindent
\bf{Claim 2:}
\rm
\vspace{1.0mm}
Let $P_s[x_{i}',z']$ be an alternating extended path. Then $V(L) \cap N(x)$ is a subset of $F_1$.
So if $z'\in W$ then $z' \notin J_a.$
If $s \notin \cal{K} (F')$, then $z'\in W~-~J_a$.
\vspace{2.0mm}
\noindent Proof of claim 2:
Let $z_1,z_2,...,z_t,..,z_p $ be the successive vertices of $V(R_s)\cap N(x)$
and $z'=z_p.$
We show by induction on $t$, that $z_t$ is in $F_1$.
Suppose that $z_{t-1}\in F_{1}(x_{i}^{l})$ for some $l$. So there is a path
$P_{t-1}[x_{i}^{l},y_{i}^{(t-1)}]$.
Composing it with $R_{s}(y_{r}^{(t-1)}, z_t)$, we get a path $Q[x^{(l)}, z_t]$.
If $z_t \notin F_1$, we do $\tau (i,t)$ from $x_{i}^{l'}$ for any $l'$, where
$\tau (i,t)$ means exchanging the colors $i$ and $t$ in $\cal G$. No color-dominating vertex
loses color $t$ even if it is an extremity of an alternating path $P_t$ in
this later case it is neighbour of $z_t$.
Some color-dominating neighbours of
$N_t(x)$ may lose color $i$. We recolor each $v\in X'_i$ by a missing color.
No color-dominating vertex of color $i$ is created by uniqueness of the color-dominating of color $t$.
We get a coloring by $b(G)-1$ colors. A contradiction. So $z_t \in F_1$ and $z_t \notin J_1$.
If $s \notin \cal{K} (F')$, then no vertex of $C_s$ is in $F_1$. So $z'\in W$.
As $F_1 \cap J_a=\emptyset$, then $z'\in W-J_a .$
\noindent
\vspace{2.0mm}
\bf{Claim 3:}
\rm
$\cal{K}$$(F')$ contains $I_a$
\vspace{2.0mm}
\noindent Proof of claim 3:
\vspace{2.0mm}
It is by contradiction. We suppose that there is a color $s$ such that
$s \in I_a \setminus \cal{K} (F')$. Thus no vertex of color $s$ belongs to $F_1$.
\vspace{2mm}
{\it Case 1:} There is no path $P_s[x_{i}^{r},z]$ with $x_{i}^{r}\in \cal{G}$ and
$z\in N(x)$.\\
We do $\tau (i,s)$ along the pseudo-alternating paths $P_s[x_{i}^{r},u]$,
$u\notin N(x)$ simultaneously.
No color-dominating vertex loses color.
We recolor the remaining vertices of $C_i$ in $\cal{G}$ and this leads to a
contradiction with the assumption.
\vspace{2mm}
{\it Case 2:} There exists $x_{i}^{r}$ in $\cal{G}$ and a path $P_s[x_{i}^{r},z_t]$ with $z_t \in N(x)$, $x_{i}^{r} \in \cal{G}$.
\vspace{2mm}
This case will be divided in two sub-cases:
\vspace{2mm}
Case 2.1: There exists $ P_s[x_{i}^{r},z_t]$ with $z_t \notin F_1$
\vspace{2mm}
By claim 1, $F'$ does not contain $z'_{t}$, with $z'_{t}\neq z_{t}$. We do $\tau (i,t)$ simultaneously in all alternating $P_t[x_{i}^{l},z]$, with $x_{i}^{l}$ in $\cal{G}$.
As $z_t \in F'$,by Claim1,
the color-dominating vertices contained in $F'$ do not lose color $t$,
they may lose color $i$. The color-dominating vertices which may lose a color are the
color-dominating vertices preterminal in $P_t[x_{i}^{l},z_t]$; these color-dominating vertices may lose
color $i$. We recolor $C_i$ by missing colors in $\cal{G}$.
\vspace{2mm}
Case 2.2: For any $P_s$, the extremity contained in $N(x)$ is in $F_1$.\\
So this extremity does not belong to $J_a$.\\
If for any $x_{i}^{l} \in X_i$, there is no alternating path $P_s[x_{i}^{l},z]$
with color $s$ preceding $z$, then we do
$\tau (i,s)$ in all the pseudo-alternating paths and the paths $P_s$.
We have the same conclusion as in case 2.1.
\vspace{2mm}
If for some $l, z$ is preceded by a vertex of color $s$ in $P_s[x_{i}^{l},z]$
we consider the extended paths $R_s$. We do
$\tau (i,s)$ simultaneously in all alternating and pseudo-alternating paths
$P_s$ and the extended paths $R_s$.
The color-dominating vertices which may lose a color are either in $N(x)$ or
in $N^{2}(x)$, they are among terminal vertices and preterminal vertices of
the alternating paths $P_s$ and $R_s$; and they may lose color $i$. We then
recolor each remaining vertex of $C_i \cap {\cal G}$ by a missing color.
\vspace{2.0mm}
In each case we have a contradiction with the definition of the component
$\cal G$. So $I_a \subset \cal K$$(F') \bullet$
\vspace{2.0mm}
By claim 1 and claim 3 there is a clique containing $x$ and $F'$. So we have
$\omega(G)\geq \vert F'\vert+1 \geq \vert I_a \vert +1$, and this finishes the
proof of the theorem.
\begin{flushright}
$\blacksquare$
\end{flushright}
\section{Graphs with large girth}
The \textit{m-degree} of a graph $G$, denoted by $m(G)$, is the largest integer $m$ such that $G$ has $m$ vertices of degree at least $m-1$. It is known that, for any graph $G$, $b(G) \leq m(G)$ (see \cite{Irving}).
Note that A.Campos et al. \cite{Camp} have shown that graphs of girth at least
$7$ have high b-chromatic number; for each such a graph $G$ this number is at
least $m(G)-1$.
We can verify that $m(G-\{x\}) \geq m(G)-1.$ Indeed, we have three possibilities to consider:
$x$ is one of the $m$ vertices of degree $m-1$; $x$ is a neighbor of one of the $m$ vertices of degree $m-1$, or $x$ is not in any of the previous situations. In the first case, there remain $m-1$ vertices of degree at least $m-2$ and, thus $m(G-\{x\}) \geq m(G)-1.$ In the second case, there remain $m$ vertices of degree at least $m-2$, and again, $m(G-\{x\})\geq m(G)-1.$ In the latter case the m-degree does not change, that is, $m(G-\{x\})= m(G)$.
So $b(G-\{x\}) \geq b(G)-2$ for graphs of girth at least $7$. No particular bound is known for graphs of girth $5$ or $6$.
In this work we show the following.
\begin{theorem}
Let $G$ be a graph of girth al least $5$.
For each vertex $x$, \\
$b(G-\{x\}) \geq b(G)-2$
\end{theorem}
\proof
Let $\cal B$ be a b-coloring of $G$ and let $i$ be the color of the deleted
vertex $x$.
Let $W$ be the set of color-dominating vertices of colors different from $i$ in $G$.
Let $W_k$ be the subset of those of color $k$.\\
We may suppose that there is a set of color-dominating vertices
$X_i$ of color $i$, different from $x$.
For each vertex $u$ let $K_1(u)$ be the set of colors with at least a color-dominating vertex
in $N(u)$, let us set $K_2(u)= \{1,2,...,b\}- (K_1(u) \cup \lbrace i\rbrace).$
We use the notations of the previous section. We may suppose $|I_1| \geq 3.$
Note that for $u\neq x$, as the girth is at least $5$, $K_1(u)$ does not
contain $I_1$. So $K_2(u)$ is not empty as it intersects $I_1$.
By definition of $I_1$, the color $i$ is a missing color for
each vertex of $J_1$ in $G- \{x\}.$
So for any $x' \in X_i$, $K_1(x')$ does not intersect $I_1$.
(a) Let $x' \in X_i$. Let $k \in K_2(x')$ be fixed.
\\
($\bf P_a$) \rm
If $G-N(N_k(x'))$ intersects $W_p$ for each color $p$ different from $i$,
\rm
\\
we color $x'$ by $k$, each vertex of $N_k(x')$ by a missing color.
So $|X_i|$ decreases.
The color-dominating vertices of $K_1(x')$ may lose color $i$ in their neighborhood.\\
\indent
(b) As long as there exists a vertex $x''$ of $X_i$ satisfying ($\bf P_a$)
\rm
we do a recoloring.
\\
From now we may suppose that $X_i$ is not empty and no vertex of $X_i$
satisfies (a).
\begin{lemma}
Let $x'$ be a fixed vertex of $X_i$.
For each $k \in K_2(x')$ there exists exactly one $j_k$ such that $W_{j_k}$
is contained in $N(N_k(x'))$ and $N(N_k(x')) \cap W_j=\emptyset$ for
any other $j$.
\end{lemma}
\proof
We know that the property $(a)$ is not satisfied. As $g(G)\geq5$, if a vertex
$w_j$ is in $N(N_k(x'))$ then $w_j$ is not neighbour of $N_r(x')$ for $r$
different from $k$.
As $(a)$ is not satisfied, it follows that for each $k\in K_2(x')$, $N_k(x')$ is
neighbour of any vertex of a set $W_j$ for some $j$. As $g(G) \geq 5$, $j$ is in
$K_2(x')$ and $W_j$ is not neighbour of $N_t(x')$ for $t \neq k$.
It follows that for $k$ fixed, $j$ is unique.
This finishes the proof of the lemma.
\\
\vspace{2mm}
Let $s $ be a fixed element of $I_1$. Let $x' \in X_i$.
We know that the set $I_1$, by definition, is a subset of $K_2(x').$ By the
precedent Lemma there is exactly one $k$ such that $W_s \subset N(N_k(x'))$.
We color each vertex of $N_k(x')$ by a missing color
and $x'$ by $k$. If $N_k(x')$ meets $N_k(x'')$ for some $x'' \in X_i$, by the
precedent Lemma, we have $W_s \subset N(N_k(x''))$; we color $x''$ by $k$ as
well and we recolor $ N_k(x'')$ by missing colors different from $i$.
We recolor so each vertex of $X_i$. Then we recolor each vertex of color $i$
by a missing color.
If,finally,the color $s$ has no vertex dominating the colors $\{1,..,b\}-\{i\}$,
we recolor each vertex $u_s$ by a missing color different from $i$.
\\
After this recoloring of color $i$ and eventually color $s$, we get a b-coloring of $G$ by at least $b(G)-2$ colors.
\begin{flushright}
$\blacksquare$
\end{flushright}
\vspace{5mm}
The first author acknowledges partial support by CAPES and CNPq.
|
\section{Introduction}
The Ising model has been largely applied in statistical physics, quantum field theory, economy and biophysics~\cite{Ising,KinrossSachdev,Tsai,Stauffer,Balatsky,Gesualdo,Alejandra,Michael, Grimmett,Bravyi,Elliott,Rudolf,Creswick,Neto1,Neto2,Kaizoji,Queiros,Burns,Owerre,Atas,Dmitriev,Takahashi1,Takahashi2,Takahashi3,Arovas,Dotsenko}. In condensed matter physics, this model can be used to study magnetic phases and its phase transitions. In particular, to study quantum phase transitions in the Ising model is necessary to add an external magnetic transverse field.
Generally, the transverse field induces a quantum phase transition from a ferromagnetic (F) or antiferromagnetic (AF) phase to a polarized one, In which the majority of the spins is along the direction of the field. In this case, the ground state before the transition (F/AF) can not be adiabatically connected with the ground state of the polarized phase.
Another intriguing result is the existence of Bosonic Dirac Materials (BDM). As pointed by Balatsky et al ~\cite{Balatsky}, the concept of fermionic Dirac materials can be extended to a bosonic version BDM. To perform this concept to a BDM is enough to show the following points; A linear characteristic of the energy dispersion in the vicinity of a point where the energy gap goes to zero; The robustness of these degenerated point against a weak disorder potential and a relative stability of these points against higher expansions of the spin wave theory.
In this work, we will demonstrate that a simple Ising model defined on the honeycomb lattice and subjected to the presence of transverse and longitudinal external field fulfils all these rules and therefore can be used as a prototype for BDM. Theoretically, the model can be written as
\begin{equation}\label{model}
\mathcal{H}= J\sum_{<i,j>}S^{z}_iS^{z}_j -h_x\sum_{i}S_{i}^{x}-h_z\sum_{i}S_{i}^{z},
\end{equation}
such that, $J$ is the nearest neighbor exchange interaction, $S_ {i}^{z}$ is the $z$ spin component in the site $i$, $h_x$ and $h_z$ are the transverse and longitudinal magnetic fields, respectively. The Eq.~\ref{model}, was subjected to several methods and different lattices~\cite{KinrossSachdev,Grimmett, Bravyi, Elliott}.
Recent works point out that, the F and AF Heisenberg model on a honeycomb lattice exhibits degenerated points in the first Brillouin zone. At these points, the spectrum can be linearized and associated with BDP. Beside that, these BDP are robust against higher order Magnon-Magnon interaction, which are responsible to make a shift in the spectrum~\cite{Balatsky}.
The main difference between our results and the results from ref.~\cite{Balatsky} is the presence of the Bosonic Dirac points only in the polarized phase. Therefore, is possible turn on or turn off these Bosonic Dirac Point by adjusting the external magnetic fields.
We find that by a fine tuning of the magnetic field, is possible to generate BDP located at the corners of the first Brillouin zone of the honeycomb lattice. From this, follows that is possible to transform a non-Bosonic Dirac phase into a Bosonic Dirac phase and vice verse. This is fascinating because at these points the magnons are massless, insensible to a weak disorder scattering potentials and can be propagated with very high speed $v_B$. A directly application of this result can be found in the generation of efficient spin transport~\cite{Pires}. It is worth to note, that some mechanism of the creation, localization and vanishing of these Dirac-like points, has attracted theoretical and experimental researchers~\cite{Rechtsman,Lu1,Lu2,Wang,Wang2,Kahanikaev,Dietl,Montambaux,Tarruell,Bellec}.
In addition, we determined the effects of the competition between the transverse and longitudinal fields, the influence of
on-site potential disorder over the BDP, edge states behavior and temperature effects.
For this purpose, we used two different methods, linear spin-wave theory (LSWT) and effective-field theory 1 (EFT-1).
The LSWT has been used to build the ground state and the dispersion in the real and momentum space.
In the real space, with open boundary conditions promoting a zig zag termination, we showed the edge states for a clean and disorder affected systems.
On the other side, EFT-1 has been used to compute zero and finite temperature effects over the sublattice magnetization, as a generalization of the pervious works~\cite{Creswick,Neto1,Neto2}.
Experimentally, in order to test our results we suggest the scanning tunneling microscopy (STM) technique. Firstly, STM can be used to create a $2D$ honeycomb magnetic lattice~\cite{Gomes} by depositing atoms or molecules in substrates. Secondly, the STM is able to change continuously the magnetic exchange interaction simply manipulating the position of the absorbantes (atoms, molecules etc..) which are deposited in a subtraction. Such that, for the present purpose an external field can be added to verify the controlling over the BDP.
The paper is organized as follows; In the section~\ref{PhaseDiagram}, we will discuss about the relevant ground states; The excitation spectrum critical exponents, Bosonic Dirac Point will be discussed in the section~\ref{LinearBehaviour}, while the impurity effects and edge states has been calculated in the sections \ref{Impurities} and \ref{Realspace}, respectively. The magnetic properties and finite temperature effects will be showed in section~\ref{FiniteTemperature}. Finally, in the last section~\ref{conclusion}, we summarize our results.
\section{Phase Diagram and ground state}
\label{PhaseDiagram}
Using LSWT and EFT-1 (appendix \ref{spinwave} and \ref{EfectiveFieldTheory}, respectively), we determined the phase diagram of the system in plane $h_x-h_z$, see Fig.~\ref{phasediagram}. The critical lines separates a canted antiferromagnetic (CAF) (region below the critical line) phase from polarized phases (PP)
(region above the critical line), see Fig.~\ref{angles}. The black and red lines were obtained by LSWT, while the blue line is a result from EFT-1.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{phasediagram.pdf}
\caption{(color online) Phase diagram at zero temperature in plane $(h_x,h_z)$. Each critical line represents a phase transition from the canted antiferromagnetic to a polarized phase. Black circles, red squares and blue triangles were obtained by LSWT-1, LSWT-2 and EFT-1, respectively. The purple, yellow, orange, red, blue and green big solid circles has been used to calculate the magnon energy dispersion in the Fig.~\ref{Dispersionplane}.}
\label{phasediagram}
\end{figure}
\begin{figure}[ht!]
\centering
\includegraphics[scale=0.9]{Angles}
\caption{(color online) Spin configurations. a) Antiferromagnetic states in absence of the magnetic fields, where $\phi_A=\phi_B=0$. b) Canted antiferromagnetic phase, where $\phi_A \neq \phi_B$. c) Polarized phase, where $\phi_A=\phi$ and $\phi_B=\pi-\phi$ or $\phi_A+\phi_B=\pi$.}
\label{angles}
\end{figure}
We obtained the black line minimizing the classical energy $E_g=E_c$ with respect to angles $\phi_A$ and $\phi_B$ (LSWT-1). On the other side, adding the quantum correction $E_q$ in the the energy, we get the red line (LSWT-2). Note that, at the critical line and in the region above the critical line $\phi_A+\phi_B = \pi$ (Fig.~\ref{angles} (c)).
Still about the angle spin configuration, see the quantum phase transition that occurs at $h_{x} =1. 5$ for $h_{z}=0$ (red big solid circle in Fig.~\ref{phasediagram}). Above $h_{x} \geq 1.5$ the spins are totally along the x-axis. Following, for $h_x=1.5$ and increasing the longitudinal field from zero to $h_{z}=0.5$ (red big solid circle) the spin configuration of the polarized phase is no more totally along the x-axis, getting an effective angle $\phi$ as in Fig.~\ref{angles} (c). Along the critical line, this angle changes and finally for $h_{x}=0$ and $h_{z}=1.5$, $\phi=0$.
The results from LSWT-2 are qualitatively equal to LSWT-1, in the sense that it describes a continuous transition between CAF and P phases. However, one can observe a considerable reduction of the CAF region for LSWT-2 in comparison to LSWT-1, see Fig~\ref{phasediagram} (red line and triangle). This is a new result, in particular in the case of a Honeycomb lattice and corroborates with the idea that the spin wave theory tends to overestimate the ordered phase region in a first approach.
The critical line from EFT-1 (blue line and triangles) was also included Fig.\ref{phasediagram} and seems to be in great agreement with the critical line obtained from LSWT-2 (red squares). Here, the EFT-1 method takes into account a cluster of one spin interacting with their vicinity. We considered a staggered magnetization as a order parameter and the quantum critical point can be founded throughout the Eqs.~\ref{B15}-~\ref{B16}. It is worth to communicate that LSWT-1 provides results quantitative equal to a first approximation of Bogoliobov Mean Field Theory (MFA-1) for a cluster with one spin \cite{Neto2}.
\begin{figure}[t]
\centering
\includegraphics[width =3.5in]{Dirac.png}
\caption{(color online) Quasiparticle excitations and Bosonic Dirac cones.
a) Bosonic Dirac points within the polarized phase $h_x=1.5,h_z=0.5$ (top) and at the critical point $h_x=1.5,h_z=0.0$ (bottom).
b) First BZ and the locations of the Bosonic Dirac cones at the corners $K$ (bottom).
Here $\Gamma=(0,0)$ (soft mode) , $M=( \frac{2\pi}{3a},0)$ and $K=( \frac{2 \pi}{3a}, \frac{2\pi}{3\sqrt{3}a})$ are points in the BZ for a lattice parameter $a$, which define the path shown in the BZ.}
\label{DispersionBZ}
\end{figure}
\section{Linear Behaviour, Bosonic Dirac points and critical exponents}
\label{LinearBehaviour}
The linear behaviour can be identified around the special points called by $\Gamma=(0,0)$ and $K=( \frac{2 \pi}{3a}, \frac{2\pi}{3\sqrt{3}a})$ within the first Brillouin zone, see Fig.~\ref{DispersionBZ}. Only the $K$ points are possible candidate to be ``Bosonic Dirac points", following the same nomenclature adopted by the Balatsky et al. \cite{Balatsky}.
The low energy description of the magnon spectrum provides a linear dispersion around $\Gamma$ equal to $\omega(\mathbf{\Gamma}) \sim D \sqrt{ \Delta^{2\nu z } + k^{2z} }$. Where the gap function $\Delta$ goes to zero at the critical point as well as $\nu$ and $z$ are the correlation and dynamical critical exponents. Note that, $D$ and $\Delta$ are independent of $k$, they are given by $D=\sqrt{ \frac{{\tilde{A} J z S {\sin(\phi_A)}^2}}{4}}$ and $\Delta=\frac{\sin(\phi_A)^2}{D^2} \left( h_{x} - h_{xc} \right)$ (since we choose to fix $h_{z}$), respectively.
The critical value can be written as $h_{xc}=\frac{JzS-h_{z}\cos(\phi_{A})}{\sin(\phi_{A})}$ where the gap vanishes, as shown in more details in appendix~\ref{Lowenergy}. These calculations leading to $\nu=\frac{1}{2}$ and $z=1$ for the critical exponents, the last ensure the linear behaviour.
For the critical exponent of the correlation length, we found $\nu=\frac{1}{2}$ as expected for a mean field approach. It is worth mentioning, for a 2D Ising model with a transverse field the value gets $\nu=0.63$~\cite{Elliott,Stinchcombe}.
Following, around the $K$ the dispersion gets a linear behavior as $\omega(\mathbf{K}) \sim \tilde{A} \pm \frac{3}{4} J S \sin(\phi_A)^2 \delta k$. It is worth to note that in this case the gap function is $ \Delta = (\omega_1(\mathbf{K})-\omega_2(\mathbf{K}))$, in particular, for $h_z=0$, $\Delta= (h_{xc}- h_x )^{\nu z}$~\cite{Stinchcombe,Croo} with $\nu z =1$.
The $z=1$ result for the Ising model in a presence of transverse, is known and works for any regular lattice since the dimension of the system be an integer~\cite{SSachdev}. In addition, this is also independent of the quantum critical point as demonstrated for the $\Gamma$ and $K$ points.
We are reporting in this work that theses linear behaviours appear in spectrum as a function of the external fields. The dominant set of critical exponents is dictated by the soft point at $\Gamma$.
The number and location of these special points depends on criticality and the magnetic phase, see Fig.~\ref{Dispersionplane} and Fig.~\ref{phasediagram}.
These results can be summarized as;
In the CAF phase there is no Bosonic Dirac Points, sice the system is gapped. At the critical line, the energy dispersion becomes gapless and six Bosonic Dirac points emerge at corners of the Brillouin zone ($K$) and there is a linear soft mode at the center ($\Gamma$). These six Bosonic Dirac point at the corners of BZ persist in the polarized phase.
Our report is quite different from the case discussed in ref.~\cite{Balatsky}, since the existence of the Bosonic Dirac Points depends on the magnetic phase. For the present case, the controlling of these Dirac points resides in a simple adjusting of the magnetic fields instead of the exchange interaction $J$.
In Fig.~\ref{Dispersionplane}, we explore the projection of the magnon excitation along of the path defined by $\Gamma-M-K-\Gamma$. This figure shows the spots of the Bosonic Dirac points in the BZ and how they are affected by the magnetic fields.
\begin{figure}[t]
\centering
\includegraphics[scale=0.46]{Projection}
\caption{(color online) Magnon quasi-particle excitations $\omega_{1}$ (solid) and $\omega_{2}$ (dashed) along the path $\Gamma-M-K-\Gamma$.
a) We have fixed the transverse field at $h_x=1.5$. Curves for longitudinal field $h_z=0.0$ (red), $h_z=0.5$ (blue) and $h_z=1.0$ (green).
In the case of red curve, the system is gapless and we can see a linear behaviour that emerges at the vicinity of the $\Gamma$ and $K$ points.
Otherwise, in case of blue and green curves, the linearity in dispersion only occurs near the $K$ point.
b) Now, the transverse field has been fixed at $h_x=1.0$ and we have calculated the magnon dispersion for $h_z=0.2$ (purple), $h_z=1.2871$ (yellow) and $h_z=1.5$ (orange). }
\label{Dispersionplane}
\end{figure}
The color scheme follows the big solid circles in Fig.~\ref{phasediagram} for each pair $h_{x}-h_{z}$. The red line in the Fig.~\ref{Dispersionplane} a), is the quasiparticle excitations at the critical point $(h_x=1.5,h_z=0)$. For a fixed $h_x=1.5$ and varying $h_z$ from zero to 0.5, we get the point $(h_x=1.5,h_z=0.5)$ (polarized phase), where the dispersion is linear only around the $K$ points. The same behavior occur at the point $(h_x=1.5,h_z=1.0)$ (green curve). A similar behavior can be observed, but for a fixed $h_x=1.0$, see Fig.~\ref{Dispersionplane} b). Therefore, the Bosonic Dirac points seems to be a particular characteristic of the polarized phase.
\section{Impurity effects and impurity resonance}
\label{Impurities}
Are these degenerate Bosonic Dirac points robust against impurities? To answer this question, we investigated the effects of an impurity
|
scattering. The impurity simulates a local defect created by any removing/modification in the spin over the honeycomb lattice. For this reason, we considered a defect given by $H_{imp}=\sum_{i} V_0 a^{\dagger}_i a_i $.
In order to present the physical conditions that the impurity scattering should obey to generate an impurity resonance, exactly at the Bosonic Dirac point $K$ and linear point $\Gamma$, we need to calculate the Green's function of the system from the Dyson equation. The absence of an impurity resonance at these points implies robustness against the scattering potential $V_0$. So, this robustness need to be checked in order to make the association with ``Bosonic Dirac point".
We write the impurity Green's function using the T-matrix approach. The single magnon Green's function for our problem is given by
\begin{equation}\label{SingleGreensfunction}
G(k,\omega) = g_0(k,\omega) + g_0(k,\omega) T_{imp} g_0(k,\omega),
\end{equation}
where, $g_0=( \omega+ i\delta - \mathcal{H} )^{-1}$, $T_{imp}=\frac{1}{2 (V_0^{-1} + \bar{g}_0) } \left( \sigma_{0} \otimes \sigma_{0} + \sigma_{0} \otimes \sigma_{z} \right)
$, $\sigma_i $ are the usual Pauli matrix and $\bar{g}_0 (\sigma_{0} \otimes \sigma_{0})=\frac{1}{N}\sum_k g_0(k,\omega)$.
To determine the impurity effects over the phase diagram in Fig.~\ref{phasediagram}, we chose two points A and B; The A point is given by $(h_x=1.0,h_z=1.28)$, which corresponds to a critical point. The B point $(h_x=1.5,h_z=1.0)$ within the polarized region. Using the A point, we calculated the density of states for a clean case $\rho_0(\omega)=-\frac{1}{\pi} Im(g_0(\omega))$ (dashed blue lines of Fig.~\ref{Densityofstates}) and the impurity correction $\delta \rho(\omega) = -\frac{1}{\pi} Im(g_0(\omega) T_{imp} g_0(\omega) ) $( solid lines of Fig.~\ref{Densityofstates}). So, in the Fig.~\ref{Densityofstates} an impurity resonance might emerges only in the case of very strong values of scattering potential $V_0$. This means that the correction in $\rho_0$ is precisely zero for low values of potential scattering $V_0$. Note yet, that the pick of $\delta \rho $ approaches from the resonance limit only when $V_0$ goes to infinity. This behaviour was observed along all critical line, ergo the critical line is robusts against this kind of defect.
Furthermore, the Fig.~\ref{DensityofstatesPolarized} shows the effects of an impurity scattering over de Bosonic Dirac point $K$ in the polarized phase. Again, low values of the impurity potential $V_0$ can not generate a resonance impurity. These results can be used to guarantee that these special degenerated points $\Gamma$ and $K$ are robust against a weak impurity potential in a similar way what as it occurs with their cousin, Fermionic Dirac points.
In the strong limit of scattering potential, the impurity resonance emerges only in the sublattice that does not hosts the scattering center. As discussed in ref.~\cite{Balatsky}, the scattering induces Friedel oscillations and the local magnon density $\rho(r,\omega)$ around the impurity describes waves emanating from the scattering center with the standard asymptotic decay proportional to $r^{-1}$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.45]{Densityofstates}
\caption{(color online) Density of state $\rho_0(\omega)$ and impurity correction $\delta \rho(\omega)$ at critical point A $(h_x=1.0,h_z=1.28)$. Dashed blue and solid lines represents $\rho_0(\omega)$ and $\delta \rho(\omega)$, respectively. The calculations were performed
a) around $\Gamma$ point and b) around K point. In both cases, the scattering potential assumes $V_0=10,20,50,10^2,10^4$ following the solid lines from top to bottom order according with the curves (red, green, dark red, mangueta and black curves).}
\label{Densityofstates}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{PolarizedKImpu}
\caption{(color online) Density of state $\rho_0(\omega)$ and impurity correction $\delta \rho(\omega)$ in the polarized phase, B point $(h_x=1.5,h_z=1.0)$ around the K. The solid lines were obtained for $V_0=10,20,50,10^2,10^4$ and follow the same color scheme of Fig.~\ref{Densityofstates}.}
\label{DensityofstatesPolarized}
\end{figure}
\section{Real space and edge states}
\label{Realspace}
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{RealCleanB}
\caption{(color online) Eigenvalues and edge states for $L_x=100$ as a function of $k_y$. The energy of the edge state is represented by red lines, while the blue lines the bosonic bulk states. a) In the CAF phase at point $h_x=1.0$ and $h_z=0.2$, b) at the critical point $h_x=1.0$ e $h_z=1.28$ and c) in the polarized phase at $h_x=1.5$ e $h_z=1.0$. d) A pictorial representation of the zigzag termination and the location of the edges states (red gradient).}
\label{RealCleanfig}
\end{figure}
An interesting feature that arises in the finite honeycomb lattice is the presence of edges states. In this section we are devoted to understand the nature of these edges states and how a disorder potential can destroys it. The presence of the Bosonic Dirac Points at $K$ points in this finite lattice is also a relevant point to be observed.
We have written the Hamiltonian eq~\ref{modelSpinwave} in the real space with open boundary conditions along the $x$ direction, while in the $y$ direction we considered a periodic boundary condition (see appendix \ref{HRealSpace}). Thus, the lattice becomes a kind of a finite cylinder with zigzag terminations, in a very similar way to a nanoribbon in graphene context. The edge states are localized around the zigzag terminations of this cylinder, the highlighted red region, as depicted in Fig.~\ref{RealCleanfig} d).
Without disorder, the eigenvalues of the clean Hamiltonian in real space Eq.~\ref{HbarkyZZ} are shown in the Fig.~\ref{RealCleanfig} a), b) and c) for $L_{x}=100$. Where $L_{x}$ counts the number of unit cell, containing a pair of sublattice A and B, along the finite $x$ direction.
In Fig.~\ref{RealCleanfig}, we have the bulk bosonic states (blue lines) and the zigzag edge states (red lines) of all magnetic phases.
The edges states from CAF and polarized phase are shown in the parts a) and c), respectively. In the part b), we showed how these edges states behaves at the critical line.
The location of the edge states are depicted in the part d). They are majority distributed around the terminations of the lattice as indicated by the red gradient color.
\begin{figure}[t]
\centering
\includegraphics[scale=0.5]{RealDisorderB}
\caption{(color online) Eigenvalues and edge states for $L_x=100$ as a function of $k_y$ for a potential disorder ensemble with 30 realisations. Calculations performed in the polarized phase at point $h_x=1.5$ and $h_z=1.0$. The energy of the edge state are represented by red lines, while the blue lines are bosonic bulk states.
a) Disorder at bulk and edges. b) Disorder only at the edges (zigzag) of the honeycomb lattice.}
\label{RealCleanfigDisorder}
\end{figure}
A coupling with a disorder potential can simulates an interaction with a substrate. The adding of this disorder potential allow us to investigate
its effect over these Bosonic Dirac points and edge states within the polarized phase. Also, providing an another way to determine the robustness of the BDP against disorder inside a finite lattice study.
After this disorder calculations, we present the result of Fig.~\ref{RealCleanfigDisorder} for an ensemble with 30 realisations.
As we can see, a weak on-site potential disorder for both bulk and edges, and only concentrated at the edges, respectively Fig.~\ref{RealCleanfigDisorder} a) and b), can not open a gap around the points K. So, even in this finite lattice study, these $K$ points are insensible in relation to this kind of disorder and so far they prove to be robust.
\section{The temperature effects}
\label{FiniteTemperature}
Beyond the zero temperature results and in order to design a more complete study, we also investigate the behavior of the ground state in the presence of thermal fluctuations. Through the sublattice magnetization $m^z_ {A, B} $ as a function of the reduced temperature $T=k_ {B} T/J$ and the magnetic fields we developed nonzero phase diagrams.
Taking into account only the critical behavior, we present the critical reduced temperature as a function of the $h_ {x}$ for several values of $h_ {z}$ obtained by LSWT, and vise versa in Fig.~\ref{TCvSHXEPS} and Fig.~\ref{TCvSHZEPS}, respectively. These results are able to be compared with Fig.~\ref{TcVHxEFTEPS} and Fig.~\ref{TcVHzEFTEPS}, that comes from EFT-1.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{LSW-Tc-Hx.pdf}
\caption{(color online) Critical temperature as a function of $h_x$ for several values of the $h_z$ obtained by LSWT-1,2.
The LSWT-1 results for sublattice $A$ and $B$ are represented by solid and doted lines, respectively. The LSWT-2 results are signed by curves with geometrical forms solid for $A$ and opened for $B$ sublattices. The longitudinal field assumes the fixed values, $h_z=0.1$ (red), $h_z=0.5$ (blue) and $h_z=1$ (black).}
\label{TCvSHXEPS}
\end{figure}
Looking to the Fig.~\ref{TCvSHXEPS}, we found that the critical temperature $T_c$ of the sublattice magnetization $m^{z}_A$ is identically to the \textit{melting temperature}. So, above $T_c$ the system is paramagnetic without any sublattice magnetization. The EFT-1 results shown in the Fig.~\ref{TcVHxEFTEPS} are qualitative equal to results of the LSWT-1, therefore both methods are in excellent agreement.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.8]{LSW-Tc-Hz-novo.pdf}
\caption{(color online) The same of Fig.~\ref{TCvSHXEPS}, but varying $h_z$ for $h_{x}=0.1$ $h_{x}=0.5$ and $h_{x}=1$ fixed values.}
\label{TCvSHZEPS}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{EFT-Tc-Hx-novo.pdf}
\caption{(color online) Critical Temperature as a function of the $h_x$, obtained by EFT. For $h_z=0.5$ (red), $h_z=1$ (blue) and $h_z=2$ (black) fixed values. To compare this results with Fig.~\ref{TCvSHXEPS} the values of fields must be renormalized by $1/2$.}
\label{TcVHxEFTEPS}
\end{figure}
On the other hand, analyzing the figures Fig.~\ref{TCvSHZEPS} and Fig.~\ref{TcVHzEFTEPS}, which shows the behavior of the $T_c$ as a function of the $h_z$ for a fixed $h_x$, we can see a substantial difference between these two methods. Notice that, while the EFT-1 method found a monotonic behaviour for magnetization, the LSWT-1,2 suggests a more intriguing situation. The Fig.~\ref{TCvSHZEPS} indicates that the critical temperature of the two sublattice magnetizations possesses different behaviors, which depends on whether $h_z$ is less or greater than critical value $h_{xc}$, where the sublattice magnetizations go to zero. Below the critical point, the sublattice magnetization increases again and this occur because the polarized phase is always induced by the increasing of the $h_{z}$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{EFT-Tc-Hz-novo.pdf}
\caption{(color online) The same of Fig.~\ref{TcVHxEFTEPS}, but varying $h_z$. For $h_x=0.5$ (red), $h_x=1$ (blue) and $h_x=1.5$ (black) fixed values. To compare this results with Fig.~\ref{TCvSHZEPS} the values of fields must be renormalized by $1/2$.}
\label{TcVHzEFTEPS}
\end{figure}
\section{Conclusion}
\label{conclusion}
In this paper, we investigated the quantum phase transitions and the existence of Bosonic Dirac points in the Ising model on a honeycomb lattice. The quantum phase transitions are induced by external fields.
Using two methods, LSWT and EFT-1, we determined the phase diagram in plane $(h_x,h_z)$ at zero temperature. From LSWT, we determine the bosonic quasiparticle excitation.
Depending of the magnetic phase, degenerated points can emerge within the spectrum at the $K$ points in the Brillouin zone. These points always occur along the critical line or within the polarized magnetic region. We demonstrated that around these special points the spectrum is linear and is robust against a weak impurity and disorder. The dynamical critical exponent was also investigated leading to $z=1$, that indicates a linear crossing around these points.
These results often characterize Fermionic Dirac Materials and therefore can be used to classify our model as a Bosonic Dirac Material.
Following this, it is fascinate how the present simple model (inside the polarized phase) became a very strong candidate to describe a Bosonic Dirac material. As we see, this phase can be reached by a fine tune of the longitudinal and transverse fields. Which is very interesting, because in principle it makes possible to turn a non Bosonic Dirac material into a Bosonic Dirac material only adjusting the external parameters like magnetic fields. Beside that, the Bosonic Dirac materials should be robust against the magnon-magnon interaction as pointed by ref.~\cite{Balatsky}, from this perspective the existence of this Bosonic Dirac points depends only on the symmetry of the honeycomb lattice. Furthermore, differently from the graphene fermionic case, our bosonic quasiparticle excitation can also exhibit one additional linear behavior at the center of the BZ, a soft mode.
In the real space we found edge states for a lattice with zigzag termination, these edge states are a peculiar characteristic of the honeycomb lattice and can be observed in the AF and polarized phase. They proved to be robust against potential disorder and provides one more evidence for a Bosonic Dirac Material description.
To determine the temperature dependence and the possible quantum corrections, we calculating the magnetic properties using the LSWT-2 and EFT-1, which allow us to conclude that the methods are in great agreement.
It is worth to highlight yet the importance of the Bosonic Dirac points due its connection with production of spin wave excitations~\cite{Pires}, which can be used to breed efficient spin currents. Accordingly, these excitations could find application inside of Spintronics and optical lattices~\cite{Offidani,Baltz,Gregersen}.
The real compound $Ba_2 Co Te O_6$ can be described by an effective Ising model on a honeycomb lattice, and accordingly the present study,
is a strong candidate to be a Bonsonic Dirac material \cite{Chanlert}.
Despite of the fact that this compound presents an approachable next-nearest-neighbor interaction $J_2$, the symmetry of honeycomb is preserved.
Another promise compound was syntetized by \cite{Okabe}.
It is a single crystal of the verdazy radical which also presents a Neel order that can be described by a spin 1/2 antiferromagntic Heisenberg model in a honeycomb lattice.
We expect to motivate experimental efforts to study the nature of the \emph{Magnon Bosonic Dirac points} and its relation with the quantum phase transitions in this simple model.
\section{Acknowledgment}
We would like to thank you the prof. Mucio Continentino and prof. Octavio D. Rodriguez Salmon for all assistance as well as the Brazilian agencies CNPq, Capes and Fapeam for providing all supporting.
|
\section{Introduction}
\label{sec:introduction}
Tunneling in systems with several degrees of freedom is an
exceptionally rich subject of
investigation~\cite{Creagh:1998,Tomsovic:2001}.
The features and probability of multidimensional tunneling
depend crucially on the properties of underlying system,
or rather on the degree of complexity of its classical dynamics. In
particular, expressions for the tunnel splittings of energy levels
are
qualitatively different in the cases of
integrable~\cite{Miller:2001,Meyer:1991,Creagh:1994}
and near-integrable~\cite{Wilkinson:1986,Takada:1994,Creagh:2001,
Creagh:2006} dynamics. The other drastically different case, tunneling
in irregular (chaotic or mixed) systems, has been a subject of
continuous theoretical~\cite{Bohigas:1993,Doron:1995,Shudo:1995,
Creagh:1999,Mouchet:2001,Ribeiro:2004,Levkov:2007e,
Backer:2008} and experimental~\cite{Dembowski:2000,Hensinger:2001,
Steck:2001,Backen:2008} research for the last few decades.
The basic concept in multidimensional tunneling is {\it dynamical
tunneling}~\cite{Miller,Heller:1981}.
It is related to the classical dynamics and reflects the fact
that transitions of a multidimensional
system between the in- and out- regions of phase space may be classically
forbidden even if there is {\it no} energy barrier separating the
regions. In this case the quantum probability ${\cal P}$ of transition
is on general grounds exponentially suppressed,
\begin{equation}
\label{eq:9}
{\cal P} = A\, \mathrm{e}^{-F/\hbar}\;,
\end{equation}
where $F$ and $A$ are the suppression exponent and prefactor
respectively. The transition itself is called dynamical
tunneling~\cite{Heller:1981}, since the reasons for its exponential
suppression are hidden in the particularities of classical
dynamics.
A new mechanism of dynamical tunneling has been independently
discovered in Refs.~\cite{Onishi:2003,Takahashi:Ikeda}
and~\cite{Bezrukov:2003yf}. It governs tunneling in non--separable
systems with multiple degrees of freedom at energies exceeding certain
{\it critical energy} $E_c$. The value of the latter energy depends
on the details of the system dynamics but is always greater than the
height of the potential barrier between the in- and out- states of the
process. The new mechanism is {\it general}:
it is relevant for tunneling in
regular~\cite{Bezrukov:2003yf,Levkov:2007a,Levkov:2007prl} and
irregular~\cite{Onishi:2003,Levkov:2007e} scattering problems, for
transitions in time-dependent one--dimensional
potentials~\cite{Takahashi:Ikeda,Takahashi:2006,Takahashi:2008}, in
the case of chaotic tunneling\footnote{In chaotic case the new
mechanism implies anomalously weak falloff of particle wave function
in some parts of classically forbidden region (``plateau
structure''~\cite{Shudo:1995,Onishi:2003}). This behavior leads to
anomalously large tunneling probabilities, the effect
known as {\it chaos--assisted
tunneling}~\cite{Bohigas:1993}. }~\cite{Shudo:1995,Shudo:2008}.
Another example emerges in field theory where the new mechanism is
generically inherent in the processes of collision--induced tunneling
at high energies~\cite{Bezrukov:2003er,Levkov:2004tf}.
The defining characteristics of the new mechanism have been given
within the semiclassical approach. It was noted that the semiclassical
trajectories describing tunneling transitions acquire qualitatively
new properties at $E>E_c$. Instead of connecting directly the in-
and out- regions of the process, the trajectories end up performing
unstable motion on the boundary between the regions. In the simplest
case of two degrees of freedom this unstable motion proceeds along the
periodic orbit describing oscillations on top of the saddle point of
the potential. Following Ref.~\cite{Bezrukov:2003yf}, we call the
latter orbit {\it sphaleron}\footnote{This term is standard in
field theory~\cite{Klinkhamer:1984di}; it is based on classic Greek
adjective $\sigma\varphi\alpha\lambda\epsilon\rho o \varsigma$ ---
``ready to fall.''} (or simply {\it unstable periodic
orbit}).
In general case of systems with more than two degrees of freedom the
boundary between the in- and out- regions is normally hyperbolic
invariant manifold (NHIM)~\cite{Wiggins:2001}; the trajectories in the
new mechanism get attracted to this manifold. In this case we use the
term sphaleron in the sense equivalent to NHIM.
Due to the above property of the
trajectories, tunneling at $E>E_c$ proceeds in two stages. First, the
long-living sphaleron ``state'' gets created. Second, the sphaleron
decays into the final asymptotic region with the probability of order
one. The overall transition remains exponentially suppressed, since
creation of the sphaleron costs exponentially small
probability factor. We call the overall transition {\it
sphaleron--driven tunneling}.
The aim of the present paper is twofold. First, we analyze the
possibility of direct experimental observation of the the mechanism of
sphaleron--driven tunneling. To the best of our knowledge, such
observation has not been performed so far. We
study two signatures of the new mechanism which may be
helpful in future experiments. Second, we systematically develop
modified semiclassical method for the calculation of tunneling
probability in the sphaleron--driven case.
We discuss two experimental signatures of the new tunneling mechanism. In
Ref.~\cite{Levkov:2007prl} we have found that the probability of
sphaleron--driven tunneling contains additional power-law dependence
on $\hbar$ as compared to the ordinary case of potential tunneling.
The additional factor is $\hbar^{1/2}$ in the case of inclusive
tunneling processes, i.e. processes without specification of the
out-state. In this paper we review the result of
Ref.~\cite{Levkov:2007prl} and extend the analysis to the new case
of exclusive processes, i.e. processes with fixed out-state quantum
numbers. We show that the additional factor is $\hbar$ in this
case. For example, consider two--dimensional inclusive
processes. Then, the prefactor $A$ in Eq.~(\ref{eq:9})
is proportional to $\hbar^{1/2}$
and $\hbar$ in the cases of potential and sphaleron--driven tunneling
respectively. For exclusive processes this dependence
is $\hbar$ ($\hbar^2$) in the potential (sphaleron--driven) case.
It is important to stress that the dependence of the tunneling
probability on $\hbar$ can, in principle, be studied
experimentally. Indeed, the semiclassical parameter, which we denote
by $\hbar$ for convenience, is in fact a certain dimensionless
combination of the Planck constant and parameters characterizing
the system. Changing the latter parameters one varies the value of
effective $\hbar$.
The second manifestation of the new mechanism is
spreading of the out-state of the tunneling process over an
anomalously wide range of quantum numbers. This effect was originally
observed in Ref.~\cite{Takahashi:Ikeda} in the case of
a one-dimensional system with time--dependent potential; here we
show that it is present in the multidimensional case,
cf. Ref.~\cite{Takahashi:2008}. Physically, the widening of the
out-state is related to the fact that the intermediate sphaleron
orbit is classically unstable; thus, classical trajectories
describing sphaleron decay spread exponentially over phase space. In
quantum case this corresponds to the final state wave function which
is almost constant in some region of quantum numbers.
In the second part of this paper we develop the modified semiclassical
technique which is essential in the case of sphaleron--driven tunneling.
The motivation for the new technique becomes clear if we try to apply
the standard method of complex trajectories to the problem of
inclusive
sphaleron--driven tunneling in the scattering setup. Since
the overall time interval of the scattering problem is infinite,
one generically finds {\it two} different trajectories corresponding
to the two stages of the new tunneling mechanism: one trajectory
starts in the in-region and tends to the sphaleron orbit as
$t\to +\infty$, and the second trajectory starts at the sphaleron
at $t\to -\infty$ and arrives into the out-region. The first of these
trajectories is unstable: it can be destroyed by infinitesimally
small changes in the initial Cauchy data.\footnote{Below we always
refer to this kind of instabilities.} It is problematic to
find unstable trajectories numerically. Besides, one wonders how to join
the two trajectories in order to describe the overall two--stage
process. Finally, it is not clear how to calculate the prefactor $A$
of the tunneling probability. Indeed, the standard formula for the
prefactor deals with the linear perturbations above the tunneling
trajectory. When the trajectory in question is unstable these
perturbations grow exponentially with time. Then the standard formula
gives $A = 0$, which is obviously incorrect.
Our modified semiclassical method overcomes the above difficulties.
The main idea of the method was proposed in
Refs.~\cite{Bezrukov:2003yf,Levkov:2007prl}; here we present its
detailed derivation. The modified method is summarized as
follows. We evaluate the Feynman path integral for the tunneling
amplitude in two steps. First, we restrict the integral to paths
which arrive into the out-region in a {\it fixed} time interval
$\tau$. Second, we integrate
over $\tau$. The integration at step $1$ can be done by the standard
saddle-point method, since all trajectories at finite $\tau$ are
stable and interpolate between the in- and out- regions. On the other
hand, the ordinary integral over $\tau$ at step 2 should be evaluated
with care. In particular, we find that in the case of
sphaleron--driven tunneling this integral is saturated in
the region $\tau \to +\infty$, rather than at the saddle point at
finite $\tau$.
The above manipulations with the path integral lead to a
notably simple semiclassical description of sphaleron--driven
tunneling. Namely, we show that the constraint in the path
integral leads to the deformation of the semiclassical equations of motion
with the {\it imaginary} term proportional to the small parameter
$\epsilon = \epsilon(\tau)$. The evaluation of the integral over
$\tau$ corresponds to taking the limit $\epsilon \to +0$ in both
cases of stable and unstable trajectories. However, the resulting
expressions for the tunneling probability are different in the two
cases, since the integral over $\tau$ is saturated in two different
regions. In particular, the probability formula in the case of
sphaleron--driven tunneling involves additional factor
$\hbar^{1/2}$ mentioned above. We call the modified semiclassical
technique by the {\it method of $\epsilon$--regularization}.
The new mechanism of tunneling
is relevant only at sufficiently high energies, $E>E_c$.
Below $E_c$ transitions proceed via the ordinary mechanism
of potential tunneling. In accordance with our results, the semiclassical
expression for the prefactor $A$ changes discontinuously across the
critical energy. In particular, in the inclusive case in two
dimensions $A\propto
\hbar^{1/2}$ and $\hbar$ at $E<E_c$ and $E>E_c$ respectively. This
implies that both expressions break down in a small vicinity of
$E_{c}$, where the correct {\it uniform} approximation should be
invoked. In the present paper we derive the required formula, which is
continuous and applicable in the entire energy range. At $|E-E_{c}|
\gg \hbar^{1/2}$ this formula coincides with the respective
``potential'' and ``sphaleron--driven'' semiclassical expressions.
In this regard it is similar to the uniform
approximation~\cite{Creagh:2006} for the tunnel level splitting at the
point of transition from integrable to near-integrable systems.
Next, we study semiclassically exclusive tunneling processes in the
sphaleron--driven case. We find that the new mechanism leads to
proliferation
of complex
trajectories describing a given exclusive process. These trajectories
form an infinite sequence and have the following structure: they
get attracted to the sphaleron orbit, follow it for an integer number
of periods and then slide away. The tunneling amplitude is the sum of
the contributions of all these trajectories. In analogy to the
case of inclusive probability
the sum is saturated by the trajectories which spend an
infinite time at the sphaleron. It is worth noting
that, in contrast to the inclusive case, the individual trajectories
describing exclusive process are stable. Thus, a priori, there is
no need for the modified semiclassical technique in the
case of exclusive transitions. Still, in this paper we demonstrate
that our modified technique turns out to be useful in finding and
organizing the tunneling trajectories. It also provides the link
between the semiclassical descriptions of inclusive and exclusive
processes.
Finally, for the sake of completeness
we study the processes of tunneling from low-lying
in-states. Naively, such states and hence the corresponding
tunneling processes cannot be described semiclassically; still, we show
that the probabilities of these processes are given by the
semiclassical formula~(\ref{eq:9}). In addition, we show that the
suppression exponent and
prefactor of tunneling from the low--lying states can be obtained as
certain limits of the corresponding quantities in the case of highly
excited states. The limiting relation for the suppression exponent is
known in field theory as the Rubakov--Son--Tinyakov
conjecture~\cite{Rubakov:1992ec}; it plays an important role in the
semiclassical description of collision-induced
tunneling~\cite{induced}. We
prove this conjecture in quantum mechanical setup. Our
limiting formula for the prefactor shows that the
probability of tunneling from the low--lying states contains
a factor $\hbar^{-1/2}$ as compared to the
case of highly excited in-states.
We illustrate our findings by considering tunneling transitions in
a simple model with two degrees of freedom. For this model we compare
predictions of the modified semiclassical technique with the exact
quantum mechanical results. The latter are extracted from the
numerical solution of the stationary Schr\"odinger equation. We find
perfect agreement between the two sets of results.
The outline of the paper is as follows. After presenting the model in
Sec.~\ref{sec:model} we summarize the experimental signatures of
sphaleron--driven tunneling in
Sec.~\ref{sec:summary-results}. In Sec.~\ref{sec:modif-semicl-techn}
we introduce the modified semiclassical technique: we
review the standard semiclassical method in Sec.~\ref{sec:semicl-form-tunn},
introduce $\epsilon$--regularization in
Sec.~\ref{sec:unst-traj} and derive the uniform formula in
Sec.~\ref{sec:unif-appr}. Application of the modified semiclassical
method to the exclusive tunneling processes is discussed
in Sec.~\ref{sec:nf}. Finally, we study tunneling from low--lying
in-states in
Sec.~\ref{sec:limit-small-quantum}. Section~\ref{sec:discussion}
contains discussion. Technical details are described in appendices.
\section{The model}
\label{sec:model}
We start by introducing the scattering model of
Refs.~\cite{Bonini:1999kj,Bezrukov:2003yf}.
It will be used throughout the paper for illustrative purposes.
The model describes motion of a particle with unit mass in the potential
\begin{equation}
\label{eq:24}
V(x,y) = \omega^2 y^2/2 + \mathrm{e}^{-(x+y)^2/2}.
\end{equation}
The potential represents two--dimensional harmonic waveguide extended
along the $x$ direction and intersected at an angle by the
potential barrier. The contour plot of the potential is shown in
Fig.~\ref{fig:0}a. In this and other figures we use the value
$\omega=1/2$ for the waveguide frequency. Note that potentials similar
to (\ref{eq:24}) typically arise in the studies of collinear chemical
reactions~\cite{Miller}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.5\textwidth]{fig0.eps}
\includegraphics[width=0.5\textwidth]{fig1.eps}}
\hspace{4cm}(a)\hspace{8cm}(b)
\caption{\label{fig:0} (a) The contour plot of the potential
(dashed lines) and the real part of the tunneling trajectory
at $E=1.3$, $E_y=0.05$ (solid line). The saddle point
is marked by the thick dot. (b) Time evolution of
$\mathrm{Re}\, x$ for two complex trajectories with $E_y = 0.05$ and
different values of total energy, $E = 0.9$ and $1.3$.
Note that $E_c(E_y=0.05) \approx 1.1$.}
\end{figure}
We are interested in tunneling transitions of quantum particle
between the asymptotic regions $x\to -\infty$ and $x\to +\infty$ of
the potential (in- and out- regions respectively). In the in-region
the particle evolves with constant momentum in the $x$ direction
oscillating along the $y$ axis. The corresponding in-state
is fixed by the total energy $E$ and the energy of $y$ oscillations
$E_y$. Similarly, the out-state can be fully
characterized by $E$ and $E_y^f$, where $E_y^f$ is the final oscillator
energy. In what follows we will often omit
the specification of the out-state and consider the total (inclusive)
probability of tunneling into the region $x\to +\infty$.
The height of the potential barrier separating the in- and
out- regions is $V_0 = 1$. It is given by the value of the potential
at the saddle point $(x,y) = (0,0)$. At $E<V_0$ the classical transitions
between the regions
are forbidden energetically, and their underlying mechanism is
potential tunneling. On the other hand, it is shown
in Ref.~\cite{Bonini:1999kj} that classical over--barrier
transitions between the asymptotic regions take place
at $E>E_b(E_y)$, where $E_b(E_y)$ is larger than $V_0$. Hence, at
intermediate energies $V_0 < E < E_b(E_y)$ the transitions are in the
regime of dynamical tunneling, which we are interested in.
As we have already discussed in the Introduction, the multidimensional
processes of dynamical tunneling, such as ours, generically proceed via
sphaleron--driven mechanism at sufficiently high energies. Let us
illustrate the new mechanism in the model (\ref{eq:24}) comparing the
behavior of semiclassical solutions at low and high energies
\cite{Bezrukov:2003yf}. Consider the inclusive tunneling transition
from the state $| E,E_y\rangle$ into the out-region $x\to +\infty$.
We postpone the consistent formulation of the semiclassical method till
Sec.~\ref{sec:modif-semicl-techn}. The only fact we need here is that any
tunneling process is specified by a certain complex trajectory ---
solution to the (complexified) classical equations of motion. The
latter should interpolate between the in- and out- regions of the
process.
Fig.~\ref{fig:0}b shows the complex trajectories describing
tunneling transitions at $E_y = 0.05$ and two values of total energy,
$E=0.9$ and $1.3$. [The real part of the trajectory with $E=1.3$ is
also depicted in Fig.~\ref{fig:0}a.] The behavior of the two
trajectories is drastically different. While the low--energy
trajectory interpolates between the asymptotic regions $x\to
\pm \infty$, the solution with $E=1.3$ gets stuck at finite $x$
approaching the unstable periodic orbit as $t\to +\infty$. The
latter orbit is precisely the sphaleron discussed
in the Introduction; it describes oscillations around the saddle point
of the potential, see Fig.~\ref{fig:0}a. Clearly, the high--energy
trajectory of
Fig.~\ref{fig:0} describes only half of the transition
process, since it does not arrive into the out-region. Trajectory
corresponding to the other half can be
obtained by adding to the unstable periodic orbit infinitesimally
small momentum in the direction of
the out-region and evolving the
system classically. Thus constructed, the overall semiclassical
evolution involves {\it two} trajectories which describe
creation and subsequent decay of the sphaleron\footnote{There is
another way to visualize the semiclassical
evolution~\cite{Takahashi:Ikeda}. One
introduces stable and unstable manifolds of the sphaleron
orbit. These are formed respectively by the trajectories arriving at
the sphaleron at $t\to +\infty$ and trajectories starting from it
at $t\to -\infty$. Then, the evolution describing
sphaleron--driven tunneling is guided in turn by trajectories
belonging to the stable and unstable manifolds of the sphaleron.}.
This evolution corresponds to the mechanism of sphaleron--driven
tunneling.
One finds \cite{Bezrukov:2003yf}
that there exists the critical value $E=E_c(E_y)$ of
total energy which separates the regions of qualitatively different
behavior of tunneling trajectories. Namely, the trajectories
interpolate between the in- and out- regions at $E<E_c(E_y)$ and
approach the sphaleron orbit at $E
|
_c(E_y)< E < E_b(E_y)$. This means
that the mechanism of transition changes from potential to
sphaleron--driven tunneling as the energy crosses the critical
value. From the physical viewpoint $E_c(E_y)$ can be understood as the
energy of ``phase transition'' between the two regimes of tunneling.
We remark that the energies of sphaleron
orbits and thus the critical energy for the sphaleron--driven
tunneling exceed the height of the potential barrier. Therefore, the
new mechanism is relevant only in the case of dynamical tunneling.
\begin{figure}
\centerline{\includegraphics[width=0.5\textwidth]{fig23.eps}}
\caption{\label{fig:23} Regions in the plane of in--state quantum
numbers corresponding to the potential and sphaleron--driven
tunneling mechanisms.}
\end{figure}
The region $E_c(E_y) < E < E_b(E_y)$ corresponding to the
sphaleron--driven tunneling in the model (\ref{eq:24}) is shown in
Fig.~\ref{fig:23}. The value of $E_c(E_y)$ is found numerically by
computing the complex trajectories at different energies and
investigating their stability.
\section{Experimental signatures}
\label{sec:summary-results}
In this Section we show that the mechanism of sphaleron--driven
tunneling leads to two observable effects which in principle can be
used for identification of the new mechanism in future
experiments. Both effects are related to the fact
that the relevant semiclassical solutions are unstable. We illustrate
our findings in the model (\ref{eq:24}) using the exact quantum
mechanical results. The exact calculations of this and the subsequent
sections are based on the numerical solution of time--independent
Schr\"odinger equation, see
Refs.~\cite{Bonini:1999kj,Levkov:2007e,f90code} for the numerical
method and Fortran 90 code.
The first signature of sphaleron--driven tunneling is the direct
consequence of the semiclassical analysis which will be presented in
Sec.~\ref{sec:modif-semicl-techn}. We find that the sphaleron--driven
mechanism changes the power--law dependence of the transmission
probability on $\hbar$ compared to the case of potential tunneling.
To be concrete, let us discuss inclusive
tunneling transitions in the model (\ref{eq:24}).
Then, the prefactor $A$ of the probability is
proportional to $\hbar^{1/2}$ and $\hbar$ in the cases of potential
and sphaleron--driven tunneling respectively.
The physics behind the additional power--law suppression becomes clear if
one uses the qualitative analogy with the classically allowed creation
of unstable state. The latter process considered at the classical
level requires fine tuning of the Cauchy data. As a
consequence, only a small part of the in-state wave function
contributes into the amplitude of the process. This results in the
additional suppression of the probability. On general
grounds one expects similar formal suppression in the case of
sphaleron--driven tunneling.
Experimentally, one can try to observe the unusual power--law
dependence on $\hbar$ by analyzing the probability graph ${\cal
P}(\hbar)$.
Note that the value of the semiclassical parameter which
we denote by $\hbar$ is, in principle, adjustable in
experiments. Indeed, the magnitude of quantum fluctuations
is measured by the dimensionless ratio of the Planck constant to
a certain combination of parameters characterizing the
system. Changing the latter parameters in an appropriate way, one
alters the value of the semiclassical parameter $\hbar$ without
affecting the classical dynamics of the system.
To illustrate this point consider the system (\ref{eq:24}). The
key quantity which enters into the semiclassical expansion is the
ratio of the action of the system to the Planck
constant. Restoring the dimensionful units we obtain
\[
\frac{S}{\hbar_0}=\frac{1}{\hbar_0}\int dt
\left(\frac{M{\dot{\boldsymbol{x}}}^2}{2}-\frac{M\omega_0^2y^2}{2}
-V_0{\rm e}^{-(x+y)^2/2L^2}\right)\;,
\]
where $\hbar_0$ stands for the physical Planck constant. In terms of
dimensionless variables this expression reads
\[
\frac{S}{\hbar_0}=\frac{1}{\hbar}\int d\tilde t
\left(\frac{{\dot{\tilde{\boldsymbol{x}}}}^2}{2}
-\frac{\omega^2\tilde y^2}{2}-{\rm e}^{-(\tilde x+\tilde y)^2/2}
\right)\;,
\]
where $\hbar=\hbar_0/\sqrt{MV_0L^2}$,
$\omega^2=ML^2\omega_0^2/V_0$. The effective frequency $\omega$
completely determines the classical dynamics. On the other hand, the
effective Planck constant $\hbar$ is given by an independent
combination of parameters.
One can hardly hope to extract directly the additional factor $\hbar^{1/2}$
from the experimental data on transmission probability: it is almost
impossible to identify the weak power--law dependence on top of the
leading semiclassical exponent. We suggest an indirect method. Namely,
consider the quantity
$$
F_{QM} = - \hbar \log({\cal P}/\hbar^{1/2}).
$$
In the regime of potential tunneling ($A\propto \hbar^{1/2}$)
$F_{QM}$ is almost independent of $\hbar$ at small values of the
latter. On the other hand, $F_{QM} \simeq -\frac12\hbar
\log \hbar + \mbox{const}$ whenever
the new tunneling mechanism is involved. The difference between the
two cases is seen in Fig.~\ref{fig:2}, where the dependences of
$F_{QM}$ on the total energy $E$ are shown for several values of
$\hbar$. The graphs in Fig.~\ref{fig:2} coincide at energies somewhat
smaller than $E_c$ (say, at $E\lesssim 1$), while at $E>E_c$ a clear
difference between the graphs appears. We remark that the change
in the behavior of the exact tunneling probability is gradual, in
spite of the fact that the complex trajectories have distinct
structure at $E<E_c$ and $E>E_c$. We discuss this point and
derive the appropriate uniform formula in Sec.~\ref{sec:unif-appr}.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\textwidth]{fig2.eps}}
\caption{\label{fig:2} The probability logarithm $F_{QM} = - \hbar
\log({\cal P}/\hbar^{1/2})$ plotted as a function of total energy for
several values of $\hbar$ and $E_y = 0.05$. Points represent
the exact quantum mechanical results; the interpolating lines
are drawn for convenience. The critical energy is shown by
dashed vertical line.}
\end{figure}
Another signature of the sphaleron--driven mechanism
was first pointed out in
Refs.~\cite{Takahashi:Ikeda,Takahashi:2008}.
One notes that the second stage of sphaleron--driven transition,
the decay of the sphaleron orbit, proceeds classically and does not
affect
the leading suppression exponent $F$ of the probability. In addition,
the sphaleron, being unstable, can evolve at the classical level into
the out-states with {\it different} values of oscillator energy
$E_y^{f}$. Classical trajectories corresponding to these
evolutions are obtained by adding small momentum in the direction
of the out-region at different points of the sphaleron orbit. One
concludes that in the case of sphaleron--driven tunneling the
distribution over final oscillator energies is almost constant in
some region $E_{y,1}^{f} < E_y^f < E_{y,2}^f$. The latter region
corresponds to the decays of the sphaleron along different
classical trajectories.
Note that the above feature is in sharp contrast with the properties of
final states in the standard case of potential tunneling. Namely, in
a typical situation the complex trajectory describing transmission
through the barrier is unique, and the corresponding out-state wave
function forms sharply peaked Gaussian distribution around some
optimal value $E_y^f = \langle E_y^f \rangle$.
To illustrate explicitly the effect of anomalously wide final
states in the case of sphaleron--driven tunneling,
we consider transitions between the {\it exclusive} in- and out-
states which have definite energies of $y$ oscillator, $E_y$ and
$E_y^{f}$ respectively, and the same total energy $E$. Then, we fix
the initial state ($E$ and $E_y$) and analyze the dependence of the
exact exclusive probability ${\cal P}_{e}$ on $E_y^f$. This
dependence is shown in Fig.~\ref{fig:3} in logarithmic scale for
several values of $E$. One
immediately sees in Fig.~\ref{fig:3}a that the width of the
out-state distribution grows as the value of total energy
approaches $E_c(E_y)$ from below. In particular, a flat plateau
gradually develops in the right side of the distribution. At
energies higher than critical the plateau is wide
and corresponds to the maximum probability of tunneling. Moreover,
the graphs become flatter as the value of $\hbar$
decreases, see Fig.~\ref{fig:3}b.
One sees another feature of
the new tunneling mechanism: the short--scale fluctuations in the
right and left parts of the plateaux in Fig.~\ref{fig:3}b. This is the
hallmark of quantum interference phenomena, which seem to be important
for complete understanding of exclusive processes at $E>E_c(E_y)$.
We discuss this point in Sec.~\ref{sec:nf}.
\begin{figure}[htb]
\centerline{\includegraphics[width=0.5\textwidth]{fig3.eps}
\includegraphics[width=0.5\textwidth]{fig4.eps}}
\hspace{4.2cm}(a)\hspace{8cm}(b)
\caption{\label{fig:3} Distributions of the logarithm of exclusive
tunneling probability over the out-state quantum number $E_y^f$. The graphs
are plotted at $E_y = 0.05$ and: (a) $\hbar = 1/40$ and
four values of total energy; (b) $E =
1.2$ and different values of $\hbar$. Note that $E_c(E_y=0.05)
\approx 1.1$.}
\end{figure}
\section{Modified semiclassical technique}
\label{sec:modif-semicl-techn}
In this section we describe the semiclassical technique
adapted to the analysis of sphaleron--driven tunneling.
We start by reviewing the path integral derivation of
the standard method of complex trajectories~\cite{Miller}.
Then, we manipulate with the path integral and obtain
the modified semiclassical expressions in the case of
sphaleron--driven tunneling.
For simplicity we assume that the system undergoing tunneling
transition is similar to the model of Sec.~\ref{sec:model}. Throughout
this section we consider tunneling between the asymptotic in- and out-
regions of two--dimensional waveguide potential, where the in-state of
the process $|E,\, E_y\rangle$ is fixed and the final state is
inclusive. It is worth noting that both the standard and modified
semiclassical methods are completely general and the semiclassical
formulas of this section can be generalized to other systems. In
particular, the modified method was applied to the case of chaotic
tunneling in Ref.~\cite{Levkov:2007e} and to field theory in
Ref.~\cite{Bezrukov:2003er}.
\subsection{The standard method}
\label{sec:semicl-form-tunn}
Semiclassical calculations within the method of complex
trajectories proceed as follows. One reduces the problem of computing
the tunneling probability to a problem of finding the complex
trajectory $\boldsymbol{x}^{(s)}(t)$ --- complex solution to the
classical equations of motion with certain boundary conditions. In
practice this solution is obtained numerically. Then, tunneling
probability is given by Eq.~(\ref{eq:9}) where $F$ and $A$ are certain
functionals of $\boldsymbol{x}^{(s)}(t)$. In this section we derive
the boundary conditions for $\boldsymbol{x}^{(s)}(t)$ and expressions
for the functionals $F$, $A$ in the standard case of potential
tunneling.
In order to compute the inclusive tunneling probability we first
obtain the semiclassical expression for the final state of the
tunneling process. The wave function $\Psi_f$ of the final state has
the form,
\begin{equation}
\label{eq:1}
\Psi_f(\boldsymbol{x}_f) = \langle \boldsymbol{x}_f |\,
\mathrm{e}^{-iH (t_f - t_i)/\hbar}\, | E,\,E_y \rangle =
\int d\boldsymbol{x}_i \, \langle \boldsymbol{x}_f |\,
\mathrm{e}^{-iH (t_f - t_i)/\hbar} \,| \boldsymbol{x}_i \rangle
\,\Psi_i(\boldsymbol{x}_i) \;,
\end{equation}
where $\boldsymbol{x} = (x,\, y)$, while $\Psi_i(\boldsymbol{x}_i) =
\langle \boldsymbol{x}_i | E,\, E_y\rangle$ is the in-state
wave function. Below we assume implicitly
that $\Psi_i$ and $\Psi_f$ have support in the in- and out-
asymptotic regions respectively. One uses the path integral
representation for the quantum propagator in Eq.~(\ref{eq:1}) and
writes,
\begin{equation}
\label{eq:7}
\Psi_f(\boldsymbol{x}_f) = \left. \int
d\boldsymbol{x}_i \, \Psi_i(\boldsymbol{x}_i)\,
\int [d\boldsymbol{x}]\right|_{\boldsymbol{x}_i}^{\boldsymbol{x}_f} \,
\mathrm{e}^{iS[\boldsymbol{x}]/\hbar}\;,
\end{equation}
where $S$ stands for
the classical action of the system. One observes that at small $\hbar$
the integrand in Eq.~(\ref{eq:7}) contains fast--oscillating exponent;
thus, the respective integral can be evaluated by the saddle--point
method. To keep the discussion short, we defer the details of the
saddle--point integration to appendix~\ref{sec:eval-pre-expon}; here
we quote the result. One finds the extremum of the leading exponent in
Eq.~(\ref{eq:7}), which is
represented by the trajectory $\boldsymbol{x}^{(s)}(t)$ going between
the in- and out- regions. This trajectory is generically complex. It
satisfies the
classical equations of motion $\delta S/\delta \boldsymbol{x}(t) = 0$
and arrives at a given point $\boldsymbol{x}=\boldsymbol{x}_f$ at $t =
t_f$. The boundary conditions at $t = t_i$ for
$\boldsymbol{x}^{(s)}(t)$ are obtained from the saddle--point
integration over $\boldsymbol{x}_i$; they fix the values of
in-state quantum numbers,
\begin{equation}
\label{eq:6}
E_y = (\dot{y}_i^2 + \omega^2 y_i^2)/2\;, \qquad \qquad
E = \dot{x}_i^2/2 + E_y\;,
\end{equation}
where the subscript $i$ marks the quantities evaluated at $t = t_i$.
For brevity we omit the superscript $(s)$
of the semiclassical trajectory in Eq.~(\ref{eq:6}) and in what follows.
As the result of integration in Eq.~(\ref{eq:7}), one finds the
semiclassical wave function of the final state,
\begin{equation}
\label{eq:8}
\Psi_f(\boldsymbol{x}_f) =
D^{-1/2}\cdot\exp\left\{{\frac{i}{\hbar}(S[\boldsymbol{x}]
+ B_i[\boldsymbol{x}]) + \frac{i\pi}{4}}\right\}\;,
\end{equation}
where
$B_i$ is the in-state contribution to the exponent and $D$
represents the prefactor determinant, see Eqs.~(\ref{eq:3}),
(\ref{eq:B5}) in appendix~\ref{sec:eval-pre-expon} for explicit
expressions.
Note that the leading
exponent $S+B_i$ in Eq.~(\ref{eq:8}) is evaluated on the saddle--point
trajectory $\boldsymbol{x}(t)$.
The inclusive probability ${\cal P}$ of transmission is equal to the
total flux\footnote{We use the in-state with the unit flux
normalization.} of the out-wave~(\ref{eq:8}) through the distant line
$x_f = x_f^{(0)}$, where $x_f^{(0)}$ is large
and positive. Semiclassically, one writes,
\begin{equation}
\label{eq:19}
{\cal P} = \int dy_f \, |\Psi_f(\boldsymbol{x}_f)|^2 \,\mathrm{Re}\,
\dot{x}_f\;,
\end{equation}
where we used the fact that $\partial S/\partial x_f =
\dot{x}_f$. The integral in the above expression is again computed
by the saddle--point technique. In appendix~\ref{sec:eval-pre-expon} we
show that the extremum of the leading exponent in Eq.~(\ref{eq:19}) is
achieved when
\begin{equation}
\label{eq:20}
x_f = x_f^{(0)}\;, \qquad\qquad \mathrm{Im}\, \dot{y}_f =
\mathrm{Im}\, y_f = 0\;.
\end{equation}
Equations~(\ref{eq:20}) fix the boundary conditions at $t = t_f$
for the semiclassical trajectory.
After the saddle--point integration in Eq.~(\ref{eq:19}) one finally
arrives at the familiar semiclassical expression~(\ref{eq:9}) for the
tunneling probability, where the leading exponent is
\begin{equation}
\label{eq:26}
F_{pot} = 2\, \mathrm{Im}(S + B_i)\;.
\end{equation}
Note that we mark all the standard
semiclassical expressions with the subscript $pot$ which stands for ``potential
tunneling''.
The prefactor $A_{pot}$ is computed
as follows (see appendix~\ref{sec:eval-pre-expon} for the
derivation).
One finds two independent perturbations $\delta
\boldsymbol{x}^{(1)}(t)$ and $\delta \boldsymbol{x}^{(2)}(t)$ in the
background of the complex trajectory $\boldsymbol{x}(t)$. These
perturbations satisfy the linearized classical equations of motion,
\begin{equation}
\label{eq:4}
\delta\ddot{\boldsymbol{x}}^{(n)} + \hat{V}'' (\boldsymbol{x}(t)) \delta
\boldsymbol{x}^{(n)} = 0\;, \qquad \qquad n=1,2
\end{equation}
with certain Cauchy data\footnote{First, the perturbations are real at
$t = t_f$. Second, they do not change the value of total energy,
$\delta E[\delta\boldsymbol{x}^{(n)}] = 0$. Third, $\Omega(\delta
\boldsymbol{x}^{(1)}, \delta \boldsymbol{x}^{(2)}) = 1$, where
$\Omega$ is the canonical symplectic form.} at $t = t_f$. After
evolving $\delta \boldsymbol{x}^{(n)}(t)$ back in time from $t = t_f$
to $t = t_i$, one computes the prefactor by the
formula\footnote{As discussed in appendix~\ref{sec:eval-pre-expon},
this formula is canonically covariant.}
\begin{equation}
\label{eq:21}
A_{pot} = \frac{\hbar^{1/2}\omega }{\sqrt{4 \pi\,
\mathrm{Im} (\delta E_y[\delta\boldsymbol{x}^{(1)}]\cdot \delta E_y^*
[\delta\boldsymbol{x}^{(2)}] ) }}\;,
\end{equation}
where the linear functional
\begin{equation}
\label{eq:67}
\delta E_y[\delta \boldsymbol{x}] = \dot{y}_i \delta \dot{y}_i +
\omega^2 y_i \delta y_i
\end{equation}
measures the change in the initial oscillator energy $E_y$ due to
the perturbation $\delta \boldsymbol{x}(t)$. We stress that $\delta
E_y[\delta \boldsymbol{x}^{(n)}]$ involves perturbations in the
in-region, while the
Cauchy data for $\delta \boldsymbol{x}^{(n)}(t)$ are set at $t = t_f$.
We also note that the prefactor~(\ref{eq:21}) is explicitly
proportional to $\hbar^{1/2}$; this fact was used in the previous
section.
The standard semiclassical calculation is summarized as
follows. One finds the complex trajectory $\boldsymbol{x}(t)$
satisfying the classical equations of motion with the boundary
conditions~(\ref{eq:6}),~(\ref{eq:20}). Our numerical method for
finding the trajectory is presented in
appendix~\ref{sec:numerical-method}. The suppression exponent
$F_{pot}$ of the probability is given by the value of the functional
(\ref{eq:26}) on the trajectory $\boldsymbol{x}(t)$. Then, one
considers the linear perturbations around the semiclassical trajectory
and finds the prefactor $A_{pot}$ using the expression (\ref{eq:21}).
\begin{
|
paths}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{node{-}l}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{t}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxx}}}\RktPn{(}\RktSym{append}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{paths}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{node{-}n}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{t}\RktPn{)}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{paths}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{node{-}r}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{t}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{]}\RktPn{)}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}} \\
\hbox{\RktPn{(}\RktSym{check{-}equal{\hbox{\texttt{?}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{paths}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{node}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{1}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{node}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{2}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{leaf}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{3}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{leaf}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{4}\RktPn{)}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{leaf}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{5}\RktPn{)}\RktPn{)}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxxxxx}}}\RktVal{{\textquotesingle}}\RktVal{(}\RktVal{(}\RktVal{3}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{2}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{1}\RktVal{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{(}\RktVal{4}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{2}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{1}\RktVal{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{(}\RktVal{5}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{1}\RktVal{)}\RktVal{)}\RktPn{)}}\end{tabular}\end{RktBlk}\end{bigtabular}\end{SInsetFlow}
\noindent \raisebox{-0.5999999999999943bp}{\makebox[396.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_5.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22markx2dpathsx22x29x29}\textsf{Fig.}~\textsf{5}. }{t:x28counter_x28x22figurex22_x22markx2dpathsx22x29x29}\textsf{Recording paths in a tree with continuation marks}}}\end{Figure}
Figure~\hyperref[t:x28counter_x28x22figurex22_x22markx2dpathsx22x29x29]{\FigureRef{5}{t:x28counter_x28x22figurex22_x22markx2dpathsx22x29x29}} illustrates the working of
continuation marks with a function that traverses binary
trees and records paths from roots to leaves. The top half
of the figure shows the code that performs the
traversal. Whenever the function reaches an internal node,
it leaves a continuation mark recording that node{'}s value.
When it reaches a leaf, it collects those marks, adds the
leaf to the path and returns the completed path. A trace of
the continuation mark stack is shown in the bottom half of
the figure. It highlights the execution points where the
stack is reported to the user.
Continuation marks are extensively used in the Racket
ecosystem, e.g., the generation of error messages in
the DrRacket IDE\Autobibref{~(\hyperref[t:x28autobib_x22Robert_Bruce_Findlerx2c_John_Clementsx2c_Cormac_Flanaganx2c_Matthew_Flattx2c_Shriram_Krishnamurthix2c_Paul_Stecklerx2c_and_Matthias_FelleisenDrSchemex3a_a_programming_environment_for_SchemeJornal_of_Functional_Programming_12x282x29x2c_ppx2e_159x2dx2d1822002x22x29]{\AutobibLink{Findler et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Robert_Bruce_Findlerx2c_John_Clementsx2c_Cormac_Flanaganx2c_Matthew_Flattx2c_Shriram_Krishnamurthix2c_Paul_Stecklerx2c_and_Matthias_FelleisenDrSchemex3a_a_programming_environment_for_SchemeJornal_of_Functional_Programming_12x282x29x2c_ppx2e_159x2dx2d1822002x22x29]{\AutobibLink{2002}})}, an algebraic stepper
\Autobibref{~(\hyperref[t:x28autobib_x22John_Clementsx2c_Matthew_Flattx2c_and_Matthias_FelleisenModeling_an_algebraic_stepperIn_Procx2e_European_Symposium_on_Programmingx2c_ppx2e_320x2dx2d3342001x22x29]{\AutobibLink{Clements et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22John_Clementsx2c_Matthew_Flattx2c_and_Matthias_FelleisenModeling_an_algebraic_stepperIn_Procx2e_European_Symposium_on_Programmingx2c_ppx2e_320x2dx2d3342001x22x29]{\AutobibLink{2001}})}, the DrRacket debugger, for thread{-}local
dynamic binding\Autobibref{~(\hyperref[t:x28autobib_x22Rx2e_Kent_DybvigChez_Scheme_Version_8_Userx27s_GuideCadence_Research_Systems2009x22x29]{\AutobibLink{Dybvig}} \hyperref[t:x28autobib_x22Rx2e_Kent_DybvigChez_Scheme_Version_8_Userx27s_GuideCadence_Research_Systems2009x22x29]{\AutobibLink{2009}})},
for exception handling, and even serializable continuations in the
PLT web server\Autobibref{~(\hyperref[t:x28autobib_x22Jay_McCarthyThe_twox2dstate_solutionx3a_native_and_serializable_continuations_accordIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_567x2dx2d5822010x22x29]{\AutobibLink{McCarthy}} \hyperref[t:x28autobib_x22Jay_McCarthyThe_twox2dstate_solutionx3a_native_and_serializable_continuations_accordIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_567x2dx2d5822010x22x29]{\AutobibLink{2010}})}.
Beyond Racket, continuation marks have also been
added to Microsoft{'}s CLR\Autobibref{~(\hyperref[t:x28autobib_x22Greg_Pettyjohnx2c_John_Clementsx2c_Joe_Marshallx2c_Shriram_Krishnamurthix2c_and_Matthias_FelleisenContinuations_from_generalized_stack_inspectionIn_Procx2e_International_Conference_on_Functional_Programmingx2c_ppx2e_216x2dx2d2272005x22x29]{\AutobibLink{Pettyjohn et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Greg_Pettyjohnx2c_John_Clementsx2c_Joe_Marshallx2c_Shriram_Krishnamurthix2c_and_Matthias_FelleisenContinuations_from_generalized_stack_inspectionIn_Procx2e_International_Conference_on_Functional_Programmingx2c_ppx2e_216x2dx2d2272005x22x29]{\AutobibLink{2005}})} and
JavaScript\Autobibref{~(\hyperref[t:x28autobib_x22John_Clementsx2c_Ayswarya_Sundaramx2c_and_David_HermanImplementing_continuation_marks_in_JavaScriptIn_Procx2e_Scheme_and_Functional_Programming_Workshopx2c_ppx2e_1x2dx2d102008x22x29]{\AutobibLink{Clements et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22John_Clementsx2c_Ayswarya_Sundaramx2c_and_David_HermanImplementing_continuation_marks_in_JavaScriptIn_Procx2e_Scheme_and_Functional_Programming_Workshopx2c_ppx2e_1x2dx2d102008x22x29]{\AutobibLink{2008}})}.
Other languages provide similar mechanisms, such as stack reflection in
Smalltalk and the stack introspection used by the GHCi
debugger\Autobibref{~(\hyperref[t:x28autobib_x22Simon_Marlowx2c_Josxe9_Iborrax2c_Bernard_Popex2c_and_Andy_GillA_lightweight_interactive_debugger_for_HaskellIn_Procx2e_Haskell_Workshopx2c_ppx2e_13x2dx2d242007x22x29]{\AutobibLink{Marlow et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Simon_Marlowx2c_Josxe9_Iborrax2c_Bernard_Popex2c_and_Andy_GillA_lightweight_interactive_debugger_for_HaskellIn_Procx2e_Haskell_Workshopx2c_ppx2e_13x2dx2d242007x22x29]{\AutobibLink{2007}})} for Haskell.
\Ssubsection{Feature{-}specific Data Gathering : The Protocol}{Feature{-}specific Data Gathering : The Protocol}\label{t:x28part_x22instrx2dflatx22x29}
The stack{-}sample analysis requires that a feature implementation places a
marker with a certain key on the control stack when it
begins to evaluate feature{-}specific code.
\Ssubsubsectionstarx{Marking}{Marking}\label{t:x28part_x22Markingx22x29}
Feature authors who wish to enable feature{-}specific profiling for
their features must change the implementation of the feature
so that instances mark their dynamic extents with \textit{feature marks}.
It suffices to wrap the relevant code with
\RktSym{with{-}continuation{-}mark}. These marks, added to the call stack, allow the
profiler to observe whether a thread is currently executing
code related to a feature.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29]{\FigureRef{6}{t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29}} shows an excerpt from the
instrumentation of type assertions in Typed Racket, a
variant of Racket that is statically type
checked\Autobibref{~(\hyperref[t:x28autobib_x22Sam_Tobinx2dHochstadt_and_Matthias_FelleisenThe_design_and_implementation_of_Typed_SchemeIn_Procx2e_Principles_of_Programming_Languagesx2c_ppx2e_395x2dx2d4062008x22x29]{\AutobibLink{Tobin{-}Hochstadt and Felleisen}} \hyperref[t:x28autobib_x22Sam_Tobinx2dHochstadt_and_Matthias_FelleisenThe_design_and_implementation_of_Typed_SchemeIn_Procx2e_Principles_of_Programming_Languagesx2c_ppx2e_395x2dx2d4062008x22x29]{\AutobibLink{2008}})}. The underlined conditional is
responsible for performing the actual assertion. The mark{'}s
key should uniquely identify the construct. In this case, we
use the symbol \RktVal{{\textquotesingle}}\RktVal{TR{-}assertion} as the key. Unique
choices avoid false reports and interference by distinct
features. In addition, choosing unique keys also permits the
composition of arbitrary features. As a consequence, the
analysis component of the FSP can present a unified report
to users; it also implies that users need not select in
advance the constructs they deem problematic.
The mark value{---}or \textit{payload}{---}can be anything that identifies the
feature instance to which the cost should be assigned.
In figure~\hyperref[t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29]{\FigureRef{6}{t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29}}, the payload is the source location of a
specific assertion in the program, which allows the profiler to compute the
cost of individual instances of \RktSym{assert}.
Annotating features is simple and involves only
non{-}instrusive, local code changes, but it does require
access to the implementation for the feature of interest.
Because it does not require any specialized profiling
knowledge, however, it is well within the reach of the
authors of linguistic constructs.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\begin{SInsetFlow}\begin{bigtabular}{@{\hspace{\stabLeft}}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}}
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{1}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{2}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{3}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{4}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{5}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{6}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{7}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{8}}}\end{tabular}\end{RktBlk} &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\RktPn{(}\RktSym{define{-}syntax}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{assert}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xx}}}\RktPn{(}\RktSym{syntax{-}case}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktPn{[}\RktPn{(}\RktSym{assert}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{v}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{p}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktCmt{;}\RktCmt{~}\RktCmt{the compiler rewrites this to{\hbox{\texttt{:}}}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxx}}}\RktPn{(}\RktSym{quasisyntax}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxx}}}\RktPn{(}\RktSym{let}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktPn{[}\RktSym{val}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{v}\RktPn{]}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{[}\RktSym{pred}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{p}\RktPn{]}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktPn{(}\RktSym{with{-}continuation{-}mark}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{TR{-}assertion}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxx}}}\RktPn{(}\RktSym{unsyntax}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{source{-}location}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\RktPn{)}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxx}}}\RktPn{(}\RktSym{if}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{pred}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{val}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{val}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{error}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{"Assertion failed{\hbox{\texttt{.}}}"}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{]}\RktPn{)}\RktPn{)}}\end{tabular}\end{RktBlk}\end{bigtabular}\end{SInsetFlow}
\noindent \identity{\vspace{0.1em}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29}\textsf{Fig.}~\textsf{6}. }{t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29}\textsf{Instrumentation of assertions (excerpt)}}}\end{Figure}
\Ssubsubsectionstarx{Antimarking}{Antimarking}\label{t:x28part_x22Antimarkingx22x29}
Features are
seldom {``}leaves{''} in a program; i.e., they usually run
user code whose execution time may not have to count towards
the time spent in the feature. For example, the profiler
must not count the time spent in function bodies towards the
cost of the language{'}s function call protocol.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\begin{SInsetFlow}\begin{bigtabular}{@{\hspace{\stabLeft}}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}}
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{1}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{2}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{3}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{4}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{5}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{6}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{7}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{8}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{9}}} \\
\hbox{\Smaller{\Scribtexttt{10}}}\end{tabular}\end{RktBlk} &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\RktPn{(}\RktSym{define{-}syntax}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{lambda/keyword}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xx}}}\RktPn{(}\RktSym{syntax{-}case}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktPn{[}\RktPn{(}\RktSym{lambda/keyword}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{formals}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{body}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktCmt{;}\RktCmt{~}\RktCmt{the compiler rewrites this to{\hbox{\texttt{:}}}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxx}}}\RktPn{(}\RktSym{quasisyntax}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxx}}}\RktPn{(}\RktSym{lambda}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{unsyntax}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{handle{-}keywords}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{formals}\RktPn{)}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxx}}}\RktPn{(}\RktSym{with{-}continuation{-}mark}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{kw{-}protocol}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}}}\RktPn{(}\RktSym{unsyntax}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\RktSym{source{-}location}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{stx}\RktPn{)}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxx}}}\Scribtexttt{$\cdots$parse keyword arguments, compute default values$\cdots$}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxx}}}\RktPn{(}\RktSym{with{-}continuation{-}mark}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{kw{-}protocol}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{antimark}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxxxx}}}\RktSym{body}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{)}\RktPn{]}\RktPn{)}\RktPn{)}\mbox{\hphantom{\Scribtexttt{x}}}\RktCmt{;}\RktCmt{~}\RktCmt{body is use{-}site code}}\end{tabular}\end{RktBlk}\end{bigtabular}\end{SInsetFlow}
\noindent \raisebox{-0.5999999999999943bp}{\makebox[396.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_6.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22kwx2dantimarkx2dcodex22x29x29}\textsf{Fig.}~\textsf{7}. }{t:x28counter_x28x22figurex22_x22kwx2dantimarkx2dcodex22x29x29}\textsf{Use of antimarks in instrumentation}}}\end{Figure}
To account for user code, features place \textit{antimarks} on
the stack. Such antimarks are continuation marks with a
distinguished value, a payload of \RktVal{{\textquotesingle}}\RktVal{antimark}, that
delimit a feature{'}s code. The analysis phase recognizes
antimarks and uses them to cancel out feature marks. Cost is
attributed to a feature only if the most recent mark is a
feature mark. If it is an antimark, the program is currently
executing user code, which should not be counted. An
antimark only cancels marks for its original feature. Marks
and antimarks, for the same or different features can be
nested.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22kwx2dantimarkx2dcodex22x29x29]{\FigureRef{7}{t:x28counter_x28x22figurex22_x22kwx2dantimarkx2dcodex22x29x29}} illustrates the idea with code
that instruments a simplified version of Racket{'}s optional
and keyword argument protocol \Autobibref{~(\hyperref[t:x28autobib_x22Matthew_Flatt_and_Eli_BarzilayKeyword_and_Optional_Arguments_in_PLT_SchemeIn_Procx2e_Workshop_on_Scheme_and_Functional_Programming2009x22x29]{\AutobibLink{Flatt and Barzilay}} \hyperref[t:x28autobib_x22Matthew_Flatt_and_Eli_BarzilayKeyword_and_Optional_Arguments_in_PLT_SchemeIn_Procx2e_Workshop_on_Scheme_and_Functional_Programming2009x22x29]{\AutobibLink{2009}})}. The
simplified implementation appears in the top half of the
figure and a sample trace of a function call using keyword
arguments is displayed in the bottom half. When the function
call begins, a \RktVal{{\textquotesingle}}\RktVal{kw{-}protocol} mark is placed on
the stack (annotated in \intextrgbcolor{0.37254901960784315,0.37254901960784315,0.37254901960784315}{DARK GRAY})
with a source location as its payload. Once evaluation of
the function begins, an antimark is placed on the stack
(annotated in \intextrgbcolor{0.8745098039215686,0.8745098039215686,0.8745098039215686}{LIGHT GRAY}). Once the
antimark has been removed from the stack, cost accounting is
again attributed towards keyword arguments.
In contrast, the assertions from figure~\hyperref[t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29]{\FigureRef{6}{t:x28counter_x28x22figurex22_x22flatx2dinstrx2dcodex22x29x29}} do
not require antimarks because user code evaluation happens
exclusively outside the marked region (line 8). Another feature that
has this behavior is program output, which also never calls
user code from within the feature.
\Ssubsubsectionstarx{Sampling}{Sampling}\label{t:x28part_x22Samplingx22x29}
During program
execution, the FSP{'}s sampling thread periodically collects
and stores continuation marks from the main thread. The
sampling thread knows which keys correspond to features it
should track, and collects marks for all features at once.\NoteBox{\NoteContent{In
general, the sampling thread could additionally
collect samples of all marks and sort the marks in the
analysis phase.}}
\Ssubsection{Analyzing Feature{-}specific Data}{Analyzing Feature{-}specific Data}\label{t:x28part_x22analysisx2dflatx22x29}
After the program execution terminates, the analysis
component processes the data collected by the sampling
thread to produce a feature cost report. The tool analyses
each feature separately, then combines the results into a
unified report.
\Ssubsubsectionstarx{Cost assignment}{Cost assignment}\label{t:x28part_x22Costx5fassignmentx22x29}
The profiler uses a standard sliding window
technique to assign a time cost to each sample based on the elapsed time
between the sample, its predecessor and its successor.
Only samples with a feature mark as the most recent mark contribute time
towards features.
\Ssubsubsectionstarx{Payload grouping}{Payload grouping}\label{t:x28part_x22Payloadx5fgroupingx22x29}
Payloads identify
individual feature instances. Our accounting algorithm
groups samples by payload and adds up the cost of each
sample; the sums correspond to the cost of each feature
instance. Payloads can be grouped in arbitrary equivalence
classes. Our profiler currently groups them based on
equality, but library authors can implement grouping
according to any criteria they desire. The FSP then
generates reports for each feature, using payloads as keys
and time costs as values.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.7552083333333255bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_7.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22circlex2dprofilex22x29x29}\textsf{Fig.}~\textsf{8}. }{t:x28counter_x28x22figurex22_x22circlex2dprofilex22x29x29}\textsf{Feature Profiler Results for Circle Properties}}}\end{Figure}
\Ssubsubsectionstarx{Report composition}{Report composition}\label{t:x28part_x22Reportx5fcompositionx22x29}
Finally, after generating individual feature reports, the
FSP combines them into a unified report. Constructs absent
from the program and those inexpensive enough to never be
sampled are pruned to avoid clutter. The report lists
features in descending order of cost. Likewise, each feature
instance is listed in descending order grouped by their
associated feature.
\vspace{2ex}\phantomsection\noindent\stepcounter{subsubsection}\label{t:x28part_x22archx2dhiddenx22x29}Figure~\hyperref[t:x28counter_x28x22figurex22_x22circlex2dprofilex22x29x29]{\FigureRef{8}{t:x28counter_x28x22figurex22_x22circlex2dprofilex22x29x29}} shows a program that uses the
\Scribtexttt{utils{\hbox{\texttt{.}}}rkt} library shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22typedx2duntypedx22x29x29]{\FigureRef{2}{t:x28counter_x28x22figurex22_x22typedx2duntypedx22x29x29}}. Specifically, the program
prints the radius, area, and circumference for 1,000,000
circles of increasing size. The right half of the figure
also gives a profile report for this program. Most of the
execution time is spent printing the circles{'} properties
(lines 7{-}11), and thus appears first in the feature list.
Specifically, printing the circle{'}s circumference (line 9)
takes the most time (18 s). Finally, the second item,
contract verification, has a relatively small cost compared
to output for this program (4 s).
\Ssection{Profiling Complex Features}{Profiling Complex Features}\label{t:x28part_x22richx22x29}
The feature{-}specific protocol in the preceding section
assumes that there is a one{-}to{-}one correspondence from the
placement of a feature to the location where it incurs a
run{-}time cost. This process, however, does not apply to
features whose instances have costs appear either in
multiple places or in different places than than their
syntactic location suggests. These are features with \textit{non{-}local costs}, because a feature instance and its cost
are separated. Higher{-}order contracts illustrate this idea
particularly well because they are specified in one place
yet incur costs at many others.
In other cases, several different instances of a feature
contribute to a single cost center, such as a concurrent
program that wants to attribute a cost to the program as a
whole as well as the particular thread or actor running
associated with it. These features have \textit{conflated
costs}.
While the creator of features with non{-}local or conflated
costs can use the FSP protocol to measure some aspects of
their costs, adopting a better protocol produces better
results when evaluating such features. This section shows
both how to extend the FSP{'}s analysis component with
feature{-}specific plug{-}ins and how to adapt the communication
protocol appropriately. It is divided into two parts. First,
we discuss custom payloads, values that the authors of
features use to describe their non{-}local or conflated costs
(\SecRef{\SectionNumberLink{t:x28part_x22implx2drichx22x29}{5.1}}{Custom Payloads}). Using custom payloads, an analysis
plug{-}in may convert the information into a form that
programmers can digest and act on (\SecRef{\SectionNumberLink{t:x28part_x22analysisx2drichx22x29}{5.2}}{Analyzing Complex{-}Cost Features}). We use
three running examples to demonstrate non{-}local and
conflated features and their payloads: contracts,
actor{-}based concurrency, and parser backtracking.
\Ssubsection{Custom Payloads}{Custom Payloads}\label{t:x28part_x22implx2drichx22x29}
The instrumentation for features with complex{-}cost
accounting, non{-}local or conflated, makes use of arbitrary
values to mark payloads instead of source locations. These
payloads must contain enough information to identify a
feature{'}s cost center and to distinguish specific instances.
Contracts, actor{-}based concurrency and parser backtracking
are three cases where features benefit from having such
custom payloads.
Although storing precise and detailed data in payloads is
attractive, developers must also avoid excessive computation
or allocation when constructing their payloads. After all,
payloads are constructed every time feature code is
executed, whether or not the sampler observes it.
\Ssubsubsectionstarx{Contracts}{Contracts}\label{t:x28part_x22contracts2x22x29}
As discussed in \ChapRef{\SectionNumberLink{t:x28part_x22contractsx22x29}{3}}{Profiling Racket Contracts}, higher{-}order behavioral
contracts have non{-}local costs. Rather than using source
locations as cost{-}centers, a contract uses \textit{blame
objects}. The latter tracks the parties to a contract so
that its possible to poinpoint the faulty party in case of a
violation. Every time an object traverses a higher{-}order
contract boundary, the contract system attaches a blame
object. This blame object holds enough
information to reconstruct a complete picture of contract
checking events{---}the contract to check, the name of the
contracted value, and the names of the components that
agreed to the contract.
\Ssubsubsectionstarx{Actor{-}Based Concurrency}{Actor{-}Based Concurrency}\label{t:x28part_x22Actorx2dBasedx5fConcurrencyx22x29}
Marketplace is a DSL for writing programs in terms of actor{-}based\Autobibref{~(\hyperref[t:x28autobib_x22Carl_Hewittx2c_Peter_Bishopx2c_and_Richard_SteigerA_Universal_Modular_ACTOR_Formalism_for_Artificial_IntelligenceIn_Procx2e_International_Joint_Conference_on_Artificial_Intelligence1973x22x29]{\AutobibLink{Hewitt et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Carl_Hewittx2c_Peter_Bishopx2c_and_Richard_SteigerA_Universal_Modular_ACTOR_Formalism_for_Artificial_IntelligenceIn_Procx2e_International_Joint_Conference_on_Artificial_Intelligence1973x22x29]{\AutobibLink{1973}})}
concurrency\Autobibref{~(\hyperref[t:x28autobib_x22Tony_Garnockx2dJonesx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenThe_network_as_a_language_constructIn_Procx2e_European_Symposium_on_Programming_Languagesx2c_ppx2e_473x2dx2d4922014x22x29]{\AutobibLink{Garnock{-}Jones et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Tony_Garnockx2dJonesx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenThe_network_as_a_language_constructIn_Procx2e_European_Symposium_on_Programming_Languagesx2c_ppx2e_473x2dx2d4922014x22x29]{\AutobibLink{2014}})}. Programs that use
Marketplace features have conflated costs. The cost{-}centers
of these programs are attributed in terms of the processes
the language uses, rather than the functions that an
individual process runs. To handle this, Marketplace uses
process identifiers as payloads. Since
\RktSym{current{-}continuation{-}marks} gathers all the marks
currently on the stack, the sampling thread can gather
\textit{core samples}.\NoteBox{\NoteContent{In analogy to geology, a core
sample includes marks from the entire stack, rather than
the top most mark.}} Because Marketplace VMs are spawned and
transfer control using function calls, these core samples
include not only the current process but also all its
ancestors{---}its parent VM, its grandparent, etc.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.22812499999998836bp}{\makebox[394.4000000000001bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_8.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29}\textsf{Fig.}~\textsf{9}. }{t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29}\textsf{Module graph and by{-}value views of a contract boundary}}}\end{Figure}
\Ssubsubsectionstarx{Parser Backtracking}{Parser Backtracking}\label{t:x28part_x22parserx2drich1x22x29}
The Racket ecosystem includes a parser generator named
Parsack. A parser{'}s cost{-}centers are the particular parse
path that it follows, rather than any particular production
rule that the parser happens to be using. In particular, a
feature{-}specific approach shines when determining on which
paths the parser eventually backtracks. This allows a
programmer to improve a program{'}s performance by
reordering production rules when possible. To accommodate
this, payloads for Parsack combine three values into a
payload: the source location of the current production rule
disjunction, the index of the active branch within the
disjunction, and the offset in the input where the parser is
currently matching. Because parsing a term may require
recursively parsing sub{-}terms, a Parsack payload includes
core samples that allow the plugin to to attribute time to
all active non{-}terminals.
\Ssubsection{Analyzing Complex{-}Cost Features}{Analyzing Complex{-}Cost Features}\label{t:x28part_x22analysisx2drichx22x29}
Even if payloads contain enough information to uniquely
identify a feature instance{'}s cost{-}center, programmers
usually cannot directly digest the complex information in
the corresponding payloads. When a feature uses such
payloads, its creator is encouraged to implement an analysis plug{-}in that
generates user{-}facing reports.
\Ssubsubsectionstarx{Contracts}{Contracts}\label{t:x28part_x22contracts1x22x29}
The goal of the contract plug{-}in is to report which pairs of
parties impose contract checking and how much this checking
costs. A programmer can act only after identifying the
relevant components. Hence, the analysis aims to provide an
at{-}a{-}glance overview of the cost of each contract and
boundary.
To this end, the contract analysis generates a \textit{module graph} view of
contract boundaries. This graph shows modules as nodes, contract boundaries as
edges and contract costs as labels on edges.
Because typed{-}untyped boundaries are an important source of contracts,
the module graph distinguishes typed modules (in \intextrgbcolor{0.37254901960784315,0.37254901960784315,0.37254901960784315}{DARK GRAY}) from untyped modules
(in \intextrgbcolor{0.8745098039215686,0.8745098039215686,0.8745098039215686}{LIGHT GRAY}).
To generate this view, the analysis extracts component names from blame objects.
It then groups payloads that share pairs of parties and computes costs as
discussed in \SecRef{\SectionNumberLink{t:x28part_x22analysisx2dflatx22x29}{4.3}}{Analyzing Feature{-}specific Data}.
The top{-}right part of figure~\hyperref[t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29]{\FigureRef{9}{t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29}} shows the module graph
for a program that constructs two random matrices and multiplies them.
This latter code resides in an untyped module, but the matrix functions of
the \Scribtexttt{math} library reside in a typed module.
Hence linking the client and the library introduces a contract boundary between
them.
In addition to the module graph, an FSP can provides other
views as well. For example, the bottom portion of
figure~\hyperref[t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29]{\FigureRef{9}{t:x28counter_x28x22figurex22_x22matrixx2dmodulex2dviewx22x29x29}} shows the \textit{by{-}value}
view, which provides fine{-}grained information about the cost
of individual contracted values.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\identity{\begin{minipage}[c]{0.68\textwidth}} \raisebox{-0.8718749999999997bp}{\makebox[272.80000000000007bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_9.pdf}}}
\identity{\vspace{0.1em}\end{minipage}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22echox2dprocessx2daccountingx22x29x29}\textsf{Fig.}~\textsf{10}. }{t:x28counter_x28x22figurex22_x22echox2dprocessx2daccountingx22x29x29}\textsf{Marketplace process accounting (excerpt)}}}\end{Figure}
\Ssubsubsectionstarx{Actor{-}Based Concurrency}{Actor{-}Based Concurrency}\label{t:x28part_x22actorx2dbasedx2dconcurrency1x22x29}
The goal of the
Marketplace analysis plug{-}in is to assign costs to individual Marketplace processes
and VMs, as opposed to the code they execute.
Marketplace feature marks use the names of processes and VMs as payloads, which
allows the plug{-}in to distinguish separate processes executing the same functions.
The plug{-}in uses full core samples to attribute costs to VMs
based on the costs of their children. These core samples
record the entire ancestry of processes in the same way the
call stack records the function calls that led to a certain
point in the execution. We exploit that similarity and reuse
standard edge profiling techniques\NoteBox{\NoteContent{VM cost assignment
is simpler than edge profiling because VM/process graphs are
in fact trees. Edge profiling techniques still apply,
though, which allows us to reuse part of the Racket edge
profiler{'}s implementation.}} to attribute costs to the entire
ancestry of a process. To disambiguate between similar
processes in its reports, the plug{-}in uses a process{'}s full
ancestry as an identity.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22echox2dprocessx2daccountingx22x29x29]{\FigureRef{10}{t:x28counter_x28x22figurex22_x22echox2dprocessx2daccountingx22x29x29}} shows the accounting from a
Marketplace{-}based echo server. The first entry of the profile shows the ground
VM, which spawns all other VMs and processes.
The rightmost column shows how execution time is split across the ground VM{'}s
children.
Of note are the processes handling requests from two clients.
As reflected in the profile, the client on port 53588 is sending ten times as
much input as the one on port 53587.
The plug{-}in also reports the overhead of the Marketplace library itself.
Any time attributed directly to a VM; i.e., not to any of its children{---}is
overhead from the library. In our echo server example, 32.3\% of the total
execution time is reported as the ground VM{'}s \textit{self time}, which
corresponds to the library{'}s overhead.\NoteBox{\NoteContent{The echo server
performs no actual work which, by comparison, increases the
library{'}s relative overhead.}}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.23645833333331412bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_10.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22backtrackx22x29x29}\textsf{Fig.}~\textsf{11}. }{t:x28counter_x28x22figurex22_x22backtrackx22x29x29}\textsf{An example Parsack{-}based parser and its backtracking profile}}}\end{Figure}
\Ssubsubsectionstarx{Parser backtracking}{Parser backtracking}\label{t:x28part_x22parserx2drich2x22x29}
The feature{-}specific analysis for Parsack determines how much time is spent
backtracking for each branch of each production rule disjunction.
The source locations and input offsets in the payload allows the plug{-}in
to identify each unique visit that the parser makes to each disjunction during
parsing.
The plug{-}in detects backtracking as follows. Because disjunctions are
ordered, the parser must backtrack from early branches
in the disjuction before it reaches a production rule that
parses. Therefore, whenever the analysis observes a sample
from the matching branch at a given input
location, it attributes backtracking cost to the preceding
branches. It computes that cost from the samples taken in
these branches at the same input location. As with the
Marketplace plug{-}in, the Parsack plug{-}in uses core samples
and edge profiling to handle the recursive structure of the
process.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22backtrackx22x29x29]{\FigureRef{11}{t:x28counter_x28x22figurex22_x22backtrackx22x29x29}} shows a simple parser that first attempts to parse a
sequence of \Scribtexttt{b}s followed by an \Scribtexttt{a}, and in case of failure, backtracks
in order to parse a sequence of \Scribtexttt{b}s.
The right portion of figure~\hyperref[t:x28counter_x28x22figurex22_x22backtrackx22x29x29]{\FigureRef{11}{t:x28counter_x28x22figurex22_x22backtrackx22x29x29}} shows the output of the FSP when
running the parser on a sequence of 9,000,000 \Scribtexttt{b}s. It confirms that the
parser had to backtrack from the first branch after spending almost half of the
program{'}s execution attempting it.
Swapping the \RktSym{\$a} and \RktSym{\$b} branches in the disjunction eliminates
this backtracking.
\Ssection{Controlling Profiler Costs}{Controlling Profiler Costs}\label{t:x28part_x22costsx22x29}
Features that implement the feature{-}specific protocol insert continuation
marks regardless of whether a programmer wishes to profile the program. For
features where individual instances perform a significant amount of work,
such as contracts, the overhead of marks is usually not observable as shown
in \SecRef{\SectionNumberLink{t:x28part_x22resultsx2doverheadx22x29}{7.3}}{Overhead}. For other features, such as fine{-}grained
console output, where the aggregate cost of individually inexpensive
instance annotations are significant, the overhead of marks can be
problematic. In such cases, programmers want to choose when marks are
applied on a by{-}execution basis.
In addition, programmers may also want to control when
mark insertions take place to avoid reporting costs in code
that they wish to ignore or cannot modify. For instance,
reporting that the plot library heavily relies on
pattern{-}matching in its implementation is useless to most
programmers; they cannot fix it.
It makes sense only if they are prepared to
replace the plotting library altogether.
To establish control over when and where continuation marks are added, a
profiler must support two kinds of marks: active and latent. We refer to
the marks described in the previous sections as active marks A latent mark
is an annotation that can be turned into an active mark as needed. An
implementation may employ a preprocessor for this purpose. We distinguish
between \textit{syntactically latent marks} for use with compile{-}time
meta{-}programming and \textit{functional latent marks} for use with library or
run{-}time functions.
\Ssubsection{Syntactically Latent Marks}{Syntactically Latent Marks}\label{t:x28part_x22Syntacticallyx5fLatentx5fMarksx22x29}
Syntactically latent marks exist as annotations on the intermediate
representation (IR) of a program. To add a latent mark, the feature
implementation leaves tags\NoteBox{\NoteContent{Many compilers have means to attach
information to nodes in the IR. Our Racket prototype uses syntax properties
\Autobibref{~(\hyperref[t:x28autobib_x22Rx2e_Kent_Dybvigx2c_Robert_Hiebx2c_and_Carl_BruggemanSyntax_Abstracton_in_SchemeIn_Procx2e_Lisp_and_Symbolic_Computation1993x22x29]{\AutobibLink{Dybvig et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Rx2e_Kent_Dybvigx2c_Robert_Hiebx2c_and_Carl_BruggemanSyntax_Abstracton_in_SchemeIn_Procx2e_Lisp_and_Symbolic_Computation1993x22x29]{\AutobibLink{1993}})}.}} on the residual program{'}s IR instead of directly
inserting feature marks and antimarks. These tags are discarded after
compilation and thus have no run{-}time effect on the program execution.
Other meta{-}programs or the compiler can observe latent marks and turn them
into active marks.
A feature{-}specific profiler can rely on a dedicated compiler pass to convert
syntactic latent marks into active ones. Many compilers have some mechanism
to modify a program{'}s pre{-}compiled source. Racket, for example, uses the
language{'}s \textit{compilation handler} mechanism to interpose this activation
pass. The pass traverses the input program, replacing every relevant
syntactic latent mark it finds with an active mark. As this mechanism
relies on the compiler, a programmer using latent marks must recompile the
user{'}s code. The library code, however, does not need to be re{-}compiled,
which make syntactic latent marks practical for large environments.
This implementation method applies only to features implemented using
meta{-}programming such as the sntactic extensions used in many Racket or R
programs. Thus many of these features use syntactically latent
marks. Languages without any meta{-}programming facilities can still support
latent marks with external tools that emulate meta{-}programming.
\Ssubsection{Functional Latent Marks}{Functional Latent Marks}\label{t:x28part_x22Functionalx5fLatentx5fMarksx22x29}
Functional latent marks offer an alternative to syntactically latent
marks. Instead of tagging the programmer{'}s code, a preprocessor recognizes
calls to feature{-}related functions and rewrites the program{'}s code to wrap
such calls with active marks. Like syntactic latent marks, functional
latent marks require recompilation of code that uses the relevant
functions. Also like syntactic latent marks, they do not require
recompiling libraries that \textit{provide} feature{-}related functions, which
makes them appropriate for functions provided as runtime primitives.
As an example, Racket{'}s output feature uses functional
latent marks instead of active marks. Functional latent
marks are appropriate here because a program may contain
many instances of the output feature, each having little
overhead. The output feature includes a list of runtime and
standard library functions that emit output and adds feature
marks around all calls to those functions, as well as
antimarks around their arguments to avoid measuring their
evaluation.
\Ssection{Evaluation: Profiler Results}{Evaluation: Profiler Results}\label{t:x28part_x22resultsx22x29}
Our evaluation of the Racket feature{-}specific profiler addresses three
promises: that measuring in a feature{-}specific way supplies useful insights
into performance problems; that it is easy to add support for new features;
and that the run{-}time overhead of profiling manageable. This section first
presents case studies that demonstrate how feature{-}specific profiling
improves the performance of programs. Then it reports on the effort required
to mark features and implement plug{-}ins. Finally, it discusses the run{-}time
overhead imposed by the profiler.
\begin{Figure}\begin{Leftfigure}\begin{FigureInside}\begin{SCentered}\raisebox{-0.19999999999998863bp}{\makebox[367.42968750000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_11.pdf}}}\end{SCentered}
\raisebox{-0.1999999999999993bp}{\makebox[8.0bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_12.pdf}}}
\begin{smaller}Results are the mean of 30 executions on a 6{-}core 64{-}bit Debian GNU/Linux
system with 12GB of RAM.
\noindent Because Shill supports only FreeBSD, results for \textit{grade} are from a 6{-}core
FreeBSD system with 6GB of RAM.
\noindent Error bars are one standard deviation on either side.\end{smaller}\end{FigureInside}\end{Leftfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22casex2dstudiesx2dplotx22x29x29}\textsf{Fig.}~\textsf{12}. }{t:x28counter_x28x22figurex22_x22casex2dstudiesx2dplotx22x29x29}\textsf{Execution time after profiling and improvements (lower is better)}}}\end{Figure}
\Ssubsection{Case Studies}{Case Studies}\label{t:x28part_x22casex2dstudiesx22x29}
To be useful, a profiler must accurately identify feature use
costs and provide \textit{actionable} information to
programmers. Ideally, it identifies specific feature uses
that are responsible for significant performance costs in a
given program. When it finds such instances, the profiler
must point programmers towards solutions. Additionally, it
must also provide \textit{negative} information, i.e., confirm
that some uses of language constructs need not be
investigated.
Here we present five case studies. Each one describes a
program, summarizes the profiler{'}s feedback, and explains
the changes that directly follow from the report.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22casex2dstudiesx2dplotx22x29x29]{\FigureRef{12}{t:x28counter_x28x22figurex22_x22casex2dstudiesx2dplotx22x29x29}} displays a concise overview
of the performance after incorporating this feedback. These
case{-}studies range in size from 1 to 15 modules, the
difference in size did not affect the effectiveness of the
project.
\vspace{2ex}\phantomsection\noindent\label{t:x28part_x22synthx22x29}\textit{Sound Synthesis Engine}
This case study concerns a sound synthesis engine written by
St{-}Amour. The engine uses the \Scribtexttt{math} library{'}s
arrays to represent sound signals. It consists of a \Scribtexttt{mixer} module that handles most of the interaction with the
\Scribtexttt{math} library as well as a number of specialized
synthesis modules that interface with the mixer, such as
function generators, sequencers, and a drum machine. Unlike
the engine, the \Scribtexttt{math} library is written in
Typed Racket. To ensure a sound interaction between the
languages, a contract boundary separates it from the untyped
synthesis engine. For scale, the synthesis engine spans 452
lines of code, and we profile it with ten seconds of music.\NoteBox{\NoteContent{The synthesized song is {``}Funky Town{''}, by Lipps Inc.}}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.59375bp}{\makebox[397.6000000000001bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_13.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29}\textsf{Fig.}~\textsf{13}. }{t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29}\textsf{Feature profile (excerpt) and module
graph view for the synthesizer}}}\end{Figure}
Racket{'}s traditional statistical profiler reports that around
40\% of total execution time is spent in two functions from
the \Scribtexttt{math} library:
\noindent \begin{SCentered}\raisebox{-0.3343749999999912bp}{\makebox[233.2bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_14.pdf}}}\end{SCentered}
\noindent Such profiling results suggest a problem with
the \Scribtexttt{math} library. Rewriting or avoiding
it altogether would be a significant undertaking.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29]{\FigureRef{13}{t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29}} shows the FSP{'}s take
of the same program. According to its report, almost three
quarters of the program{'}s execution time is spent checking
contracts, the most expensive being attached to the \Scribtexttt{math} library{'}s array functions. Consequently, any
significant performance improvements must come from
those contracts. Since the \Scribtexttt{math}
library{'}s contracts are automatically generated by Typed
Racket, improving their performance directly is not
practical. Reducing the use of contracts is more
likely to be profitable. Because contract generation happens
only at the boundary of typed and untyped code, modifying a
few modules that create this boundary may lower the imposed
cost.
In order to determine how to move a boundary, the programmer
turns to the module graph view in the lower portion of
figure~\hyperref[t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29]{\FigureRef{13}{t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29}}. This graph is provided
by our feature{-}specific analysis for contracts.
Almost half the total execution time lies between the
untyped interface to the \Scribtexttt{math} library used by the \Scribtexttt{mixer} module (in
\intextrgbcolor{0.8745098039215686,0.8745098039215686,0.8745098039215686}{LIGHT GRAY}) and the typed portions of the library (in \intextrgbcolor{0.37254901960784315,0.37254901960784315,0.37254901960784315}{DARK GRAY}).
This suggests converting the \Scribtexttt{mixer} module to Typed Racket;
a 15{-}minute effort that improves performance by$\sim$48\%.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29]{\FigureRef{13}{t:x28counter_x28x22figurex22_x22funkyx2dtownx2dmultix2dprofilex22x29x29}} also shows that generic sequence
operations, while often expensive, do not impose a significant cost in this
program, despite their pervasive use.
Manually specializing sequences would be a waste of time.
Similarly, since the report does not feature file output costs, optimizing how
the generated signal is emitted as a WAVE file would also be a waste of time.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-1.7010416666666686bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_15.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22mazex2dbeforex2dafterx22x29x29}\textsf{Fig.}~\textsf{14}. }{t:x28counter_x28x22figurex22_x22mazex2dbeforex2dafterx22x29x29}\textsf{Fusing output operations in the maze generator}}}\end{Figure}
\vspace{2ex}\phantomsection\noindent\label{t:x28part_x22mazex22x29}\textit{Maze Generator} The second case study
employs a version of a maze generator written by Olin Shivers. The program
is 758 lines of Racket; it generates a maze on a hexagonal grid, ensures
that it is solvable, and prints it.
The top portion of the output of an FSP shows 55\% of
the execution time is spent on output:
\noindent \begin{SCentered}\raisebox{-0.25937499999999547bp}{\makebox[259.6000000000001bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_16.pdf}}}\end{SCentered}
\noindent Three calls to \RktSym{display}, each responsible for
printing part of the bottom of hexagons, stand out as
especially expensive. Printing each part separately results
in a large number of single{-}character output operations.
This report suggests fusing all three output operations into
one. The result of this reorganization is shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22mazex2dbeforex2dafterx22x29x29]{\FigureRef{14}{t:x28counter_x28x22figurex22_x22mazex2dbeforex2dafterx22x29x29}}. Following this advice
results in a \identity{1.39$\times$} speedup.
The profiler reports that a dynamic cast inside an inner loop has no effect
on performance. This result deviates from the more intuitive thought that
such a cast would be costly. Programmers can use this information to keep the
benefits of the cast.
\vspace{2ex}\phantomsection\noindent\label{t:x28part_x22shillx22x29}\textit{Shill{-}Based Grading Script} Our third
case study involves a grading script, written by Scott Moore, that tests
students{'} OCaml code. The script is 330 lines of Shill\Autobibref{~(\hyperref[t:x28autobib_x22Scott_Moorex2c_Christos_Dimoulasx2c_Dan_Kingx2c_and_Stephen_ChongSHILLx3a_a_secure_shell_scripting_languageIn_Procx2e_USENIX_Symposium_on_Operating_Systems_Design_and_Implementation2014httpsx3ax2fx2fwwwx2eusenixx2eorgx2fconferencex2fosdi14x2ftechnicalx2dsessionsx2fpresentationx2fmoorex22x29]{\AutobibLink{Moore et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Scott_Moorex2c_Christos_Dimoulasx2c_Dan_Kingx2c_and_Stephen_ChongSHILLx3a_a_secure_shell_scripting_languageIn_Procx2e_USENIX_Symposium_on_Operating_Systems_Design_and_Implementation2014httpsx3ax2fx2fwwwx2eusenixx2eorgx2fconferencex2fosdi14x2ftechnicalx2dsessionsx2fpresentationx2fmoorex22x29]{\AutobibLink{2014}})} code; Shill is a
least{-}privilege shell scripting language written in Racket.
According to the FSP, contracts for security permissions
account for more than 66\% of execution time:
\noindent \begin{SCentered}\raisebox{-0.534374999999994bp}{\makebox[343.20000000000005bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_17.pdf}}}\end{SCentered}
\noindent Overhead from calling external programs causes the
most slowdown. Unlike the sound synthesis example, Shill
uses contracts and a kernel extension to ensure external
programs do not violate Shill{'}s security properties. The
script contains three external programs, one being OCaml and
the other two being text manipulation utilities.
Reimplementing the two text manipulation utilities in Shill
reduces the time spent in permission checking, resulting in
a 32\% improvement in the script{'}s performance.
The results of this profile also contain useful negative information. Shill
uses an ambient language to interface between traditional operating system
permission models and Shill{'}s capability language. The FSP shows that
capability code accounts for 98\% of the time spent inside of the Racket
environment. This demonstrates that the transition layer imposed by the
ambient language has little overhead.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.7562499999999659bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_18.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29}\textsf{Fig.}~\textsf{15}. }{t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29}\textsf{Profiling results for the SSH server (excerpt, top) module graph view of SSH server (bottom)}}}\end{Figure}
\vspace{2ex}\phantomsection\noindent\label{t:x28part_x22sshx22x29}\textit{Marketplace{-}Based SSH Server}
The fourth case study involves an SSH
server\NoteBox{\NoteContent{\href{https://github.com/tonyg/marketplace-ssh}{\Snolinkurl{https://github.com/tonyg/marketplace-ssh}}}} in Marketplace.
The SSH server is 3,762 lines of untyped Marketplace
code and Marketplace itself is 4,801 lines of Typed Racket code.
To exercise it, a driver script starts the server, connects to it, launches
a Racket read{-}eval{-}print{-}loop on the local host, evaluates the expression
\RktPn{(}\RktSym{+}\Scribtexttt{ }\RktVal{1}\Scribtexttt{ }\RktVal{2}\Scribtexttt{ }\RktVal{3}\Scribtexttt{ }\RktVal{4}\Scribtexttt{ }\RktVal{5}\Scribtexttt{ }\RktVal{6}\RktPn{)}, disconnects and terminates the server.
As figure~\hyperref[t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29]{\FigureRef{15}{t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29}} shows, the profiler brings out two useful
facts. First, two \textit{spy} processes{---}the \Scribtexttt{tcp{-}spy} process and the
boot process of the \Scribtexttt{ssh{-}session} VM{---}account for 25\% of execution time.
In Marketplace, spies are processes that observe other processes for logging
purposes. The SSH server spawns these spy processes even when logging is
ignored, resulting in unnecessary overhead. Second, contracts account for
close to 67\% of the running time. The module view, shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29]{\FigureRef{15}{t:x28counter_x28x22figurex22_x22sshx2dmultix2dprofilex22x29x29}}, shows that the majority of these contracts
lie at the boundary between the typed Marketplace library and the untyped
SSH server. We can selectively remove these contracts in one of two ways:
by adding types to the SSH server or by disabling typechecking in
Marketplace. Disabling spy processes and type{-}induced contracts results in
a speedup of around \identity{4.41$\times$}. In addition, the report provides
negative information. First, pattern matching again shows to have little
cost despite its pervasive use. Additionally, Racket data structures can be
implicitly coerced to a sequence that a program is capable of iterating
over. This coercion has a runtime cost, but we show it is small.
\vspace{2ex}\phantomsection\noindent\label{t:x28part_x22markdownx22x29}\textit{Markdown Parser}
Our last case study involves a Parsack{-}based Markdown
parser\NoteBox{\NoteContent{\href{https://github.com/greghendershott/markdown}{\Snolinkurl{https://github.com/greghendershott/markdown}}}}
written by Greg Hendershott.
The Markdown parser is 4,058 lines of Racket code that
we run on 1,000 lines of sample text.\NoteBox{\NoteContent{The sample text
is {``}The Time Machine{''}, by H. G. Wells. \href{http://www.gutenberg.org/ebooks/35}{\Snolinkurl{http://www.gutenberg.org/ebooks/35}}}}
The FSP{'}s feedback shows one interesting result.
Specifically, backtracking from three branches takes noticeable
time and accounts for 34\%, 2\%, and 2\% of total execution
time, respectively:
\noindent \begin{SCentered}\raisebox{-0.3031249999999912bp}{\makebox[290.40000000000003bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_19.pdf}}}\end{SCentered}
\noindent Based on the tool{'}s report, moving the problematic
branches further down in their enclosing disjunction is the
appropriate action. Making this change leads to a speedup of \identity{1.40$\times$}.
For comparison, Parsack{'}s author, Stephen Chang, manually optimized the same
version of the Markdown parser using ad{-}hoc, low{-}level, and hand{-}written,
instrumentation. His application specific instrumentation leads to a speed
up of \identity{1.37$\times$}. With no knowledge of the parser{'}s internals, we
were able to achieve a similar speedup in only a few minutes of work.
\Ssubsection{Plug{-}in Implementation Effort}{Plug{-}in Implementation Effort}\label{t:x28part_x22effortx22x29}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\begin{SInsetFlow}\begin{bigtabular}{@{\hspace{\stabLeft}}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}l@{}}
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{1}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{2}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{3}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{4}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{5}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{6}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{7}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{8}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}\Smaller{\Scribtexttt{9}}} \\
\hbox{\Smaller{\Scribtexttt{10}}}\end{tabular}\end{RktBlk} &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\hbox{ } &
\begin{RktBlk}\begin{tabular}[c]{@{}l@{}}
\hbox{\RktPn{(}\inrgbcolorbox{0.9019607843137255,0.9019607843137255,0.9019607843137255}{\RktSym{define}\Scribtexttt{ }\RktSym{marketplace{-}continuation{-}mark{-}key}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xx}}}\inrgbcolorbox{0.9019607843137255,0.9019607843137255,0.9019607843137255}{\RktPn{(}\RktSym{make{-}continuation{-}mark{-}key}\Scribtexttt{ }\RktVal{{\textquotesingle}}\RktVal{marketplace}\RktPn{)}}\RktPn{)}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}} \\
\hbox{\RktPn{[}\RktSym{{\hbox{\texttt{.}}}{\hbox{\texttt{.}}}{\hbox{\texttt{.}}}}\RktPn{]}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{x}}}} \\
\hbox{\RktPn{(}\RktSym{marketplace{-}log}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{debug}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{"Entering process $\sim$v($\sim$v)"}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{debug{-}name}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{pid}\RktPn{)}} \\
\hbox{\RktPn{(}\RktSym{define}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{result}\mbox{\hphantom{\Scribtexttt{x}}}\RktPn{(}\inrgbcolorbox{0.9019607843137255,0.9019607843137255,0.9019607843137255}{\RktSym{with{-}continuation{-}mark}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxxxxxxxx}}}\inrgbcolorbox{0.9019607843137255,0.9019607843137255,0.9019607843137255}{\RktSym{marketplace{-}continuation{-}mark{-}key}\Scribtexttt{ }\RktPn{(}\RktSym{or}\Scribtexttt{ }\RktSym{debug{-}name}\Scribtexttt{ }\RktSym{pid}\RktPn{)}}} \\
\hbox{\mbox{\hphantom{\Scribtexttt{xxxxxxxxxxxxxxxxx}}}\RktSym{enclosed{-}expr}\RktPn{)}\RktPn{)}} \\
\hbox{\RktPn{(}\RktSym{marketplace{-}log}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{{\textquotesingle}}\RktVal{debug}\mbox{\hphantom{\Scribtexttt{x}}}\RktVal{"Leaving}\mbox{\hphantom{\Scribtexttt{xx}}}\RktVal{process $\sim$v($\sim$v)"}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{debug{-}name}\mbox{\hphantom{\Scribtexttt{x}}}\RktSym{pid}\RktPn{)}}\end{tabular}\end{RktBlk}\end{bigtabular}\end{SInsetFlow}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22marketplacex2dinstrx22x29x29}\textsf{Fig.}~\textsf{16}. }{t:x28counter_x28x22figurex22_x22marketplacex2dinstrx22x29x29}\textsf{Instrumentation for Marketplace (excerpt)}}}\end{Figure}
Getting a Racket library ready for feature{-}specific profiling requires
little effort, both in terms of the profilier{'}s protocol and the creation of an
optional analysis plug{-}in. It is easily within reach for library authors,
especially because it does not require advanced profiling knowledge. To
support this claim, we report anecdotal evidence and the lines of code for
adding marks to other features, as well as their plug{-}ins.
For illustrative purposes, the instrumentation for Marketplace is shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22marketplacex2dinstrx22x29x29]{\FigureRef{16}{t:x28counter_x28x22figurex22_x22marketplacex2dinstrx22x29x29}} with the added code highlighted.
Unlike other examples, which use symbols as continuation mark keys,
this code creates a fresh key using \RktSym{make{-}continuation{-}mark{-}key} to
avoid key collisions.
We report the number of lines of code for each remaining
features{'} plug{-}in in figure~\hyperref[t:x28counter_x28x22figurex22_x22profilersx2dlocx22x29x29]{\FigureRef{17}{t:x28counter_x28x22figurex22_x22profilersx2dlocx22x29x29}}. The second
column reports the number of lines that are required to
instrument the feature with marks. The third column reports
the number of lines of plug{-}in analysis code. Finally, the
fourth column reports the feature{'}s implementation size in
lines of code. The line counts for Marketplace and Parsack
do not include the roughly 500 lines of Racket{'}s edge
profiler, which are re{-}linked into the plug{-}ins. With the
exception of contract instrumentation{---}which covers
multiple kinds of contracts and is spread across about
16,000 lines of the contract system{---}instrumentation is
local and non{-}intrusive.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.09687499999999716bp}{\makebox[353.6742187500001bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_20.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22profilersx2dlocx22x29x29}\textsf{Fig.}~\textsf{17}. }{t:x28counter_x28x22figurex22_x22profilersx2dlocx22x29x29}\textsf{Instrumentation and analysis LOC per feature}}}\end{Figure}
\Ssubsection{Overhead}{Overhead}\label{t:x28part_x22resultsx2doverheadx22x29}
Our prototype imposes an acceptable overhead on program execution.
figure~\hyperref[t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29]{\FigureRef{18}{t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29}} summarizes our measurements. The results are
the mean of 30 executions with 95\% confidence error bars. The machine for
these tests is a 64{-}bit Debian GNU/Linux system with 12 core Intel Xeon CPU
clocked at 2.4 GHz and 11 GB of 1333 MHz DDR3 ram.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.27812499999998863bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_21.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29}\textsf{Fig.}~\textsf{18}. }{t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29}\textsf{Instrumentation and sampling overhead}}}\end{Figure}
We use the programs listed in figure~\hyperref[t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29]{\FigureRef{18}{t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29}} as benchmarks.
They include three of the case studies from \SecRef{\SectionNumberLink{t:x28part_x22casex2dstudiesx22x29}{7.1}}{Case Studies}, two programs that
make heavy use of contracts (lazy and ode), and six programs from the Computer
Language Benchmarks Game\NoteBox{\NoteContent{\href{http://benchmarksgame.alioth.debian.org}{\Snolinkurl{http://benchmarksgame.alioth.debian.org}}}}
that use the features supported by our prototype.
The first column of figure~\hyperref[t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29]{\FigureRef{18}{t:x28counter_x28x22figurex22_x22overheadx2dplotx22x29x29}} corresponds to programs
executing without any feature marks and serves as our baseline.
The second column reports results for programs that include only marks that are
active by default: contract marks and Marketplace marks. This bar represents
the default mode for executing programs without profiling.
The third column reports results for a program that is run with all marks activated.
The fourth column includes all of the above as well as the overhead from the
sampling thread; it is closest to the user experience when profiling.
With all marks activated, the overhead is lower than 6\% for all but two
programs, synth and maze, where it accounts for 16\% and 8.5\% respectively.
The overhead for marks that are active by default is only noticeable for two of
the four programs that include such marks, synth and ode, and account for 16\%
and 4.5\% respectively.
Total overhead, including sampling, ranges from 3\% to 33\%.
Based on this experiment, we conclude that instrumentation overhead is
reasonable in general. The one exception, the synth benchmark, involves a
large quantity of contract checking for cheap contracts, which is the worst
case scenario for contract instrumentation. Further engineering effort
could lower this overhead. The overhead from sampling is similar to that of
state{-}of{-}the{-}art sampling profilers \Autobibref{~(\hyperref[t:x28autobib_x22Todd_Mytkowiczx2c_Amer_Diwanx2c_Matthias_Hauswirthx2c_and_Peter_Fx2e_SweeneyEvaluating_the_accuracy_of_Java_profilersIn_Procx2e_Programming_Langauges_Design_and_Implementationx2c_ppx2e_187x2dx2d1972010x22x29]{\AutobibLink{Mytkowicz et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Todd_Mytkowiczx2c_Amer_Diwanx2c_Matthias_Hauswirthx2c_and_Peter_Fx2e_SweeneyEvaluating_the_accuracy_of_Java_profilersIn_Procx2e_Programming_Langauges_Design_and_Implementationx2c_ppx2e_187x2dx2d1972010x22x29]{\AutobibLink{2010}})}.
This evaluation has one threat to validity. Because instrumentation is
localized to feature code, its overhead is also localized. That is to say,
the act of profiling a feature makes that feature slightly slower compared
to the rest of the program. This may cause feature execution time to be
overestimated. However, we conjecture that this is not a problem in
practice because these overheads are low in general. In contrast, sampling
overhead is uniformily\NoteBox{\NoteContent{Assuming random sampling, which we did not
verify.}} distributed across a program{'}s execution and should not introduce
such biases.
\Ssection{Broader applicability: Profiling R}{Broader applicability: Profiling R}\label{t:x28part_x22otherlangsx22x29}
The applicability of feature{-}specific profiling is not limited to a
particular language. Clearly linguistic features with complex costs are not
unique to Racket, and many languages support some sort of user{-}defined
features. Specifically, languages with first{-}class functions,
macros, or facilities for embedding DSLs tend to come with complex{-}cost
features and can therefore benefit from our idea.
This section demonstrates the feasibility of implementing a feature{-}specific
profiler for the R programming language. For a straightforward adaptation
of the Racket prototype, a language must have a sampling profiler and a
stack annotation mechanism. While sampling profilers have been implemented
for many languages, stack annotations are less commonly
supported. In particular, R lacks them. Fortunately, adding continuation marks to a language such as R
takes only a few lines of code.
\Ssubsection{A Sample Feature in R}{A Sample Feature in R}\label{t:x28part_x22rx2dfeaturex22x29}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.28020833333333117bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_22.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}\textsf{Fig.}~\textsf{19}. }{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}\textsf{Looping Constructs}}}\end{Figure}
Like most programming languages, R provides
looping and mapping constructs such as \RktMeta{}\RktMeta{for}\RktMeta{}, \RktMeta{}\RktMeta{while}\RktMeta{},
and \RktMeta{lapply}.\NoteBox{\NoteContent{\RktMeta{lapply} is similar to \Scribtexttt{map} in
functional languages.}} Unfortunately, R implementers and
users have different opinions on the performance of loops. Folklore
in the R community suggests that looping constructs
are slow and should be avoided in favor of
vectorized operations. By contrast, R implementers claim that
loops run reasonably fast and are slow only
because of \textit{secondary effects}. That is, loops are slow
because of effects that are a by{-}product of using a feature
but are not caused by using the feature directly.
A profiler can help decide which
of the common beliefs matters.
The left{-}hand side of figure~\hyperref[t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29]{\FigureRef{19}{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}}
shows two \RktMeta{}\RktMeta{for}\RktMeta{} loop instances, the first on line
2 and the second on line 5. These loops have an
accumulator whose costs must be attributed to the feature
and a body of user code whose costs must \textit{not} be attributed
to the feature.
The right{-}hand side of figure~\hyperref[t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29]{\FigureRef{19}{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}} shows a run
of these loops with a feature{-}specific profiler. As with the Racket
prototype, a sampling profiler collects marks and antimarks, and an analyzer
converts the data into information for programmers. The resulting display
shows that no time is spent on the looping constructs. That is, the output
(figure~\hyperref[t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29]{\FigureRef{19}{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}}) shows no samples collected during
code associated with looping constructs. While this one run is not
conclusive evidence, it supports the R implementers{'} claim that the direct
overhead of looping constructs is not significant. R code that uses loops
may still be slow, but the slowdown is not directly caused by the loop
construct.
\Ssubsection{Implementation}{Implementation}\label{t:x28part_x22rx2dcontinuationx2dmarksx22x29}
Only a few modifications to R{'}s implementation were required to support
feature{-}specific profiling. We implemented continuation marks in 134 lines
of C. The extension to Rprof to inspect the new continuation marks accounted
for 105 lines of code. Finally, we created a library to implement the
analysis tool in 136 lines of R code. The implementation was created over a
week with no prior experience with the R language or its internals. These
results suggest that implementing feature{-}specific profiling may be possible
even when the host language does support continuation marks or stack
annotations.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\begin{SCodeFlow}\begin{RktBlk}\begin{SingleColumn}\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{1}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{SEXP}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{attribute{\char`\_}hidden}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{do{\char`\_}for}\RktPn{(}\RktMeta{SEXP}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{call}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{SEXP}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{op}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{SEXP}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{args}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{SEXP}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{rho}\RktPn{)}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{\char`\{}}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{2}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{}\RktPn{[}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{]}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{3}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{R{\char`\_}AddMark}\RktPn{(}\RktMeta{FOR}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{call}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktVal{TRUE}\RktPn{)}\RktPn{;}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{4}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{}\RktMeta{for}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{(}\RktMeta{i}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{=}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktVal{0}\RktPn{;}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{i}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{{\fontencoding{T1}\selectfont<}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{n}\RktPn{;}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{i}\RktPn{+}\RktPn{+}\RktPn{)}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{\char`\{}}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{5}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktMeta{switch}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{(}\RktMeta{val{\char`\_}type}\RktPn{)}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{\char`\{}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xx}}}\RktMeta{}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{\char`\}}}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{6}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktMeta{}\RktPn{[}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{]}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{7}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktMeta{R{\char`\_}AddMark}\RktPn{(}\RktMeta{FOR}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{ANTIMARK}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktVal{TRUE}\RktPn{)}\RktPn{;}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{8}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktMeta{eval}\RktPn{(}\RktMeta{body}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{rho}\RktPn{)}\RktPn{;}\RktMeta{}
\Smaller{\Smaller{\mbox{\hphantom{\Scribtexttt{x}}}\Scribtexttt{9}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxxxxxx}}}\RktMeta{R{\char`\_}AddMark}\RktPn{(}\RktMeta{FOR}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{call}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktVal{TRUE}\RktPn{)}\RktPn{;}\RktMeta{}
\Smaller{\Smaller{\Scribtexttt{10}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{}\RktPn{{\char`\}}}\RktMeta{}
\Smaller{\Smaller{\Scribtexttt{11}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{}\RktPn{[}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{{\hbox{\texttt{.}}}}\RktPn{]}\RktMeta{}
\Smaller{\Smaller{\Scribtexttt{12}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\mbox{\hphantom{\Scribtexttt{xxxx}}}\RktMeta{return}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{R{\char`\_}NilValue}\RktPn{;}\RktMeta{}
\Smaller{\Smaller{\Scribtexttt{13}\mbox{\hphantom{\Scribtexttt{x}}}}}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{}\RktPn{{\char`\}}}\RktMeta{}\end{SingleColumn}\end{RktBlk}\end{SCodeFlow}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22forx2dloopx2dimplementationx22x29x29}\textsf{Fig.}~\textsf{20}. }{t:x28counter_x28x22figurex22_x22forx2dloopx2dimplementationx22x29x29}\textsf{For{-}loop implementation with marks (excerpt)}}}\end{Figure}
\Ssubsubsectionstarx{Continuation Marks}{Continuation Marks}\label{t:x28part_x22Continuationx5fMarksx22x29}
Although R does not support continuation marking directly, R programs can
inspect and manipulate the call stack. It is possible to extend the frames
in the call stack to support continuation marks with modifications to the
R{'}s engine, namely, by extending frames to store marks in a hash map with
unique keys and multiple payloads; by teaching the garbage collector how to
track these maps; and by adding primitives to add and inspect continuation
marks.
The capability to add marks to the stack must be accessible from both R and
C, as R features are written in both languages. While supporting
continuation marks does add to the complexity of the R code base, that
complexity is localized. Marks also do not affect the performance of
programs when they are disabled.\NoteBox{\NoteContent{With our modifications, R can be
compiled with and without continuation marks. While this may seem like a
questionable design, it is actually a standard practice for many R
tools\Autobibref{~(\hyperref[t:x28autobib_x22Florxe9al_Morandatx2c_Brandon_Hillx2c_Leo_Osvaldx2c_and_Jan_VitekEvaluating_the_Design_of_the_R_LanguageIn_Procx2e_European_Conference_on_Objectx2dOriented_Programming2012httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d642x2d31057x2d7x5f6x22x29]{\AutobibLink{Morandat et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Florxe9al_Morandatx2c_Brandon_Hillx2c_Leo_Osvaldx2c_and_Jan_VitekEvaluating_the_Design_of_the_R_LanguageIn_Procx2e_European_Conference_on_Objectx2dOriented_Programming2012httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d642x2d31057x2d7x5f6x22x29]{\AutobibLink{2012}})}.}}
The API for continuation marks in R is similar to its Racket variant:
\noindent \begin{itemize}\atItemizeStart
\item \RktMeta{add}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{mark}\RktPn{(}\RktMeta{key}\RktPn{,}\RktMeta{}\mbox{\hphantom{\Scribtexttt{x}}}\RktMeta{value}\RktPn{)}\RktMeta{}, which imperatively adds
(\RktMeta{key},\RktMeta{value}) to the call stack.
\item \RktMeta{marks}\RktPn{(}\RktMeta{key}\RktPn{)}\RktMeta{}, which walks the call stack and
retrieves all marks that match \RktMeta{key}.\end{itemize}
\noindent The API for Racket and R differ in primarily one
aspect. The function to add a mark in Racket takes an
expression, which is missing in the R variant. Unlike in
Racket, \RktMeta{add}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{mark} places the continuation mark on
the stack; the mark is implicitly removed when the current stack frame
is popped.
R features that are implemented in C use the \RktMeta{R{\char`\_}AddMark}
and \RktMeta{R{\char`\_}Marks} functions to manipulate continuation marks.
These functions behave identically to their R equivalents.
As an example, figure~\hyperref[t:x28counter_x28x22figurex22_x22forx2dloopx2dimplementationx22x29x29]{\FigureRef{20}{t:x28counter_x28x22figurex22_x22forx2dloopx2dimplementationx22x29x29}} shows
the marks in R{'}s implementation of \RktMeta{}\RktMeta{for}\RktMeta{}. The modified implementation
places a mark at the beginning of the loop and replaces it
with an antimark when the call to \RktMeta{eval} begins executing
the loop{'}s body. Once finished, the run{-}time removes the
frame for \RktMeta{do}\RktPn{{-}}\RktMeta{for}\RktMeta{} from the call stack, which also removes the mark.
\Ssubsubsectionstarx{Sampling Profiler}{Sampling Profiler}\label{t:x28part_x22rx2dsamplingx2dprofx22x29}
Our prototype profiler uses Rprof, which is R{'}s built{-}in sampling profiler.
This profiler uses Unix interrupts to sample the call stack during
execution. These samples are written to a file for post{-}processing. We
modified Rprof to capture marks in addition to local variables To enables
continuation marks, one must set \RktMeta{marks}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{profiling}, as shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29]{\FigureRef{19}{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}}. Modifying Rprof to track
continuation marks rather than using R{'}s native stack inspection mechanism
allows programmers to use other Rprof features, such as disabling the
profiler during portions of the computation.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-1.0489583333333425bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_23.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29}\textsf{Fig.}~\textsf{21}. }{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29}\textsf{Dynamic Dispatch (top) and profile output (excerpt, bottom)}}}\end{Figure}
\Ssubsubsectionstarx{Analysis Pass}{Analysis Pass}\label{t:x28part_x22Analysisx5fPassx22x29}
Similar to the analysis pass in Racket, the R analysis pass shows four
pieces of information: (1) the execution time; (2) number of samples
collected; (3) a detailed list of every feature under analysis; (4) as well
as the time spent in that feature and its instances. Programmers run the
analysis pass by giving the Rprof trace to the \RktMeta{feature}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{profile} function,
as shown in figure~\hyperref[t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29]{\FigureRef{19}{t:x28counter_x28x22figurex22_x22loopingx2dconstructsx2dsamplex22x29x29}} line 9. Processing each
feature happens again in the same three steps that the Racket analysis
performs.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29]{\FigureRef{21}{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29}} shows a report. It presents the cost
dynamic dispatch for one of R{'}s object systems. The analysis lists feature
instances by method name rather than the source location. The data is
particularly interesting because, like behavioral contracts, dynamic
dispatch has dispersed costs. The source of dynamic dispatch is where the
method definition is, but the cost manifests itself at the method{'}s call
sites. Because the continuation mark payloads store the name of the method,
we can attribute the cost of dynamic dispatch to the proper
source.
\Ssubsection{Use Cases}{Use Cases}\label{t:x28part_x22rx2dcorpusx22x29}
Next we present four small case studies of features that demonstrate how our
profiler can help programmers. The case studies range over a wide spectrum
of features: dynamic dispatch, parameter{-}naming function applications,
copy{-}on{-}write parameter passing, and vector
subsetting\Autobibref{~(\hyperref[t:x28autobib_x22Hadley_WickhamAdvanced_RFirst_editionx2e_Chapman_and_Hallx2fCRC2014httpx3ax2fx2fadvx2drx2ehadx2ecox2enzx2fx22x29]{\AutobibLink{Wickham}} \hyperref[t:x28autobib_x22Hadley_WickhamAdvanced_RFirst_editionx2e_Chapman_and_Hallx2fCRC2014httpx3ax2fx2fadvx2drx2ehadx2ecox2enzx2fx22x29]{\AutobibLink{2014}})}.\NoteBox{\NoteContent{Called slicing in other languages.}}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.5864583333333369bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_24.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dfixedx2dsamplex22x29x29}\textsf{Fig.}~\textsf{22}. }{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dfixedx2dsamplex22x29x29}\textsf{Dynamic Dispatch (fixed, top) and profile output (excerpt, bottom)}}}\end{Figure}
\Ssubsubsectionstarx{Dynamic Dispatch}{Dynamic Dispatch}\label{t:x28part_x22Dynamicx5fDispatchx22x29}
R{'}s S4 object system supports multiple dispatch. Any R function, including
primitives, can be transformed into the default implementation of an S4
method. When a method is called, it executes the implementation whose
arguments best match the parameter types. The run{-}time system calls the
default version of the function if no arguments match the required input
types.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29]{\FigureRef{21}{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29}} depicts the method \RktMeta{}\RktPn{\%in\%}\RktMeta{}, used here
as a part of Kruskal{'}s algorithm to find a minimum spanning tree of a
graph. This version uses dynamic dispatch recursively until it finds the
desired node or the list is empty. The variant of this code in
figure~\hyperref[t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dfixedx2dsamplex22x29x29]{\FigureRef{22}{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dfixedx2dsamplex22x29x29}} uses dynamic dispatch \textit{once}
and thereafter calls a static function. Both variants of this method have
equivalent behavior when the list is a homogeneous list of nodes. The
recursive use of dynamic dispatch causes the first definition to be slower
than the second. Conventional profilers identify the use of dynamic
dispatch as having a major performance impact in the program.
Unfortunately, they cannot identify which specific use of dynamic dispatch
is causing the performance problems, as they point to the S4 implementation
but do not trace the costs back to calls. A feature{-}specific profile, as
shown in figure~\hyperref[t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29]{\FigureRef{21}{t:x28counter_x28x22figurex22_x22dynamicx2ddispatchx2dsamplex22x29x29}}, not only identifies dynamic
dispatch as a major problem in the program, but it also points to the
\RktMeta{}\RktPn{\%in\%}\RktMeta{} method as the culprit
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.26770833333337096bp}{\makebox[398.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_25.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29}\textsf{Fig.}~\textsf{23}. }{t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29}\textsf{Function Application (top) and Profile Output (bottom)}}}\end{Figure}
\Ssubsubsectionstarx{Function Application}{Function Application}\label{t:x28part_x22Functionx5fApplicationx22x29}
Function calls in R may use named arguments in addition to traditional
positional arguments. Named arguments at call sites are matched with named
parameters. When a function is called and an argument is passed with a name,
the argument is bound to the parameter whose name has the longest matching
prefix of the name given for the argument. Thus, every function used with
named arguments must perform run{-}time string comparisons. Additionally, such
a function application succeeds even if the number of arguments does not
coincide with the number of parameters. Execution halts only when a
parameter without a value is evaluated. As a result, function calls are
difficult to optimize, and thus programmers consider them to be slow. An
profiler can help identify which function calls cause the most runtime
overhead and which are not cause for concern.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29]{\FigureRef{23}{t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29}} shows the skeleton of two
functions: \RktMeta{serve} and \RktMeta{respond}. The former has a computationally
simple and fast function body compared with a complicated slow calling
interface. The latter has a complicated and slow function body but fast and
simple calling interface. Traditional profilers find similar execution times
for each function, because the combined running time of both the function
body and calling interface are the same. While both timings are similar, \RktMeta{serve} spends more time in the calling interface than required. As shown in
figure~\hyperref[t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29]{\FigureRef{23}{t:x28counter_x28x22figurex22_x22functionx2dapplicationx2dsamplex22x29x29}}, our profiler identifies the
primary bottleneck for \RktMeta{serve}{'}s calling interface. Thus, the program{'}s
performance can be improved by inlining \RktMeta{serve} or simplifiying its
interface, which programmers can do in response to the FSP{'}s actionable
report.
\Ssubsubsectionstarx{Copy{-}on{-}Write}{Copy{-}on{-}Write}\label{t:x28part_x22Copyx2donx2dWritex22x29}
Conceptually, the semantics of R requires a deep copy of every argument
passed into a function. In reality, the implementation only duplicates
objects when absolutely necessary. Operations such as mutation force the
duplication, creating copies. If no such operation occurs, then objects
are never duplicated. This so{-}called copy{-}on{-}write policy can lead
to unpredictable performance effects.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-1.1177083333333369bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_26.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22copyx2donx2dwritex2dsamplex22x29x29}\textsf{Fig.}~\textsf{24}. }{t:x28counter_x28x22figurex22_x22copyx2donx2dwritex2dsamplex22x29x29}\textsf{Copy{-}on{-}Write (top) and profile output (bottom)}}}\end{Figure}
The \RktMeta{array}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{duplicate} function in figure~\hyperref[t:x28counter_x28x22figurex22_x22copyx2donx2dwritex2dsamplex22x29x29]{\FigureRef{24}{t:x28counter_x28x22figurex22_x22copyx2donx2dwritex2dsamplex22x29x29}}
illustrates the surprising impact of copy{-}on{-}write. It duplicates the
vector only if the second parameter is true. The program has two loops: a
slow loop that causes the duplication of the array and a fast loop that does
not duplicate the array. Traditional profilers correctly identify
\RktMeta{array}\RktPn{{\hbox{\texttt{.}}}}\RktMeta{duplicate} as a bottleneck. Our profiler identifies array
duplication as the problem and furthermore identifies the duplication of a
specific vector.
\Ssubsubsectionstarx{Vector Subset}{Vector Subset}\label{t:x28part_x22Vectorx5fSubsetx22x29}
Vectors are the basic data structures in R. Even a number such as 42 is a
vector, which allows functions to operate over both vectors and other
objects seamlessly. The vector{-}subset feature retrieves elements from a
vector based on a vector of indices. Subset occurs frequently and some of
their uses are more expensive than others. The syntax for subset uses
square brackets, similar to array indexing. Traditional indexing is a
special case of subsetting where the argument is a singleton vector. For
example, the expression \RktMeta{c}\RktPn{(}\RktVal{2}\RktPn{,}\RktVal{4}\RktPn{,}\RktVal{6}\RktPn{)}\RktPn{[}\RktVal{2}\RktPn{]}\RktMeta{}, which uses the function \RktMeta{c} to
create a vector, evaluates to \RktMeta{}\RktVal{4}\RktMeta{}.
Figure~\hyperref[t:x28counter_x28x22figurex22_x22vectorx2dsubsetx2dsamplex22x29x29]{\FigureRef{25}{t:x28counter_x28x22figurex22_x22vectorx2dsubsetx2dsamplex22x29x29}} shows a code snippet with two subset
operations. The first retrieves every second element from the given
vector. The other retrieves every third element; it occurs roughly one
fourth as often as the first. Traditional profilers identify vector
subsetting as the primary bottleneck in the program. Unfortunately, these
profilers point to the implementation of subset, which is not enough
information to identify which subset operation is costly. Our profiler
instead indicates that the first subset operation is the primary cost center.
\Ssubsection{Profiling Overhead}{Profiling Overhead}\label{t:x28part_x22rx2devalx22x29}
Figure~\hyperref[t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29]{\FigureRef{26}{t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29}} reports the overhead our prototype imposes
on several benchmarks. These results are the mean of 30 executions on a
machine running OS X Yosemite with a 4 core Intel Core i7 clocked at 2.5 GHz
and 16 GB of 1600 MHz DDR3 ram. The error bars show the 95\% confidence
interval. The samples are collected with R build r69166,\NoteBox{\NoteContent{\href{https://github.com/LeifAndersen/R}{\Snolinkurl{https://github.com/LeifAndersen/R}}}} and the sampling interval is 20ms.
The benchmark programs are described in figure~\hyperref[t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29]{\FigureRef{26}{t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29}}.
They include two benchmarks from the
\href{http://benchmarksgame.alioth.debian.org/}{Computer Language
Benchmark Game} that use features our prototypes supports, the
five feature samples used earlier in the paper, and Oliver Keyes{'}s
{``}\href{https://github.com/oliver-papers/GoingPostel}{GoingPostel}{''}, a
program that aggregates information about IETF RFCs.
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.7427083333333369bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_27.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22vectorx2dsubsetx2dsamplex22x29x29}\textsf{Fig.}~\textsf{25}. }{t:x28counter_x28x22figurex22_x22vectorx2dsubsetx2dsamplex22x29x29}\textsf{Vector Subset (top) and profile output (bottom)}}}\end{Figure}
We report runs of each program in three configurations:
\noindent \begin{itemize}\atItemizeStart
\item The first configuration corresponds to the program executing without
continuation marks or profiler
in a build of R with all required packages installed.
\item The second configuration corresponds to the program executing in a
build of R with continuation marks.
All of features that our profiler supports annotate
the stack with continuation marks, but the sampling is turned off.
\item The third configuration is like the second, but with profiling turned on.\end{itemize}
\begin{Figure}\begin{Centerfigure}\begin{FigureInside}\raisebox{-0.9156249999999773bp}{\makebox[392.00000000000006bp][l]{\includegraphics[trim=2.4000000000000004 2.4000000000000004 2.4000000000000004 2.4000000000000004]{pict_28.pdf}}}\end{FigureInside}\end{Centerfigure}
\Centertext{\Legend{\FigureTarget{\label{t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29}\textsf{Fig.}~\textsf{26}. }{t:x28counter_x28x22figurex22_x22rx2dperformancex22x29x29}\textsf{Instrumentation and Sampling Performance of the Going Postel (Left),Computer Language Benchmark Game Benchmarks (Center),and Feature Samples (Right)}}}\end{Figure}
With continuation marks and profiling, the overhead is lower than 20\% for
half of the programs and larger for the other half (85\%, 100\%, 42\%, and
59\%). The latter four programs, however, are feature samples, which
essentially perform no work except exercise the relevant feature, and
therefore represent pathological worst cases. In all cases the cost of
sampling is less than 2\%. The primary cause of overhead comes from
continuation marks rather than the modified sampling profiler. A threat to
validity comes from the fact that continuation mark overhead is concentrated
at feature annotations, which causes features to appear slower than they
are, thus skewing results. Nevertheless, we consider this experiment to
validate the viability of feature{-}specific profiling. While the overheads
are greater than in Racket, performance of the R profiler remains
acceptable. We conjecture that this prototype could be improved to match the
performance of the Racket implementation with careful tuning of the
implementation.
\Ssection{Limitations}{Limitations}\label{t:x28part_x22limitationsx22x29}
Our approach to feature{-}specific profiling applies to some linguistic
features. This section discusses limitations. We believe they are not
fundamental to the idea of feature{-}specific profiling and that they could be
addressed by different approaches to data gathering.
Because our instrumentation strategy relies on continuation marks, it does
not support features that interfere with marks. This rules out non{-}local
control features that unroll the stack, e.g. exception raising. This also
prevents us from profiling continuation marks themselves.
\identity{~\\}
The sampler must be able to observe a feature in order to profile it. This
rules out uninterruptible features, e.g., allocation or FFI calls, which do
not allow the sampling thread to be scheduled during their execution. Other
obstacles to observability include sampling
bias\Autobibref{~(\hyperref[t:x28autobib_x22Todd_Mytkowiczx2c_Amer_Diwanx2c_Matthias_Hauswirthx2c_and_Peter_Fx2e_SweeneyEvaluating_the_accuracy_of_Java_profilersIn_Procx2e_Programming_Langauges_Design_and_Implementationx2c_ppx2e_187x2dx2d1972010x22x29]{\AutobibLink{Mytkowicz et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Todd_Mytkowiczx2c_Amer_Diwanx2c_Matthias_Hauswirthx2c_and_Peter_Fx2e_SweeneyEvaluating_the_accuracy_of_Java_profilersIn_Procx2e_Programming_Langauges_Design_and_Implementationx2c_ppx2e_187x2dx2d1972010x22x29]{\AutobibLink{2010}})} and instances that execute too quickly to
be sampled reliably.
Some non{-}syntactic language features, such as garbage collection, have costs
that cannot be attributed to a single source location in the program.
Frequently, these features have costs that are small and spread out, and are
thus difficult to capture with a sampling profiler. An event{-}based
approach, such as \hyperref[t:x28autobib_x22Florxe9al_Morandatx2c_Brandon_Hillx2c_Leo_Osvaldx2c_and_Jan_VitekEvaluating_the_Design_of_the_R_LanguageIn_Procx2e_European_Conference_on_Objectx2dOriented_Programming2012httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d642x2d31057x2d7x5f6x22x29]{\AutobibLink{Morandat et al\Sendabbrev{.}}}{'}s (\hyperref[t:x28autobib_x22Florxe9al_Morandatx2c_Brandon_Hillx2c_Leo_Osvaldx2c_and_Jan_VitekEvaluating_the_Design_of_the_R_LanguageIn_Procx2e_European_Conference_on_Objectx2dOriented_Programming2012httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d642x2d31057x2d7x5f6x22x29]{\AutobibLink{2012}}), would fare better.
While our prototype profiles concurrent programs such as the Marketplace
described in \ChapRef{\SectionNumberLink{t:x28part_x22richx22x29}{5}}{Profiling Complex Features}, it cannot handle parallel programs. We
conjecture that our approach could be extended to handle multi{-}threaded
programs but we have not tried.
Features have both direct costs and indirect costs. Direct costs come from
using a feature, while indirect costs are not imposed by the feature itself
but by lost opportunities due to a feature{'}s use. Profiliers only track
direct costs.
Finally, it is up to the feature authors to work out the
correctness of their annotations. While feature authors can
clearly make mistakes when annotating their libraries, in
our experience and that of our users, we have not found this
to be an issue at all. Because authors are familiar with
their libraries, they also tend to have a reasonable idea of
where adding annotations will be \textit{useful}.
\Ssection{Related Work}{Related Work}\label{t:x28part_x22relatedx22x29}
Programmers already have access to a wide variety of
complementary performance tools. This section compares
feature{-}specific profiling to those approaches that
|
are
closely related.
Profilers have been successfully used to diagnose performance issues for
decades. They most commonly report on the consumption of time, space and
I/O resources. Traditional profilers group costs according to program
organization, be it static{---}e.g., per function definition{---}or
dynamic{---}e.g., per HTTP request. Each of these views is useful in
different contexts. For example, a feature{-}specific profiler{'}s view is most useful when
non{-}local feature costs make up a significant portion of a program{'}s running
time. In contrast, traditional profilers may detect a broader range of
issues than feature{-}specific profilers, such as inefficient algorithms, which are
invisible to feature{-}specific profilers.
A vertical profiler\Autobibref{~(\hyperref[t:x28autobib_x22Matthias_Hauswirthx2c_Peter_Fx2e_Sweeneyx2c_Amer_Diwanx2c_and_Michael_HindVertical_profilingIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_251x2dx2d2692004x22x29]{\AutobibLink{Hauswirth et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Matthias_Hauswirthx2c_Peter_Fx2e_Sweeneyx2c_Amer_Diwanx2c_and_Michael_HindVertical_profilingIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_251x2dx2d2692004x22x29]{\AutobibLink{2004}})} attempts to see through the use of
high{-}level language features. It therefore gathers information from
multiple layers{---}hardware performance counters, operating system, virtual
machine, libraries{---}and correlates them into a gestalt of performance.
Vertical profiling focuses on helping programmers understand how the
interaction between different layers of abstraction affects their program{'}s performance.
By comparison, feature{-}specific profiling focuses on helping them understand the cost of features
per se.
Feature{-}specific profiling also presents information in terms of features and feature instances,
which is accessible to non{-}expert programmers, whereas vertical profilers
report low{-}level information, which requires some understanding of the
compiler and run{-}time system.
Hauswirth et al.{'}s work introduces the notion of \textit{software performance
monitors}, which are analogous to hardware performance monitors but record
software{-}related performance events.
These monitors could possibly be used to implement feature{-}specific profiling by tracking the
execution of feature code.
\goAway{\Autobibref{~(\hyperref[t:x28autobib_x22Jeremy_Singer_and_Chris_KirkhamDynamic_analysis_of_Java_program_concepts_for_visualization_and_profilingScience_of_Computer_Programming_70x282x2d3x29x2c_ppx2e_111x2dx2d1262008x22x29]{\AutobibLink{Singer and Kirkham}} \hyperref[t:x28autobib_x22Jeremy_Singer_and_Chris_KirkhamDynamic_analysis_of_Java_program_concepts_for_visualization_and_profilingScience_of_Computer_Programming_70x282x2d3x29x2c_ppx2e_111x2dx2d1262008x22x29]{\AutobibLink{2008}})}}
\goAway{\Autobibref{~(\hyperref[t:x28autobib_x22Juan_Mx2e_Tamayox2c_Alex_Aikenx2c_Nathan_Bronsonx2c_and_Mooly_SagivUnderstanding_the_behavior_of_database_operations_under_program_controlIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_983x2dx2d9962012x22x29]{\AutobibLink{Tamayo et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Juan_Mx2e_Tamayox2c_Alex_Aikenx2c_Nathan_Bronsonx2c_and_Mooly_SagivUnderstanding_the_behavior_of_database_operations_under_program_controlIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_983x2dx2d9962012x22x29]{\AutobibLink{2012}})}}
A number of profilers offer alternative views to the traditional attribution of
time costs to program locations.
Most of these views focus on particular aspects of program performance and are
complementary to the view offered by a feature{-}specific profiler.
Some recent examples include
Singer and Kirkham{'}s (2008) profiler, which assigns
costs to programmer{-}annotated code regions, listener latency
profiling\Autobibref{~(\hyperref[t:x28autobib_x22Milan_Jovic_and_Matthias_HauswirthListener_latency_profilingScience_of_Computer_Programming_19x284x29x2c_ppx2e_1054x2dx2d10722011x22x29]{\AutobibLink{Jovic and Hauswirth}} \hyperref[t:x28autobib_x22Milan_Jovic_and_Matthias_HauswirthListener_latency_profilingScience_of_Computer_Programming_19x284x29x2c_ppx2e_1054x2dx2d10722011x22x29]{\AutobibLink{2011}})}, which reports high{-}latency
operations, and Tamayo et al.{'}s (2012) tool, which provides
information about the cost of database operations.
One notable example, MAJOR\Autobibref{~(\hyperref[t:x28autobib_x22Walter_Binderx2c_Danilo_Ansalonix2c_Alex_Villazxf3nx2c_and_Philippe_MoretFlexible_and_efficient_profiling_with_aspectx2doriented_programmingIn_Procx2e_Concurrency_and_Computationx3a_Practice_and_Experiencex2c_ppx2e_1749x2dx2d17732011httpsx3ax2fx2fdoix2eorgx2f10x2e1002x2fcpex2e1760x22x29]{\AutobibLink{Binder et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Walter_Binderx2c_Danilo_Ansalonix2c_Alex_Villazxf3nx2c_and_Philippe_MoretFlexible_and_efficient_profiling_with_aspectx2doriented_programmingIn_Procx2e_Concurrency_and_Computationx3a_Practice_and_Experiencex2c_ppx2e_1749x2dx2d17732011httpsx3ax2fx2fdoix2eorgx2f10x2e1002x2fcpex2e1760x22x29]{\AutobibLink{2011}})}, uses Aspect
Oriented Programming with inter{-}advice communication to
create these complementary views.
Dynamic instrumentation frameworks such as
Valgrind\Autobibref{~(\hyperref[t:x28autobib_x22Nicholas_Nethercote_and_Julian_SewardValgrindx3a_A_framework_for_heavyweight_dynamic_binaryx5cninstrumentationIn_Procx2e_Programming_Langauges_Design_and_Implementation2007httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1273442x2e1250746x22x29]{\AutobibLink{Nethercote and Seward}} \hyperref[t:x28autobib_x22Nicholas_Nethercote_and_Julian_SewardValgrindx3a_A_framework_for_heavyweight_dynamic_binaryx5cninstrumentationIn_Procx2e_Programming_Langauges_Design_and_Implementation2007httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1273442x2e1250746x22x29]{\AutobibLink{2007}})} or
Javana\Autobibref{~(\hyperref[t:x28autobib_x22Jonas_Maebex2c_Dries_Buytaertx2c_Lieven_Eeckhoutx2c_and_Koen_De_BosschereJavanax3a_A_System_for_Building_Customized_Java_Program_Analysis_ToolsIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applications2006httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1167515x2e1167487x22x29]{\AutobibLink{Maebe et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Jonas_Maebex2c_Dries_Buytaertx2c_Lieven_Eeckhoutx2c_and_Koen_De_BosschereJavanax3a_A_System_for_Building_Customized_Java_Program_Analysis_ToolsIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applications2006httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1167515x2e1167487x22x29]{\AutobibLink{2006}})}
serve as the basis for profilers and other kinds of performance tools.
These frameworks resemble the use of continuation marks in our framework and
could potentially be used to build feature{-}specific profilers.
These frameworks are much more heavy{-}weight than continuation marks and, in turn,
allow more thorough instrumentation, e.g., of the memory
hierarchy, of hardware performance counters, etc. They have not been
used to measure the cost of individual linguistic features.
Like a feature{-}specific profiler, an optimization coach\Autobibref{~(\hyperref[t:x28autobib_x22Vincent_Stx2dAmourx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenOptimization_coachingx3a_optimizers_learn_to_communicate_with_programmersIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_163x2dx2d1782012x22x29]{\AutobibLink{St{-}Amour et al\Sendabbrev{.}}} \hyperref[t:x28autobib_x22Vincent_Stx2dAmourx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenOptimization_coachingx3a_optimizers_learn_to_communicate_with_programmersIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_163x2dx2d1782012x22x29]{\AutobibLink{2012}})} focuses on
enabling compiler optimizations through a feedback loop that involves the
developer. The two are complementary. Optimization coaches operate at
compile time whereas feature{-}specific profilers, like other profilers, operate at run
time. Because of this, feature{-}specific profilers require representative program input
to operate, whereas coaches do not. Then again, by having access to run
time data, feature{-}specific profilers can target actual program hot spots, while existing
optimization coaches must rely on static heuristics to prioritize reports.
An important tool for measuring R programs is \Scribtexttt{tracemem}. It is included
with the R tool suite, but requires programmers to rebuild R. This tool
serves to track uses of copy{-}on{-}write during the execution of R programs.
It tracks the memory that is being copied, and the source location that is
responsible for causing the copy. Also, it allows programmers to tag
individual objects they care about tracking, while ignoring everything else.
\Ssection{Conclusion}{Conclusion}\label{t:x28part_x22conclusionx22x29}
\vspace{2ex}\phantomsection\noindent\stepcounter{subsection}\label{t:x28part_x22x22x29}Feature{-}specific profiling is a novel profiling technique
that supplements traditional cost{-}centers with
language{-}specific ones. These cost centers give a new
perspective on program performance, enabling developers to
tune their programs. Feature{-}specific profiling is
especially useful when programs use language features with
dispersed or non{-}local costs. Additionally, feature{-}specific
profiling is useful with languages that allow for the
programmatic creation of new features such as Racket, R, or
even C++. The implementation of a feature{-}specific profiler
is straightforward. If the host language supports stack
annotations and inspection, such as Racket, then
implementing is as simple as that of a sampling profiler.
Languages without this support, such as R, must be extended
by adding stack annotations. This paper shows that
modifications required are practical.
While using a feature{-}specific profiler requires little effort, it does
require more setup than traditional profilers. Either library authors must
add support for their code, or developers must modify the library{'}s source.
Fortunately, adding support is simple and generally requires only a few
lines of code. The information provided by the profiler has the same
limitations as that of stack{-}based sampling profilers. This means that
language features that do not show up on the call stack cannot be measured.
The sampling nature of our profiler also means that it can only profile
interruptible features. Other profile designs, such as an event based
profiler, trade these limitations for a different set. The idea of
feature{-}specific profiling itself is not limited to the architecture
designed in this paper. We conjecture that other architectures can also
support feature{-}specific profiling.
\Ssubsubsectionstarx{Acknowledgements}{Acknowledgements}\label{t:x28part_x22Acknowledgementsx22x29}
Tony
Garnock{-}Jones implemented the Marketplace plug{-}in and helped
with the SSH case study. Stephen Chang assisted with the
Parsack plug{-}in and the Markdown case study. Christos
Dimoulas and Scott Moore collaborated on the Shill plug{-}in
and the grading script experiment. Robby Findler provided
assistance with the contract system. Oliver Keyes
implemented Going Postel. We thank Eli Barzilay, Matthew
Flatt, Asumu Takikawa, Sam Tobin{-}Hochstadt, Benjamin Chung,
Helena Kotthaus, Tomas Kalibera, Oli Fl\"{u}ckiger, Kyle Bemis,
Olga Vitek, and Luke Tierney for helpful discussions. This
work was partially supported by the National Science
Foundation (NSF) under Grants SHF 1544542 and 1518{-}844, as
well as the European Research Council (ERC) under the
European Union{'}s Horizon 2020 research and innovation
program (grant agreement 695412), and finally the Office of
Navel Research (ONR) award 503353. Any opinions, findings,
and conclusions expressed in this material may be those of
the authors and likely do not reflect the views of our
funding agencies.
\Ssectionstarx{Bibliography}{Bibliography}\label{t:x28part_x22docx2dbibliographyx22x29}
\begin{AutoBibliography}\begin{SingleColumn}\label{t:x28autobib_x22Gene_Mx2e_AmdahlValidity_of_the_Single_Processor_Approach_to_Achieving_Large_Scale_Computing_CapabilitiesIn_Procx2e_Spring_Joint_Computer_Conference1967x22x29}\Autobibentry{Gene M. Amdahl. Validity of the Single Processor Approach to Achieving Large Scale Computing Capabilities. In \textit{Proc. Spring Joint Computer Conference}, 1967.}
\label{t:x28autobib_x22Walter_Binderx2c_Danilo_Ansalonix2c_Alex_Villazxf3nx2c_and_Philippe_MoretFlexible_and_efficient_profiling_with_aspectx2doriented_programmingIn_Procx2e_Concurrency_and_Computationx3a_Practice_and_Experiencex2c_ppx2e_1749x2dx2d17732011httpsx3ax2fx2fdoix2eorgx2f10x2e1002x2fcpex2e1760x22x29}\Autobibentry{Walter Binder, Danilo Ansaloni, Alex Villaz\'{o}n, and Philippe Moret. Flexible and efficient profiling with aspect{-}oriented programming. In \textit{Proc. Concurrency and Computation: Practice and Experience}, pp. 1749{--}1773, 2011. \href{https://doi.org/10.1002/cpe.1760}{\Snolinkurl{https://doi.org/10.1002/cpe.1760}}}
\label{t:x28autobib_x22John_Clementsx2c_Matthew_Flattx2c_and_Matthias_FelleisenModeling_an_algebraic_stepperIn_Procx2e_European_Symposium_on_Programmingx2c_ppx2e_320x2dx2d3342001x22x29}\Autobibentry{John Clements, Matthew Flatt, and Matthias Felleisen. Modeling an algebraic stepper. In \textit{Proc. European Symposium on Programming}, pp. 320{--}334, 2001.}
\label{t:x28autobib_x22John_Clementsx2c_Ayswarya_Sundaramx2c_and_David_HermanImplementing_continuation_marks_in_JavaScriptIn_Procx2e_Scheme_and_Functional_Programming_Workshopx2c_ppx2e_1x2dx2d102008x22x29}\Autobibentry{John Clements, Ayswarya Sundaram, and David Herman. Implementing continuation marks in JavaScript. In \textit{Proc. Scheme and Functional Programming Workshop}, pp. 1{--}10, 2008.}
\label{t:x28autobib_x22Rx2e_Kent_DybvigChez_Scheme_Version_8_Userx27s_GuideCadence_Research_Systems2009x22x29}\Autobibentry{R. Kent Dybvig. \textit{Chez Scheme Version 8 User{'}s Guide}. Cadence Research Systems, 2009.}
\label{t:x28autobib_x22Rx2e_Kent_Dybvigx2c_Robert_Hiebx2c_and_Carl_BruggemanSyntax_Abstracton_in_SchemeIn_Procx2e_Lisp_and_Symbolic_Computation1993x22x29}\Autobibentry{R. Kent Dybvig, Robert Hieb, and Carl Bruggeman. Syntax Abstracton in Scheme. In \textit{Proc. Lisp and Symbolic Computation}, 1993.}
\label{t:x28autobib_x22Robert_Bruce_Findlerx2c_John_Clementsx2c_Cormac_Flanaganx2c_Matthew_Flattx2c_Shriram_Krishnamurthix2c_Paul_Stecklerx2c_and_Matthias_FelleisenDrSchemex3a_a_programming_environment_for_SchemeJornal_of_Functional_Programming_12x282x29x2c_ppx2e_159x2dx2d1822002x22x29}\Autobibentry{Robert Bruce Findler, John Clements, Cormac Flanagan, Matthew Flatt, Shriram Krishnamurthi, Paul Steckler, and Matthias Felleisen. DrScheme: a programming environment for Scheme. \textit{Jornal of Functional Programming} 12(2), pp. 159{--}182, 2002.}
\label{t:x28autobib_x22Robert_Bruce_Findler_and_Matthias_FelleisenContracts_for_Higherx2dorder_FunctionsIn_Procx2e_International_Conference_on_Functional_Programming2002httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f581478x2e581484x22x29}\Autobibentry{Robert Bruce Findler and Matthias Felleisen. Contracts for Higher{-}order Functions. In \textit{Proc. International Conference on Functional Programming}, 2002. \href{https://doi.org/10.1145/581478.581484}{\Snolinkurl{https://doi.org/10.1145/581478.581484}}}
\label{t:x28autobib_x22Matthew_Flatt_and_Eli_BarzilayKeyword_and_Optional_Arguments_in_PLT_SchemeIn_Procx2e_Workshop_on_Scheme_and_Functional_Programming2009x22x29}\Autobibentry{Matthew Flatt and Eli Barzilay. Keyword and Optional Arguments in PLT Scheme. In \textit{Proc. Workshop on Scheme and Functional Programming}, 2009.}
\label{t:x28autobib_x22Matthew_Flatt_and_PLTReferencex3a_RacketPLT_Incx2ex2c_PLTx2dTRx2d2010x2d12010httpx3ax2fx2fracketx2dlangx2eorgx2ftr1x2fx22x29}\Autobibentry{Matthew Flatt and PLT. Reference: Racket. PLT Inc., PLT{-}TR{-}2010{-}1, 2010. \href{http://racket-lang.org/tr1/}{\Snolinkurl{http://racket-lang.org/tr1/}}}
\label{t:x28autobib_x22Tony_Garnockx2dJonesx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenThe_network_as_a_language_constructIn_Procx2e_European_Symposium_on_Programming_Languagesx2c_ppx2e_473x2dx2d4922014x22x29}\Autobibentry{Tony Garnock{-}Jones, Sam Tobin{-}Hochstadt, and Matthias Felleisen. The network as a language construct. In \textit{Proc. European Symposium on Programming Languages}, pp. 473{--}492, 2014.}
\label{t:x28autobib_x22Matthias_Hauswirthx2c_Peter_Fx2e_Sweeneyx2c_Amer_Diwanx2c_and_Michael_HindVertical_profilingIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_251x2dx2d2692004x22x29}\Autobibentry{Matthias Hauswirth, Peter F. Sweeney, Amer Diwan, and Michael Hind. Vertical profiling. In \textit{Proc. Object{-}oriented Programming, Systems, Languages, and Applications}, pp. 251{--}269, 2004.}
\label{t:x28autobib_x22Carl_Hewittx2c_Peter_Bishopx2c_and_Richard_SteigerA_Universal_Modular_ACTOR_Formalism_for_Artificial_IntelligenceIn_Procx2e_International_Joint_Conference_on_Artificial_Intelligence1973x22x29}\Autobibentry{Carl Hewitt, Peter Bishop, and Richard Steiger. A Universal Modular ACTOR Formalism for Artificial Intelligence. In \textit{Proc. International Joint Conference on Artificial Intelligence}, 1973.}
\label{t:x28autobib_x22Milan_Jovic_and_Matthias_HauswirthListener_latency_profilingScience_of_Computer_Programming_19x284x29x2c_ppx2e_1054x2dx2d10722011x22x29}\Autobibentry{Milan Jovic and Matthias Hauswirth. Listener latency profiling. \textit{Science of Computer Programming} 19(4), pp. 1054{--}1072, 2011.}
\label{t:x28autobib_x22Jonas_Maebex2c_Dries_Buytaertx2c_Lieven_Eeckhoutx2c_and_Koen_De_BosschereJavanax3a_A_System_for_Building_Customized_Java_Program_Analysis_ToolsIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applications2006httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1167515x2e1167487x22x29}\Autobibentry{Jonas Maebe, Dries Buytaert, Lieven Eeckhout, and Koen De Bosschere. Javana: A System for Building Customized Java Program Analysis Tools. In \textit{Proc. Object{-}oriented Programming, Systems, Languages, and Applications}, 2006. \href{https://doi.org/10.1145/1167515.1167487}{\Snolinkurl{https://doi.org/10.1145/1167515.1167487}}}
\label{t:x28autobib_x22Simon_Marlowx2c_Josxe9_Iborrax2c_Bernard_Popex2c_and_Andy_GillA_lightweight_interactive_debugger_for_HaskellIn_Procx2e_Haskell_Workshopx2c_ppx2e_13x2dx2d242007x22x29}\Autobibentry{Simon Marlow, Jos\'{e} Iborra, Bernard Pope, and Andy Gill. A lightweight interactive debugger for Haskell. In \textit{Proc. Haskell Workshop}, pp. 13{--}24, 2007.}
\label{t:x28autobib_x22Jay_McCarthyThe_twox2dstate_solutionx3a_native_and_serializable_continuations_accordIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_567x2dx2d5822010x22x29}\Autobibentry{Jay McCarthy. The two{-}state solution: native and serializable continuations accord. In \textit{Proc. Object{-}oriented Programming, Systems, Languages, and Applications}, pp. 567{--}582, 2010.}
\label{t:x28autobib_x22Scott_Moorex2c_Christos_Dimoulasx2c_Dan_Kingx2c_and_Stephen_ChongSHILLx3a_a_secure_shell_scripting_languageIn_Procx2e_USENIX_Symposium_on_Operating_Systems_Design_and_Implementation2014httpsx3ax2fx2fwwwx2eusenixx2eorgx2fconferencex2fosdi14x2ftechnicalx2dsessionsx2fpresentationx2fmoorex22x29}\Autobibentry{Scott Moore, Christos Dimoulas, Dan King, and Stephen Chong. SHILL: a secure shell scripting language. In \textit{Proc. USENIX Symposium on Operating Systems Design and Implementation}, 2014. \href{https://www.usenix.org/conference/osdi14/technical-sessions/presentation/moore}{\Snolinkurl{https://www.usenix.org/conference/osdi14/technical-sessions/presentation/moore}}}
\label{t:x28autobib_x22Florxe9al_Morandatx2c_Brandon_Hillx2c_Leo_Osvaldx2c_and_Jan_VitekEvaluating_the_Design_of_the_R_LanguageIn_Procx2e_European_Conference_on_Objectx2dOriented_Programming2012httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d642x2d31057x2d7x5f6x22x29}\Autobibentry{Flor\'{e}al Morandat, Brandon Hill, Leo Osvald, and Jan Vitek. Evaluating the Design of the R Language. In \textit{Proc. European Conference on Object{-}Oriented Programming}, 2012. \href{https://doi.org/10.1007/978-3-642-31057-7_6}{\Snolinkurl{https://doi.org/10.1007/978-3-642-31057-7_6}}}
\label{t:x28autobib_x22Todd_Mytkowiczx2c_Amer_Diwanx2c_Matthias_Hauswirthx2c_and_Peter_Fx2e_SweeneyEvaluating_the_accuracy_of_Java_profilersIn_Procx2e_Programming_Langauges_Design_and_Implementationx2c_ppx2e_187x2dx2d1972010x22x29}\Autobibentry{Todd Mytkowicz, Amer Diwan, Matthias Hauswirth, and Peter F. Sweeney. Evaluating the accuracy of Java profilers. In \textit{Proc. Programming Langauges Design and Implementation}, pp. 187{--}197, 2010.}
\label{t:x28autobib_x22Nicholas_Nethercote_and_Julian_SewardValgrindx3a_A_framework_for_heavyweight_dynamic_binaryx5cninstrumentationIn_Procx2e_Programming_Langauges_Design_and_Implementation2007httpsx3ax2fx2fdoix2eorgx2f10x2e1145x2f1273442x2e1250746x22x29}\Autobibentry{Nicholas Nethercote and Julian Seward. Valgrind: A framework for heavyweight dynamic binary
instrumentation. In \textit{Proc. Programming Langauges Design and Implementation}, 2007. \href{https://doi.org/10.1145/1273442.1250746}{\Snolinkurl{https://doi.org/10.1145/1273442.1250746}}}
\label{t:x28autobib_x22Greg_Pettyjohnx2c_John_Clementsx2c_Joe_Marshallx2c_Shriram_Krishnamurthix2c_and_Matthias_FelleisenContinuations_from_generalized_stack_inspectionIn_Procx2e_International_Conference_on_Functional_Programmingx2c_ppx2e_216x2dx2d2272005x22x29}\Autobibentry{Greg Pettyjohn, John Clements, Joe Marshall, Shriram Krishnamurthi, and Matthias Felleisen. Continuations from generalized stack inspection. In \textit{Proc. International Conference on Functional Programming}, pp. 216{--}227, 2005.}
\label{t:x28autobib_x22R_Development_Core_TeamR_Language_DefinitionR_Development_Core_Teamx2c_3x2e3x2e12016httpx3ax2fx2fwebx2emitx2eedux2fx7erx2fcurrentx2farchx2famd64x5flinux26x2flibx2fRx2fdocx2fmanualx2fRx2dlangx2epdfx22x29}\Autobibentry{R Development Core Team. R Language Definition. R Development Core Team, 3.3.1, 2016. \href{http://web.mit.edu/~r/current/arch/amd64_linux26/lib/R/doc/manual/R-lang.pdf}{\Snolinkurl{http://web.mit.edu/~r/current/arch/amd64_linux26/lib/R/doc/manual/R-lang.pdf}}}
\label{t:x28autobib_x22Jeremy_Singer_and_Chris_KirkhamDynamic_analysis_of_Java_program_concepts_for_visualization_and_profilingScience_of_Computer_Programming_70x282x2d3x29x2c_ppx2e_111x2dx2d1262008x22x29}\Autobibentry{Jeremy Singer and Chris Kirkham. Dynamic analysis of Java program concepts for visualization and profiling. \textit{Science of Computer Programming} 70(2{-}3), pp. 111{--}126, 2008.}
\label{t:x28autobib_x22Vincent_Stx2dAmourx2c_Leif_Andersenx2c_and_Matthias_FelleisenFeaturex2dspecific_ProfilingIn_Procx2e_International_Conference_on_Compiler_Construction2015httpsx3ax2fx2fdoix2eorgx2f10x2e1007x2f978x2d3x2d662x2d46663x2d6x5f3x22x29}\Autobibentry{Vincent St{-}Amour, Leif Andersen, and Matthias Felleisen. Feature{-}specific Profiling. In \textit{Proc. International Conference on Compiler Construction}, 2015. \href{https://doi.org/10.1007/978-3-662-46663-6_3}{\Snolinkurl{https://doi.org/10.1007/978-3-662-46663-6_3}}}
\label{t:x28autobib_x22Vincent_Stx2dAmourx2c_Sam_Tobinx2dHochstadtx2c_and_Matthias_FelleisenOptimization_coachingx3a_optimizers_learn_to_communicate_with_programmersIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_163x2dx2d1782012x22x29}\Autobibentry{Vincent St{-}Amour, Sam Tobin{-}Hochstadt, and Matthias Felleisen. Optimization coaching: optimizers learn to communicate with programmers. In \textit{Proc. Object{-}oriented Programming, Systems, Languages, and Applications}, pp. 163{--}178, 2012.}
\label{t:x28autobib_x22Juan_Mx2e_Tamayox2c_Alex_Aikenx2c_Nathan_Bronsonx2c_and_Mooly_SagivUnderstanding_the_behavior_of_database_operations_under_program_controlIn_Procx2e_Objectx2doriented_Programmingx2c_Systemsx2c_Languagesx2c_and_Applicationsx2c_ppx2e_983x2dx2d9962012x22x29}\Autobibentry{Juan M. Tamayo, Alex Aiken, Nathan Bronson, and Mooly Sagiv. Understanding the behavior of database operations under program control. In \textit{Proc. Object{-}oriented Programming, Systems, Languages, and Applications}, pp. 983{--}996, 2012.}
\label{t:x28autobib_x22Sam_Tobinx2dHochstadt_and_Matthias_FelleisenThe_design_and_implementation_of_Typed_SchemeIn_Procx2e_Principles_of_Programming_Languagesx2c_ppx2e_395x2dx2d4062008x22x29}\Autobibentry{Sam Tobin{-}Hochstadt and Matthias Felleisen. The design and implementation of Typed Scheme. In \textit{Proc. Principles of Programming Languages}, pp. 395{--}406, 2008.}
\label{t:x28autobib_x22Hadley_WickhamAdvanced_RFirst_editionx2e_Chapman_and_Hallx2fCRC2014httpx3ax2fx2fadvx2drx2ehadx2ecox2enzx2fx22x29}\Autobibentry{Hadley Wickham. Advanced R. First edition. Chapman and Hall/CRC, 2014. \href{http://adv-r.had.co.nz/}{\Snolinkurl{http://adv-r.had.co.nz/}}}\end{SingleColumn}\end{AutoBibliography}
\end{document}
|
\section{Introduction}
\label{sec:intro}
The first theoretical proposal of a Chern band insulator
came from a pioneering work of Haldane in 1988 \cite{haldane1988model}.
In that paper, Haldane introduced a spinless tight-binding model on a
honeycomb lattice with broken time-reversal symmetry that even without
an external source of magnetic field displays a quantum Hall effect.
The emergence of this distinct insulating quantum Hall phase derives
from the topologically nontrivial electronic band structure of the
Haldane model: the nonzero Chern numbers \cite{thouless1982}
of these electronic bands yield a finite Hall conductivity at half-filling, i.e.,
the system exhibits the so-called anomalous quantum Hall effect
\cite{qi11,review-aqhe}.
The Haldane model on a honeycomb lattice was later geneneralized by
Kane and Mele \cite{kane-mele05,kane2005quantum},
providing the first microscopic model for a topological insulator
\cite{hasan2010colloquium,kane13}.
Here the spin degree of freedom is explicitly included and, in contrast with
Haldane model, time-reversal symmetry is preserved.
Although at half filling time-reversval symmetry yields a vanishing
total Chern number, such a system may exhibit a quantum spin Hall
effect \cite{kane-mele05,kane2005quantum,zhang06}.
Indeed, the Kane-Mele model is an example of
a Z$_2$ topological insulator, a system which is characterized by a Z$_2$
invariant that distinguishes between the trivial insulator phase and
the topologically nontrivial one \cite{hasan2010colloquium,kane13}.
In spite of the fact that the Kane-Mele model
is not experimentally realized so far, the quantum spin Hall effect
was theoretically predicted \cite{bernevig2006quantum}
and later experimentally observed \cite{konig2007quantum}
in HgTe/CdTe quantum wells at low temperatures.
Interestingly, experimental implementations of topological
insulators using ultracold atoms in optical lattices have also been considered
\cite{zoller16,review-adv-phys18,rmp-cooper19}.
Correlation effects on topological insulators have also been
receiving some attention in recent years \cite{rachel2018, hohenadler2013}.
An interesting example of a correlated topological insulator on a
honeycomb lattice is the Kane-Mele-Hubbard model
\cite{rachel2010, assad2011,xie2011,zheng2011,hohenadler2012,hung13,lang13,hung14,laubach14,klein21},
which is a generalization of the Kane-Mele model with the
electron-electron interaction being described by an on-site Hubbard repulsion term.
The phase diagram of the
model has been determined \cite{rachel2010,zheng2011,hohenadler2012} at half filling.
In particular, quantum Monte Carlo simulations have been
performed \cite{zheng2011,hohenadler2012},
since, in this case, the so-called fermion sign problem is absent, a
feature that is related to the fact that the model preserves
particle-hole symmetry at half filling \cite{zheng2011}.
It was shown that, apart from some possible intermediate phases,
a $Z_2$ topological band insulator phase survives
for small to moderate values of the on-site repulsion energy $U$
and that the system enters a magnetically ordered phase
above a critical on-site repulsion energy $U_c$.
An analytical description of such Mott transition was recently performed
\cite{klein21}.
Another set of interacting topological systems that has been recently
gaining some attention is made out of lattice models that display (nearly)
flat and topologically nontrivial electronic bands in the noninteracting limit
\cite{katsura2010,neupert2012topological,doretto2015flat,su2018topological,
su2019ferromagnetism,gu2019itinerant,gu21,leite2021}.
In a sense, these papers transport the long discussed subject of
flat-band ferromagnetism
\cite{kusakabe1994,tasaki1996,flat-fm} to the realm of lattice models
with topologically nontrivial free-electronic bands.
Indeed, the merging of these two subjects was motivated by a series of
papers \cite{neupert2011fractional, sun2011nearly,tang2011high} that
describe tight-binding models, specially in two dimensions, with only
short-range hoppings and whose parameters, once fine tuned, may yield
nearly flat and topologically nontrivial electronic bands.
In particular, in Ref.~\cite{leite2021}, we studied the flat-band
ferromagnetic (FM) phase of a correlated Chern insulator on a honeycomb
lattice described by a Haldane-Hubbard model.
We considered the model at $1/4$ filling (half filling of the lower
and doubly degenerated free-electronic band) and in the vicinity of
a suitable choice of the model parameters
\cite{neupert2011fractional}, that yields nearly flat
noninteracting bands.
In order to describe such a flat-band FM phase, we employed
a bosonization scheme for flat-band correlated Chern insulators \cite{doretto2015flat}, that
was developed by one of us.
Such a formalism allows us to map the Haldane-Hubbard model
to an effective interacting boson model:
We considered the effective boson model within a harmonic approximation
and determined the spin-wave spectrum;
it was found that the excitation spectrum has one gapped and one
gapless excitation branches, with a Goldstone mode at the center of
the first Brillouin zone (a feature that indicates the stability of
the flat-band FM phase) and Dirac points
at the $K$ and $K'$ points of the first Brillouin zone (BZ).
In the present paper, we extend our previous study \cite{leite2021}
about the flat-band FM phase of a correlated Chern
insulator on a honeycomb lattice, by considering a similar,
but now time-reversal symmetric, topological
Hubbard (THM) model on a honeycomb lattice.
The noninteracting term of such correlated Z$_2$ topological insulator
is given by a spinfull version of the Haldane model \cite{haldane1988model}
that preserves time-reversal symmetry.
Similarly to Ref.~\cite{leite2021}, we consider the THM
at $1/4$ filling of its noninteracting limit and in the
vicinity of the nearly flat band limit \cite{neupert2011fractional}
of its lower free-electronic band.
The flat-band FM phase of the time-reversal symmetric
THM is described within a bosonization scheme for
flat-band correlated Z$_2$ topological insulators, a formalism that was introduced
in Ref.~\cite{doretto2015flat} and is based on the bosonization formalism \cite{doretto2005}
proposed to study the quantum Hall system at filling factor $\nu = 1$.
Again, the THM is mapped to an effective
interacting boson model. We define boson operators
[Eq.~\eqref{eq:bosons}] associated with two distinct spin-flip
excitations that are termed
mixed-lattice [Eq.~\eqref{eq:projS1}] and
same-lattice [Eq.~\eqref{eq:projS2}] excitations.
In both cases, we find that the spin-wave excitation spectrum is
gapped and constituted by two bands completely separated from each
other, a feature that contrasts with the spin-wave spectrum of the correlated Chern
insulator \cite{leite2021}, whose bands touch at the corners of the first BZ.
Interestingly, in contrast with the square lattice correlated
topological insulator \cite{doretto2015flat},
whose flat-band FM phase is characterized by mixed-lattice excitations,
here, for the correlated topological insulator on a honeycomb lattice,
we find that the same-lattice ones are indeed the correct mode, which
furnishes the lowest-energy excitations [see Figs.~\ref{figEspectro}(a)-(f)].
Finally, we also find some indications that the spin-wave excitation
bands for the same-lattice excitations might be topologically
nontrivial, since the corresponding Chern numbers are nonzero.
As far as we know, this is the first calculation of the spin-wave spectrum
for the flat-band FM phase of a correlated $Z_2$ topological
insulator on a honeycomb lattice described by a Haldane-Hubbard like
model.
Our paper is organized as follows.
In Sec.~\ref{sec:TBmodel}, we introduce the time-reversal symmetric
THM on a honeycomb lattice.
In Sec.~\ref{sec:boso}, the bosonization formalism for flat-band Z$_2$
topological insulators \cite{doretto2015flat} is briefly reviewed.
In Sec.~\ref{sec:flatferromagnetism}, the effective interacting boson
model, that allows us to described the flat-band FM
phase of the correlated topological insulator, is presented;
the boson model is considered within the harmonic approximation:
the spin-wave spectrum is determined for homogeneous and
sublattice dependent on-site Hubbard repulsion energies.
Section~\ref{sec:summary} contains a brief summary of our main results.
Some details of the bosonization formalism and additional results are
presented in the five Appendices.
\begin{figure*}[t]
\centerline{
\includegraphics[width=4.5cm]{figLatticea.pdf}
\hskip1.5cm
\includegraphics[width=3.0cm]{figLatticeb.pdf}
\hskip1.5cm
\includegraphics[width=5.0cm]{figExcitation.pdf}
}
\caption{
(a) Schematic representation of the THM
model \eqref{eqHH} on an honeycomb lattice. Red and blue circles
respectively represent the sites of the (triangular) sublattices $A$ and $B$.
The nearest-neighbor and next-nearest-neighbor hopping
energies are given by $t_1$ and $t_2 e^{\pm i \phi}$
(positive sign follows arrow direction), respectively,
while $U_A$ and $U_B$ indicate the sublattice dependent on-site
Hubbard repulsion energies.
The nearest-neighbor \eqref{deltavectors}
and next-nearest-neighbor \eqref{tauvectors} vectors are indicate by
$\mbox{\boldmath $\delta $}_i$ and $\mbox{\boldmath $\tau $}_i$, respectively.
(b) The first BZ, where
$\mathbf{K} = ( 4\pi/3\sqrt{3}, 0 )$,
$\mathbf{K'} = ( 2\pi/3\sqrt{3}, 2\pi/3 )$,
$\mathbf{M}_1 = ( \pi/\sqrt{3}, \pi/3)$, and
$\mathbf{M}_2 = ( 0, 2\pi/3 )$,
with the nearest-neighbor distance of the honeycomb lattice $a = 1$.
(c) Schematic representation of the noninteracting electronic bands
\eqref{eq:omega} in the nearly-flat band limit \eqref{optimal-par} of
the lower bands $c$. At $1/4$ filling, the ground state is the
FM state \eqref{eq:FM} and low energy excitations are
particle-hole pairs (spin flips) within the lower bands.
Although the noninteracting bands $c$ and $d$ are doubly degenerated
with respect to the spin degree of freedom, we introduce an offset between the
$\sigma = \uparrow$ and $\downarrow$ bands for clarity.
The Chern numbers \eqref{eqCn} of each band are
also shown on the right side.}
\label{fig:Lattice}
\end{figure*}
\section{The time-reversal symmetric Haldane-Hubbard model}
\label{sec:TBmodel}
In this section, we introduce a time-reversal symmetric Haldane-Hubbard
model on a honeycomb lattice.
Our discussion closely follows the lines of Sec.~II from Ref.~\cite{leite2021},
where such a Haldane-Hubbard model with broken time-reversal symmetry
is described.
\subsection{The fermionic interacting model}
\label{sec:model}
Let us consider $N_e$ spin-$1/2$ electrons on a honeycomb lattice
described by a Haldane-Hubbard model, whose Hamiltonian assumes the
form
\begin{equation}
H = H_0 + H_U,
\label{eqHH}
\end{equation}
where the noninteracting term is given by
\begin{align}
H_0 &= t_1 \sum_{i \in A, \delta, \sigma} \left( c_{i A \sigma}^{\dagger} c_{i + \delta B \sigma}
+ {\rm H.c.} \right)
\nonumber \\
&+ t_2 \sum_{i \in A, \tau, \sigma}
\left( e^{-i\gamma_\sigma\phi} c_{i A \sigma}^{\dagger} c_{i + \tau A \sigma} + {\rm H.c.} \right)
\nonumber \\
&+ t_2 \sum_{i \in B, \tau, \sigma}
\left( e^{+i\gamma_\sigma\phi} c_{i B \sigma}^{\dagger} c_{i + \tau B \sigma} + {\rm H.c.} \right),
\label{eqHH0}
\end{align}
while the interacting one is an on-site Hubbard repulsion term,
\begin{equation}
H_U = \sum_i \sum_{a = A,B} U_a \hat{\rho}_{i a \uparrow} \hat{\rho}_{i a \downarrow}.
\label{eqHHU}
\end{equation}
Here the operator $c_{i a \sigma}^{\dagger}$ ($c_{i a \sigma}$) creates (destroys)
an electron with spin $\sigma = \uparrow, \downarrow$ on
the $i$-th site of the (triangular) sublattice $a = A$, $B$ of the
honeycomb lattice.
The nearest-neighbor and next-nearest-neighbor hopping amplitudes
are both positive and given by $t_1$ and $t_2$, respectively
[see Fig.~\ref{fig:Lattice}(a)].
Indeed, the next-nearest-neibhbor hopping is complex,
$t_2e^{\pm i \gamma_\sigma\phi}$, which indicates that the electron
acquires a (spin-dependent) $+\gamma_\sigma\phi$ phase
and a $-\gamma_\sigma\phi$ phase as it moves, respectively,
in the same and opposite directions of the arrows associated with the
dashed lines in Fig.~\ref{fig:Lattice}(a)
(see also note \cite{comment01}).
The complex next-nearest-neibhbor hopping yields a fictitious flux
pattern with zero net flux per unit cell \cite{neupert2011fractional}.
Importantly, time-reversal invariance requires that
$\gamma_\uparrow = - \gamma_\downarrow = 1$,
which implies that the spin $\uparrow$ electrons and the
spin $\downarrow$ electrons experience an opposite fictitious flux
pattern (see also Sec.~II from Ref.~\cite{doretto2015flat}).
The index $\delta$ indicates the nearest-neighbor vectors
\begin{align}
\mbox{\boldmath $\delta $}_1 &= -a\hat{y},
\quad\quad
\mbox{\boldmath $\delta $}_{2,3} = \pm\frac{a}{2}\left( \sqrt{3}\hat{x} \pm \hat{y} \right),
\label{deltavectors}
\end{align}
as illustrated in Fig.~\ref{fig:Lattice}(a),
and $\tau$ corresponds to the next-nearest-neighbor vectors
$\mbox{\boldmath $\tau $}_1 = \mbox{\boldmath $\delta $}_2 - \mbox{\boldmath $\delta $}_3$,
$\mbox{\boldmath $\tau $}_2 = \mbox{\boldmath $\delta $}_3 - \mbox{\boldmath $\delta $}_1$, and
$\mbox{\boldmath $\tau $}_3 = \mbox{\boldmath $\delta $}_1 - \mbox{\boldmath $\delta $}_2$:
\begin{align}
\mbox{\boldmath $\tau $}_1 &= a\sqrt{3}\hat{x},
\quad\quad
\mbox{\boldmath $\tau $}_{2,3} = -\frac{a}{2}\left( \sqrt{3}\hat{x} \mp 3\hat{y} \right).
\label{tauvectors}
\end{align}
Hereafter, we set the nearest-neigbhor distance $a = 1$.
One should mention that, for $\phi = \pi/2$,
the tight-binding model \eqref{eqHH0} corresponds to the
Kane-Mele model in the absence of the Rashba term \cite{kane-mele05}.
Finally, $\hat{\rho}_{i a\sigma}$ is the density operator for
spin $\sigma$ electrons at site $i$ of sublattice $a$,
\begin{equation}
\hat{\rho}_{i a\sigma} = c_{i a \sigma}^{\dagger} c_{i a \sigma},
\label{dens-op}
\end{equation}
and $U_a > 0$ are the on-site and sublattice-dependent repulsion
energies.
\subsection{Diagonalization of the noninteracting Hamiltonian}
\label{sec:Diagonalization}
In order to diagonalize the noninteracting model \eqref{eqHH0},
one considers the Fourier transform
\begin{equation}
c_{i a \sigma}^{\dagger} = \frac{1}{\sqrt{N_a}} \sum_{{\bf k} \in {\rm BZ}}
e^{i \mathbf{k} \cdot \mathbf{R}_i} c_{ \mathbf{k} \, a \, \sigma}^{\dagger} ,
\label{eq:Fourier}
\end{equation}
where $N_a = N$ is the number of sites of the sublattice $a$
and the momentum sum runs over the first BZ
associated with the underline triangular Bravais lattice,
see Fig.~\ref{fig:Lattice}(b).
The noninteracting Hamiltonian \eqref{eqHH0} can then be written in a
matrix form, i.e.,
\begin{equation}
H_{0} = \sum_{\mathbf{k}} \Psi_{\mathbf{k} }^{\dagger} H_{\bf k} \Psi_{\mathbf{k} },
\label{eqH0k}
\end{equation}
where the $4 \times 4$ $H_{\bf k}$ matrix reads
\begin{equation}
H_{\bf k} = \left(\begin{array}{cc}
h^{\uparrow}_{\mathbf{k}} & 0 \\
0 & h^{\downarrow}_{\mathbf{k}}
\end{array} \right)
\label{Hmatrix}
\end{equation}
and the four-component spinor $\Psi_{\mathbf{k}} $ is defined as
\begin{equation}
\Psi_{\mathbf{k}} = \left(
c_{ \mathbf{k} A \uparrow} \;\;
c_{ \mathbf{k} B \uparrow} \;\;
c_{ \mathbf{k} A \downarrow} \;\;
c_{ \mathbf{k} B \downarrow} \right)^T.
\end{equation}
The $2 \times 2$ matrices $h^\uparrow_{\bf k}$ and $h^\downarrow_{\bf k}$
associated with each spin sector in Eq.~\eqref{Hmatrix} can be written
in terms of the identity matrix $\tau_0$ and the vector
$\hat{\tau} = (\tau_1$, $\tau_2$, $\tau_3)$, whose
components are Pauli matrices, i.e.,
\begin{equation}
h_{\bf k}^\sigma = B_{0, \mathbf{k} }^\sigma \tau_0 + \mathbf{B}_{\mathbf{k} }^\sigma \cdot \hat{\tau} ,
\end{equation}
where $\mathbf{B}^\sigma_{\mathbf{k} } = ( B^\sigma_{1,\mathbf{k} }, B^\sigma_{2, \mathbf{k} }, B^\sigma_{3, \mathbf{k} })$ and
\begin{align}
B^\sigma_{0,\mathbf{k} } &= B_{0,\mathbf{k} }
= 2 t_2 \cos(\phi)\sum_{\mathbf{\tau}} \cos(\mathbf{k} \cdot \mbox{\boldmath $\tau $} ) ,
\nonumber \\
B^\sigma_{1,\mathbf{k} } &= B_{1,\mathbf{k} }
= t_1 \sum_{\mathbf{\delta}} \cos(\mathbf{k} \cdot \mbox{\boldmath $\delta $}) ,
\nonumber \\
B^\sigma_{2,\mathbf{k} } &= B_{2,\mathbf{k} }
= t_1 \sum_{\mathbf{\delta}} \sin(\mathbf{k} \cdot \mbox{\boldmath $\delta $} ) ,
\label{eqBs} \\
B^\sigma_{3,\mathbf{k} } &= \gamma_\sigma B_{3,\mathbf{k} }
= \gamma_\sigma(-2 t_2) \sin(\phi ) \sum_{\mathbf{\tau}} \sin(\mathbf{k} \cdot \mbox{\boldmath $\tau $} ),
\nonumber
\end{align}
with $\gamma_\uparrow = - \gamma_\downarrow = 1$ and
the indices $\delta$ and $\tau$ corresponding to the
nearest-neighbor \eqref{deltavectors} and next-nearest-neighbor
\eqref{tauvectors} vectors, respectively.
Although the two matrices associated with each spin sector are different,
they are not independent, since time-reversal symmetry
yields $h^\uparrow_{\bf k} = h^{ \downarrow \, *}_{-{\bf k}} $
(see Appendix A from Ref.~\cite{doretto2015flat} for further details).
It is possible to diagonalize the Hamiltonian \eqref{eqH0k} with the aid of
the canonical transformation
\begin{align}
&c_{ \mathbf{k} A \uparrow} = u_{\mathbf{k}}^* d_{ \mathbf{k} \uparrow} + v_{\mathbf{k}} c_{ \mathbf{k} \uparrow},
\;\;
c_{ \mathbf{k} A \downarrow} = u_{-\mathbf{k}} d_{ \mathbf{k} \downarrow} + v_{-\mathbf{k}}^* c_{ \mathbf{k} \downarrow},
\nonumber \\
&c_{ \mathbf{k} B \uparrow} = v_{\mathbf{k}}^{*} d_{\mathbf{k} \uparrow} - u_{\mathbf{k}} c_{ \mathbf{k} \uparrow},
\;\;
c_{ \mathbf{k} B \downarrow} = v_{-\mathbf{k}} d_{ \mathbf{k}\downarrow} - u_{-\mathbf{k}}^* c_{ \mathbf{k} \downarrow},
\label{eq:BogoTransf}
\end{align}
where the coefficients $u_{\bf k}$ and $v_{\bf k}$ are given by
\begin{align}
|u_{\mathbf{k}}|^2, |v_{\mathbf{k}}|^2 &= \frac{1}{2} \left(1 \pm \hat{B}_{3, \mathbf{k}} \right),
\nonumber \\
u_{\mathbf{k}} v_{\mathbf{k}}^{*} &= \frac{1}{2} \left( \hat{B}_{1, \mathbf{k}} + i \hat{B}_{2, \mathbf{k}} \right),
\label{eq:Bogocoef}
\end{align}
with $\hat{B}_{i,{\bf k}}$ being the $i$-th component of the unit
vector $\hat{\mathbf{B}}_{\bf k} = \mathbf{B}_{\bf k}/|\mathbf{B}_{\bf k}|$.
The diagonalized Hamiltonian reads
\begin{align}
H_{0} = \sum_{\mathbf{k} \sigma }
\omega_{\mathbf{k}}^c c_{\mathbf{k} \sigma}^{\dagger} c_{\mathbf{k} \sigma}
+ \omega_{\mathbf{k}}^d d_{\mathbf{k} \sigma}^{\dagger} d_{\mathbf{k} \sigma},
\label{eq:Hfree}
\end{align}
where the dispersions of the lower band $c$ ($-$ sign)
and the upper band $d$ ($+$ sign) are given by
\begin{align}
\omega^{d/c}_{\mathbf{k}} = B_0 \pm \sqrt{B_{1, \mathbf{k}}^2 + B_{2, \mathbf{k}}^2 + B_{3, \mathbf{k}}^2 } .
\label{eq:omega}
\end{align}
Notice that both $c$ and $d$ free-electronic bands are doubly
degenerated with respect to the spin degree of freedom.
For more details,
we refer the reader to Fig.~2 from Ref.~\cite{leite2021}, where the
free-electronic bands \eqref{eq:omega} are plotted
for different values of the parameters $t_2/t_1$ and $\phi$.
As discussed in detail in Refs.~\cite{neupert2011fractional,leite2021},
the noninteracting band structure \eqref{eq:omega} have quite interesting
properties when the model parameters $t_2/t_1$ and $\phi$ are fine tunned.
For instance, for (nearly flat band limit)
\begin{equation}
\phi=0.656 \quad \text{and} \quad t_2 = 0.3155 t_1,
\label{optimal-par}
\end{equation}
the lower band $c$ and the upper band $d$ are separated by an energy
gap and the lower band $c$ is almost flat.
Away from the nearly flat band limit \eqref{optimal-par}, the lower
band $c$ gets more dispersive, and, in particular, for $\phi = 0$ or $t_2=0$, the
energy gap closes at the $K$ and $K'$ points of the first BZ
(see Fig.~2(a) from Ref.~\cite{leite2021}).
In vicinity of the nearly flat band limit \eqref{optimal-par},
the free-electronic bands \eqref{eq:omega} are also topologically
nontrivial. Indeed, for tight-binding models of the form
\eqref{eqH0k}, one shows that the Chern numbers
of the upper and lower bands assume the form
\cite{kane13,review-adv-phys18,rmp-class-top}
\begin{equation}
C_{\sigma}^{c/d} =\pm \gamma_\sigma\frac{1}{4 \pi} \int_{BZ} d^2k \,
\hat{\mathbf{B}}_{\bf k} \cdot (\partial_{k_x} \hat{\mathbf{B}}_{\bf k} \times \partial_{k_y} \hat{\mathbf{B}}_{\bf k} ).
\label{eqCn}
\end{equation}
In particular, for the noninteracting model \eqref{eqHH0},
one finds
$C^d_\uparrow = -C^d_\downarrow = -1$ and
$C^c_\uparrow = -C^c_\downarrow = +1$.
As discussed in Sec.~IV from Ref.~\cite{doretto2015flat},
at half-filling, the so-called
charge Chern number
$C_{\rm charge} = (C^c_\uparrow + C^c_\downarrow)/2 = 0$
while the spin Chern number
$C_{\rm spin} = (C^c_\uparrow - C^c_\downarrow)/2 = 1$.
Since the tight-binding model \eqref{eqHH0} conserves the
$z$-component of the total spin (see Sec.~II.A from Ref.~\cite{doretto2015flat}),
the Z$_2$ topological invariants \cite{kane13, review-adv-phys18}
for the free-electronic bands
$
\nu_{c/d} = C^{c/d}_{\rm spin} \; {\rm mod} \; 2 = \pm 1,
$
i.e., the tight-binding model \eqref{eqHH0} is indeed a Z$_2$
topological insulator. At half filling, such a system should display
the quantum spin Hall effect \cite{kane-mele05,kane2005quantum,zhang06}
with the spin Hall conductivity $\sigma^{SH}_{xy} = e C^c_{\rm spin}/2 \pi$.
\subsection{Interaction term in momentum space}
\label{sec:hubbard}
To find the expression of the on-site Hubbard repulsion term
\eqref{eqHHU} in momentum space, we start writing the Fourier
transform of the electron density operator \eqref{dens-op},
\begin{equation}
\hat{\rho}_{i a \sigma} = \frac{1}{N} \sum_{{\bf q} \in {\rm BZ}}
e^{i \mathbf{q} \cdot \mathbf{R}_i} \hat{\rho}_{a \sigma}({\bf q}).
\label{eq:Fourier-rho}
\end{equation}
After substituting Eq.~\eqref{eq:Fourier-rho} into Eq.~\eqref{eqHHU},
we obtain
\begin{equation}
H_U = \frac{1}{N}\sum_{a = A,B} \sum_{\bf q} U_a
\hat{\rho}_{a \uparrow}(-{\bf q}) \hat{\rho}_{a \downarrow}({\bf q}).
\label{hu-k}
\end{equation}
In terms of the fermion operators $c_{ {\bf k} \, a \, \sigma}^{\dagger}$
[see Eq.~\eqref{eq:Fourier}],
the electron density operator $\hat{\rho}_{a \sigma}({\bf q})$ reads
\begin{equation}
\hat{\rho}_{a \sigma}({\bf q}) = \sum_{\bf p} c^\dagger_{{\bf p}-{\bf q}\, a\,\sigma}c_{{\bf p}\, a\,\sigma}.
\label{density-op-2}
\end{equation}
Substituting Eq.~\eqref{eq:BogoTransf} into \eqref{density-op-2}
and neglecting the terms that contain the fermions
$d_{{\bf k}\,\sigma}$, one finds the expression of the electron density
operator \eqref{density-op-2} {\sl projected} into the lower noninteracting
bands $c$
(see Eq.~(28) from Ref.~\cite{doretto2015flat})
\begin{equation}
\bar{\rho}_{a\, \sigma}({\bf q}) = \sum_{\bf p}
G_{a\, \sigma}({\bf p},{\bf q})c^\dagger_{{\bf p}-{\bf q}\,\sigma}c_{{\bf p}\,\sigma},
\label{proj-dens-op}
\end{equation}
where the $G_{a\,\sigma}({\bf p},{\bf q})$ functions are given by
\begin{align}
G_{a\, \sigma}({\bf p}, {\bf q}) &=
\delta_{a,A} \left( \delta_{\sigma,\uparrow}v^*_{{\bf p} - {\bf q}} v_{\bf p}
+ \delta_{\sigma,\downarrow}v_{-{\bf p} + {\bf q}} v^*_{-{\bf p}} \right)
\nonumber \\
&+ \delta_{a,B} \left( \delta_{\sigma,\uparrow}u^*_{{\bf p} - {\bf q}} u_{\bf p}
+ \delta_{\sigma,\downarrow}u_{-{\bf p} + {\bf q}} u^*_{-{\bf p}} \right),
\label{eq:ga-sigma}
\end{align}
with $u_{\bf k}$ and $v_{\bf k}$ being the coefficients \eqref{eq:Bogocoef}.
Finally, we quote the expression of the on-site Hubbard
term \eqref{hu-k} projected into the lower noninteracting bands $c$, which
follows from Eq.~\eqref{hu-k}
with $\hat{\rho}_{a \sigma}({\bf q}) \rightarrow \bar{\rho}_{a \sigma}({\bf q})$:
\begin{equation}
\bar{H}_U = \frac{1}{N}\sum_{a = A,B} \sum_{\bf q} U_a
\bar{\rho}_{a \uparrow}(-{\bf q}) \bar{\rho}_{a \downarrow}({\bf q}).
\label{hu-k-bar}
\end{equation}
\section{Bosonization formalism for flat-band Z$_2$ topological insulators}
\label{sec:boso}
Here we summarize the bosonization formalism for a Z$_2$
topological insulator introduced by one of us in Ref.~\cite{doretto2015flat}
for the description of the flat-band FM phase of a
square lattice correlated Z$_2$ topological insulator.
Our discussion follows the lines of Sec.~III from Ref.~\cite{leite2021}.
In order to introduce the bosonization scheme, one needs to define
a reference state.
Let us consider a spinfull topological insulator on a bipartite lattice whose
Hamiltonian assumes the form \eqref{eqH0k}, choose the model
parameters such that (at least) the lower band $c$ is (nearly)
flat, and focus on the $1/4$ filling of the electronic bands:
the number of electrons
$N_e = N_A = N_B = N$, with $N_A$ and $N_B$ being, respectively, the
number of sites of the sublattices $A$ and $B$.
Assuming that the lower band $c \, \uparrow\,$
is completely occupied [see Fig.~\ref{fig:Lattice}(c)],
the ground state of the noninteracting system
(the {\sl reference} state) is completely spin polarized:
\begin{equation}
|{\rm FM} \rangle = \prod_{\mathbf{k} \in BZ} c_{\mathbf{k} \uparrow}^{\dagger} |0 \rangle.
\label{eq:FM}
\end{equation}
Excited states are generated by spin-flips: As illustrated in
Fig.~\ref{fig:Lattice}(c), the lowest-energy neutral excitations
above the reference state \eqref{eq:FM} are particle-hole pairs
within the lower bands $c$, since the lower flat bands $c$ are
separated from the upper ones $d$ by an energy gap; such an excited state
with well-defined momentum can be written as
$| \Psi_{\mathbf{k}} \rangle \propto S_{\bf k}^{-} | {\rm FM} \rangle$.
Interestingly, it is possible to define boson operators that are
associated with such spin-flip excitations
(see Ref.~\cite{doretto2015flat} for details),
\begin{align}
b_{\alpha,\mathbf{q}} &= \frac{\bar{S}_{-\mathbf{q},\alpha}^{+}}{F_{\alpha\alpha,\mathbf{q}}}
= \frac{1}{F_{\alpha\alpha,{\bf q}}} \sum_{\mathbf{p}} g_{\alpha}^* (-\mathbf{p}, \mathbf{q})
c_{\mathbf{p+q}\uparrow}^{\dagger} c_{\mathbf{p}\downarrow},
\nonumber \\
b_{\alpha,\mathbf{q}}^{\dagger} &= \frac{\bar{S}_{\mathbf{q},\alpha}^{-}}{F_{\alpha\alpha,\mathbf{q}}}
= \frac{1}{F_{\alpha\alpha,{\bf q}} } \sum_{\mathbf{p}} g_{\alpha} (\mathbf{p}, \mathbf{q})
c_{\mathbf{p-q}\downarrow}^{\dagger} c_{\mathbf{p}\uparrow},
\label{eq:bosons}
\end{align}
with $\alpha = 0,1$, that obey the commutation relations
\begin{align}
[b_{\alpha,\mathbf{k}} , b_{\beta, \mathbf{q}}^{\dagger} ] &= \delta_{\alpha, \beta} \; \delta_{\mathbf{k}, \mathbf{q}},
\nonumber \\
[b_{\alpha,\mathbf{k}} , b_{\beta,\mathbf{q}} ] &= [b_{\alpha,\mathbf{k}}^{\dagger} , b_{\beta,\mathbf{q}}^{\dagger} ] = 0.
\label{eq:BComutations}
\end{align}
Concerning the definition of the projected spin operators $\bar{S}^\pm_{\mathbf{q}, \alpha}$
in Eq.~\eqref{eq:bosons}, we consider {\sl two distinct} proposals: \\
(i) {\bf Mixed-lattice excitations}: Motivated by previous results \cite{doretto2015flat}
concerning a correlated Z$_2$ topological insulator on a square lattice,
we define $\bar{S}^\pm_{\mathbf{q}, \alpha}$ as
\begin{equation}
\bar{S}^\pm_{{\bf q}, \alpha} = \bar{S}^\pm_{{\bf q}, AB}
+ (-1)^\alpha \bar{S}^\pm_{{\bf q}, BA},
\label{eq:projS1}
\end{equation}
where
\begin{equation}
\bar{S}^\pm_{{\bf q},ab} = \bar{S}^x_{{\bf q},ab} \pm i \bar{S}^y_{{\bf q},ab}.
\label{linearcomb}
\end{equation}
The operator $\bar{S}^\lambda_{{\bf q},ab}$,
with $\lambda = x,y,z$ and $a,b = A$, $B$,
is the spin operator $S^\lambda_{{\bf q},ab}$ {\sl projected}
into the lower noninteracting bands $c$.
The spin operator $S^\lambda_{{\bf q},ab}$ is indeed the Fourier transform of
the operator
\begin{equation}
S^\lambda_{i, ab} = \frac{1}{2}\sum_{\mu,\nu=\uparrow,\downarrow}
c_{i a \mu}^\dagger \sigma^\lambda_{\mu\,\nu} c_{i b \nu},
\label{eq:abrikosov}
\end{equation}
where $\sigma^\lambda_{\mu\,\nu} $ is the matrix element of the Pauli
matrix $\sigma^\lambda$.
The spin operators \eqref{eq:projS1} are indeed related
with spin-flip excitations that also change the sublattice index.
Due to such a feature,
we denote the excitations defined by
the boson operators \eqref{eq:bosons} and the spin operator \eqref{eq:projS1}
as {\sl mixed-lattice} (ML) excitations.
The $F^2_{\alpha\beta, {\bf q}}$ function reads
\begin{align}
F^2_{\alpha \beta, {\bf q}} = \sum_{\bf p} g_\alpha({\bf p}, {\bf q}) g_\beta^*(-{\bf p}+{\bf q},{\bf q}),
\label{eq:F2abhigh}
\end{align}
with $g_\alpha(\mathbf{p},\mathbf{q})$ defined in terms
of the coefficients \eqref{eq:Bogocoef},
\begin{align}
g_\alpha({\bf p},{\bf q}) = -u_{\bf p} v_{-{\bf p}+{\bf q}} - (-1)^\alpha v_{\bf p} u_{-{\bf p} + {\bf q}}.
\label{eq:g1}
\end{align}
Interestingly, the $F^2_{\alpha \beta, {\bf q}}$ function can be explicitly expressed
in terms of the $B_{i, \mathbf{k}}$ functions \eqref{eqBs}, see Eq.~\eqref{eq:F2high}. \\
(ii) {\bf Same-lattice excitations}: Motivated by our previous study \cite{leite2021}
about a honeycomb lattice correlated Chern insulator,
we also consider spin-flip excitations that preserve the sublattice
index. In this case, one defines
\begin{equation}
\bar{S}^\pm_{\mathbf{q}, \alpha} = \bar{S}^\pm_{\mathbf{q}, A}+ (-1)^\alpha \bar{S}^\pm_{\mathbf{q}, B},
\label{eq:projS2}
\end{equation}
where $\bar{S}^\pm_{{\bf q}, a}$ is also
given by Eqs.~\eqref{linearcomb} and \eqref{eq:abrikosov}
with $a = b$, i.e., $\bar{S}^\pm_{{\bf q}, a} = \bar{S}^\pm_{{\bf q}, aa}$;
again, boson operators are defined as done in Eq.~\eqref{eq:bosons},
with $F^2_{\alpha \beta, {\bf q}}$ also given by Eq.~\eqref{eq:F2abhigh},
but now $g_\alpha(\mathbf{p},\mathbf{q})$ assumes
the form
\begin{align}
g_\alpha({\bf p},{\bf q}) = v_{-{\bf p}+{\bf q}} v_{\bf p} + (-1)^\alpha u_{-{\bf p} + {\bf q}} u_{\bf p},
\label{eq:g2}
\end{align}
with $u_{\bf k}$ and $v_{\bf k}$ being the coefficients \eqref{eq:Bogocoef}.
Since the spin operators \eqref{eq:projS2} preserve the sublattice
index, we denote such excitations as {\sl same-lattice} (SL) excitations.
The expression of the $F^2_{\alpha \beta, {\bf q}}$ function in terms of the
$B_{i, \mathbf{k}}$ functions \eqref{eqBs} is shown in
Appendix~\ref{sec:ap-details-same},
see Eq.~\eqref{eq:F2low}.
Finally, one should note that, for both ML and SL excitations,
\begin{equation}
b_{\alpha,{\bf q}} | {\rm FM} \rangle = 0,
\label{vacuum}
\end{equation}
which indicates that the spin-polarized (reference) state
\eqref{eq:FM} is indeed the vacuum for the boson operators
\eqref{eq:bosons}, regardless the definition of the projected spin operators.
As discussed in detail in Ref.~\cite{doretto2015flat},
it is possible to find the bosonic representation of
any operator that is written in terms of the fermions
$c^\dagger_{{\bf k}\sigma}$ and $c_{{\bf k}\sigma}$.
For instance, in terms of the boson operators \eqref{eq:bosons}
(either defined in terms of the ML or the SL excitations),
the bosonic representation of the projected electron density operator
\eqref{proj-dens-op} reads
\begin{equation}
\bar{\rho}_{a \sigma}(\mathbf{k}) = \frac{1}{2}N\delta_{\sigma, \uparrow}\delta_{\mathbf{k}, 0}
+ \sum_{\alpha,\beta,{\bf q}} \: \mathcal{G}_{\alpha \beta a \sigma}(\mathbf{k}, \mathbf{q})
b_{\beta,\mathbf{k}+\mathbf{q} }^{\dagger} b_{\alpha, \mathbf{q}},
\label{eq:rhoBoson}
\end{equation}
where the $\mathcal{G}_{\alpha \beta a \sigma}({\bf k},{\bf q})$ function is
defined by Eq.~\eqref{Gcal}.
Similar to the $F^2_{\alpha \beta, q}$ function \eqref{eq:F2abhigh},
$\mathcal{G}_{\alpha \beta a \sigma}({\bf k},{\bf q})$
can also be written in terms of the coefficients \eqref{eqBs},
see Eqs.~\eqref{Gcal2} and \eqref{Gcal3} for ML and SL
excitations, respectively.
As discussed in the next section, both the
Hamiltonian \eqref{eqHH0} and the interaction term
\eqref{eqHHU}, projected into the lower noninteracting bands $c$,
can also be expressed in terms of the boson operators \eqref{eq:bosons}.
Apart from the expressions of $F^2_{\alpha \beta, q}$ and
$\mathcal{G}_{\alpha\beta a \sigma}({\bf k},{\bf q})$,
the bosonic representation \eqref{eq:rhoBoson} of the density operator
\eqref{proj-dens-op} and the effective boson model [see Eq.~\eqref{eq:Heffective} below]
derived from the THM \eqref{eqHH} are equal,
regardless the nature of the excitations considered (ML or SL ones);
due to such a feature, we employ the same notation for the boson
operators \eqref{eq:bosons} for both ML \eqref{eq:projS1}
and SL \eqref{eq:projS2} excitations.
Finally, it is important to emphasize that, for the square lattice $\pi$-flux model
\cite{doretto2015flat}, only the ML excitations \eqref{eq:projS1}
yield two sets of independent bosons operators $b_0$ and $b_1$.
Such a feature distinguishes the time-reversal symmetric square lattice
$\pi$-flux model from the generalized Haldane one [Eq.~\eqref{eqHH0}],
which, in principle, allows us to define boson operators from both
ML \eqref{eq:projS1} and SL \eqref{eq:projS2} excitations.
Interestingly, for the generalized square lattice $\pi$-flux model
\cite{doretto2015flat} and the generalized Haldane model \cite{leite2021},
both with broken time-reversal symmetry,
the SL excitations \eqref{eq:projS2} are the
lowest-energy excitations of the corresponding correlated Chern
insulators.
\section{Flat-band ferromagnetism in the topological Hubbard model}
\label{sec:flatferromagnetism}
In this section, we study the flat-band FM phase of the THM \eqref{eqHH}.
We consider the model at $1/4$ filling of its corresponding
noninteracting limit and assume that the noninteracting lower bands
$c$ are in the vicinity of the nearly flat band limit \eqref{optimal-par}.
We focus on the determination of the dispersion
relation of the elementary particle-hole pair excitations
(spin-waves) above the (flat-band) FM ground state \eqref{eq:FM}:
ML [Eq.~\eqref{eq:projS1}] and SL [Eq.~\eqref{eq:projS2}]
excitations are discussed separately, since they are two distinct
proposals for the definition of the boson operators \eqref{eq:bosons};
most
|
importantly, we find that the SL excitations \eqref{eq:projS2} are
indeed the lowest-energy excitations.
\subsection{Effective interacting boson model}
\label{sec:ChernInsu}
Here we derive an effective interacting boson model from
the THM \eqref{eqHH}
within the bosonization formalism summarized in Sec.~\ref{sec:boso}.
Our presentation closely
follows the lines of Sec.~IV.A from Ref.~\cite{leite2021}
and more details can be found in Ref.~\cite{doretto2015flat}.
First of all, we project the Hamiltonian \eqref{eqHH} into the
lower noninteracting bands $c$
(such a restriction is justified, once the on-site repulsion energies
$U_a$ fullfil some conditions, see comment above Eq.~(35) from
Ref.~\cite{leite2021}),
\begin{align}
H \rightarrow \bar{H} &= \bar{H}_{0} + \bar{H}_U,
\label{H-projected}
\end{align}
where the projected noninteracting Hamiltonian $\bar{H}_{0}$ is
obtained from Eq.~\eqref{eq:Hfree},
\begin{align}
\bar{H}_{0} = \sum_{\mathbf{k} \sigma } \omega^c_{ \mathbf{k}}
c_{\mathbf{k} \sigma}^{\dagger} c_{\mathbf{k} \sigma},
\label{eq:omegaProje}
\end{align}
and $\bar{H}_U$ is given Eq.~\eqref{hu-k-bar}.
In terms of the boson operators \eqref{eq:bosons},
the noninteracting (kinetic) term $\bar{H}_{0}$ reads
\begin{equation}
\bar{H}_{0,B} = E_0 + \sum_{\alpha \beta} \sum_{\mathbf{q} \in BZ} \bar{\omega}^{\alpha \beta}_{\mathbf{q} }
b_{\beta, \mathbf{q}}^{\dagger} b_{\alpha, \mathbf{q}},
\label{eq:H0B1}
\end{equation}
where $E_0 = \sum_{\bf k} \omega^c_{\bf k}$ is a constant related to
the action of the Hamiltonian $\bar{H}_0$ into the reference state
\eqref{eq:FM} and
\begin{align}
\bar{\omega}^{\alpha \beta}_{\mathbf{q}} &=
\sum_{\mathbf{p}}
\left( \omega^c_{\mathbf{p-q}} - \omega^c_{\mathbf{p}} \right)
\frac{g_{\alpha} (\mathbf{p}, \mathbf{q}) g^*_{\beta}(-{\bf p}+{\bf q}, {\bf q})}
{F_{\alpha\alpha, \mathbf{q}} F_{\beta\beta, \mathbf{q}}},
\label{eq:omegaBar}
\end{align}
with $F_{\alpha\beta, {\bf q}}$ given by
Eqs.~\eqref{eq:F2high} (ML excitations) and
\eqref{eq:F2low} (SL excitations) and $g_{\alpha}(\mathbf{p}, \mathbf{q})$
given by Eqs.~\eqref{eq:g1} (ML excitations) and \eqref{eq:g2} (SL excitations).
The on-site Hubbard term $\bar{H}_U$ can be cast into its bosonic
representation with the aid of Eqs.~\eqref{hu-k-bar} and \eqref{eq:rhoBoson};
after normal-ordering the resulting expression,
one arrives at \cite{doretto2015flat}
\begin{align}
\bar{H}_{U,B} &= \bar{H}_{U,B}^{(2)}+ \bar{H}_{U,B}^{(4)},
\end{align}
where the quadratic and quartic terms are given by
\begin{align}
\bar{H}_{U,B}^{(2)} &= \sum_{\alpha \beta} \sum_{\mathbf{q} } \epsilon^{\alpha \beta}_{\mathbf{q} }
b_{\beta, \mathbf{q}}^{\dagger} b_{\alpha, \mathbf{q}},
\label{H42} \\
\bar{H}_{U,B}^{(4)} &= \frac{1}{N} \sum_{\mathbf{k} , \mathbf{q}, \mathbf{p}} \sum_{\alpha \beta \alpha' \beta'}
V^{\alpha \beta \alpha' \beta'}_{\mathbf{k}, \mathbf{q}, \mathbf{p} }
b_{\beta', \mathbf{p+k}}^{\dagger} b_{\beta, \mathbf{q-k}}^{\dagger} b_{\alpha \mathbf{q}} b_{\alpha' \mathbf{p}},
\label{H44}
\end{align}
with the coefficient $\epsilon^{\alpha \beta}_{\bf q} $ assuming the form
\begin{align}
\epsilon^{\alpha \beta}_{\bf q} &= \frac{1}{2} \sum_{a}
U_a\mathcal{G}_{\alpha \beta a \downarrow} (0, \mathbf{q})
\nonumber \\
&+ \frac{1}{N} \sum_{a,\alpha', \mathbf{k}}
U_a\mathcal{G}_{\alpha' \beta a \uparrow}(-\mathbf{k}, \mathbf{k+q})
\mathcal{G}_{\alpha \alpha' a \downarrow}(\mathbf{k}, \mathbf{q}),
\label{eq:Epsilon}
\end{align}
and the boson-boson interaction being defined by
\begin{align}
V^{\alpha \beta \alpha' \beta'}_{\mathbf{k}, \mathbf{q}, \mathbf{p} } &= \frac{1}{N} \sum_a
U_a\mathcal{G}_{\alpha \beta a \uparrow}(-\mathbf{k}, \mathbf{q})
\mathcal{G}_{\alpha' \beta' a \downarrow} (\mathbf{k}, \mathbf{p}).
\label{eq:Vkq}
\end{align}
One should recall that, in terms of the coefficients \eqref{eqBs}, the
$\mathcal{G}_{\alpha \beta a \sigma}({\bf k},{\bf q})$ functions are given by
Eqs.~\eqref{Gcal2} and \eqref{Gcal3} for ML and SL
excitations, respectively.
In summary, the effective {\sl interacting} boson model,
which allows us to describe the flat-band FM phase of the
THM \eqref{eqHH}, reads
\begin{align}
\bar{H}_B = \bar{H}_{0,B} + \bar{H}^{(2)}_{U,B} + \bar{H}^{(4)}_{U,B}.
\label{eq:Heffective}
\end{align}
It is important to emphasize that the effective boson model
\eqref{eq:Heffective} is quite general, since, in principle,
it can describe the flat-band FM phase of a correlated Z$_2$
topological insulator described by a THM
on a bipartite lattice, as long as its corresponding noninteracting term assumes
the form \eqref{eqH0k} and its free-electronic bands can be
made almost dispersionless by carefully choosing the model
parameters (see Sec.~V from Ref.~\cite{leite2021} for more details):
recall that, all terms of the Hamiltonian \eqref{eq:Heffective}
can be written in terms of the functions \eqref{eqBs}, which
completely characterize tight-binding models of the form \eqref{eqH0k}.
\subsection{Spin-wave spectrum}
\label{sec:spin-wave}
We now determine the spin-wave spectrum of the flat-band
FM phase of the THM \eqref{eqHH} with the aid of
effective boson model \eqref{eq:Heffective}. In the
lowest-order (harmonic) approximation,
the Hamiltonian \eqref{eq:Heffective} reads
\begin{align}
\bar{H}_B \approx \bar{H}_{0,B} + \bar{H}^{(2)}_{U,B}.
\label{eq:Heffective2}
\end{align}
The Hamiltonian \eqref{eq:Heffective2} can be diagonalized with the
aid of the following canonical transformation
\begin{equation}
b_{0, {\bf q} } = u^*_{\bf q} a_{+, {\bf q}} + v_{\bf q} a_{-, {\bf q}},
\quad
b_{1, {\bf q} } = v^*_{\bf q} a_{+, {\bf q}} - u_{\bf q} a_{-, {\bf q}}.
\label{eq:BogoTransf2}
\end{equation}
One then easily shows that
\begin{equation}
\bar{H}_B = E_0 + \sum_{\mu = \pm } \sum_{{\bf q} \in BZ }
\Omega_{\mu, {\bf q}} a_{\mu, \mathbf{q}}^{\dagger} a_{\mu, \mathbf{q}},
\label{eq:HFinalFlat}
\end{equation}
where the constant
$E_0 = \sum_{\bf k} \omega^c_{\bf k} = (-1.69\,t_1)N$ for the nearly flat band
limit \eqref{optimal-par}, the dispersion relation $\Omega_{\mu, {\bf q}}$
of the bosons $a_\pm$ (the spin-wave spectrum) is given by
\begin{equation}
\Omega_{\pm,{\bf q}} = \frac{1}{2}\left( \epsilon^{00}_{\bf q} + \epsilon^{11}_{\bf q} \right)
\pm \epsilon_{\bf q},
\label{omega-b}
\end{equation}
with
$\epsilon_{\bf q} = \frac{1}{2}\sqrt{ \left( \epsilon^{00}_{\bf q} - \epsilon^{11}_{\bf q} \right)^2
+ 4 \epsilon^{01}_{\bf q} \epsilon^{10}_{\bf q}}$,
and the coefficients $u_{\bf q}$ and $v_{\bf q}$ satisfy the relations
\begin{align}
|u_{\bf q}|^2, |v_{\bf q}|^2 &= \frac{1}{2} \pm
\frac{1}{4\epsilon_{\bf q}}\left( \epsilon^{00}_{\bf q} - \epsilon^{11}_{\bf q} \right),
\nonumber \\
u_{\bf q} v_{\bf q}^{*} &= \frac{\epsilon^{01}_{\bf q}}{4\epsilon_{\bf q}},
\quad
v_{\bf q} u_{\bf q}^{*} = \frac{\epsilon^{10}_{\bf q}}{4\epsilon_{\bf q}}.
\label{eq:Bogocoef2}
\end{align}
Note that the vacuum state for the bosons $a_\pm$ is the ground state of the
Hamiltonian \eqref{eq:HFinalFlat}. Indeed, due to the form of the canonical
transformation \eqref{eq:BogoTransf2}, one sees that the vacuum for
the bosons $a_\pm$ corresponds to the spin-polarized ferromagnet state
\eqref{eq:FM}, which is the vacuum
(reference) state for the bosons $b_0$ and $b_1$ [see Eq.~\eqref{vacuum}].
Such a result points to the stability of a flat-band FM
phase for the THM \eqref{eqHH}.
The spin-wave spectra \eqref{omega-b} for the ML excitations
\eqref{eq:projS1} are shown in Figs.~\ref{figEspectro}(a)--(c),
while the results for the SL ones \eqref{eq:projS2} are displayed in
Figs.~\ref{figEspectro}(d)--(f) and \ref{fig:espectro-phi}(a) and (b).
In the following, we concentrate on the spin-wave spectrum for the SL
excitations, since they are the lowest-energy excitations
that characterize the flat-band FM phase of the THM
\eqref{eqHH}. A detailed discussion about the ML excitations
can be found in Appendix~\ref{sec:ap-spin-wave}.
\subsubsection*{SL excitations}
\label{sec:spin-waveII}
In order to determine the spin-wave spectrum
\eqref{omega-b} for the SL excitations \eqref{eq:projS2},
one needs to calculate the kinetic coefficients \eqref{eq:omegaBar}
and the coefficients \eqref{eq:Epsilon}
associated with the quadratic term \eqref{H42}.
In this case, one should consider the expressions of the
$g_\alpha({\bf p}, {\bf q})$, $F_{\alpha\beta, {\bf q}}$, and
$\mathcal{G}_{\alpha \beta a \sigma}({\bf p},{\bf q})$
functions given by Eqs.~\eqref{eq:g2},
\eqref{eq:F2low}, and \eqref{Gcal3}, respectively.
Differently from the ML excitations (see Appendix~\ref{sec:ap-spin-wave}),
for the SL excitations, one finds that the kinetic coefficients
\eqref{eq:omegaBar} vanishes, $\bar{\omega}^{\alpha \beta}_{\mathbf{q}} = 0$.
Moreover, the quadratic term \eqref{H42} of the effective boson model
\eqref{eq:Heffective} is Hermitian, since the coefficients
$\epsilon^{\alpha \alpha}_{\bf q} $ are real quantities
while $\epsilon^{01}_{\bf q}$ and $\epsilon^{10}_{\bf q}$ are complex
ones with $\epsilon^{01}_{\bf q} = (\epsilon^{10}_{\bf q})^*$
[see Eq.~\eqref{eq:Epsilon} and Fig.~\ref{fig:F2low}(c)];
such a feature is distinct from the ones found for the ML excitations
(see Appendix~\ref{sec:ap-spin-wave}) and for the correlated
Chern insulator \cite{leite2021}, whose corresponding
quadratic Hamiltonians \eqref{H42} are non-Hermitian.
Finally, similarly to the ML excitations (see Fig.~\ref{fig:F2high})
and the correlated Chern insulator \cite{leite2021},
one finds that the condition
\begin{equation}
F_{\alpha \beta,{\bf q}} = \delta_{\alpha,\beta}F_{\alpha \alpha,{\bf q}}
\label{conditionF}
\end{equation}
is not fulfilled for all momenta within the first BZ
[see Figs.~\ref{fig:F2low} (a) and (b)];
the validity of the condition \eqref{conditionF} is an important
ingredient for the definition \eqref{eq:bosons}
of the two sets of {\sl independent} boson operators $b_0$ and $b_1$;
for a detailed discussion about this important issue, we refer the
reader to Appendix~\ref{sec:ap-spin-wave}
and to Appendix~B from Ref.~\cite{leite2021}.
\begin{figure*}[t]
\centerline{\includegraphics[width=6.1cm]{spectrumMagh.pdf}
\hskip2.0cm \includegraphics[width=6.1cm]{spectrumMagl.pdf}}
\vskip0.2cm
\centerline{\includegraphics[width=6.1cm]{spectrumUBh08.pdf}
\hskip2.0cm \includegraphics[width=6.1cm]{spectrumUBl08.pdf}}
\vskip0.2cm
\centerline{\includegraphics[width=6.1cm]{spectrumUBh06.pdf}
\hskip2.0cm \includegraphics[width=6.1cm]{spectrumUBl06.pdf}
}
\caption{Dispersion relation \eqref{omega-b} (spin-wave spectrum) of
the effective boson model \eqref{eq:Heffective2} in the harmonic
approximation for the nearly flat band limit \eqref{optimal-par}
along paths in the first BZ [Fig.~\ref{fig:Lattice}(b)].
Solid and dashed lines respectively represent the real part of $\Omega_{\pm,{\bf q}}$
and the imaginary part of $\Omega_{+,{\bf q}} = -\Omega_{-,{\bf q}} $, where
the latter is multiplied by a factor of 20 for clarity.
The spin-wave spectrum (solid magenta line) for the ML excitations \eqref{eq:projS1} are
shown in panels (a), (b), and (c), while
panels (d), (e), and (f) correspond to the spin-wave spectrum (solid
green line) for the SL excitations \eqref{eq:projS2}.
The on-site Hubbard repulsion energies are
$U_A = U_B = U$ [(a) and (d)],
$U_B = 0.8\, U_A = 0.8\, U$ [(b) and (e)], and
$U_B = 0.6\, U_A = 0.6\, U$ [(c) and (f)]. }
\label{figEspectro}
\end{figure*}
\begin{figure}[t]
\centerline{\includegraphics[width=5.5cm]{spectrumphila-v2.pdf}}
\vskip0.5cm
\centerline{\includegraphics[width=5.5cm]{spectrumphild-v2.pdf}
}
\caption{SL excitations \eqref{eq:projS2}: Spin-wave spectrum \eqref{omega-b}
along paths in the first BZ for on-site repulsion energies
$U_A = U_B = U$ and the next-nearest-neighbor hopping amplitude
$t_2$ given by $\cos(\phi) = t_1/ (4 t_2)$.
Solid and dashed lines respectively represent the real part of $\Omega_{\pm,{\bf q}}$
and the imaginary part of $\Omega_{+,{\bf q}} = -\Omega_{-,{\bf q}} $.
Phase
$\phi = 0.4$ [blue line in (a)],
$\phi = 0.5$ [green line in (a)],
$\phi = 0.656$ (magneta line),
$\phi = 0.7$ [green line in (b)], and
$\phi = 0.8$ [blue line in (b)]. }
\label{fig:espectro-phi}
\end{figure}
The dispersion relation \eqref{omega-b}
[the spin-wave spectrum for the SL excitations \eqref{eq:projS2}]
for the nearly flat band limit \eqref{optimal-par} and on-site
repulsion energies $U_A = U_B = U$ is shown in Fig.~\ref{figEspectro}(d).
Notice that, instead of the nearest-neighbor hopping energy $t_1$,
the energy scale of the spin-wave spectrum is given by the
on-site repulsion energy $U$, since the kinetic coefficients
\eqref{eq:omegaBar}
(associated with the noninteracting bands $c$) are neglected.
Similarly to the ML excitations [Fig.~\ref{figEspectro}(a)],
the spin-wave spectrum for the SL
excitations is also gapped and has two branches:
the gap of the lower branch is at the $\Gamma$ point of
the first BZ while the gap of
upper one is at the $K$ and $K'$ points.
In contrast with the correlated Chern insulator \cite{leite2021},
whose spin-wave spectrum has a Goldstone mode at the $\Gamma$ point
related with a continuous SU(2) symmetry that is spontaneously broken,
the flat-band FM phase of the
correlated topological insulator \eqref{eqHH} has a gapped spectrum:
such a feature, that is properly described by the bosonization
formalism, is due to the fact that both the Hamiltonian \eqref{eqHH}
and the ground state \eqref{eq:FM} preserve a U(1) spin rotation symmetry
(see Sec.~II.A from Ref~\cite{doretto2015flat}
and Ref.~\cite{neupert2012topological} for more details).
Differently from the corresponding correlated Chern insulator \cite{leite2021},
whose spin-wave spectrum has Dirac points at the $K$ and $K'$ points,
here one finds an energy gap between the lower and upper bands at the
$K$ and $K'$ points,
\begin{equation}
\Delta^{(K)} = \Omega_{+,K} - \Omega_{-,K} = 4.96 \times 10^{-2}\, U;
\label{gap-SL}
\end{equation}
such a gap is large than the one [Eq.~\eqref{gap-ML}] found for the ML excitations.
Interestingly, apart from the energy gaps at the $\Gamma$, $K$, and
$K'$ points, the spin-wave spectrum shown in Fig.~\ref{figEspectro}(d)
qualitatively resembles the one of the correlated Chern
insulator on the honeycomb lattice that we have previously studied
(see Fig.~6(a) from Ref.~\cite{leite2021}).
Finally, since the quadratic boson term \eqref{H42} is Hermitian,
the spin-wave excitations \eqref{omega-b} are real quantities, i.e.,
the decay rates of the spin-wave excitations
vanish, in contrast with the behaviour of the ML excitations, which
display a quite small decay rate [see Figs.~\ref{figEspectro}(a) and (d)].
Importantly, for each momentum within the first BZ,
the excitation energy associated with the upper band of the SL case
is lower than the corresponding value of the ML case,
a feature also found for the lower bands
[see Figs.~\ref{figEspectro}(a) and (d)]. Therefore,
the SL excitations are indeed the lowest-energy excitations
that characterize the flat-band FM
phase of the THM \eqref{eqHH}, a feature that contrasts
with the square lattice correlated Z$_2$ topological insulator
\cite{doretto2015flat}, whose elementary excitations of the
corresponding flat-band FM phase are of the ML type.
In addition to the THM \eqref{eqHH} with
homogeneous on-site repulsion energies $U_A = U_B = U$, the spin-wave spectrum
with a sublattice dependent on-site energy $U_a$ was also determined.
We show the spin-wave spectrum \eqref{omega-b}
for the nearly flat-band limit \eqref{optimal-par} and
$U_B = 0.8\, U_A = 0.8\, U$ and
$U_B = 0.6\, U_A = 0.6\, U$ in Figs.~\ref{figEspectro}(e) and (f), respectively.
Similarly to the ML excitations [Figs.~\ref{figEspectro}(b) and (c)],
we find that a finite $\Delta U = U_A - U_B$ modifies the spin-wave spectrum
as compare to the homogeneous case $U_A = U_B = U$.
In particular, it breaks the symmetry at the $K$ and $K'$ points
displayed by the spin-wave spectrum in the homogeneous case.
Such an asymmetry at the $K$ and $K'$ points of the
spin-wave spectrum as $\Delta U$ increases was also found for
the correlated Chern insulator \cite{leite2021}
and it might be related to the fact that a Hubbard
term with $U_A \not= U_B$ breaks inversion symmetry.
Notice that, as the diference $\Delta U$ increases:
The energies of the spin-wave excitations decrease;
the energy gap between the lower and upper bands at the $K$ point decreases,
\begin{align}
\Delta^{(K)} &= 1.82 \times 10^{-2}\, U
\quad {\rm for} \quad \Delta U = 0.2\,U,
\nonumber \\
\Delta^{(K)} &= 1.34 \times 10^{-2}\, U
\quad {\rm for} \quad \Delta U = 0.4\,U,
\nonumber
\end{align}
while the one at the $K'$ point increases,
\begin{align}
\Delta^{(K')} &= 7.11 \times 10^{-2}\, U
\quad {\rm for} \quad \Delta U = 0.2\,U,
\nonumber \\
\Delta^{(K')} &= 9.26 \times 10^{-2}\, U
\quad {\rm for} \quad \Delta U = 0.4\,U.
\nonumber
\end{align}
For $U_B > U_A$ (not shown here), similar features are observed, but now the energy
gap at the $K$ point increases instead of the one at $K'$ point.
Again, similarly to the homogeneous case, the energies of spin-wave
spectrum of the SL case are lower than the corresponding ones of
the ML case for a fixed $\Delta U$.
We also investigate how the spin-wave spectrum \eqref{omega-b}
modifies as the THM \eqref{eqHH} is tuned away from
the nearly flat band limit \eqref{optimal-par}, once
the next-nearest-neighbhor hopping amplitude $t_2$ and
the phase $\phi$ are modified while the on-site Hubbard energies $U_a$
are kept fixed. As mentioned in Sec.~\ref{sec:Diagonalization}
(see also Fig.~2 from Ref.~\cite{leite2021}),
the noninteracting electronic bands $c$ [Eq.~\eqref{eq:omega}]
become more dispersive as the model \eqref{eqHH} moves away from the
nearly flat band limit \eqref{optimal-par}.
In the following, we describe the effects on the spin-wave spectrum
only due to variations of the parameters $t_2$ and $\phi$.
We refer the reader to Appendix~\ref{sec:mass-term} for a similar
discussion concerning the effects of a finite staggered on-site energy term
in the Hamiltonian \eqref{eqHH}.
In Fig.~\ref{fig:espectro-phi}(a), it is shown
the spin-wave spectrum \eqref{omega-b} for $\phi = 0.656$, $0.7$, and $0.8$,
hopping amplitude $t_2$ determined by $\cos(\phi) = t_1/(4 t_2)$,
and on-site repulsion energies $U_A = U_B = U$.
We find that the spin-wave spectrum
(in units of the on-site Hubbard energy $U$) for $\phi = 0.7$ and $0.8$ is
rather similar to the one for the nearly-flat band limit \eqref{optimal-par},
which corresponds to $\phi = 0.656$.
As the parameter $\phi$ increases, one sees that only
the excitation energies of the lower band in the vicinity of the
$\Gamma$ point increases while the rest of the spectrum remains almost
the same as compared with the one obtained for the nearly flat band limit
\eqref{optimal-par}.
Indeed, for $\phi=0.8$, the energy gap of the lower band moves from
the $\Gamma$ to the $M_i$ points.
On the other hand, as shown in Fig.~\ref{fig:espectro-phi}(b), a
decreasing of the parameter $\phi$ from $\phi = 0.656$ yields:
A decreasing of the excitation energies of the lower band in the
vicinity of the $\Gamma$ point;
an increasing of the excitation energies of the upper band around the same point;
and a small decreasing in the energy gap between the lower and upper
bands at the $K$ and $K'$ points.
Indeed, one finds that such energy gap
$\Delta^{(K)} = \Omega_{+,K} - \Omega_{-,K} = 3.81 \times 10^{-2}\, U$ ($\phi =0.4$),
$4.35 \times 10^{-2}\, U$ ($\phi =0.5$),
$4.96 \times 10^{-2}\, U$ ($\phi =0.656$),
$5.19 \times 10^{-2}\, U$ ($\phi =0.7$), and
$5.34 \times 10^{-2}\, U$ ($\phi =0.8$).
We believe that such rather small modifications in the spin-wave
spectrum as the model \eqref{eqHH} is tuned away from the nearly flat
band limit \eqref{optimal-par} might be due to the fact that
the main effects associated with the dispersion of the lower
noninteracting band $c$, that are encoded in the kinetic coefficients
\eqref{eq:omegaBar}, are not properly taken into account by the
bosonization scheme.
\begin{table}[t]
\centering
\caption{Chern numbers of the lower spin-wave bands \eqref{omega-b}
for both the ML ($C_{ML}$) and the SL ($C_{SL}$) excitations
at the nearly flat band limit \eqref{optimal-par}.}
\begin{tabular}{lcccc}
\hline\hline
& & $U_A = U_B = U$ & & $U_B = 0.8\, U_A = 0.8\, U$ \\
\hline
$C_{ML}$ &\quad \quad\quad\quad & $\pm 0.29$ & \quad\quad\quad & $\pm 0.18$ \\
$C_{SL}$ &\quad \quad\quad\quad & $\pm 1.17$ & \quad\quad\quad & $\pm 1.08$ \\
\hline\hline
\end{tabular}
\label{tb-chern}
\end{table}
Concerning the topological properties of the spin-wave bands,
we find some evidences that the spin-wave bands for the SL excitations
\eqref{eq:projS2} might be topologically nontrivial. In Table~\ref{tb-chern},
we present the Chern numbers $C_{SL}$ of the lower spin-wave bands \eqref{omega-b}
for the SL excitations shown in Figs.~\ref{figEspectro}(d) and (e)
[the corresponding Chern numbers $C_{ML}$ for the ML excitations
shown in Figs.~\ref{figEspectro}(a) and (b) are also included for comparison].
Such a feature contrasts with the one found for the corresponding
correlated Chern insulator on a honeycomb lattice
\cite{gu2019itinerant,leite2021}, whose spin-wave
bands are topologically trivial in the completely flat band limit.
For more details about the topological properties of the spin-wave bands,
we refer the reader to Appendix~\ref{sec:chernNumber2}.
\section{Summary}
\label{sec:summary}
In summary, in this paper we studied the flat-band FM
phase of a correlated Z$_2$ topological insulator on a honeycomb
lattice described by a topological Hubbard model, whose noninteracting
limit is given by a generalization of the spinless Haldane model
\cite{haldane1988model}.
Such a study complements our previous one \cite{leite2021} concerning
the flat-band FM phase of a correlated Chern insulator
described by a Haldane-Hubbard model.
We considered the model at $1/4$ filling of its noninteracting limit
and study the system within a bosonization scheme for flat-band
correlated Z$_2$ topological insulators.
Our main result [Figs.~\ref{figEspectro}(d)] is the
calculation of the spin-wave excitation
spectrum for the nearly flat band limit \eqref{optimal-par} of the
noninteracting lower bands and equal on-site repulsion
energies associated with the sublattices $A$ and $B$
($U_A =U_B = U$).
Moreover, we also determined the spin-wave spectrum
when an offset in the on-site repulsion energies is
introduced ($U_A \not= U_B$), and when the width of the lower
noninteracting bands increases due to changes in the parameters of the
noninteracting electronic Hamiltonian.
Differently from the correlated Chern insulator \cite{leite2021},
for the correlated topological insulator \eqref{eqHH}, one can,
in principle, define two sets of boson
operators $b_0$ and $b_1$ as done in Eq.~\eqref{eq:bosons} considering
both the spin-flip excitations \eqref{eq:projS1}, that changes the
sublattice index (ML excitations),
and the spin-flip excitations \eqref{eq:projS2}, that preserves the
sublattice index (SL excitations).
We found that the spin-wave spectrum for both ML and SL
excitations are gapped and have two branches, with an energy gap
between the lower and upper bands at the $K$ and $K'$ points of the
first BZ. Such features are in contrast with the ones found for
the correlated Chern insulator on a honeycomb lattice
\cite{leite2021}, whose spin-wave spectrum has a
Goldstone mode at the center of the BZ
($\Gamma$ point) and Dirac points at the $K$ and $K'$ points.
Mostly important, the lowest-energy excitations are the SL ones, a
feature that is distinct from the one found for the square lattice
$\pi$-flux model \cite{doretto2015flat}, whose flat-band FM
phase is characterized by ML excitations: while both correlated Chern
insulators on the square \cite{doretto2015flat} and honeycomb
\cite{leite2021} lattices are characterize by the SL excitations, such
a common feature seems to be not shared by the corresponding
topological insulators.
Finally, our findings indicated that the spin-wave
bands for the SL excitations might be topologically nontrivial, even
in the completely flat band limit, a feature that also contrasts with
the behaviour of the corresponding correlated Chern insulator \cite{gu2019itinerant}.
\acknowledgments
We thank E. Miranda for helpful discussions and
L.S.G.L. kindly acknowledges the financial support of the Conselho
Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico (CNPq) under
the Grant No.~162323/2017-4.
|
\section{\label{sec:level1}First-level heading:\protect\\ The line
{\label{sec:level1}}
\section{Introduction}
In recent years quantum entanglement has attracted more and more
attentions because it is not only a key resource in quantum
information theory but also an essential concept in condensed matter
physics \cite{book}. It is well known that a great deal of
properties of quantum many-body systems are closely related with the
entanglement between particles, and the study of which is very
important for both the system composed of distinguishable particles
and the system composed of identical particles. However, unlike the
system of distinguishable particles, for which there are various
quantities to define and measure entanglement, the definition and
quantification of entanglement between identical particles is still
not very clear. Fortunately, the von Neumann entropy introduced by
reduced density matrix is a good quantity to describe quantum
entanglement for systems consisting of identical particles
\cite{Law, You, Zhou, pra66042113,prb63085311,pra64022303,Long}. The
entanglement entropy of two identical bosons or fermions in abstract
wave functions has been studied by the Schmidt decomposition
\cite{You,Long,prb63085311,pra64022303}, while the entanglement
between two identical interacting trapped atoms in a continuous
system was studied in Refs.\cite{Law, Zhou, pra66042113}. It is
shown that the statistical properties of identical particles play
very important roles in their entanglement behaviors.
As a natural generalization of boson and fermion, anyon was proposed
to describe the particle obeying fractional statistics and has been
a subject of great interest in the past decades
\cite{book2,laughlin1,laughlin2, prl661529,prl521583,2D1,2D2}.
Although the concept of anyon arises originally from two-dimensional
systems \cite{prl521583,2D1,2D2} related to fractional quantum Hall
effect (FQH) and high-temperature superconductivity
\cite{book2,laughlin1,laughlin2}, the study of 1D anyonic model has
attracted great theoretical interest
\cite{prl661529,Kundu,1D1,1D2,Wang,Girardeau,Patu,del
Campo,junction,Santachiara08,Guan2,Guan3,Guan1,Hao1,Hao2,Calabrese09}.
Motivated by possible experiments with cold atoms to simulate the
creation and manipulation of anyons
\cite{Paredes_anyon,Aguado,Jiang}, and the possibility of performing
topological quantum computation \cite{tqc}, the properties of 1D
anyons are under current research focus. Several studies have been
devoted to 1D anyons with a $\delta$-function potential interaction
\cite{Kundu,Guan1,Guan2,Guan3,Hao1,1D1} and the limiting cases of
hard-core anyons \cite{1D2,Santachiara08,Girardeau,Patu,del
Campo,Calabrese09,Hao2}. Particularly, the 1D interacting anyon
model is exactly solvable by the Bethe-ansatz method as firstly
found by Kundu \cite{Kundu}. Despite the intensive studies, the
entanglement properties in anyonic systems are rarely studied except
for the model in the hard-core limit \cite{1D2}.
In this work, we investigate the entanglement properties of a
continuous system composed of $N$ anyonic particles with repulsive
contact interaction on a ring of length $L$ by calculating the von
Neumann entropy of the single-particle reduced density matrix. In
general, it is hard to calculate the von Neumann entropy of a
continuous many-body system analytically. So far, most of the
studies focus on the two-particle system \cite{Law, Zhou,
pra66042113}. The integrability of the exactly solvable many-body
system provides us the possibility to study the entanglement
properties of a many-body system analytically. Based on the Bethe
ansatz solution of the interacting anyonic model \cite{Kundu,Guan1},
we evaluate the one-particle reduced density matrix firstly and then
obtain the von Neumann entropy for the arbitrary statistical
parameter in the full interacting regime.
This paper is organized as follows. In section II, we first give a
brief introduction to the model and formulate the method. In section
III, we first consider the Bose limit and focus on the effect of
interacting strength on the von Neumann entropy. In section IV, we
deal with the general anyonic case and discuss the effect of
statistical parameter $\kappa$ on the entanglement properties by
calculating the von Neumann entropy for different $\kappa$. A brief
summary is given in Section V.
\section{models and methods}
The second quantized Hamiltonian for the one-dimensional anyonic
system is formulated as
\begin{eqnarray}
\mathcal{H}_A &=&-\frac{\hbar ^2}{2m}\int_0^Ldx \Psi _A^{\dagger }
\frac{\partial^2 }{\partial x^2}\Psi _A \nonumber \\
&&+\frac {g_{1D}}{2} \int_0^Ldx\Psi _A^{\dagger }\left( x\right)
\Psi _A^{\dagger }\left( x\right) \Psi _A\left( x\right) \Psi
_A\left( x\right), \label{H}
\end{eqnarray}
in which $m$ is the mass of anyons and $g_{1D}$ denotes interacting
strength between anyons \cite{Kundu,Guan1}. Here the field operators
obey anyonic commutation relations $ \Psi ^{\dag }_{A}(x_{1})\Psi
^{\dag }_{A}(x_{2})= e^{i\kappa\pi\epsilon (x_{1}-x_{2})}\Psi ^{\dag
}_{A}(x_{2})\Psi ^{\dag}_{A}(x_{1})$, and $ \Psi _{A}(x_{1})\Psi
^{\dag }_{A}(x_{2})=\delta (x_{1}-x_{2})+e^{-i\kappa \pi \epsilon
(x_{1}-x_{2})}\Psi ^{\dag }_{A}(x_{2})\Psi _{A}(x_{1})$ with
$\epsilon(x-y)=1, -1, 0$ for $x>y, x<y$, and $x=y$ respectively. The
model (\ref{H}) is known to be exactly solvable \cite{Kundu}. The
parameter $\kappa$ characterizes the statistical property of the
anyonic system with $\kappa =0$ and $\kappa =1.0$ corresponding to
Bose statistics and Fermi statistics respectively. The dependence of
entanglement properties on both the interaction constant $g_{1D}$
and statistical parameter $\kappa$ ($0\leq\kappa\leq1$) will be
considered.
The eigenvalue problem of Hamiltonian (\ref{H}) can be reduced to
the quantum mechanical problem of $N$ anyons with $\delta$
interaction \cite{Kundu,Guan1}
\begin{eqnarray}
H\psi(x_{1},...,x_{N})=E\psi(x_{1},...,x_{N})
\end{eqnarray}
with
\begin{eqnarray}
H=-\sum\limits_{i=1}^{N}\frac{\partial ^{2}}{\partial x_{i}^{2}}
+2c\sum\limits_{1\leq i\leq j\leq N}^{N}\delta
(x_{i}-x_{j}),\label{H2}
\end{eqnarray}
where the natural unit is used and $c=mg_{1D}/\hbar^2$ ($c>0$) is a
dimensionless interaction constant.
In terms of the exact ground-state wavefunction $\psi$, the
single-particle reduced density matrix is defined as
\begin{equation}
\hat{\rho}_1 = \mathrm{Tr}_{2,3,...,N}|\psi\rangle \langle \psi|,
\end{equation}
where the trace means to do integrations over all the position
coordinates except one of them. The single-particle entanglement is
quantified by the von Neumann entropy as
\begin{equation}
S =-\mathrm{Tr} (\hat{\rho}_1 \log _{2} \hat{\rho}_1) ,
\end{equation}%
where $\hat{\rho}$ is the one-particle reduced density matrix with
the normalization condition Tr$\hat{\rho} =1$.
In coordinate representation, the one-particle reduced density
matrix is expressed as
\begin{eqnarray}
& & \rho_1(x,x^{\prime}) \nonumber \\
&=& \frac{\int_{0}^{L}dx_{2}...dx_{N}
[\psi^{\ast}(x,x_{2},...x_{N})\psi(x^{\prime},x_{2},...,x_{N})]}
{\int_{0}^{L}dx_{1}...dx_{N}|\psi(x_{1},x_{2},...,x_{N})|^{2}}.~~~~
\label{density matrix}
\end{eqnarray}
Obviously the (\ref{density matrix}) is Hermitian, i.e., we have
$\rho_1^{\ast}(x,x^{\prime})=\rho_1(x^{\prime},x)$. The eigen-
equation of the one-particle reduced density matrix $(\ref{density
matrix})$ is
\begin{eqnarray}
\int_{0}^{L}dx^{\prime}\rho_1(x,x^{\prime})\phi_{\eta}(x^{\prime})=
\lambda_{\eta}\phi_{\eta}(x),\label{eigenquation}
\end{eqnarray}
where $\lambda_{\eta}$ are the occupation numbers for natural
orbitals $\phi_{\eta}(x)$ which form a complete and orthonormal set
of functions. The one-particle reduced density matrix is diagonal in
the basis of natural orbitals and
$\sum_{\eta=1}^{\infty}\lambda_{\eta}=1$. In terms of eigenvalues
$\lambda_{\eta}$ and the eigenfunctions $\phi_{\eta}(x)$, we can
rewrite
\[
\rho_1(x,x^{\prime})=\sum_{\eta=1}^{\infty}\lambda_{\eta}
\phi_{\eta}(x) \phi_{\eta}^{\ast}(x^{\prime})
\]
and the von Neumann entropy then reads
\begin{equation}
S=-\sum_{\eta=1}^{\infty}\lambda_{\eta}\log_{2}\lambda_{\eta} .
\end{equation}
From the above scheme, we can understand the difficulties which
prevent us from studying the entanglement properties of a many-body
system. First, the calculation of the ground-state wavefunction for
a many-body system is generally difficult except for some exactly
solvable systems. Furthermore, even though the exact many-body wave
function is constructed, the calculation of the reduced density
matrix for a large system remains a difficult task due to the time
consuming to calculate multidimensional integrals.
{\label{sec:level1}}
\section{The dependence of entanglement on the interaction strength}
In this section, we shall
|
focus on the Bose limit and study the
dependence of entanglement on the interaction strength. The effect
of fractional statistics will be discussed in the next section. In
the Bose limit ($\kappa=0$), the model is reduced to the well-known
Lieb-Linger model \cite{1963}. In this case the field operators
$\Psi_A ^{\dag }(x)$ and $\Psi_A (x)$ obey boson commutation
relations and the wavefunction $\psi(x_{1},...,x_{N})$ satisfies
exchange symmetry. In the scheme of Bethe-ansatz method \cite{1963},
the many-particle wave function can be formulated as
\begin{eqnarray}
\psi(x_{1},...,x_{N}) &=&\sum_{Q}\theta(x_{q_{N}}-x_{q_{N-1}})
...\theta(x_{q_{2}}-x_{q_{1}})\nonumber\\
&&\times
\varphi_{Q}(x_{q_{1}},x_{q_{2}},...,x_{q_{N}}),\label{wavefunction}
\end{eqnarray}
where $Q$ labels the region $0\leq x_{q_{1}}\leq x_{q_{2}}...\leq
x_{q_{N}}\leq L$, in which $q_{1},q_{2},...,q_{N}$ is one of the
permutations of $1,2,...,N$, $\Sigma_{Q}$ sums over all permutations
and $\theta(x-y)$ is the step function. Here
$\varphi_{Q}(x_{q_{1}},x_{q_{2}},...,x_{q_{N}})$ takes the
Bethe-ansatz type
\begin{eqnarray}
\varphi_{Q}(x_{q_{1}},...,x_{q_{N}})=\sum\limits_{P}[A_{p_{1}p_{2}...p_{N}}\exp(i\Sigma_{j}(k_{p_{j}}x_{q_{j}}))]\label{wavefunction1}
\end{eqnarray}
with
$A_{p_{1}p_{2}...p_{N}}=\varepsilon_{P}\prod_{j<l}^{N}(ik_{p_{l}}-ik_{p_{j}}+c)$
and ${k_{p_{j}}}$ is a set of quasi-momentums determined by the
Bethe-ansatz equations. Here $p_{1},p_{2},...,p_{N}$ means one of
permutations of $1,2,...,N$, and $\varepsilon_{P}$ denotes a $+$
$(-)$ sign associated with even (odd) permutation of $P$. Using the
periodical boundary condition, we can obtain Bethe-ansatz equations
\cite{1963} whose logarithmic forms are formulated as
\begin{eqnarray}
k_{j}L=2n_{j}\pi-\sum\limits_{l=1(l\neq j)}^{N}2\arctan
\Big(\frac{k_{j}-k_{l}}{c}\Big),\label{bae}
\end{eqnarray}
where ${n_{j}}$ is a set of integers to determine the eigenstates
and for the ground state $n_{j}=(N+1)/2-j$ $(1\leq j\leq N)$. The
energy of the system is $E=\sum_{j=1}^{N}k_{j}^{2}$ and the total
momentum is $k=\sum_{j=1}^{N}k_{j}$.
\begin{figure}
\includegraphics[height=6cm,width=\linewidth]{fig1}
\caption{\label{fig:epsart} The occupation numbers of the
interacting Bose system for different interaction strength $c$.}
\end{figure}
\begin{figure}
\includegraphics[height=7cm,width=\linewidth]{fig2}
\caption{\label{fig:epsart} The entanglement entropy $S$ versus the
logarithm of the repulsive interaction constant $c$ for a system
with $4$ particles. One can see that $S$ monotonically and smoothly
increases from zero to about 1.846 with the growth of $c$ from zero
to the infinity limit.}
\end{figure}
By numerically solving the Bethe-ansatz equations (\ref{bae}), we
can obtain the exact ground-state wavefunction, and then the
one-particle reduced density matrix (\ref{density matrix}). Solving
the eigenvalue problem (\ref{eigenquation}) numerically, we can get
a series of $\lambda_i$ and then determine the entanglement entropy.
For simplicity, we shall discuss the many-body system with $N=4$ in
the following context. In Fig. 1, we show the occupation numbers for
the interacting Bose system composed of four identical bosons versus
different interaction strength $c$. At $c=0$, only the lowest nature
orbital is occupied which means all the bosons condensate to the
ground state. The occupation number of the lowest natural orbital
$\lambda_1$ decreases with the increase of the interaction strength,
accompanying with the increase of occupation numbers of higher
natural orbitals. Our numerical results for the dependence of
ground-state entanglement $S$ on the interaction strength are shown
in Fig. 2.
In order to exhibit the change with $c$ in a wide range, the
logarithm coordinate for c is used in this figure. It is shown that
the entanglement entropy changes monotonically with the change of
interaction constant.
When there is no interaction between bosons ($c=0$), no entanglement
exists in the system. Along with the growth of interaction constant,
the entanglement entropy $S$ increases slowly in the weakly
interacting regime, and then goes sharply to $1.74$. When the
interaction gets close to the strongly interacting regime, the
entanglement entropy $S$ slowly approaches to about $1.846$, which
is smaller than $2$. Our result is accordant with \cite{You,Zhou},
in which we note the entanglement entropy of $N$ identical
boson-particle system ranges from $S=0$ for free-boson state to a
maximum $S$ in the infinitely repulsive limit which is smaller than
$S=\log_2N$. This can be understood as follows: when there is no
interaction between particles, no correlation exists between bosons
and the occupation number of lowest-energy state is $N$; while in
the strong interaction limit ($c\rightarrow\infty$), particles will
be prevented from occupying the same state and higher-energy states
should be occupied.
{\label{sec:level1}}
\section{The dependence of entanglement on the anyonic parameter}
Now we turn to the dependence of ground state entanglement entropy
on the statistics. For anyonic system the many-body wave function
shall satisfy the generalized symmetry
\begin{eqnarray}
\psi(...,x_{i},...,x_{j},...) =
e^{-i\theta}\psi(...,x_{j},...,x_{i},...),
\end{eqnarray}
where the anyonic phase
\[
\theta=\kappa
\pi\left[\sum_{k=i+1}^{j}\epsilon(x_{i}-x_{k})-\sum_{k=i+1}^{j-1}\epsilon(x_{j}-x_{k})\right].
\]
for $i<j$. Considering the symmetry under coordinates reflection, we
confine $\kappa$ to $[0,1]$ in the present paper. The wavefunction
of anyons takes a similar form with that of bosons ($\kappa=0$)
\cite{Kundu,Guan1}
\begin{eqnarray}
\psi_{A}(x_{1},...,x_{N})&=&\sum_{Q}\theta(x_{q_{N}}-x_{q_{N-1}})
...\theta(x_{q_{2}}-x_{q_{1}})\nonumber\\
&&\times\phi_{A}\varphi_{Q}(x_{q_{1}},x_{q_{2}},...,x_{q_{N}}),\label{anyonicwf}
\end{eqnarray}
where $\phi_{A}$ is an additional anyonic phase part
\begin{eqnarray}
\phi_{A}=\exp(-i\frac{\kappa
\pi}{2}\sum_{q_i<q_j}\epsilon(x_{q_{i}}-x_{q_{j}}))
\end{eqnarray}
and $\varphi_{Q}(x_{q_{1}},x_{q_{2}},...,x_{q_{N}})$ has the same
form as that of Lieb-Liniger Bose gas (\ref{wavefunction1}) except
that now we have
$A_{p_{1}p_{2}...p_{N}}=\varepsilon_{P}\prod_{j<l}^{N}(ik_{p_{l}}-ik_{p_{j}}+c^{\prime
})$ with
\begin{eqnarray}
c^{\prime }=c/\cos(\kappa/2)\label{effective constant}.
\end{eqnarray}
Similarly, the quasi-momenta $k_i$ is determined by the Bethe ansatz
equations
\begin{eqnarray}
k_{j}L=2n_{j}\pi-\sum\limits_{l=1(l\neq j)}^{N}2\arctan
\Big(\frac{k_{j}-k_{l}}{c^{\prime}}\Big),\label{bae1}
\end{eqnarray}
under the twisted boundary condition $
\psi_{A}(0,x_{2},...,x_{N})=e^{i\kappa
\pi(N-1)}\psi_{A}(L,x_{2},...,x_{N}) $. The Bethe-ansatz equations
have the same form as that for Lieb-Liniger Boson gas (\ref{bae}) if
we replace $c$ with the renormalized interaction constant $c^{\prime
}$. According to (\ref{effective constant}) the effective
interaction between anyons depends on the statistical parameter
$\kappa$, which increases with the increase of $\kappa$ and
approaches $\infty$ in the fermionic limit ($\kappa \rightarrow
1.0$).
\begin{figure}
\includegraphics[height=6cm,width=\linewidth]{fig3}
\caption{\label{fig:epsart}The occupation numbers for different
anyonic parameter $\kappa$ with $c=1$.}
\end{figure}
\begin{figure}
\includegraphics[height=8cm,width=\linewidth]{fig4}
\caption{\label{fig:epsart}(Color Online) The entanglement entropy
versus the statistics parameter $\kappa$ for the anyonic system with
$c=1$, $c=10$ and $c=100$, respectively.}
\end{figure}
With the same procedure as presented in the above section we obtain
the one-particle entanglement entropy for various c. As a concrete
example, in Figure 3 we display the change of occupation numbers
with different statistical parameters at a fixed interaction
strength $c=1$. The dependence of entanglement entropy on the
statistical parameters $\kappa$ is displayed in Figure 4 for $c=1$,
10 and 100. It is shown that the entanglement entropy increases with
the increasing of anyonic parameter $\kappa$ and interaction
strength $c$. For the case of $c=10$, as $\kappa=0$ the system
reduces to Lieb-L
|
j_{12} \\
j_3 & j & j_{23}
\end{array}\right\}^S=
\sum_{j_{13}}(-1)^{\Theta^{R}}
\left\{\begin{array}{ccc}
j_1 & j_3 & j_{13} \\
j_2 & j & j_{23}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_2 & j_1 & j_{12} \\
j_3 & j & j_{13}
\end{array}\right\}^S\,.
\end{equation}
where
\begin{eqnarray}
& &\Theta^{R}=
[j_1+j_2-j_{12}]+[j_2+j_3-j_{23}]+[j_2+j_{13}-j]+[j_1+j_2+j_3+j] \nonumber\\
& &\qquad+2j_{12}I_{12}+2j_{23}I_{23}+2j_{13}I_{13}+2jI_{123}\,.
\end{eqnarray}
The sign factor in this sum rule is more involved
than the corresponding factor in the ordinary rotation case. However,
they almost coincide in several particular cases. For example, when
the values of the super-spins in the left-hand side of eq. (3.6) are
such that their sums in all the triangles $(j_1,j_2,j_{12})$,
$(j_2,j_3,j_{23})$, $(j_1,j_{23},j)$ and $(j_{12},j_3,j)$ are
integer, Racah sum rule takes the form
\begin{equation}
\left\{\begin{array}{ccc}
j_1 & j_2 & j_{12} \\
j_3 & j & j_{23}
\end{array}\right\}^S=
\sum_{j_{13}}(-1)^{[j_{12}+j_{23}+j_{13}]+2j_{13}I_{13}}
\left\{\begin{array}{ccc}
j_1 & j_3 & j_{13} \\
j_2 & j & j_{23}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_2 & j_1 & j_{12} \\
j_3 & j & j_{13}
\end{array}\right\}^S\,.
\end{equation}
which means that the terms in the right-hand side, for which the
triangle sums with the participation of $j_{13}$ are also integer,
have the sign factor $(-1)^{j_{12}+j_{23}+j_{13}}$, identical to
the sign factor in the rotation case.
Another particular case of Racah sum rule reads
\begin{equation}
\sum_{j_{12}}(-1)^{2(2(j_1+j_2)(j_{12}+j'_{12})+j_{12})}
\left\{ \begin{array}{ccc}
j_1 & j_2 & j_{12}\\
j_1 & j_2 & j'_{12}
\end{array}\right\}^S
=1~.
\end{equation}
\setcounter{equation}{0}
\section{Biedenharn-Elliott Identity}
\hspace*{10mm}In order to derive the Biedenharn-Elliott identity we will
consider the recoupling of four super-spins $j_1, j_2, j_3$ and $j_4$.
Using the same technique as in the derivation of eq. (3.2) we can express
the vector $\mid ((j_2 \lambda_2;j_3 \lambda_3)j_{23} \lambda_{23};
(j_1 \lambda_1;j_4 \lambda_4)j_{14} \lambda_{14}) j \lambda;\ell m\rangle$
first, as a result of two successive recouplings of three super-spins, and
second, as a result of three recouplings summing over the intermediate states
containing $j_{124}$. The calculations being completely analogous to the ones
presented in the previous section, we will omit the intermediate (~rather
tedious) calculations and will present only the results: the
Biedenharn-Elliott identity for the super-rotation
recoupling coefficients reads
\begin{eqnarray}
&&[((j_2 \lambda_2;j_3 \lambda_3)
j_{23} \lambda_{23};(j_1 \lambda_1;j_4 \lambda_4)
j_{14} \lambda_{14}) j \lambda
\!\mid(((j_2 \lambda_2;j_3 \lambda_3)
j_{23} \lambda_{23};j_1 \lambda_1)j_{123} \lambda_{123};
j_4 \lambda_4) j \lambda] \nonumber\\
&&\times[((j_1 \lambda_1;(j_2 \lambda_2;j_3 \lambda_3)
j_{23} \lambda_{23})j_{123} \lambda_{123};
j_4 \lambda_4) j \lambda
\!\mid(((j_1 \lambda_1;j_2 \lambda_2)
j_{12} \lambda_{12};j_3 \lambda_3)j_{123} \lambda_{123};
j_4 \lambda_4) j \lambda]\nonumber\\
&&=\sum_{j_{124}}(-1)^{\Theta^{BE}_{RC}}\nonumber\\
&&\times[((j_1 \lambda_1;
j_4 \lambda_4)j_{14} \lambda_{14};(j_2 \lambda_2;j_3 \lambda_3)
j_{23} \lambda_{23}) j \lambda
\!\mid(((j_1 \lambda_1;j_4 \lambda_4)j_{14} \lambda_{14};
j_2 \lambda_2)j_{124} \lambda_{124};j_3 \lambda_3) j \lambda]\nonumber\\
&&\times[((j_2 \lambda_2;(j_1 \lambda_1;j_4 \lambda_4)
j_{14} \lambda_{14})j_{124} \lambda_{124};j_3 \lambda_3) j \lambda
\!\mid(((j_2 \lambda_2;j_1 \lambda_1)j_{12} \lambda_{12};
j_4 \lambda_4)j_{124} \lambda_{124};j_3 \lambda_3)j \lambda]\nonumber\\
&&\times[(j_3 \lambda_3;((j_1 \lambda_1;j_2 \lambda_2)j_{12} \lambda_{12};
j_4 \lambda_4)j_{124} \lambda_{124})j \lambda
\!\mid((j_3 \lambda_3;(j_1 \lambda_1;j_2 \lambda_2)
j_{12} \lambda_{12})j_{123} \lambda_{123};
j_4 \lambda_4) j \lambda].\nonumber\\
&&\mbox{}
\end{eqnarray}
It is interesting to note that the phase $\Theta^{BE}_{RC}$ is
independent of the parities
\begin{eqnarray}
&&\Theta^{BE}_{RC}=I_{23}(I_{1234}+I_{123}+I_{14})+
I_{124}(I_{1234}+I_{14}+1)+I_{12}(I_{123}+1)+I_{1234}I_{14}\nonumber\\
&&+[j_1+j_2-j_{12}]+[j_{12}+j_3-j_{123}]+[j_1+j_{23}-j_{123}]\nonumber\\
&&+[j_{14}+j_2-j_{124}]+[j_{14}+j_{23}-j]+[j_{124}+j_3-j]\,.
\end{eqnarray}
As a result, the Biedenharn-Elliott identities for the $\lambda$-dependent
$S6-(j,\lambda)$ and $\lambda$-indepen\-dent super-rotation $
|
S6-j$ symbols
take the same form
\begin{eqnarray}
&&\left\{\begin{array}{ccc}
j_1 & j_2 & j_{12} \\
j_3 & j_{123} & j_{23}
\end{array}\right\}^S
\left\{\begin{array}{ccc}
j_{23} & j_1 & j_{123} \\
j_4 & j & j_{14}
\end{array}\right\}^S \nonumber\\
&&=\sum_{j_{124}}(-1)^{\Theta^{BE}}
\left\{\begin{array}{ccc}
j_2 & j_1 & j_{12} \\
j_4 & j_{124} & j_{14}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_3 & j_{12} & j_{123} \\
j_4 & j & j_{124}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_{14} & j_2 & j_{124}\\
j_3 & j & j_{23}
\end{array}\right\}^S\,.
\end{eqnarray}
where
\begin{eqnarray}
&&\Theta^{BE}=I_{12}(I_{1234}+I_{124}+I_{123}+1)+I_{14}I_{23}\nonumber\\
&&\qquad+[j_1+j_2-j_{12}]+[j_{12}+j_3-j_{123}]+[j_1+j_{23}-j_{123}]\nonumber\\
&&\qquad+[j_{14}+j_2-j_{124}]+[j_{14}+j_{23}-j]+[j_{124}+j_3-j]\nonumber\\
&&\qquad+[j_1+j_2+j_3+j_{123}]+[j_1+j_4+j_{23}+j]+
[j_1+j_2+j_4+j_{124}]\nonumber\\
&&\qquad+[j_3+j_4+j_{12}+j]+[j_2+j_3+j_{14}+j]\,.
\end{eqnarray}
Here, as well as in the case of the Racah sum rule, we see that the structure
of the Biedenharn-Elliott identity for the super-rotation $S6-j$ symbols
is exactly the same as the structure of the Biedenharn-Elliott identity [8,9]
for the rotation $6-j$ symbols. Only the sign factor $\Theta^{BE}$
is somewhat more involved in the supersymmetric case. This factor
almost coincides with the corresponding one in the rotation case when the
values of the super-spins in the left-hand side of eq. (4.3) are
such that their sums in all the triangles $(j_1,j_2,j_{12})$,
$(j_2,j_3,j_{23})$, $(j_1,j_{23},j_{123})$, $(j_{12},j_3,j_{123})$,
$(j_1,j_4,j_{14})$, $(j_{123},j_4,j)$ and $(j_{14},j_{23},j)$ are
integer
\begin{eqnarray}
&&\left\{\begin{array}{ccc}
j_1 & j_2 & j_{12} \\
j_3 & j_{123} & j_{23}
\end{array}\right\}^S
\left\{\begin{array}{ccc}
j_{23} & j_1 & j_{123} \\
j_4 & j & j_{14}
\end{array}\right\}^S \nonumber\\
&&=\sum_{j_{124}}(-1)^{[j_1+j_2+j_3+j_4+j_{12}+j_{23}+j_{14}+
j_{123}+j_{124}+j+\frac{1}{2}]} \nonumber\\
&&\qquad\times\left\{\begin{array}{ccc}
j_2 & j_1 & j_{12} \\
j_4 & j_{124} & j_{14}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_3 & j_{12} & j_{123} \\
j_4 & j & j_{124}
\end{array}\right\}^S~
\left\{\begin{array}{ccc}
j_{14} & j_2 & j_{124}\\
j_3 & j & j_{23}
\end{array}\right\}^S\,.
\end{eqnarray}
\section{Conclusion}
\hspace*{10mm}In this paper we continued the development of the
Racah-Wigner calculus for the super-rotation $osp(1|2)$
superalgebra. In particular, we have derived the Racah sum rule and the
Biedenharn-Elliott identity for the super-rotation $S6-j$ symbols.
It turned out that the structure of these relations is the same as
the corresponding structure in the rotation case. Only the sign factors are
somewhat more involved.
In a forthcoming publication we will use the Biedenharn-Elliott
identity in order to obtain recursion relations for the
super-rotation $S6-j$ symbols. These relations will permit us to develop
numerical algorithms for the evaluation of the $S6-j$ symbols
for large values of the superspins, our final goal being to
obtain a Regge-Ponzano [10] type formula for the $S6-j$ symbols.
\section*{Acknowledgements}
One of the authors (S.T.) was supported by grant from The Commission
of the European Communities under contract $n^o$ ERB3510PL920706 706
( ERB-CIPA-CT-92-2137 proposal NR.: 706 ). He
acknowledges the warm hospitality of the Laboratoire de Physique
Th\'{e}orique de l'Universit\'{e} de Bordeaux I.
\newpage
\section*{References}
\noindent
[1] A. Pais and V. Rittenberg, J. Math. Phys. {\bf 16}, 2062 (1975).
\newline
\noindent
[2] M. Scheunert, W. Nahm, and V. Rittenberg, J. Math. Phys. {\bf 18}, 146
and 155 (1977).
\newline
\noindent
[3] F. A. Berezin and V. N. Tolstoy, Commun. Math. Phys. {\bf 78}, 409 (1981).
\newline
\noindent
[4] P. Minnaert and M. Mozrzymas, J. Math. Phys. {\bf 33}, 1582
and 1594 (1992).
\newline
\noindent
[5] M. Daumens, P. Minnaert, M. Mozrzymas and S. Toshev, J. Math. Phys.
{\bf 34}, 2475 (1993).
\newline
\noindent
[6] M. Daumens, P. Minnaert, M. Mozrzymas and S. Toshev, Europhys. Lett.
{\bf 20}, 671 (1992).
\newline
\noindent
[7] L. C. Biedenharn and J. S. Louck, {\em Angular Momentum in
Quantum Physics; Encyclopedia of Mathematics and Its Applications},
Addison-Wesley, London, (1981), Vol. 8, p. 106.
\newline
\noindent
[8] L. C. Biedenharn, J. Math. and Phys. {\bf 31}, 287 (1953).
\newline
\noindent
[9] J. P. Elliott, Proc. Roy. Soc. {\bf A218}, 345 (1953).
\newline
\noindent
[10] G. Ponzano and T. Regge, in: {\em Spectroscopic and Group Theoretical
Methods in Physics (Racah Memorial Volume)}, eds. F.Bloch {\em et al.},
North-Holland, Amsterdam, (1968), p. 1.
\end{document}
|
\subsection{(Local) Incremental exponential stabilizability}
\label{sec:app_increm}
In the following we clarify the connection between incremental stabilizability properties and the terminal ingredients.
\begin{definition}
\label{def:increm_stab}
A set of reference trajectories $r$ specified by some dynamic inclusion $r(t+1)\in\mathcal{R}(r(t))$ is locally incrementally exponentially stabilizable for the system~\eqref{eq:sys}, if there exist constants $\rho\in(0,1),M,c>0$ and a control law $\kappa(x,r)$, such that for any initial condition
satisfying $\|x(0)-x_r(0)\|\leq c$, the trajectory $x(t)$ with $x(t+1)=f(x(t),\kappa(x(t),r(t)))$ satisfies $\|x(t)-x_r(t)\|\leq M\rho^t\|x(0)-x_r(0)\|$,~$\forall t\geq 0$.
\end{definition}
This definition is closely related to the concept of universal exponential stabilizability~\cite{manchester2017control}, which characterizes the stabilizability of arbitrary trajectories in continuous-time.
One of the core differences in the definitions is the treatment of constraints, i.e. we study stabilizability of classes of trajectories $r$ that satisfy certain constraints, compare Assumption~\ref{ass:ref} and Remark~\ref{rk:ref}.
This difference is crucial when discussing local versus global stabilizability and constrained control.
The following proposition shows that the conditions in Lemma~\ref{lemma:lpv} directly imply local incremental exponential stabilizability of the reference trajectory.
\begin{proposition}
\label{prop:increm}
Suppose that there exist matrices $P_f(r),~K_f(r)$ that satisfy the conditions in Lemma~\ref{lemma:lpv}.
Then the control law $k_f(x,r)=u_r+K_f(r)(x-x_r)$ locally incrementally exponentially stabilizes any reference $r$ satisfying Assumption~\ref{ass:ref}.
\end{proposition}
\begin{proof}
The following proof follows the arguments of~\cite[Prop.~1,2]{kohlernonlinear19}.
For any $\|x(0)-x_r(0)\|\leq c$ with $c=\sqrt{\alpha/c_{u}}$, we have $x(0)\in\mathcal{X}_f(r)$, with $\alpha,~c_{u}$ according to Lemma~\ref{lemma:lpv}.
Thus, the terminal cost $V_f(x,r)$ is a local incremental Lyapunov function that satisfies
\begin{align*}
V_f(x(t+1),r(t+1))\leq \rho^2 V_f(x(t),r(t)),~ \rho^2=1-\dfrac{\lambda_{\min}(Q)}{c_u},
\end{align*}
and thus
\begin{align*}
\|x(t)-x_r(t)\|\leq {\rho}^tM \|x_r(0)-x(0)\|,\quad M=\sqrt{{c_u}/{c_l}}.
\end{align*}
\end{proof}
\begin{remark}
\label{rk:increm}
This result establishes local incremental stabilizability with the incremental Lyapunov function $V_f(x,r)$ based on properties of the linearization, compare~\cite[Prop.~1]{kohlernonlinear19}.
This system property is a natural extension of previous works on incremental stability and corresponding incremental Lyapunov functions, see~\cite{angeli2002lyapunov,tran2016incremental},~\cite[Ass.~1]{kohlernonlinear19}.
This property implies stabilizability of $(A(r),~B(r))$ around any (fixed) steady-state $r^+=r$, but it does not necessarily imply stabilizability of $(A(r),~B(r))$ for arbitrary $r\in\mathcal{Z}_r$, as $P_f(r)$ might decrease along the trajectory.
For continuous-time systems, an analogous result exists based on contraction metrics and universal stabilizability~\cite{manchester2017control}.
\end{remark}
The following proposition shows that in the absence of constraints we recover non-local results similar to~\cite{manchester2017control}.
\begin{proposition}
\label{prop:univ}
Consider $\mathcal{Z}_r=\mathcal{Z}=\mathbb{R}^{n+m}$.
Suppose that there exist matices $P_f(r),~K_f(r)$ that satisfy the conditions in Lemma~\ref{lemma:lpv}.
Assume further that $c_lI\leq P_f(r)\leq c_uI$ for all $r\in\mathbb{R}^{n+m}$ with some constants $c_l,~c_u$ and $K_f(r)=K(x_r)$.
Then any reference $r$ satisfying Assumption~\ref{ass:ref} is exponentially incrementally stabilizable with the control law
\begin{align*}
\kappa(x,r)=K(x)x-K(x_r)x_r+u_r,
\end{align*}
i.e., for any initial condition $x(0)\in\mathbb{R}^n$ the state trajectory $x(t+1)=f(x(t),\kappa(x(t),r(t)))$ satisfies $\|x(t)-x_r(t)\|\leq M\rho^t\|x(0)-x_r(0)\|$.
\end{proposition}
\begin{proof}
Consider an auxiliary (pre-stabilized) system defined by $\tilde{f}(x,v)=f(x,K(x)x+v)$.
Consider a reference $r$ generated by some input trajectory $u_r$ with the system dynamics~\eqref{eq:sys} (Ass.~\ref{ass:ref}) and some initial condition $x_r(0)$ resulting in the state reference $x_r$.
Now, consider a reference $\tilde{r}$ generated by the input $v_r(t)=u_r(t)-K(x_r(t))x_r(t)$ with the system dynamics according to $\tilde{f}$ and the same initial condition.
Due to the definition of the auxiliary system we have $\tilde{x}_r(t)=x_r(t)$, $\forall t\geq 0$.
For an arbitrary, but fixed input $v$, stability of the reference trajectory is equivalent to contractivity of the nonlinear time-varying system $\tilde{f}(x,t)$.
This can be established with the contractivity metric $P_f(r(t))=P_f(x_r(t),t)$, compare~\cite{lohmiller1998contraction}.
\end{proof}
In the absence of constraints, it is crucial that $P_f$ has a constant lower and upper bound.
If the matrix $K_f$ depends on the full reference $r$ (not just $x_r$), the controller $\kappa$ in Proposition~\ref{prop:univ} is not necessarily well defined.
\begin{remark}
\label{rk:track_vs_stab}
The relation between the controller $k_f$ (Prop.~\ref{prop:increm}) and $\kappa$ (Prop.~\ref{prop:univ}), is that of reference tracking versus pre-stabilization.
The first one is more natural in the context of tracking MPC and contains existing results for the design of terminal ingredients as special cases~\cite{chen1998quasi,faulwasser2011model,aydiner2016periodic}.
The second controller $\kappa$ allows for non-local stability results and is more suited for unconstrained control problems~\cite{manchester2017control}.
For constant matrices $K$ the two controllers are equivalent, but the incremental Lyapunov functions (and thus terminal costs) are differently parameterized ($P_f(r)$, $P_f(x,u)$).
\end{remark}
\begin{remark}
\label{rk:term_other}
The problem of computing reference generic terminal ingredients is equivalent to computing an incrementally stabilizing controller and is thus strongly related to the computation of robust positive invariant (RPI) tubes in nonlinear robust MPC schemes, compare~\cite{bayer2013discrete,kohler2018novel}.
For comparison, in~\cite{yu2010robust,yu2013tube} constant matrices $P_f,~K_f$ are computed that certify incremental stability for continuous-time systems (by considering small Lipschitz nonlinearities or by describing the linearization as a convex combination of different linear systems).
This approach can be directly extended to more general nonlinear systems using the proposed terminal ingredients.
In particular, by changing the stage cost to
\begin{align*}
\ell(x,u,r)=\|u-u_r+K(x_r)x_r-K(x)x\|_R^2
\end{align*}
one can design a nonlinear version of~\cite{chisci2001systems}, compare also~\cite{kohler2018novel}.
A detailed description of a corresponding nonlinear robust tube based (tracking) MPC scheme based on incremental stabilizability can be found in Appendix~\ref{sec:app_robust}.
\end{remark}
\begin{remark}
\label{rk:non_quadratic}
In case a system is not exponentially stabilizable~\cite{muller2017quadratic}, it might be possible to make a nonlinear transformation resulting in a quadratically stabilizable system, (see for example nonlinear systems in normal form~\cite{isidori2013nonlinear}).
\end{remark}
\subsection{Robust reference tracking}
\label{sec:app_robust}
In the following, we summarize the theoretical results for robust reference tracking based on the reference generic terminal ingredients and~\cite{kohler2018novel}, where robust setpoint stabilization without terminal constraints was considered.
This method is applicable to nonlinear incrementally stabilizable systems (Sec.~\ref{sec:app_increm}) with polytopic constraints and additive disturbances and can be thought of as a nonlinear version of~\cite{chisci2001systems}.
\subsubsection{Setup}
We consider nonlinear discrete-time systems subject to additive bounded disturbances and polytopic constraints
\begin{align*}
x(t+1)=&f(x(t),u(t))+w(t),\quad \|w\|\leq \hat{w},\\
\mathcal{Z}=&\{r\in\mathbb{R}^{n+m}|~L_j r\leq 1,\quad j=1,\dots,q\}.
\end{align*}
\subsubsection{Incremental stabilizability}
\begin{assumption}
\label{ass:contract} \cite[Ass.~1]{kohler2018novel}\cite[Ass.~1]{kohlernonlinear19}
There exist a control law $\kappa:\mathbb{R}^n\times\mathcal{Z}\rightarrow\mathbb{R}^m$, an incremental Lyapunov function $V_{\delta}:\mathbb{R}^n\times \mathcal{Z}\rightarrow\mathbb{R}_{\geq 0}$, that is continuous in the first argument and satisfies $V_{\delta}(x_r,x_r,u_r)=0$ for all $(x_r,u_r)\in\mathcal{Z}$, and parameters $c_{\delta,l},~c_{\delta,u},~\delta_{\text{loc}},~c_{j}\in\mathbb{R}_{>0}$, $\rho\in(0,1)$, such that the following properties hold for all $(x,x_r,u_r)\in\mathbb{R}^n\times\mathcal{Z}$, $r^+=(x_r^+,u_r^+)\in\mathcal{Z}$ with $V_{\delta}(x,r)\leq \delta_{\text{loc}}$:
\begin{subequations}
\begin{align}
c_{\delta,l}\|x-x_r\|^2\leq V_{\delta}(x,x_r,u_r)\leq& c_{\delta,u}\|x-x_r\|^2,\\
\label{eq:lipschitz}
L_j (x-x_r,\kappa(x,x_r,u_r)-u_r)\leq& c_{j}\sqrt{V_{\delta}(x,x_r,u_r)},\\
\label{eq:contract}
V_{\delta}(x^+,x_r^+,u_r^+)\leq& \rho^2 V_{\delta}(x,x_r,u_r),
\end{align}
\end{subequations}
with $x^+=f(x,\kappa(x,x_r,u_r))$, $x_r^+=f(x_r,u_r)$, $j=1,\dots,q$.
\end{assumption}
This assumption implies incremental stabilizability (Def.~\ref{def:increm_stab}) for all feasible trajectories $r$, i.e., $r(t+1)\in\mathcal{R}(r(t))$ (Ass.~\ref{ass:ref}).
For $\kappa(x,x_r,u_r)=u_r$ this reduces to incremental stability and correspondingly the robust MPC method in~\cite{bayer2013discrete} can also be used.
This assumption can be verified by using Algorithm~\ref{alg:offline} to compute a terminal cost that is valid on $\mathcal{Z}$, compare Proposition~\ref{prop:increm}.
The contraction rate $\rho$~\eqref{eq:contract}, is used to design a generic constraint tightening to ensure robust constraint satisfaction.
The condition~\eqref{eq:lipschitz} is satisfied if the control law $\kappa$ is locally Lipschitz continuous, compare also~\cite{kohler2018novel}.
\subsubsection{Constraint tightening} The constraints are tightened using the following scalar operations
\begin{align*}
\epsilon_j=&c_{j}\sqrt{c_u}\hat{w},\quad
\epsilon_{j,k}=\dfrac{1-{\rho}^k}{1-{\rho}}\epsilon_j, k=0,\dots,N,\\
\mathcal{Z}_k=&\{r\in\mathbb{R}^n|~L_j r\leq 1-\epsilon_{j,k},\quad j=1,\dots, q\}.
\end{align*}
The following bound on the disturbance is required to ensure that the tightened constraints are non-empty, i.e., $0\in\text{int}\left(\mathcal{Z}_N\right)$:
\begin{align}
\label{eq:w_1}
\hat{w}<\dfrac{1}{\max_jc_j}\dfrac{1}{\sqrt{c_u}}\dfrac{1-\rho^N}{1-\rho},
\end{align}
\subsubsection{Terminal ingredients}
In~\cite{kohler2018novel} the robust constraint tightening is considered for an MPC scheme without terminal constraints, compare Remark~\ref{rk:withoutterm}.
Some details regarding the extension/modification of the robust MPC scheme to a setting with terminal constraints are based on~\cite{hertneck2018learning}.
\begin{assumption}
\label{ass:term_dist}
There exist matrices $K_f(r)\in\mathbb{R}^{m\times n}$, $P_f(r)\in\mathbb{R}^{n\times n}$ with $c_l I_n\leq P_f(r)\leq c_u I_n$, a terminal set $\mathcal{X}_f(r)=\{x\in\mathbb{R}^n|~V_f(x,r)\leq \alpha_w\}$ with the terminal cost $V_f(x,r)=\|x-x_r\|_{P_f(r)}^2$, such that the following properties hold for any $r\in\mathcal{Z}_r$, any $x\in\mathcal{X}_f(r)$, any $r^+\in\mathcal{R}(r)$ and any $w\in\mathcal{W}_N$
\begin{subequations}
\label{eq:term_dist}
\begin{align}
\label{eq:term_dist_dec}
V_f(x^+,r^+)\leq& V_f(x,r)- \ell(x,k_f(x,r),r),\\
\label{eq:term_dist_RPI}
V_f(x^++w,r^+)\leq& \alpha_w,\\
\label{eq:term_dist_con}
(x,k_f(x,r))\in&\mathcal{Z}_N,
\end{align}
\end{subequations}
with $x^+=f(x,k_f(x,r))$, $k_f(x,r)=u_r+K_f(r)\cdot (x-x_r)$, $\mathcal{W}_N=\{w\in\mathbb{R}^n|~\|w\|\leq \hat{w}_N= \hat{w} \rho^N\sqrt{c_{\delta,u}/c_{\delta,l}} \}$, and positive constants $c_l,~c_u,~\alpha_w$.
\end{assumption}
Compared to the nominal case (Ass.~\ref{ass:term}), we have a smaller terminal set size $\alpha_w$ due to the tightened constraints~\eqref{eq:term_dist_con} and an RPI condition that needs to be verified~\eqref{eq:term_dist_RPI}.
Due to the quadratic nature of the terminal cost and the stage cost, \eqref{eq:term_dist_dec} implies $V_f(x^+,r^+)\leq \rho_f^2V_f(x,r,)$, with some $\rho_f\in(0,1)$, e.g. $\rho_f=1-\lambda_{\min}(Q)/c_u$.
\begin{proposition}
\label{prop:RPI}
Let Assumption~\ref{ass:term} hold and assume that $\hat{w}$ satisfies~\eqref{eq:w_1}.
Then the terminal ingredients (Ass.~\ref{ass:term}) satisfy~\eqref{eq:term_dist_con} with a positive constant $\alpha_w$.
Suppose further that
\begin{align}
\label{eq:w_2}
\hat{w}\leq\sqrt{\dfrac{\alpha_w c_{\delta,l}}{c_{\delta,u}c_{u}}}\dfrac{1-\rho_f}{\rho^N}.
\end{align}
Then~\eqref{eq:term_dist_RPI} and thus Assumption~\ref{ass:term_dist} is satisfied.
\end{proposition}
\begin{proof}
Condition~\eqref{eq:term_dist_dec} directly follows fom Assumption~\ref{ass:term}.
Inequality~\eqref{eq:w_1} ensures that $0\in\text{int}(\mathcal{Z}_N)$, which in combination with the quadratic bounds on $V_f$ and linear bounds on $k_f$ ensures that~\eqref{eq:term_dist_con} is satisfied for some positive constant $\alpha_w$, compare the proof of Lemma~\ref{lemma:lpv}, Algorithm~\ref{alg:offline_alpha} and the optimization problem~\eqref{eq:alpha_2_better} for the computation of $\alpha_w$.
Using the quadratic nature of the terminal cost, a sufficient condition for~\eqref{eq:term_dist_RPI} is given by
\begin{align*}
&V_f(x^++w,r^+)\\
\leq& V_f(x^+,r^+)+2\sqrt{c_uV_f(x^+,r^+)}\hat{w}_N+c_u \hat{w}_N^2\leq \alpha_w,
\end{align*}
with $\|w\|\leq \hat{w}_N$.
Using the contraction rate $\rho_f$ to bound $V_f(x^+,r^+)\leq \rho_f^2\alpha_w$, this condition reduces to $\hat{w}_N\leq (1-\rho_f)\sqrt{\alpha_w/c_u}$.
The inequality on $\hat{w}$ follows from the definition of $\hat{w}_N$.
\end{proof}
\subsubsection{Robust tracking MPC}
The robust tracking MPC is based on the following MPC optimization problem
\begin{subequations}
\label{eq:robust_MPC}
\begin{align}
V(x(t),r(\cdot|t))=\min_{u(\cdot|t)}&J_N(x(\cdot|t),u(\cdot|t),r(\cdot|t))\\
\text{s.t. }&x(k+1|t)=f(x(k|t),u(k|t)),\\
&x(0|t)=x(t),\\
&(x(k|t),u(k|t))\in\mathcal{Z}_k,\\
&x(N|t)\in\mathcal{X}_f({r}(N|t)).
\end{align}
\end{subequations}
Compared to~\eqref{eq:MPC}, in this optimization problem the state and input constraints are tightened.
\subsubsection{Theoretical guarantees}
\begin{theorem}
\label{thm:robust}
Let Assumptions~\ref{ass:ref},~\ref{ass:contract} and~\ref{ass:term_dist} hold.
Assume further that $\hat{w}\leq \sqrt{\delta_{\text{loc}}/c_{\delta,u}}$ and that~\eqref{eq:robust_MPC} is feasible at $t=0$.
The optimization problem~\eqref{eq:robust_MPC} is recursively feasible and the tracking error $e_r=0$ is (uniformly) practically exponentially stable for the resulting closed-loop system~\eqref{eq:close}.
\end{theorem}
\begin{proof}
The proof is analogous to~\cite{kohler2018novel}, except for the satisfaction of the terminal constraint, which is guaranteed by Assumption~\ref{ass:term_dist}, compare also~\cite[Thm.~7]{hertneck2018learning}.
\end{proof}
Note that both the size of the constraint set~\eqref{eq:w_1} and the local incremental stabilizability~\eqref{eq:w_2} lead to hard bounds on the size of the disturbance $\hat{w}$, that can be considered in this approach.
This approach can also be extended to utilize a general nonlinear state and input dependent characterization of the disturbance in order to reduce the conservatism, compare~\cite{Robust_TAC_19}.
\subsection{Continuous-time dynamics}
\label{sec:app_cont}
In the following, we summarize the continuous-time analog of the reference generic offline computations in Section~\ref{sec:loc_stab}.
The nonlinear continuous-time dynamics are given by
\begin{align*}
\dfrac{d}{dt}[x]=\dot{x}=f(x,u)
\end{align*}
and $f$ is assumed to be twice continuously differentiable.
The following condition characterizes the admissible reference trajectories as the continuous-time analog of Assumption~\ref{ass:ref}.
\begin{assumption}
\label{ass:ref_cont}
The reference signal $r:\mathbb{R}\rightarrow\mathbb{R}^{n+m}$ is continuously differentiable and satisfies
\begin{align*}
r(t)\in&\mathcal{Z}_r\subseteq\text{int}(\mathcal{Z}),\\
\dot{r}(t)\in&\mathcal{R}(r(t))=\{(\dot{x}_r,\dot{u}_r)|~\dot{x}_r=f(x_r,u_r),~\|\dot{u}_r\|_{\infty}\leq u_{\max}\},
\end{align*}
for all $t\geq 0$ with some constant $u_{\max}$.
\end{assumption}
\begin{remark}
This assumption can be generalized to consider non-differentiable reference signal $r$ ($\dot{u}_r$ unbounded).
In this case, the terminal cost $P_f$ should be parameterized with parameters $\theta_i$ independent of $u_r$, i.e., $P_f(x_r)$.
\end{remark}
The following assumption characterizes the terminal ingredients, as a continuous-time analog of Assumption~\ref{ass:term}.
\begin{assumption}
\label{ass:term_cont}
There exist matrices $K_f(r)\in\mathbb{R}^{m\times n}$, $P_f(r)\in\mathbb{R}^{n\times n}$ with $c_l I_n\leq P_f(r)\leq c_u I_n$, $P_f$ continuously differentiable, a terminal set $\mathcal{X}_f(r)=\{x\in\mathbb{R}^n|~V_f(x,r)\leq \alpha\}$ with the terminal cost $V_f(x,r)=\|x-x_r\|_{P_f(r)}^2$, such that the following properties hold for any $r\in\mathcal{Z}_r$, any $x\in\mathcal{X}_f(r)$ and any $\dot{r}\in\mathcal{R}(r)$
\begin{align}
\label{eq:term_dec_cont}
\dfrac{d}{dt}[V_f(x,r)]\leq& -\ell(x,k_f(x,r),r),\\
\label{eq:term_con_cont}
(x,k_f(x,r))\in&\mathcal{Z},
\end{align}
with positive constants $c_l,~c_u,~\alpha$ and
\begin{align*}
\dot{x}=&f(x,k_f(x,r)), \quad k_f(x,r)=u_r+K_f(r)\cdot (x-x_r),\\
&\dfrac{d}{dt}V_f(x,r)\\
=&2(x-x_r)^\top P_f(r)(\dot{x}-\dot{x}_r)+ \|x-x_r\|_{\frac{d}{dt}{P}_f(r)}^2.
\end{align*}
\end{assumption}
The following Lemma provides sufficient conditions for Assumption~\ref{ass:term_cont} to be satisfied based on the linearization, as a continuous-time version of Lemma~\ref{lemma:lpv}.
\begin{lemma}
\label{lemma:lpv_cont}
Assume that there exist matrices $K_f(r)\in\mathbb{R}^{m\times n}$ continuous in $r$ and a positive definite matrix $P_f(r)\in\mathbb{R}^{n\times n}$ continuously differentiable with respect to $r$, such that for any $r\in\mathcal{Z}_r$, $\dot{r}\in\mathcal{R}(r)$, the following matrix inequality is satisfied
\begin{align}
\label{eq:lpv_cont}
&(A(r)+B(r)K_f(r))^\top P_f(r)+ P_f(r)(A(r)+B(r)K_f(r))\nonumber\\
&+\sum_{j=1}^{n+m}\dfrac{\partial P_f}{\partial r_j}\dot{r}_j +(Q+\epsilon I_n+K_f(r)^\top R K_f(r))\leq 0
\end{align}
with some positive constant $\epsilon$.
Then there exists a sufficiently small constant $\alpha$, such that $P_f,~K_f$ satisfy Assumption~\ref{ass:term_cont}.
\end{lemma}
\begin{proof}
Denote $\Delta x=x-x_r$, $\Delta u=K_f(r)\Delta x$.
Using a first order Taylor approximation at $r=(x_r,u_r)$, we get
\begin{align*}
f(x,u)={f(x_r,u_r)}+A(r)\Delta x+B(r)\Delta u+\Phi_r(\Delta x),
\end{align*}
with the remainder term $\Phi_r$.
The terminal cost satisfies
\begin{align*}
&\dfrac{d}{dt} V_f(x,r)=2(x-x_r)^\top P_f(r)(\dot{x}-\dot{x}_r)\\
&+(x-x_r)^\top \left[\sum_{j=1}^{n+m} \dfrac{\partial P_f}{\partial r_j} \dot{r_j}\right](x-x_r)\\
\stackrel{\eqref{eq:lpv_cont}}{\leq} &-\ell(x,k_f(x,r))-\epsilon\|\Delta x\|^2+2(x-x_r)^\top P_f(r) \Phi_r(\Delta x).
\end{align*}
For $\alpha$ sufficiently small, this implies~\eqref{eq:term_dec_cont} (due to the arbitrarily small local Lipschitz bound on the higher order terms $\Phi_r$).
Constraint satisfaction~\eqref{eq:term_con_cont} is guaranteed analogous to Lemma~\ref{lemma:lpv}.
\end{proof}
The following Lemma provides corresponding LMI conditions, similar to Lemma~\ref{lemma:LMI}.
\begin{lemma}
\label{lemma:LMI_cont}
Suppose that there exists a matrix $Y(r)$ continuous in $r$ and $X(r)$ continuously differentiable with respect to $r$, that satisfy the constraints in~\eqref{eq:LMI_cont} for all $r\in\mathcal{Z}_r,~\dot{r}\in\mathcal{R}(r)$.
Then $P_f=X^{-1}$, $K_f=YP_f$ satisfy~\eqref{eq:lpv_cont}.
\end{lemma}
\begin{proof}
Multiplying~\eqref{eq:lpv_cont} from left and right with $X(r)$ yields
\begin{align*}
&(A(r)X(r)+B(r)Y(r))^\top +(A(r)X(r)+B(r)Y(r)) \\
&+X(r)\dfrac{d}{dt}[X^{-1}(r)]X(r)+X(r)(Q+\epsilon I_n)X(r)\\
&+Y(r)^\top R Y(r)\leq 0.
\end{align*}
Note that the chain rule applied to the inverse of $X$ yields
\begin{align*}
X(r)\dfrac{d}{dt}\left[X^{-1}(r)\right]X(r)=-\dfrac{d}{dt}[X(r)]=-\sum_{i=1}^p X_i\dfrac{\partial \theta_i}{\partial r} \dot{r} .
\end{align*}
Applying the Schur complement results in~\eqref{eq:LMI_cont}.
\end{proof}
If a gridding approach is considered to compute the terminal ingredients, one needs to grid $r\in\mathcal{Z}_r$ and consider the $2 ^m$ vertices of $\dot{u}_r$ (since~\eqref{eq:LMI_cont} is affine in $\dot{u}_r$).
For the convex approach, polytopic sets need to be constructed such that
$(\theta,\dot{\theta})\in\Theta\times\Omega=\overline{\Theta},~\forall r\in\mathcal{Z}_r,~\dot{r}\in\mathcal{R}(r)$.
The following proposition provides the corresponding LMI conditions based on the vertices of $\overline{\Theta}$, similar to Proposition~\ref{prop:LMI_lpv}.
\begin{proposition}
\label{prop:LMI_lpv_cont}
Suppose that there exists matrices $X_i,~Y_i,~\Lambda_i,~X_{\min}$ that satisfy the constraints in~\eqref{eq:LMI_LPV_cont}.
Then the following matrices satisfy~\eqref{eq:lpv_cont}:
\begin{align*}
P_f(r)=&X^{-1}(r),\quad K_f(r)=Y(r)P_f(r).
\end{align*}
\end{proposition}
\begin{proof}
The proof is analogous to Proposition~\ref{prop:LMI_lpv}, based on multi-convexity and Lemma~\ref{lemma:LMI_cont}.
\end{proof}
\begin{remark}
\label{rk:sample_hold}
The continuous-time formulation is suitable if the online MPC optimization considers continuous-time input signals, instead of piece-wise constant inputs (as is common in many numerical implementations).
Nevertheless, if the sampling time $h$ is sufficiently small, the continuous-time terminal cost (scaled by $1/h$) might satisfy the discrete-time conditions (Ass.~\ref{ass:term}) with a piece-wise constant input.
This can be favorable since the corresponding offline optimization problem~\eqref{eq:LMI_cont} or \eqref{eq:LMI_LPV_cont} is often easier formulated and faster solved, especially if a non-trivial discretization is considered.
Algorithm~\ref{alg:offline_alpha} can be used to ensure the validity of the computed terminal ingredients with the zero-order hold input (instead of the continuous-time feedback).
\end{remark}
\subsection{Output tracking stage cost}
\label{sec:app_output}
In the following, we discuss how the derivation in Section~\ref{sec:loc_stab} can be extended to deal with an output tracking stage cost.
As an alternative to~\eqref{eq:stage}, consider the following output reference tracking stage cost
\begin{align}
\label{eq:stage_output}
\ell(x,u,r)=\|h(x,u)-h(x_r,u_r)\|_{S(r)}^2,
\end{align}
with a nonlinear twice continuously differentiable output function $h:\mathcal{Z}\rightarrow \mathbb{R}^p$ and a positive definite weighting matrix $S(r)$, which assumed to be continuous in $r$.
Such a stage cost can be used for output regulation, output trajectory tracking, output path following or manifold stabilization, compare~\cite{faulwasser2012optimization}.
We denote the Jacobian of the output $h$ around an arbitrary point $r\in\mathcal{Z}_r$ by
\begin{align}
\label{eq:C_r}
C(r)=\left.\left[\dfrac{\partial h}{\partial x}\right]\right|_{(x,u)=r},\quad D(r)=\left.\left[\dfrac{\partial h}{\partial u}\right]\right|_{(x,u)=r}.
\end{align}
The following lemma establishes sufficient conditions for Assumption~\ref{ass:term} with the stage cost~\eqref{eq:stage_output} based on the linearization, similar to Lemma~\ref{lemma:lpv}.
\begin{lemma}
\label{lemma:lpv_output}
Suppose that $f,~h$ are twice continuously differentiable.
Assume that there exists a matrix $K_f(r)\in\mathbb{R}^{m\times n}$ and a positive definite matrix $P_f(r)\in\mathbb{R}^{n\times n}$ continuous in $r$, such that for any $r\in\mathcal{Z}_r$, $r^+\in\mathcal{R}(r)$, the following matrix inequality is satisfied
\begin{align}
\label{eq:lpv_output}
&(A(r)+B(r)K_f(r))^\top P_f(r^+)(A(r)+B(r)K_f(r))- P_f(r)\\
\leq&-(C(r)+D(r)K_f(r))^\top S(r) (C(r)+D(r)K_f(r))-\tilde{\epsilon} I_n\nonumber
\end{align}
with some positive constant $\tilde{\epsilon}$.
Then there exists a sufficiently small constant $\alpha$, such that $P_f,~K_f$ satisfy Assumption~\ref{ass:term}.
\end{lemma}
\begin{proof}
A first order Taylor approximation at $r=(x_r,u_r)$ yields
\begin{align*}
&h(x,k_f(x,r))-{h(x_r,u_r)}\\
=&(C(r)+D(r)K_f(r))\Delta x+\tilde{\Phi}_{r}(\Delta x),
\end{align*}
with the remainder term $\tilde{\Phi}_{r}$ and $\Delta x=x-x_r$.
The stage cost satisfies
\begin{align}
\label{eq:output_1}
&\ell(x,k_f(x,r),r)
\\
\geq &\|(C(r)+D(r)K_f(r))\Delta x\|_{S(r)}^2+\|\tilde{\Phi}_{r}(\Delta x)\|_{S(r)}^2\nonumber\\
&- 2\|\tilde{\Phi}_{r}(\Delta x)\|_{S(r)} \|(C(r)+D(r)K_f(r))\Delta x \|_{S(r)}.\nonumber
\end{align}
Given continuity and compactness, there exists a constant
\begin{align}
\label{eq:output_2}
c_y=\max_{r\in\mathcal{Z}_r}\|(C(r)+D(r)K_f(r))\|_{S(r)}.
\end{align}
For a sufficiently small $\alpha$, the remainder term $\tilde{\Phi}_{r}$ satisfies the following (local) Lipschitz bound
\begin{align}
\label{eq:output_3}
\|\tilde{\Phi}_{r}(\Delta x)\|_{S(r)}/\|\Delta x\|=:\tilde{L}_{r,x}\leq \tilde{L}^*:=c_y-\sqrt{c_y^2-\tilde{\epsilon}/2},
\end{align}
for all $x\in\mathcal{X}_f(r)$ and all $r\in\mathcal{Z}_r$.
This implies
\begin{align*}
&\ell(x,k_f(x,r))\\
\stackrel{\eqref{eq:output_1},\eqref{eq:output_3}}{\geq}& \|(C(r)+D(r)K_f(r))\Delta x\|_{S(r)}^2+\tilde{L}_{r,x}^2\|\Delta x\|^2\\
&-2\tilde{L}_{r,x}\|\Delta x\|^2\|(Cr)+D(r)K_f(r)\|_{S(r)}\\
\stackrel{\eqref{eq:output_2}}{\geq}& \|(C(r)+D(r)K_f(r))\Delta x\|_{S(r)}^2+\tilde{L}_{r,x}(\tilde{L}_{r,x}-2c_y)\|\Delta x\|^2\\
\stackrel{\eqref{eq:output_3}}{\geq}& \|(C(r)+D(r)K_f(r))\Delta x\|_{S(r)}^2+\tilde{L}^*(\tilde{L}^*-2c_y)\|\Delta x\|^2\\
\stackrel{\eqref{eq:output_3}}=& \|(C(r)+D(r)K_f(r))\Delta x\|_{S(r)}^2-\tilde{\epsilon}/2\|\Delta x\|^2.
\end{align*}
The second to last step follows by using the fact that the function $L(L-2c_y)$ attains it minimum for $L\in[0,L^*]$ at $L=L^*$.
Combining the derived bound on $\ell(x,k_f(x,r))$ with~\eqref{eq:lpv_output} ensures that the terminal cost $V_f$ satisfies inequality~\eqref{eq:lpv_1} in Lemma~\ref{lemma:lpv} with the modified stage cost and with ${\epsilon}=\tilde{\epsilon}/2$.
The remainder of the proof is analogous to Lemma~\ref{lemma:lpv}.
\end{proof}
\begin{remark}
\label{rk:output_1}
For the linear output $h(x,u)=[Q^{1/2}x;R^{1/2}u]\in\mathbb{R}^{n+m}$ and $S=I$ we recover the conditions in Lemma~\ref{lemma:lpv} with the stage cost~\eqref{eq:stage}.
\end{remark}
\begin{remark}
\label{rk:output_2}
Depending on the output $h$ and the reference $r$, there may exist multiple solutions that achieve exact output tracking.
Thus, we can in general not expect asymptotic/exponential stability of the reference $r$, but instead stability of a corresponding set or manifold, compare~\cite{faulwasser2012optimization}.
Under suitable (incremental) detectability conditions on the output $h$, we can recover stability of the specific reference trajectory $r$.
\end{remark}
Based on these conditions, Lemma~\ref{lemma:LMI_output} provides LMI conditions to compute $P_f,~K_f$, similar to Lemma~\ref{lemma:LMI}.
Furthermore, if the parameters $\theta_i$ are chosen, such that
\begin{align*}
S^{-1}(r)=S_0+\sum_{i=1}^p \theta_i(r) S_i,
\end{align*}
then Proposition~\ref{prop:LMI_lpv_output} yields LMI conditions based on the vertices of $\overline{\Theta}$, similar to Proposition~\ref{prop:LMI_lpv}.
\begin{table*}
\small{
\begin{subequations}
\label{eq:LMI_cont}
\begin{align}
\min_{X(r),Y(r),X_{\min}}& - \log\det X_{\min}\\
\text{s.t. }&\begin{pmatrix}
A(r)X(r)+B(r)Y(r) +(A(r)X(r)+B(r)Y(r))^\top -\dfrac{d}{dt}[X(r)]&((Q+\epsilon)^{1/2}X(r))^\top&(R^{1/2}Y(r))^\top\\
*&-I&0\\
*&0&-I\\
\end{pmatrix}\leq 0,\\
&X_{\min}\leq X(r),\\
&\forall r\in\mathcal{Z}_r,~\dot{r}\in\mathcal{R}(r).
\end{align}
\end{subequations}
}
\label{tab:long-eq}
\end{table*}
\begin{table*}
\small{
\begin{subequations}
\label{eq:LMI_LPV_cont}
\begin{align}
\min_{X_i,Y_i,\Lambda_i,X_{\min}}& - \log\det X_{\min}\\
\text{s.t. }&\begin{pmatrix}
A(\theta)X(\theta)+B(\theta)Y(\theta) +(A(\theta)X(\theta)+B(\theta)Y(\theta))^\top -X(\dot{\theta})+X_0&((Q+\epsilon)^{1/2}X(\theta))^\top&(R^{1/2}Y(\theta))^\top\\
*&-I&0\\
*&0&-I\\
\end{pmatrix}\nonumber\\
\leq& - \begin{pmatrix}\sum_{i=1}^p\theta_i^2\Lambda_i&0\\0&0\end{pmatrix},\\
&X_{\min}\leq X(\theta),\quad \forall (\theta,\dot{\theta})\in\text{Vert}(\overline{\Theta}), \\
\label{eq:LMI_LPV2_cont}
&\Lambda_i+
(A_iX_i+B_iY_i)
+(A_iX_i+B_iY_i)^\top
\geq 0,\quad \Lambda_i\geq 0,\quad i=1,\dots,p.
\end{align}
\end{subequations}
}
\end{table*}
\begin{table*}
\begin{lemma}
\label{lemma:LMI_output}
Suppose that there exists matrices $X(r)$,~$Y(r)$ continuous in $r$, that satisfy the following constraints
\small{
\begin{subequations}
\label{eq:LMI_output}
\begin{align}
\min_{X(r),Y(r),X_{\min}}& - \log \det X_{\min}\\
\text{s.t. }&\begin{pmatrix}
X(r)&(A(r)X(r)+B(r)Y(r))^\top&(C(r)X(r)+D(r)Y(r))^\top&\sqrt{\tilde{\epsilon}} X(r)\\
*&X(r^+)&0&0\\
*&*&S^{-1}(r)&0\\
*&*&*&I
\end{pmatrix}\geq 0,\\
&X_{\min}\leq X(r),\quad
\forall r\in\mathcal{Z}_r,~r^+\in\mathcal{R}(r).
\end{align}
\end{subequations}
}
\label{tab:long-eq}
\normalsize
Then $P_f=X^{-1}$, $K_f=YP_f$ satisfy~\eqref{eq:lpv_output}.
\end{lemma}
\begin{proof}
The proof is similar to Lemma~\ref{lemma:lpv}, compare also~\cite{boyd1994linear}.
Define $X(r)=P_f(r)^{-1}$ and $Y(r)=K_f(r)X(r)$.
Multiplying~\eqref{eq:lpv_output} from left and right with $X(r)$ yields
\small{
\begin{align*}
&(A(r)X(r)+B(r)Y(r))^\top X(r^+)^{-1} (A(r)X(r)+B(r)Y(r))-X(r)
+\tilde{\epsilon} X(r) IX(r)\\
&+(C(r)X(r)+D(r)Y(r))^\top S(r)(C(r)X(r)+D(r)Y(r))\leq 0.
\end{align*}
}
\normalsize
This can be equivalently written as
\small{
\begin{align*}
X(r)-
\begin{pmatrix}
A(r)X(r)+B(r)Y(r)\\C(r)X(r)+D(r)Y(r)\\\sqrt{\tilde{\epsilon}}X(r)
\end{pmatrix}^{\top}
\begin{pmatrix}
X(r^+)^{-1}&0&0\\
0&S(r)&0\\
0&0&I
\end{pmatrix}
\begin{pmatrix}
A(r)X(r)+B(r)Y(r)\\C(r)X(r)+D(r)Y(r)\\\sqrt{\tilde{\epsilon}}X(r)
\end{pmatrix}
\geq 0
\end{align*}}
\normalsize
Using the Schur complement this reduces to~\eqref{eq:LMI_output}, which is linear in $X,~Y$.
\end{proof}
\end{table*}
\begin{table*}
\begin{proposition}
\label{prop:LMI_lpv_output}
Suppose that there exists matrices $X_i,~Y_i,~\Lambda_i,~X_{\min}$ that satisfy the following constraints
\small{
\begin{subequations}
\label{eq:LMI_LPV_output}
\begin{align}
\min_{X_i,Y_i,\Lambda_i,X_{\min}}& - \log\det X_{\min}\\
\text{s.t. }&\begin{pmatrix}
X(\theta)&(A(\theta)X(\theta)+B(\theta)Y(\theta))^\top&(C(\theta)X(\theta)+D(\theta)Y(\theta))^\top&\sqrt{\tilde{\epsilon}} X(\theta)\\
*&X(\theta^+)&0&0\\
*&*&S^{-1}(\theta)&0\\
*&*&*&I
\end{pmatrix}\geq \begin{pmatrix}\sum_{i=1}^p\theta_i^2\Lambda_i&0\\0&0\end{pmatrix},\\
&X_{\min}\leq X(\theta),\quad \forall (\theta,\theta^+)\in\text{Vert}(\overline{\Theta}), \\
\label{eq:LMI_LPV_output2}
&\begin{pmatrix}
0&(A_iX_i+B_iY_i)^\top&(C_iX_i+D_iY_i)^\top\\
(A_iX_i+B_iY_i)&0&0\\
(C_iX_i+D_iY_i)&0&0
\end{pmatrix}\leq \Lambda_i,\quad \Lambda_i\geq 0,\quad i=1,\dots,p.
\end{align}
\end{subequations}
}
\normalsize
Then $P_f=X^{-1}$ and $K_f=Y P_f$ satisfy~\eqref{eq:lpv_output}.
\end{proposition}
\begin{proof}
The proof is analogous to Proposition~\ref{prop:LMI_lpv} based on Lemma~\ref{lemma:LMI_output}.
The constraint~\eqref{eq:LMI_LPV_output2} ensures multi-convexity.
\end{proof}
\end{table*}
\section{Conclusion}
\label{sec:sum}
We have presented a procedure to compute terminal ingredients for nonlinear reference tracking MPC schemes offline.
The main novelty in this approach is that the offline computation only needs to be done once, irrespective of the setpoint or trajectory to be stabilized.
This is possible by computing parameterized terminal ingredients and approximating the nonlinear system locally as a quasi-LPV system, with the reference trajectory to be stabilized as the parameter.
Furthermore, we have shown that the reference generic offline computation enables us to design nonlinear MPC schemes that ensure optimal periodic operation despite online changing operation conditions.
We have demonstrated the applicability and advantages of the proposed procedure with numerical examples.
The extension of the proposed procedure to large scale nonlinear distributed systems using a seperable formulation is part of future work.
\subsection{Automated driving - robust reference tracking}
The following example shows the applicability of the proposed procedure to nonlinear robust reference tracking and demonstrates the performance improvement of including suitable terminal ingredients.
\subsubsection*{System model}
We consider a nonlinear kinematic bicycle model of a car
\begin{align*}
\dot{z}_1=&v\cos(\psi+\beta),\quad
\dot{z}_2=v\sin(\psi+\beta),\\
\dot{\psi}=&v/l_r\sin(\beta),\quad
\dot{v}=a,\quad
\dot{\delta}=u_{\delta},\\
\beta=&\tan^{-1}\left(\dfrac{l_r}{l_f+l_r}\tan(\delta)\right),\\
x=&[z_1,z_2,\psi,v,\delta]^{\top}\in\mathbb{R}^5,\quad u=[a,u_{\delta}]^{\top}\in\mathbb{R}^2,
\end{align*}
with the position $z_i$, the inertial heading $\psi$, the velocity $v$, the front steering angle $\delta$, the acceleration $a$ and the change in the steering angle $u_{\delta}$.
The model constants $l_f=1.4$ and $l_r=1.5$ represent the distance of the center of mass to the front and rear axle.
More details on kinematic bicycle models can be found in~\cite{kong2015kinematic}.
The (non-compact) constraint sets are given by
\begin{align*}
\mathcal{Z}_r=&\{v\in[10,50],a\in[-1,1],\delta\in[-0.4,0.4],u_{\delta}\in[-3,3]\},\\
\mathcal{Z}=&{\{v\in[5~,55],a\in[-2,2],\delta\in[-0.5,0.5],u_{\delta}\in[-6,6]\}}.
\end{align*}
\subsubsection*{Offline computations}
We consider the stage cost $Q=I_5$, $R= I_2$ and $\epsilon=0.1$ and use an Euler discretization with the step size $h=2ms$.
Computing the linearization~\eqref{eq:A_r} and using a quasi-LPV parameterization~\eqref{eq:A_paramlin} results in $\theta\in\mathbb{R}^8$, where the parameters $\theta$ consist of trigonometric functions in $\Psi,~\delta$ and are linear in the velocity $v$.
For this example, the convex approach (Prop.~\ref{prop:LMI_lpv}) is not feasible, since the simple and conservative hyperbox\footnote{%
This description does not take into account that $\sin(\psi+\beta)$ and $\cos(\psi+\beta)$ cannot be zero simultaneously.
This issue can be circumvented by considering a more detailed description of $\Theta$, e.g. using coupled ellipsoidal constraints. } description $\theta\in\Theta$ includes linearized dynamics which are not stabilizable.
For the gridding, we consider both the discrete-time and a continuous-time formulation (compare Appendix~\ref{sec:app_cont}).
In the continuous-time formulation $a$ and $u_{\delta}$ enter the LMIs affinely.
Thus, we only consider the $2^2=4$ vertices of $(a,u_{\delta})$ and grid $(\psi,v,\delta)$ using $10^3$ points.
For the discrete-time formulation~\eqref{eq:LMI} the LMIs are not affine in $u_{\delta}$ and thus we grid $(\psi,v,\delta,u_{\delta})$ using $10^3\cdot 5$ points and consider the two vertices of $a$.
The dimensions of the corresponding LMI-blocks are $(2n+m)\times (2n+m)=12\times 12$ and $(3n+m)\times (3n+m)=17\times 17$ , respectively.
The following table captures the weighting of the terminal cost and the computational effort of the proposed approach.
\begin{tabular}{c|c|c}
\multirow{2}{*}{Method}
& {Continuous-Time}&{Discrete-time}\\
& (Lemma.~\ref{lemma:LMI_cont})& (Lemma.~\ref{lemma:LMI}) \\\hline
$\#$LMIs-blocks&$10^3\cdot 2^2=4\cdot 10^3$ &$10^3\cdot 5\cdot 2=10^4$ \\\
comp.~time&14~min&33~min\\
\small{$\max_{r\in\mathcal{Z}_r}$$\lambda_{\max}(P_f(r))$}&$8.0\cdot 10^4\cdot h$&$8.4\cdot 10^4$
\end{tabular}
\begin{remark}
\label{rk:cont_discrete}
For the considered example and parameters, the continuous-time terminal cost is also valid for a zero-order hold discrete-time implementation with $h=2~ms$.
This is in general not the case.
For example if $R=10^{-4}$ or $h=10~ms$ is chosen, the terminal ingredients based on the continuous-time offline optimization are not stabilizing for the discrete-time system.
If the continuous-time offline procedure is used, the computation (and thus verification) of $\alpha$ for the discrete-time system using Algorithm~\ref{alg:offline_alpha} is crucial.
This issue is also discussed in Remark~\ref{rk:sample_hold} of Appendix~\ref{sec:app_cont}.
\end{remark}
In the following, we only consider the discrete-time terminal ingredients based on Lemma~\ref{lemma:LMI}.
Executing Algorithm~\ref{alg:offline_alpha} to ensure that $\alpha_1=10^4$ is valid takes $25$~min using $20^3\cdot 10^2\cdot 100= 8\cdot 10^7$ samples.
\subsubsection*{Robust trajectory tracking - Evasive maneuver test}
In order to demonstrate the applicability of the proposed tracking MPC scheme, we consider an evasive maneuver test (compare~ISO norm 3888-2~\cite{ISO3888}).
In this scenario a car is driving with $v=20~m/s$ and performs two consecutive lane changes to simulate the avoidance of a possible obstacle.
The basic setup, with a feasible reference trajectory $r$, additional path constraints\footnote{%
Ideally, these constraints should restrict the overall position of the vehicle.
For simplicity we treat them as (time-varying) polytopic constraints on $z_2$, that require the $z_2$ position to be within a margin of $\pm 35~cm$.
} $\mathcal{X}$ and the terminal set (projected on $z_1\times z_2$) can be seen in Figure~\ref{fig:elchtest}. The terminal set size is restricted by the input constraint on $u_\delta$ and the path constraint $\mathcal{X}$, yielding the terminal set size $\alpha=\alpha_2\approx 10^2$.
For comparison, we also computed a terminal cost for this specific \textit{given} trajectory based on an LTV description~\cite{faulwasser2011model}.
The generic offline computation results in a roughly five times larger terminal cost, which gives an indication of the conservatism.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.4\textwidth]{Plots/Reference_casadi.pdf}
\end{center}
\caption{ Evasive maneuver test: Reference trajectory $r$ (blue), terminal sets $\mathcal{X}_f(r)$ (red) and additional state constraints $\mathcal{X}$ (black).}
\label{fig:elchtest}
\end{figure}
In order to show that the proposed approach can be applied under realistic conditions, we consider additive disturbances $w(t)\in\mathbb{R}^n$ and a prediction horizon of $N=10$.
To ensure robust constraint satisfaction, we use the constraint tightening method proposed in~\cite{kohler2018novel}, which is based on the achievable contraction\footnote{%
This property is verified by computing a terminal cost, which is valid on the full constraint set $\mathcal{Z}$, compare Prop.~\ref{prop:increm} and App.~\ref{sec:app_robust}.
Analogous to the computation of $\alpha$, the numerical value of $\rho$ can be ascertained using Alg.~\ref{alg:offline_alpha}.
} rate $\rho=0.9995$.
To ensure robust recursive feasibility, the terminal set needs to be robust positively invariant, which can be ensured for $\|w(t)\|\leq \hat{w}=1.82\cdot 10^{-5}=9.1\cdot 10^{-3}h$, compare \eqref{eq:w_2} in Proposition~\ref{prop:RPI} of Appendix~\ref{sec:app_robust}.
The constraints are tightened over the prediction horizon with a scalar using the method in~\cite{kohler2018novel}
\begin{align*}
(x(k|t),u(k|t))\in(1-\epsilon_{k})\mathcal{Z},
\quad \epsilon_k=\epsilon\frac{1-{\rho}^k}{1-{\rho}},
\end{align*}
with $\epsilon=2.5\cdot 10^{-4}$.
The resulting robust tracking MPC scheme guarantees (uniform) practical exponential stability and robust constraint satisfaction, for details see Appendix~\ref{sec:app_robust} and~\cite{kohler2018novel}.
We simulated the closed-loop MPC using random disturbances $\|w(t)\|=\hat{w}$ and compared the performance to MPC without terminal constraints ($V_f=0$, UC,~\cite{kohlernonlinear19}) and MPC with terminal equality constraint ($\mathcal{X}_f(r)=x_r$, TEC).
To enable a comparison of the computational demand we fixed the number of iterations in CasADi to $1$ per time step, resulting in online computation time of approx $13$~ms for all three approaches.
The corresponding results can be seen in Figures~\ref{fig:Car_closedloop_1} and~\ref{fig:Car_closedloop_2}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.5\textwidth]{Plots/Robustclose_error_casadi.pdf}
\end{center}
\caption{ Evasive maneuver test: Closed-loop tracking stage cost for the proposed terminal constraint tracking MPC (blue,solid,QINF), a corresponding tracking MPC scheme without terminal constraints (green,dash-star,UC) and an MPC scheme with a terminal equality constraint (red,dashed,TEC)}
\label{fig:Car_closedloop_1}
\end{figure}
\begin{figure}[hbtp]
\begin{center}
\includegraphics[width=0.4\textwidth]{Plots/RobustClose_traj_casadi_comb_v2.pdf}
\end{center}
\caption{Evasive maneuver test: Closed-loop trajectory of $z_1,~z_2$ over the time interval $t\in[1.32s,1.81s]$ with the reference $r$ (black,solid), the MPC based on the proposed terminal ingredients (blue,solid,QINF), a corresponding tracking MPC scheme without terminal constraints (green,dash-star,UC) and an MPC scheme with a terminal equality constraint (red,dashed,TEC). }
\label{fig:Car_closedloop_2}
\end{figure}
The closed-loop performance (as measured by the tracking stage cost
\footnote{%
If we ignore the input tracking stage cost and only consider $\|x-x_r\|_Q^2$ as the performance, then the TEC has only $13\%$ of the tracking error of QINF and UC has $30$-times the tracking error.
If, for some reason, we would only be interested in the tracking error in the input $\|u-u_r\|_R^2$, then UC has only $48\%$ of the error of QINF and TEC has $4.5\cdot 10^3$ times the error of QINF.
}) of UC and TEC are $10$ and $3.000$ times larger than the proposed scheme with the terminal cost (QINF), compare Figure~\ref{fig:Car_closedloop_1}.
Specifically, the MPC without terminal constraints (UC) has a significant (growing) tracking error in the position (see Figure~\ref{fig:Car_closedloop_2}), since the UC with a short horizon typically leads to a slower convergence with smaller control action (as stability is not explicitly enforced).
On the other side, the terminal equality constraint MPC (TEC) has large deadbeat like input oscillations, which is a result of the terminal constraint with the short prediction horizon.
UC and TEC achieve a similar performance to QINF with $N=10$, if the prediction horizon\footnote{%
For this second comparison, we did not limit the number of iterations for UC and TEC, since we were unable to achieve a similar performance with UC using only $1$ iterations (which may be due to the lack of a good warmstart).
} is increased to $N=23$ and $N=59$, respectively.
This increases the online computational demand compared to QINF by $100\%$ and $300\%$, respectively.
The proposed MPC scheme robustly achieves a small tracking error with a short prediction horizon.
This shows that including (suitable) terminal ingredients significantly reduces the tracking error and improves the closed-loop performance, as also articulated in~\cite{mayne2013apologia}.
\subsection{Periodic reference tracking - CSTR}
\subsubsection*{System model}
We consider a continuous-time model of a continuous stirred-tank reactor (CSTR)
\begin{align*}
\begin{pmatrix}
\dot{x}_1\\\dot{x}_2\\\dot{x}_3
\end{pmatrix}
=&\begin{pmatrix}
1-x_1-10^4x_1^2\exp(\frac{-1}{x_3})-400x_1\exp(\frac{-0.55}{x_3})\\
10^4 x_1^2\exp(\frac{-1}{x_3})-x_2\\
u-x_3
\end{pmatrix},
\end{align*}
where $x_1,~x_2,~x_3$ correspond to the concentration of the reaction, the desired product, waste product and $u$ is related to the heat flux through the cooling jacket, compare~\cite{bailey1971cyclic}, \cite[Sec.~3.4]{faulwasser2018economic}.
The constraints are
\begin{align*}
\mathcal{Z}_r=&[0.05,0.45]\times[0.05,0.15]\times [0.05,0.2]\times[0.059,0.439],\\
\mathcal{Z}=&[0,1]^3\times[0.049,0.449].
\end{align*}
The discrete-time model is defined with explicit Runge-Kutta discretization of order $4$ and a sampling time\footnote{
In~\cite[Sec.~3.4]{faulwasser2018economic} a sampling time of $h=0.1$ is used.
However, with the considered fourth order explicit Runge-Kutta discretization, a sampling time of $h=0.1$ does not preserve stability of the continuous-time system.
} of $h=0.01$.
For this system, periodic operation is economically beneficial, compare~\cite{bailey1971cyclic}.
Thus, we consider the problem of tracking reachable periodic reference trajectories $r$ (Assumption~\ref{ass:ref}), corresponding to the economic operation of the plant.
\subsubsection*{Offline computations}
In the following, we illustrate the reference generic offline computation for this system.
We consider the standard quadratic tracking stage cost with $Q=I_3$, $R=10$ and use $\epsilon=0.1$.
For the continuous-time system, the Jacobian~\eqref{eq:A_r} contains four nonlinear terms, yielding the parameters
\begin{align*}
\theta_1(x)=&400\exp(-0.55/x_3),\quad
\theta_2(x)=2\cdot 10^4 x_1\exp(-1/x_3),\\
\theta_3(x)=&10^4({x_1}/{x_3})^2\exp(-1/x_3),\\
\theta_4(x)=&400\cdot 0.55{x_1}/({x_3^2})\exp(-0.55/x_3).
\end{align*}
The input $u_r$ enters the LMIs affinely.
Thus, we only consider the two vertices of $u_r$ and grid $(x_1,x_3)$ using $10^2$ points.
For the discrete-time system, the explicit description of the nonlinear dynamics $f$ and the corresponding Jacobian $A(r),~B(r)$ is complex.
Thus, we directly define the non-constant\footnote{
The derivatives $\partial f_3/\partial r$, $\partial f_1/\partial x_2$, and $\partial f_2/\partial x_2$ are constant.
} components of the Jacobian $A,~B$ as the parameters $\theta\in\mathbb{R}^6$.
We compute the hyperbox sets $\Theta,\Omega\subseteq\mathbb{R}^6$ satisfying~\eqref{eq:Theta_set} numerically.
For the discrete-time convex approach the polytopic description $\overline{\Theta}$~\eqref{eq:overline_theta} and the hyperbox description ${\Theta}\times \Omega$ (Remark~\ref{remark:box}) are considered.
For the gridding, $(x_1,x_3,u_r,u_r^+)\in\mathbb{R}^4$ is gridded using $10^4$ points, of which approximately $8.000$ satisfy the conditions~\eqref{eq:grid_r} and are considered in the optimization problem~\eqref{eq:LMI}.
The computational demand and the performance of the different methods are detailed in Table~\ref{tab:CSTR_comp}.
As expected, the gridding approach yields the smallest and least conservative terminal cost.
For this example, the convex discrete-time approach (Prop.~\ref{prop:LMI_lpv}) seems less favorable, which is mainly due to the simple description of the parameters.
Due to the small sampling time $h$ and correspondingly small set $\Omega$, the more detailed description $\overline{\Theta}$ only marginally improves the performance but significantly increases the offline computational demand.
Furthermore, the continuous-time formulation can be computed more efficiently.
We note, that the parameters $Q,~R,~h$, are chosen, such that the continuous-time control law is also stabilizing for the discrete-time implementation.
In particular, if $R$ is decreased or $h$ increased, the terminal ingredients based on the continuous-time formulation do not satisfy Assumption~\ref{ass:term} with a piece wise constant input.
Such considerations are not necessary for the discrete-time formulation, compare Remark~\ref{rk:sample_hold} in Appendix~\ref{sec:app_cont}.
\begin{table*}
\small{
\begin{tabular}{c|c|c|c|c|cc}
\multirow{2}{*}{Method}
& \multicolumn{2}{c}{Continuous-Time}&\multicolumn{3}{c}{Discrete-time}\\
&Gridding (Lemma.~\ref{lemma:LMI_cont})&{Convex (Prop.~\ref{prop:LMI_lpv_cont})}&Gridding (Lemma.~\ref{lemma:LMI})&
\multicolumn{2}{c}{Convex (Prop.~\ref{prop:LMI_lpv})}\\\hline
$\#$LMIs&$200$& $4^4=256$ &$7994$&$4^6=4096$&$6^6= 46.656$\\
computational time &12~s&10~s&783~s&356~s&3h~18min\\
\small{$\max_{r\in\mathcal{Z}_r}$$\lambda_{\max}(P_f(r))$}&$3.3\cdot10^3\cdot h$ &$8.3\cdot 10^4\cdot h$&$3.5\cdot 10^3$& $4\cdot 10^7$&$3.8\cdot 10^7$\\
\end{tabular}
\caption{Computational demand and conservatism of different offline computations - CSTR. }
\label{tab:CSTR_comp}
}
\end{table*}
In the following, we consider the discrete-time terminal ingredients based on Lemma~\ref{lemma:LMI}.
Computing $\alpha_2=0.02$ using~\eqref{eq:alpha_2_better} requires $30$~s.
Executing Algorithm~\ref{alg:offline_alpha} to ensure that $\alpha=0.02$ is valid takes $10$~min using $2\cdot 20^4\cdot 100=3.2\cdot 10^7$ samples.
In Figure~\ref{fig:CSTR_periodic} we can see an exemplary periodic trajectory and the corresponding terminal set\footnote{%
If $\alpha$ would be recomputed for the specific trajectory $r$, we would get $\alpha=0.1$.
This conservatism is a result of the fact, that the previously computed value $\alpha$ needs to be valid for \textit{every} reachable reference trajectory (Ass.~\ref{ass:ref}).
}.
The period length is $T=1144$, which corresponds to $11.44~s$, compare~\cite[Sec.~3.4]{faulwasser2018economic}.
\begin{figure}[hbtp]
\begin{center}
\includegraphics[scale=0.5]{Plots/Periodic_ref.pdf}
\end{center}
\caption{Periodic trajectory - CSTR: Reference trajectory $r$ (blue) with terminal sets $\mathcal{X}_f(r)$ (red ellipses). }
\label{fig:CSTR_periodic}
\end{figure}
We wish to emphasize that this offline computation is only done once and requires no explicit knowledge of the specific trajectory or its period length $T$.
This is in contrast to the existing methods, such as~\cite{faulwasser2011model,aydiner2016periodic} which would compute terminal ingredients for a specific reference trajectory and thus could not deal with online changing operation conditions (e.g. due to changes in the price signal~\cite{ferramosca2014economic}).
\section{Numerical examples}
\label{sec:num}
The following examples show the applicability of the proposed method to nonlinear systems and the closed-loop performance improvement when including suitable terminal ingredients.
We first illustrate the basic procedure at the example of a periodic reference tracking task for a continuous stirred-tank reactor (CSTR).
Then we demonstrate the advantages of using suitable terminal ingredients with (robust) trajectory tracking and an evasive maneuver test for a car.
Additional examples, including tracking of periodic output signals (Sec.~\ref{sec:gen_output}) with a nonlinear ball and plate system can be found in~\cite{kohler2018mpc}.
In the following examples, the offline computation is done with an Intel Core i7 using the semidefinite programming (SDP) solver SeDuMi-1.3~\cite{sturm1999using} and the online optimization is done with CasADi~\cite{andersson2019casadi}.
The offline computation can be done using both the discrete-time formulation (Sec.~\ref{sec:loc_stab}) or the continuous-time formulation (Appendix~\ref{sec:app_cont}).
Hence, we also compare the performance of these different formulations.
\input{Example_CSTR_periodic}
\input{Example_Car_Casadi}
\subsection{Nonlinear periodic tracking MPC subject to changing exogenous output references}
\label{sec:gen_output}
We assume that at time $t$ an exogenous $T$-periodic output reference signal $y_{e}(\cdot|t)\in\mathbb{R}^{p\times T}$ is given.
For some $T$-periodic reference $r(\cdot|t)=(x_r(\cdot|t),u_r(\cdot|t))\in\mathbb{R}^{(n+m)\times T}$, we define the tracking cost with respect to this output signal $y_e$ by
\begin{align*}
J_T(r(\cdot|t),y_e(\cdot|t))=&\sum_{j=0}^{T-1} \|\underbrace{h(r(j|t))}_{={y_r(j|t)}}-y_e(j|t)\|^2,
\end{align*}
with a bounded nonlinear output function $h:\mathcal{Z}_r\rightarrow\mathbb{R}^p$.
The objective is to stabilize the feasible $T$-periodic reference trajectory $r$, that minimizes $J_T$.
In~\cite{limon2008mpc,limon2018nonlinear} the issue of stabilizing the optimal setpoint for piece-wise constant output signals has been investigated.
In~\cite{limon2016mpc} periodic trajectories have been considered for the special case of linear systems.
By combining these methods with the proposed terminal ingredients, we can design a nonlinear MPC scheme that stabilizes the optimal periodic\footnote{
In the case of setpoint tracking ($T=1$), the MPC scheme reduces to~\cite{limon2018nonlinear}.
As discussed in Section~\ref{sec:increm_setpoint}, the proposed procedure can be used to design suitable terminal ingredients for setpoints.
} trajectory for periodic output reference signals, compare~\cite{kohler2018mpc}.
The scheme is based on the following optimization problem
\begin{subequations}
\label{eq:limon}
\begin{align}
&W_T(x(t),y_e(\cdot|t))\\
=&\min_{u(\cdot|t),r(\cdot|t)}J_N(x(\cdot|t),u(\cdot|t),r(\cdot|t))
+J_T(r(\cdot|t),y_e(\cdot|t))\nonumber\\
\text{s.t. }&x(k+1|t)=f(x(k|t),u(k|t)),\quad x(0|t)=x(t),\\
&(x(k|t),u(k|t))\in\mathcal{Z},\quad
x(N|t)\in\mathcal{X}_f({r}(N|t)),\\
&r(j+1|t)\in\mathcal{R}(r(j|t))\subseteq\mathcal{Z}_r,\\
&r(l+T|t)=r(l|t), ~ l=0,\dots,\max\{0,N-T\},\\
&j=0,\dots,T-1,\quad k=0,\dots,N-1. \nonumber
\end{align}
\end{subequations}
This scheme is recursively feasible, independent of the output reference signal $y_e$.
Furthermore, if the exogenous signal $y_e$ is $T$-periodic the closed-loop system is stable.
Additionally, if a convexity and continuity condition on the set of feasible periodic orbits and the output function $h$ is satisfied~\cite[Ass.~5]{kohler2018mpc}, then the optimal reachable periodic trajectory is (uniformly) exponentially stable for the resulting closed-loop system.
Thus, the terminal ingredients enable us to implement a nonlinear version of the tracking scheme in~\cite{limon2008mpc,limon2016mpc}, that ensures exponential stability of the optimal (periodic) operation.
More details on the theoretical properties and numerical examples can be found in~\cite{kohler2018mpc}.
Although the consideration of general non-periodic trajectories is still an open issue, we conjecture that the approach can be extended to any class of finitely parameterized reference trajectories.
\section{Nonlinear MPC subject to changing operation conditions}
\label{sec:ext}
Many control problems are more general than the reference tracking considered in Section~\ref{sec:MPC}.
One challenge includes tracking and output regulation with exogenous signals in order to accommodate online changing operation conditions.
For this set of problems, the reference $r$ might not satisfy Assumption~\ref{ass:ref} (due to sudden changes and unreachable signals), compare~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear}.
More generally, the minimization of a possibly online changing and non-convex economic cost is a (non-trivial) control problem which is often encountered, compare~\cite{fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic}.
One promising method to solve these problems is the simultaneous optimization of an artificial reference, as done in~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear,fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic}.
Compared to a standard reference tracking MPC formulation such as~\eqref{eq:MPC}, these schemes ensure recursive feasibility despite changes in exogenous signals (such as the desired output reference or the economic cost).
In this section, we show how the reference generic terminal ingredients can be used to design nonlinear MPC schemes that reliably operate under changing operating conditions, as an extension and combination of the ideas in~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear,fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic}.
In particular, we present a scheme that exponentially stabilizes the periodic trajectory which best tracks an exogenous output signal.
The extension of the economic MPC schemes~\cite{fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic} to periodic artificial trajectories based on the reference generic terminal ingredients is beyond the scope of this work and part of current research.
\input{Generic_Operation_1_new}
\subsection{(Local) Incremental exponential stabilizability}
\label{sec:loc_stab_0}
In the following we clarify the connection between incremental stabilizability properties and the terminal ingredients.
\begin{definition}
\label{def:increm_stab}
A set of reference trajectories $r$ specified by some dynamic inclusion $r(t+1)\in\mathcal{R}(r(t))$ is locally incrementally exponentially stabilizable for the system~\eqref{eq:sys}, if there exist constants $\rho\in(0,1),M,c>0$ and a control law $\kappa(x,r)$, such that for any initial condition
satisfying $\|x(0)-x_r(0)\|\leq c$, the trajectory $x(t)$ with $x(t+1)=f(x(t),\kappa(x(t),r(t)))$ satisfies $\|x(t)-x_r(t)\|\leq M\rho^t\|x(0)-x_r(0)\|$,~$\forall t\geq 0$.
\end{definition}
This definition is closely related to the concept of universal exponential stabilizability~\cite{manchester2017control}, which characterizes the stabilizability of arbitrary trajectories in continuous-time.
One of the core differences in the definitions is the treatment of constraints, i.e. we study stabilizability of classes of trajectories $r$ that satisfy certain constraints, compare Assumption~\ref{ass:ref} and Remark~\ref{rk:ref}.
This difference is crucial when discussing local versus global stabilizability and constrained control.
The following proposition shows that the conditions in Lemma~\ref{lemma:lpv} directly imply local incremental exponential stabilizability of the reference trajectory.
\begin{proposition}
\label{prop:increm}
Suppose that there exist matrices $P_f(r),~K_f(r)$ that satisfy the conditions in Lemma~\ref{lemma:lpv}.
Then the control law $k_f(x,r)=u_r+K_f(r)(x-x_r)$ locally incrementally exponentially stabilizes any reference $r$ satisfying Assumption~\ref{ass:ref}.
\end{proposition}
\begin{proof}
The following proof follows the arguments of~\cite[Prop.~1,2]{kohlernonlinear19}.
For any $\|x(0)-x_r(0)\|\leq c$ with $c=\sqrt{\alpha/c_{u}}$, we have $x(0)\in\mathcal{X}_f(r)$, with $\alpha,~c_{u}$ according to Lemma~\ref{lemma:lpv}.
Thus, the terminal cost $V_f(x,r)$ is a local incremental Lyapunov function that satisfies
\begin{align*}
V_f(x(t+1),r(t+1))\leq \rho^2 V_f(x(t),r(t)),~ \rho^2=1-\dfrac{\lambda_{\min}(Q)}{c_u},
\end{align*}
and thus
\begin{align*}
\|x(t)-x_r(t)\|\leq {\rho}^tM \|x_r(0)-x(0)\|,\quad M=\sqrt{{c_u}/{c_l}}.
\end{align*}
\end{proof}
\begin{remark}
\label{rk:increm}
This result establishes local incremental stabilizability with the incremental Lyapunov function $V_f(x,r)$ based on properties of the linearization, compare~\cite[Prop.~1]{kohlernonlinear19}.
This system property is a natural extension of previous works on incremental stability and corresponding incremental Lyapunov functions, see~\cite{angeli2002lyapunov},~\cite[Ass.~1]{kohlernonlinear19}.
This property implies stabilizability of $(A(r),~B(r))$ around any (fixed) steady-state $r^+=r$, but it does not necessarily imply stabilizability of $(A(r),~B(r))$ for arbitrary $r\in\mathcal{Z}_r$, as $P_f(r)$ might decrease along the trajectory.
For continuous-time systems, an analogous result exists based on contraction metrics and universal stabilizability~\cite{manchester2017control}.
\end{remark}
The following proposition shows that in the absence of constraints we recover non-local results similar to~\cite{manchester2017control}.
\begin{proposition}
\label{prop:univ}
Consider $\mathcal{Z}_r=\mathcal{Z}=\mathbb{R}^{n+m}$.
Suppose that there exist matices $P_f(r),~K_f(r)$ that satisfy the conditions in Lemma~\ref{lemma:lpv}.
Assume further that $c_lI\leq P_f(r)\leq c_uI$ for all $r\in\mathbb{R}^{n+m}$ with some constants $c_l,~c_u$ and $K_f(r)=K(x_r)$.
Then any reference $r$ satisfying Assumption~\ref{ass:ref} is exponentially incrementally stabilizable with the control law
\begin{align*}
\kappa(x,r)=K(x)x-K(x_r)x_r+u_r,
\end{align*}
i.e., for any initial condition $x(0)\in\mathbb{R}^n$ the state trajectory $x(t+1)=f(x(t),\kappa(x(t),r(t)))$ satisfies $\|x(t)-x_r(t)\|\leq M\rho^t\|x(0)-x_r(0)\|$.
\end{proposition}
\begin{proof}
Consider an auxiliary (pre-stabilized) system defined by $\tilde{f}(x,v)=f(x,K(x)x+v)$.
Consider a reference $r$ generated by some input trajectory $u_r$ with the system dynamics~\eqref{eq:sys} (Ass.~\ref{ass:ref}) and some initial condition $x_r(0)$ resulting in the state reference $x_r$.
Now, consider a reference $\tilde{r}$ generated by the input $v_r(t)=u_r(t)-K(x_r(t))x_r(t)$ with the system dynamics according to $\tilde{f}$ and the same initial condition.
Due to the definition of the auxiliary system we have $\tilde{x}_r(t)=x_r(t)$, $\forall t\geq 0$.
For an arbitrary, but fixed input $v$, stability of the reference trajectory is equivalent to contractivity of the nonlinear time-varying system $\tilde{f}(x,t)$.
This can be established with the contractivity metric $P_f(r(t))=P_f(x_r(t),t)$, compare~\cite{lohmiller1998contraction}.
\end{proof}
In the absence of constraints, it is crucial that $P_f$ has a constant lower and upper bound.
If the matrix $K_f$ depends on the full reference $r$ (not just $x_r$), the controller $\kappa$ in Proposition~\ref{prop:univ} is not necessarily well defined.
\begin{remark}
\label{rk:track_vs_stab}
The relation between the controller $k_f$ (Prop.~\ref{prop:increm}) and $\kappa$ (Prop.~\ref{prop:univ}), is that of reference tracking versus pre-stabilization.
The first one is more natural in the context of tracking MPC and contains existing results for the design of terminal ingredients as special cases~\cite{chen1998quasi,faulwasser2011model,aydiner2016periodic}.
The second controller $\kappa$ allows for non-local stability results and is more suited for unconstrained control problems~\cite{manchester2017control}.
For constant matrices $K$ the two controllers are equivalent, but the incremental Lyapunov functions (and thus terminal costs) are differently parameterized ($P_f(r)$, $P_f(x,u)$).
\end{remark}
\begin{remark}
\label{rk:term_other}
The problem of computing reference generic terminal ingredients is equivalent to computing an incrementally stabilizing controller and is thus strongly related to the computation of robust positive invariant (RPI) tubes in nonlinear robust MPC schemes, compare~\cite{bayer2013discrete,kohler2018novel}.
For comparison, in~\cite{yu2010robust,yu2013tube} constant matrices $P_f,~K_f$ are computed that certify incremental stability for continuous-time systems (by considering small Lipschitz nonlinearities or by describing the linearization as a convex combination of different linear systems).
This approach can be directly extended to more general nonlinear systems using the proposed terminal ingredients.
In particular, by changing the stage cost to
\begin{align*}
\ell(x,u,r)=\|u-u_r+K(x_r)x_r-K(x)x\|_R^2
\end{align*}
one can design a nonlinear version of~\cite{chisci2001systems}, compare also~\cite{kohler2018novel}.
A detailed description of a corresponding nonlinear robust tube based (tracking) MPC scheme based on incremental stabilizability can be found in~\cite[App.~A]{JK_QINFJournal_extended}.
\end{remark}
\begin{remark}
\label{rk:non_quadratic}
In case a system is not exponentially stabilizable~\cite{muller2017quadratic}, it might be possible to make a nonlinear transformation resulting in a quadratically stabilizable system.
\end{remark}
\subsection{Sufficient conditions based on the linearization}
We denote the Jacobian of $f$ evaluated around an arbitrary point $r\in\mathcal{Z}_r$ by
\begin{align}
\label{eq:A_r}
A(r)=\left.\left[\dfrac{\partial f}{\partial x}\right]\right|_{(x,u)=r},\quad B(r)=\left.\left[\dfrac{\partial f}{\partial u}\right]\right|_{(x,u)=r}.
\end{align}
The following lemma establishes local incremental properties of the nonlinear system dynamics based on the linearization.
\begin{lemma}
\label{lemma:lpv}
Suppose that $f$ is twice continuously differentiable.
Assume that there exists a matrix $K_f(r)\in\mathbb{R}^{m\times n}$ and a positive definite matrix $P_f(r)\in\mathbb{R}^{n\times n}$ continuous in $r$, such that for any $r\in\mathcal{Z}_r$, $r^+\in\mathcal{R}(r)$, the following matrix inequality is satisfied
\begin{align}
\label{eq:lpv}
&(A(r)+B(r)K_f(r))^\top P_f(r^+)(A(r)+B(r)K_f(r))\\
\leq& P_f(r)-(Q+K_f(r)^\top R K_f(r))-\epsilon I_n\nonumber
\end{align}
with some positive constant $\epsilon$.
Then there exists a sufficiently small constant $\alpha$, such that $P_f,~K_f$ satisfy Assumption~\ref{ass:term}.
\end{lemma}
\begin{proof}
The proof is very much in line with the result for setpoints in~\cite{chen1998quasi,rawlings2017model}.
First we show satisfaction of the decrease condition~\eqref{eq:term_dec} and then constraint satisfaction~\eqref{eq:term_con}. \\
\textbf{Part I:}
Denote $\Delta x:=x-x_r$ and $\Delta u:=K_f(r)\Delta x$.
Using a first order Taylor approximation at $r=(x_r,u_r)$, we get
\begin{align*}
f(x,k_f(x,r))={f(x_r,u_r)}+A(r)\Delta x+B(r)\Delta u+\Phi_r(\Delta x),
\end{align*}
with the remainder term $\Phi_r$.
The terminal cost satisfies
\begin{align}
\label{eq:lpv_1}
&V_f(x^+,r^+)=\|f(x,u)-f(x_r,u_r)\|_{P_f(r^+)}^2\nonumber\\
=&\|(A(r)+B(r)K_f(r))\Delta x+\Phi_r(\Delta x)\|_{P_f(r^+)}^2\nonumber\\
\leq&\|(A(r)+B(r)K_f(r))\Delta x\|_{P_f(r^+)}^2\nonumber
+\|\Phi_r(\Delta x)\|_{P_f(r^+)}^2\nonumber\\
&+2\|\Phi_r(\Delta x)\|_{P_f(r^+)}\|(A(r)+B(r)K_f(r))\Delta x\|_{P_f(r^+)}\nonumber\\
\stackrel{\eqref{eq:lpv}}{\leq}&V_f(x,r)-\epsilon\|\Delta x\|^2-\ell(x,k_f(x,r),r)+\|\Phi_r(\Delta x)\|_{P_f(r^+)}^2\nonumber\\
&+2\|\Phi_r(\Delta x)\|_{P_f(r^+)}{\|(A(r)+B(r)K_f(r))\Delta x\|_{P_f(r^+)}}.
\end{align}
Using the continuity of $P_f(r),~K_f(r)$ and the compactness of the constraint set $\mathcal{Z}_r$, there exist finite constants
\begin{align}
\label{eq:c_u}
&c_{u}=\max_{r\in\mathcal{Z}_r}\lambda_{\max}(P_f(r)),\quad c_{l}=\min_{r\in\mathcal{Z}_r}\lambda_{\min}(P_f(r)),\\
\label{eq:k_u}
&k_{u}=\max_{r\in\mathcal{Z}_r}\|K_f(r)\|,\\
&c_{u,2}=\max_{r\in\mathcal{Z}_r}\lambda_{\max}(P_f(r)-(\epsilon I+Q+K_f(r)^\top RK_f(r)) ). \nonumber
\end{align}
Suppose that the remainder term $\Phi_r$ is locally Lipschitz\footnote{
In line with existing procedures~\cite{chen1998quasi}, we first deriving a sufficient local Lipschitz bound $L_{\Phi}^*$ and then obtain a local region $\alpha_1$~\eqref{eq:alpha_1}.
Alternatively, it is possible to directly use the quadratic bound $\|\Phi_r(\Delta x)\|\leq c\|\Delta x\|^2$ and work with higher order terms to obtain $\alpha_1$, compare~\cite[Prop.~1]{kohlernonlinear19}.
} continuous in the terminal set with a constant $L_{\Phi,\alpha}$ satisfying
\begin{align}
\|\Phi_r(\Delta x)\|\leq L_{\Phi,\alpha}\|\Delta x\|,\nonumber\\
\label{eq:lpv_2}
L_{\Phi,\alpha}\leq L_{\Phi}^*:=\sqrt{\dfrac{c_{u,2}+\epsilon}{c_u}}-\sqrt{\dfrac{c_{u,2}}{c_u}}.
\end{align}
Then we have
\begin{align*}
&~~\|\Phi_r(\Delta x)\|_{P_f(r^+)}^2\\
&~~+2\|\Phi_r(\Delta x)\|_{P_f(r^+)}\|(A(r)+B(r)K_f(r))\Delta x\|_{P_f(r^+)}\\
&\stackrel{\eqref{eq:lpv}\eqref{eq:c_u}\eqref{eq:lpv_2}}{\leq} \left(L_{\Phi,\alpha}^2c_u+2L_{\Phi,\alpha} \sqrt{c_u}\sqrt{c_{u,2}}\right)\|\Delta x\|^2\\
&~=~\left(c_u\left(L_{\Phi,\alpha}+\sqrt{\frac{c_{u,2}}{c_u}}\right)^2-c_{u,2}\right)\|\Delta x\|^2\stackrel{\eqref{eq:lpv_2}}{\leq}\epsilon \|\Delta x\|^2,
\end{align*}
which in combination with~\eqref{eq:lpv_1} implies the desired inequality~\eqref{eq:term_dec}.
Twice continuous differentiability of $f$ in combination with compactness of $\mathcal{Z}$ implies that there exists some constant $T$ with
\begin{align*}
\|\Phi_r(\Delta x)\|\leq T\left(\|\Delta x\|^2+\|\Delta u\|^2\right)\stackrel{\eqref{eq:k_u}}{\leq} T(1+k_u^2)\|\Delta x\|^2.
\end{align*}
Using $\|\Delta x\|\leq \sqrt{\frac{\alpha}{c_l}}$ from the terminal constraint, we get~\eqref{eq:lpv_2} for all $\alpha\leq \alpha_1$ with
\begin{align}
\label{eq:alpha_1}
\alpha_1:=c_l\left(\dfrac{L_{\Phi}^*}{T(1+k_u^2)}\right)^2.
\end{align}
\textbf{Part II:} Constraint satisfaction:
The terminal constraint $\|\Delta x\|_{P_f(r)}^2\leq \alpha$ in combination with~\eqref{eq:c_u},~\eqref{eq:k_u} implies
\begin{align*}
(\Delta x,~\Delta u)\in\mathcal{B}(\alpha)=\left\{z\in\mathbb{R}^{n+m}|~\|z\|^2\leq \frac{\alpha}{c_l}\left(1+k_u^2\right)\right\}.
\end{align*}
Given $\mathcal{Z}_r\subseteq\text{Int}(\mathcal{Z})$, there exists a small enough $\alpha_2$ such that
\begin{align}
\label{eq:alpha_2}
(x,u)=r+(\Delta x,\Delta u)\subseteq \mathcal{Z}_r\oplus\mathcal{B}(\alpha)\subseteq\mathcal{Z},~ \forall \alpha\leq \alpha_2.
\end{align}
\end{proof}
As a summary, given matrices $P_f,~K_f$ satisfying~\eqref{eq:lpv}, we can compute a local Lipschitz bound~\eqref{eq:lpv_2}, which in turn implies a maximal terminal set size $\alpha_1$.
Similarly, the constraint sets $\mathcal{Z}$ and $\mathcal{Z}_r$ in combination with $K_f,~P_f$ imply an upper bound $\alpha_2$ to ensure constraint satisfaction.
Then Assumption~\ref{ass:term} is satisfied for any $\alpha\leq \min\{\alpha_1,~\alpha_2\}$.
This result is an extension of~\cite{chen1998quasi,rawlings2017model} to arbitrary dynamic references.
\subsection{Quasi-LPV based procedure}
Lemma~\ref{lemma:lpv} states that matrices satisfying inequality~\eqref{eq:lpv} also satisfy Assumption~\ref{ass:term} with a suitable terminal set size $\alpha$.
In the following, we formulate computationally tractable optimization problems to compute matrices that satisfy the conditions in Lemma~\ref{lemma:lpv}.
The following Lemma transforms the conditions in~\eqref{eq:lpv} to be linear in the arguments.
\begin{lemma}
\label{lemma:LMI}
Suppose that there exists matrices $X(r)$,~$Y(r)$ continuous in $r$, that satisfy the constraints in~\eqref{eq:LMI} for all $r\in\mathcal{Z}_r,~r^+\in\mathcal{R}(r)$.
Then $P_f=X^{-1}$, $K_f=YP_f$ satisfy~\eqref{eq:lpv}.
\end{lemma}
\begin{proof}
The proof is standard, compare~\cite{boyd1994linear} and Lemma~~\ref{lemma:LMI_output} in the Appendix.
\begin{comment}
Define $X(r)=P_f(r)^{-1}$ and multiply~\eqref{eq:lpv} from left and right with $X(r)$
\begin{align*}
(A(r)X(r)+B(r)Y(r))^\top X(r^+)^{-1} (A(r)X(r)+B(r)Y(r))-X(r)+X(r)QX(r)+Y(r)^\top RY(r)\leq 0.
\end{align*}
This can be equivalently written as
\begin{align*}
X(r)-
\begin{pmatrix}
A(r)X(r)+B(r)Y(r)\\Q^{1/2}X(r)\\R^{1/2}Y(r)
\end{pmatrix}^{\top}
\begin{pmatrix}
X(r^+)&0&0\\
0&I&0\\
0&0&I
\end{pmatrix}
\begin{pmatrix}
A(r)X(r)+B(r)Y(r)\\Q^{1/2}X(r)\\R^{1/2}Y(r)
\end{pmatrix}
\geq 0
\end{align*}
Using the Schur complement this reduces to~\eqref{eq:LMI}, which is linear in the matrices $X,~Y$.
\end{comment}
\end{proof}
The optimization problem~\eqref{eq:LMI} is convex, linear in $X,~Y$ and minimizes the worst-case terminal cost $P_f(r)\leq X_{\min}^{-1}$.
So far, the result is only conceptual, since~\eqref{eq:LMI} is an infinite programming problem (infinite dimensional optimization variables with infinite dimensional constraints).
In particular, we need a finite parameterization of $X,~Y$ and the infinite constraints need to be converted into a finite set of sufficient constraints.
\begin{remark}
\label{rk:SOS}
One solution to this problem would be sum-of-squares (SOS) optimization~\cite{parrilo2003semidefinite}.
Assuming $A,~B$ are polynomial, consider matrices $X,~Y$ polynomial in $r$ (with a specified order $d$) and ensure that the matrix in~\eqref{eq:LMI} is SOS.
A similar approach is suggested in~\cite{manchester2017control} to find a control contraction metric (CCM) for continuous-time systems (which is a strongly related problem).
This approach is not pursued here since most systems require a polynomial of high order to approximate the nonlinear dynamics and the computational complexity grows exponentially in $n^d$, thus prohibiting the practical application.
The connection between CCM and LPV gain-scheduling design is discussed in~\cite{wang2019comparison}.
\end{remark}
We approach this problem from the perspective of quasi-LPV systems and gain-scheduling~\cite{rugh2000research}.
First, write the Jacobian~\eqref{eq:A_r} as
\begin{align}
\label{eq:A_paramlin}
A(r)=A_0+\sum_{j=1}^{p} \theta_j(r) A_j,~
B(r)=B_0+\sum_{j=1}^{p} \theta_j(r) B_j,
\end{align}
with some nonlinear (continuously differentiable) parameters $\theta\in\mathbb{R}^p$.
This can always be achieved with $p\leq n(n+m)$.
We impose the same structure on the optimization variables with
\begin{align}
\label{eq:X}
X(r)=X_0+\sum_{j=1}^p \theta_j(r) X_j,~ Y(r)=Y_0+\sum_{j=1}^p \theta_j(r) Y_j.
\end{align}
\begin{remark}
\label{rk:affine}
For input affine systems of the form $f(x,u)=f_x(x)+B u$, the Jacobian~\eqref{eq:A_paramlin} and correspondingly the parameters $\theta_i$ only depend on $x_r$.
Thus, the resulting terminal ingredients are solely parameterized by the state $x_r$.
\end{remark}
Using the parameterization \eqref{eq:A_paramlin}-\eqref{eq:X}, \eqref{eq:LMI} contains only a finite number of optimization variables, but still needs to be verified for all $r\in\mathcal{Z}_r,~r^+\in\mathcal{R}(r)$.
There are two options to deal with this: convexifying the problem or gridding the constraint set.
\subsubsection{Convexify}
\begin{table*}
\small{
\begin{subequations}
\label{eq:LMI}
\begin{align}
\min_{X(r),Y(r),X_{\min}}& - \log \det X_{\min}\\
\text{s.t. }&\begin{pmatrix}
X(r)&X(r)A(r)^\top+Y(r)^\top B(r)^\top&(Q+\epsilon)^{1/2}X(r)&(R^{1/2}Y(r))^\top\\
*&X(r^+)&0&0\\
*&*&I&0\\
*&*&*&I
\end{pmatrix}\geq 0,\\
&X_{\min}\leq X(r),\\
\label{eq:LMI_r}
&\forall r\in\mathcal{Z}_r,~r^+\in\mathcal{R}(r).
\end{align}
\end{subequations}
\begin{subequations}
\label{eq:LMI_LPV}
\begin{align}
\min_{X_i,Y_i,\Lambda_i,X_{\min}}& - \log\det X_{\min}\\
\label{eq:LMI_LPV1}
\text{s.t. }&\begin{pmatrix}
X(\theta)&X(\theta)A(\theta)^\top+Y(\theta)^\top B(\theta)^\top&(Q+\epsilon)^{1/2}X(\theta)&(R^{1/2}Y(\theta))^\top\\
*&X(\theta^+)&0&0\\
*&*&I&0\\
*&*&*&I
\end{pmatrix}
- \begin{pmatrix}\sum_{i=1}^p\theta_i^2\Lambda_i&0\\0&0\end{pmatrix}\geq 0,\\
&X_{\min}\leq X(\theta),\quad \forall (\theta,\theta^+)\in\text{Vert}(\overline{\Theta}), \\
\label{eq:LMI_LPV2}
&\begin{pmatrix}
0&(A_iX_i+B_iY_i)^\top\\
(A_iX_i+B_iY_i)&0
\end{pmatrix}-\Lambda_i\leq 0,\quad \Lambda_i\geq 0,\quad i=1,\dots,p.
\end{align}
\end{subequations}
}
\end{table*}
In order to convexify~\eqref{eq:LMI}, we match the constraint sets $\mathcal{Z}_r,~\mathcal{R}(r)$ on the reference $r$ to polytopic constraint sets $\Theta,~\Omega$ on the parameters $\theta$.
The polytopic sets $\Theta,~\Omega(\theta)$ need to satisfy
\begin{align}
\label{eq:Theta_set}
\theta(r)\in&\Theta,\quad \forall r\in\mathcal{Z}_r,\\
\theta(r^+)\in&\Omega(\theta(r)),\quad \forall r^+\in\mathcal{R}(r). \nonumber
\end{align}
Computing a set $\Theta$, such that $\theta(r)\in\Theta$ for all $r\in\mathcal{Z}_r$ can be achieved by considering a hyperbox $\Theta=\{\theta\in\mathbb{R}^p|~\theta_i\in[\underline{\theta}_i,\overline{\theta}_i]\}$.
For $\Omega$, a simple approach is $\Omega(\theta)=\{\theta\}\oplus\Omega$, where $\Omega$ is a hyperbox that encompasses the maximal change in the parameters $\theta$ in one time step, i.e. $\Omega=\{\Delta \theta\in\mathbb{R}^p|~\Delta\theta_i\in[\underline{v}_i,\overline{v}_i]\}$.
We denote the joint polytopic constraint set by
\begin{align}
\label{eq:overline_theta}
(\theta,\theta^+)\in\overline{\Theta}=\{(\theta,\theta^+)\in \Theta\times\Theta|~\theta^+\in\{\theta\}\oplus\Omega\},
\end{align}
which consists of $6^p$ vertices.
The following proposition provides a simple convex procedure to compute a terminal cost, by solving a finite number of LMIs.
\begin{proposition}
\label{prop:LMI_lpv}
Suppose that there exist matrices $X_i,~Y_i,~\Lambda_i,~X_{\min}$ that satisfy the constraints in~\eqref{eq:LMI_LPV}.
Then the matrices
\begin{align*}
P_f(r)=&X^{-1}(r),\quad K_f(r)=Y(r)P_f(r),
\end{align*}
satisfy~\eqref{eq:lpv}, with $X,~Y$ according to~\eqref{eq:X}.
\end{proposition}
\begin{proof}
Due to Lemma~\ref{lemma:LMI}, it suffices to show that $X(r),~Y(r)$ satisfy the constraints in~\eqref{eq:LMI}.
Due to the definition of the set $\overline{\Theta}$~\eqref{eq:overline_theta} and $\Lambda_i\geq 0$, any solution that satisfies the constraints~\eqref{eq:LMI_LPV1} over all $(\theta,\theta^+)\in\overline{\Theta}$, also satisfies the constraints~\eqref{eq:LMI} for all $r\in\mathcal{Z}_r,~r^+\in\mathcal{R}(r)$.
It remains to show that it suffices to check the inequality on the vertices of the constraint set $\overline{\Theta}$.
This last result is a consequence of multi-convexity~\cite[Corollary 3.2]{apkarian2000parameterized}.
In particular, if a function $f$ is multi-concave along the edges of the constraint set $\overline{\Theta}$, then it attains its minimum at a vertex of $\overline{\Theta}$ and thus it suffices to verify~\eqref{eq:LMI_LPV1} over the vertices of $\overline{\Theta}$.
The edges of $\overline{\Theta}$~\eqref{eq:overline_theta} are characterized by $\{\theta_i,~\theta_i^+,~\theta_i^+-\theta_i\}$, $i=1,\dots,p$.
A function is multi-concave if the second derivative w.r.t. these directions is negative-semi-definite, compare~\cite[Corollary~3.4]{
|
apkarian2000parameterized}.
Similar to~\cite[Corollary~3.5]{apkarian2000parameterized}, the additional constraint~\eqref{eq:LMI_LPV2} ensures that the function is multi-concave.
Thus, it suffices to verify inequality~\eqref{eq:LMI_LPV1} on the vertices of the constraint set $\overline{\Theta}$.
\end{proof}
\begin{remark}
\label{remark:box}
The result in Proposition~\ref{prop:LMI_lpv} remains valid, if the set $\overline{\Theta}$ in~\eqref{eq:overline_theta} is replaced by the set $\overline{\Theta}=\Theta\times (\Theta\oplus\Omega)$.
This set has only $4^p$ vertices and the induced conservatism of this approximation is negligible if $\Omega$ is small compared to $\Theta$.
\end{remark}
\subsubsection{Gridding}
A common heuristic to ensure that parameter dependent LMIs such as~\eqref{eq:LMI} hold for all $(r,r^+)$ is to consider the constraints on sufficiently many sample points in the constraint set, compare e.g. \cite[Sec.~4.2]{apkarian2000parameterized}.
Due to continuity, the constraint is typically satisfied on the full constraint set if it holds on a sufficiently fine grid.
For this method it is crucial that satisfaction of~\eqref{eq:term_dec} is verified by using a fine grid (compare Algorithm~\ref{alg:offline_alpha}).
The gridding consists of a grid over all possible state and input combinations $(r,r^+)$, i.e., all considered points satisfy
\begin{align}
\label{eq:grid_r}
r,~r^+\in\mathcal{Z}_r, \quad r^+\in\mathcal{R}(r),\quad \mathcal{R}(r^+)\neq\emptyset.
\end{align}
For the simple structure $\mathcal{R}(r)$ in Assumption~\ref{ass:ref} this can be achieved by gridding $r$, computing $x_r^+=f(x_r,u_r)$, and considering all $u_r^+$, such that $(x_r^+,u_r^+)\in\mathcal{Z}_r$ and $(f(x_r^+,u_r^+),\tilde{u}_r)\in\mathcal{Z}_r$ with some $\tilde{u}_r$.
This approach does not introduce additional conservatism, but is computationally challenging for high dimensional systems.
As discussed in Remark~\ref{rk:ref} we can include additional constraints on the reference, which makes the offline computation less conservative.
If some parameters, e.g. $u_r$, enter the LMIs affinely and are subject to polytopic constraints, it suffices to consider the vertices of the corresponding constraint set.
The advantage of the convex procedure (compared to the gridding) is that it typically scales better with the system dimension.
This comes at the cost of additional conservatism due to the construction of the set $\overline{\Theta}$ and the additional multi-convexity constraint~\eqref{eq:LMI_LPV2}.
The computational demand can be reduced by considering (block-)diagonal multipliers $\Lambda_i=\lambda_i I$.
It can often be beneficial to consider a combination of the two approaches, i.e. grid in some dimensions and conservatively convexify in others.
The advantages and applicability of both approaches are explored in more detail in the numerical examples in Section~\ref{sec:num}.
The main result is that we can formulate the offline design procedure similar to the gain scheduling synthesis of (quasi)-LPV systems and thus can draw on a well established field to formulate\footnote
If the parameters $\theta_i$ are chosen based on a vertex representation ($\theta_i\geq 0,\sum_{i=1}^p\theta_i=1$) the multi-convexity condition~\eqref{eq:LMI_LPV2} can be replaced by positivity conditions of the polynomials, compare for example~\cite{montagner2005gain}.
In~\cite{mao2003robust} a convexification with an additional matrix is considered.
More elaborate methods to formulate LPV synthesis with finite LMIs can be found in~\cite{scherer2000linear}.
} offline LMI procedures, compare~\cite{rugh2000research}.
\subsection{Non-conservative terminal set size $\alpha$}
\label{sec:alpha}
The terminal set size $\alpha$ derived in Lemma~\ref{lemma:lpv} can be quite conservative.
In the following we illustrate how a non conservative value $\alpha$ can be computed (given $P_f$ and $K_f$).
\subsubsection{Constraint satisfaction - $\alpha_2$}
Assume that we have polytopic constraints of the form $\mathcal{Z}=\{r=(x,u)|L_r r\leq l\}$.
The constant $\alpha_2$, with the property that $\alpha\leq \alpha_2$ implies constraint satisfaction~\eqref{eq:term_con}, can be computed with
\begin{align}
\label{eq:alpha_2_better}
\alpha_2&:=\max_{\alpha}~ \alpha\\
\text{s.t. }& \|P_f(r)^{-1/2}\begin{pmatrix}I_n&K_f^\top(r)\end{pmatrix}L_{r,j}^\top\|^2\alpha\leq (l_j-L_{r,j} r)^2,\nonumber\\
& \forall r\in\mathcal{Z}_r,\quad j=1,\dots n_z. \nonumber
\end{align}
This problem can be efficiently solved by girdding the constraint set $\mathcal{Z}_r$, solving the resulting linear program (LP) for each point $r$ and taking the minimum.
In the special case that $P_f,~K_f$ are constant this reduces to one small scale LP.
\subsubsection{Local Stability - $\alpha_1$}
Determining a non-conservative constant $\alpha_1$, related to the local Lyapunov function $V_f$ can be significantly more difficult.
For comparison, in the setpoint stabilization case a non-convex optimization problem is formulated to check whether~\eqref{eq:term_dec} holds for a specific value of $\alpha_1$, compare~\cite[Rk.~3.1]{chen1998quasi}.
In a similar fashion, we consider the following algorithm\footnote
Algorithm~\ref{alg:offline_alpha} can be thought of as a sampling based strategy to solve this non-convex optimization problem considered in~\cite[Rk.~3.1]{chen1998quasi}.
Using standard convex solvers, like sequential quadratic programming (SQP), yield a faster solution, but can get stuck in local minima.
This is dangerous for this problem, since the local minima correspond to values $\alpha$ that do not satisfy Assumption~\ref{ass:term}.
Alternatively, nonlinear Lipschitz-like bounds can be used to reduce the conservatism, compare~\cite{griffith2018robustly} (which, however, also use sampling).
} to determine whether~\eqref{eq:term_dec} holds for all $\alpha\leq \alpha_1$:
\begin{algorithm}[H]
\caption{Offline computation - Local stability $\alpha_1$}
\label{alg:offline_alpha}
\begin{algorithmic}[1]
\Statex Given a candidate constant $\alpha_1$:
\Statex \textbf{Grid: } {Select $(r,r^+)$ satisfying~\eqref{eq:grid_r}}
\State Evaluate $P_f(r),P_f(r^+),K_f(r)$ using~\eqref{eq:X}.
\State Generate random vectors $\Delta x_i$: with $\|\Delta x_i\|_{P_f(r)}^2\leq \alpha_1$.
\State Check if $x_i=x_r+\Delta x_i$ satisfies~\eqref{eq:term_dec}.
\end{algorithmic}
\end{algorithm}
Starting with $\alpha_1=\alpha_2$, the value $\alpha_1$ is iteratively decreased until all considered combination ($r,r^+,x_i$) satisfy~\eqref{eq:term_dec}.
The overall offline procedure to compute the terminal ingredients (Ass.~\ref{ass:term}) is summarized as follows:
\begin{algorithm}[H]
\caption{Offline computation}
\label{alg:offline}
\begin{algorithmic}[1]
\State Define $\theta$ corresponding to the linearization~\eqref{eq:A_paramlin}.
\State LMI computation using gridding or convexification:
\Statex \textbf{Convex}: Determine hyperbox sets $\Theta$, $\Omega$ satisfying~\eqref{eq:Theta_set}.
\Statex \quad \quad Solve~\eqref{eq:LMI_LPV} using $\overline{\Theta}$ according to~\eqref{eq:overline_theta} or Remark~\ref{remark:box}.
\Statex \textbf{Gridding}: Select $(r_i,r_i^+)$ satisfying~\eqref{eq:grid_r}.
\Statex \quad\quad Solve~\eqref{eq:LMI} for all $(r_i,~r_i^+)$.
\State Compute size of the terminal set $\alpha=\min\{\alpha_1,\alpha_2\}$:
\Statex \quad a):compute $\alpha_1$ using Algorithm~\ref{alg:offline_alpha} (or~\eqref{eq:alpha_1}),
\Statex \quad b):compute $\alpha_2$ using~\eqref{eq:alpha_2_better} (or~\eqref{eq:alpha_2}).
\end{algorithmic}
\end{algorithm}
The presented offline procedure is considerably more involved than for example the computation for one specific setpoint~\cite{chen1998quasi}.
We emphasize that this procedure only has to be completed once and we need \textit{no} repeated offline computations to account for changing operation conditions.
Furthermore, the applicability to nonlinear systems with the corresponding computational effort offline is detailed with numerical examples in Section~\ref{sec:num}.
\subsection{Setpoint tracking}
\label{sec:increm_setpoint}
Now we discuss setpoint tracking, which is included in the previous derivation as a special case with $\mathcal{Z}_r$ such that $(x_r,u_r)\in\mathcal{Z}_r$ implies $x_r=f(x_r,u_r)$ and $\mathcal{R}(r)=r$.
Note, that both presented approaches significantly simplify in this case.
For the gridding approach it suffices to grid along the steady-state manifold $\mathcal{Z}_r$ which is typically low dimensional.
In the convex approach (Prop.~\ref{prop:LMI_lpv}) we have $\theta^+=\theta$ and thus we only consider the $2^p$ vertices of $\Theta$.
Compared to the dynamic reference tracking problem, the problem of tracking a setpoint has received a lot of attention in the literature and many solutions have been suggested.
One of the first attempts to solve this issue is the usage of a pseudo linearization in~\cite{findeisen2000nonlinear}.
There, a nonlinear state and input transformation is sought, such that the linearization of the transformed system around the setpoints is constant and thus constant terminal ingredients can be used.
This approach seems unpractical, since there is no easy or simple method to compute such a pseudo linearization.
In~\cite{limon2018nonlinear,wan2003efficient,wan2003offline} the steady-state manifold $\mathcal{Z}_r$ is partitioned into sets.
In each set the nonlinear system is described as an LTV system and a constant terminal cost and controller are computed.
Correspondingly, in closed-loop operation under changing setpoints~\cite{limon2018nonlinear} the terminal cost matrix $P_f$ is piece-wise constant.
This might cause numerical problems in the optimization, since the cost is not differentiable with respect to the reference $r$.
Furthermore, the (manual) partitioning of the steady-state manifold seems difficult for general MIMO systems (if the dimension of the steady-state manifold is larger than one).
In comparison, Algorithm~\ref{alg:offline} yields continuously parameterized terminal ingredients, thus avoiding the need for user defined partitioning and piece-wise definitions.
In~\cite[Remark~8]{muller2014performance} it was proposed to compute a continuously parameterized controller $K_f(r)$ by analytically using a pole-placement formula and solving the corresponding Lyapunov\footnote
In~\cite{muller2014performance}, the terminal cost $V_f$ is computed for a (differentiable) economic stage cost $\ell(x,u)$ (not necessarily quadratic), compare also~\cite{fagiano2013generalized}.
The computation of the terminal cost is decomposed into a linear and quadratic term, compare~\cite{amrit2011economic}.
Computing the quadratic term of this economic terminal cost is equivalent to computing a quadratic terminal cost for a quadratic stage cost (Ass~\ref{ass:term}).
} equation to obtain $P_f(r)$.
The resulting terminal ingredients are quite similar to the proposed ones.
However, this procedure cannot be directly translated into a simple optimization problem and might hence not be tractable.
\section{Reference generic offline computations}
\label{sec:loc_stab}
This section provides a reference \textit{generic} offline computation to design terminal ingredients for nonlinear reference tracking MPC.
In Lemma~\ref{lemma:lpv} we provide sufficient conditions for the terminal ingredients based on properties of the linearization.
Then, two approaches based on LMI computations are described to compute the terminal ingredients, based on Lemma~\ref{lemma:LMI} and Proposition~\ref{prop:LMI_lpv}.
After that, a procedure to obtain a non conservative terminal set size $\alpha$ is discussed.
Finally, the overall offline procedure is summarized in Algorithm~\ref{alg:offline}.
For the special case of setpoint tracking, existing methods are discussed in relation to the proposed procedure.
In Appendix~\ref{sec:app_cont} and \ref{sec:app_output}, these results are extended to continuous-time dynamics and output tracking stage costs, respectively.
\input{Increm_Stab_1}
\input{Increm_Stab_2}
\input{Increm_Stab_3}
\input{Increm_Stab_4}
\section{Introduction}
Model Predictive Control (MPC)~\cite{rawlings2017model} is a well established control method, that computes the control input by repeatedly solving an optimization problem online.
The main advantages of MPC are the ability to cope with general nonlinear dynamics, hard state and input constraints, and the inclusion of performance criteria.
In MPC (theory), recursive feasibility and closed-loop stability of a desirable setpoint are usually ensured by including suitable terminal ingredients (terminal set and terminal cost) in the optimization problem~\cite{mayne2000constrained}.
In many applications, the control goal goes beyond the stabilization of a pre-determined setpoint.
These practical challenges include tracking of changing reference setpoints, stabilization of dynamic trajectories, output regulation and general economic optimal operation.
There exist many promising ideas to tackle these issues in MPC, for example by simultaneously optimizing an artificial reference~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear,fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic}.
However, most of these approaches are limited in some form to linear systems and/or setpoint stabilization.
The computation of suitable terminal ingredients seems to be a bottleneck for the practical extension of these methods to nonlinear systems and dynamic trajectories.
We bridge this gap, by providing a reference \textit{generic} offline computation for the terminal ingredients.
Thus, we can provide practical schemes for nonlinear systems subject to changing operating conditions.
\IEEEpubidadjcol
\subsection*{Related work}
For linear stabilizable systems, a terminal set and terminal cost can be computed based on the linear quadratic regulator (LQR) and the maximal output admissible set~\cite{gilbert1991linear}.
For the purposes of stabilizing a given setpoint, a suitable design procedure for nonlinear systems with a stabilizable linearization has been provided in~\cite{chen1998quasi,rawlings2017model}.
In practice, the setpoint to be stabilized can change and thus procedures independent of the setpoint are necessary.
In~\cite{findeisen2000nonlinear}, the issue of finding a setpoint independent terminal cost has been investigated based on the concept of pseudo linearizations.
While in principle very appealing, the computation of such a pseudo linearization for general nonlinear systems seems unpractical.
In~\cite{magni2005solution}, a locally stabilizing controller is assumed and the terminal cost and constraints are defined implicitly based on the infinite horizon tail cost.
The main drawback of this method is the implicit description of the terminal cost, which can significantly increase the online computational demand.
In~\cite{limon2018nonlinear} the feasible setpoints are partitioned into disjoint sets and for each such set a fixed stabilizing controller and terminal cost are designed using the methods in~\cite{wan2003efficient,wan2003offline} based on a local linear time-varying (LTV) system description.
This method is mainly limited to systems with a one dimensional steady-state manifold, due to the otherwise complex and difficult partitioning.
In addition, the piece-wise definition can also lead to numerical difficulties since the terminal cost is not differentiable with respect to the setpoint.
There are many applications in which we want to stabilize some dynamic trajectory or periodic orbit.
The nonlinear system along this trajectory can be locally approximated with an LTV system.
In~\cite{faulwasser2011model}, this is used to compute a (time-varying) terminal cost for asymptotically constant trajectories.
In~\cite{aydiner2016periodic} periodic trajectories are considered and a (periodic) terminal cost is computed based on linear matrix inequalities (LMIs).
A significant practical restriction for these methods is the fact that the offline computation is accomplished for a \textit{specific} (a priori known) trajectory.
In general, the existing procedures to compute terminal ingredients for MPC are mainly focused on computing a terminal cost for a \textit{specific} reference point or reference trajectory.
Thus, online changes in the setpoint or trajectory cannot be handled directly and necessitate repeated offline computations.
\subsection*{Contribution}
In this work, we provide a reference \textit{generic} offline procedure to compute a parameterized terminal cost.
This procedure is applicable to both setpoint or trajectory stabilization.
The feasibility of this approach requires local incremental stabilizability of the nonlinear dynamics.
The existing design procedures~\cite{chen1998quasi,faulwasser2011model,aydiner2016periodic} use the linearization around the considered setpoint or trajectory to locally establish properties of the nonlinear systems.
In a similar spirit, we consider the linearization of the nonlinear system dynamics around all possible points in the constraint set and describe the dynamics analogous to quasi-linear parameter-varying (LPV) systems.
With this description, we formulate the desired properties on the linearized dynamics and provide suitable LMIs to compute the parameter dependent terminal cost and controller.
In closed-loop operation we have a quadratic terminal cost with an ellipsoidal terminal constraint directly available.
This provides a generalization of the offline computations in~\cite{chen1998quasi,faulwasser2011model,aydiner2016periodic} to \textit{generic} references.
We employ the proposed method in an evasive maneuver test for a car and show that the design of suitable reference generic terminal ingredients can significantly improve the control performance compared to MPC schemes with terminal equality constraints or without terminal constraints.
Given these terminal ingredients, we can extend existing tracking MPC schemes, such as~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear,fagiano2013generalized,muller2013economic,muller2014performance,ferramosca2014economic} to nonlinear system dynamics and optimal periodic operation, which is a fundamental step towards practical nonlinear MPC schemes.
In particular, we provide a nonlinear periodic tracking MPC scheme for exogenous output signals as an extension to~\cite{limon2008mpc,limon2016mpc,limon2018nonlinear}.
\subsection*{Outline}
The remainder of this paper is structured as follows:
Section~\ref{sec:MPC} presents the reference tracking MPC scheme based on the proposed parameterized terminal ingredients.
Section~\ref{sec:loc_stab} provides a constructive procedure to design parametric terminal ingredients independent of the considered reference.
Section~\ref{sec:ext} shows how the resulting parameterized terminal ingredients can be used to extend existing MPC schemes for changing operation conditions to nonlinear system dynamics and periodic operation.
Section~\ref{sec:num} shows the practicality of this procedure with numerical examples.
Section~\ref{sec:sum} concludes the paper.
In the appendix, these results are extended to \textit{robust} trajectory tracking (App.~\ref{sec:app_robust}), continuous-time dynamics (App.~\ref{sec:app_cont}), and output tracking stage costs (App.~\ref{sec:app_output}).
In addition, the connection between the generic terminal ingredients and incremental system properties is discussed (App.~\ref{sec:app_increm}).
\section{Reference tracking model predictive control}
\label{sec:MPC}
\subsection{Notation}
The quadratic norm with respect to a positive definite matrix $Q=Q^\top$ is denoted by $\|x\|_Q^2=x^\top Q x$.
The minimal and maximal eigenvalue of a symmetric matrix $Q=Q^\top$ is denoted by $\lambda_{\min}(Q)$ and $\lambda_{\max}(Q)$, respectively.
The identity matrix is $I_n\in\mathbb{R}^{n\times n}$.
The interior of a set $\mathcal{X}$ is denoted by $\text{int}(\mathcal{X})$.
The vertices of a polytopic set $\Theta$ are denoted by $\theta_i\in\text{Vert}(\Theta)$.
\subsection{Setup}
We consider the following nonlinear discrete-time system
\begin{align}
\label{eq:sys}
x(t+1)&=f(x(t),u(t))
\end{align}
with the state $x\in\mathbb{R}^n$, control input $u\in\mathbb{R}^m$, and time step $t\in\mathbb{N}$.
The extension of the following derivation to continuous-time dynamics is detailed in Appendix~\ref{sec:app_cont}.
We impose point-wise in time constraints on the state and input
\begin{align}
\label{eq:constraint}
(x(t),u(t))\in \mathcal{Z},
\end{align}
with some compact\footnote
The derivations can be extended to time-varying constraint sets $\mathcal{Z}(t)$ and dynamics $f(x,u,t)$.
The consideration of non-compact constraint sets may require additional uniformity conditions on the nonlinear dynamics.
} set $\mathcal{Z}$.
We consider the following assumption regarding the reference signal $r=(x_r,u_r)\in\mathbb{R}^{n+m}.$
\begin{assumption}
\label{ass:ref}
The reference signal $r$ satisfies $r(t)\in\mathcal{Z}_r$, $\forall t\geq 0$, with some set $\mathcal{Z}_r\subseteq\text{int}(\mathcal{Z})$.
Furthermore, the evolution of the reference signal is restricted by $r(t+1)\in\mathcal{R}(r(t))$, with $\mathcal{R}(r)=\{(x_r^+,u_r^+)\in\mathcal{Z}_r|~x_r^+=f(x_r,u_r)\}$.
\begin{comment}
There exists a reference constraint set $\mathcal{Z}_r\subset\text{int}(\mathcal{Z})$ and a set valued map $\mathcal{R}:\mathcal{Z}_r\rightarrow\mathcal{Z}_r$, such that the reference signal $r$ satisfies
\begin{align*}
r(t)\in&\mathcal{Z}_r,\quad \forall t\geq 0,\\
r(t+1)\in&\mathcal{R}(r(t))\subseteq\mathcal{Z}_r.
\end{align*}
\end{comment}
\end{assumption}
This assumption characterizes that the reference trajectory $r$ is reachable, i.e., follows the dynamics $f$ and lies (strictly) in the constraint set $\mathcal{Z}$.
If the reference trajectory is not reachable it is possible to enforce these constraints on an artificial reference trajectory which can be included in the MPC optimization problem, compare Section~\ref{sec:ext}.
\begin{remark}
\label{rk:ref}
The set $\mathcal{R}(r)$ can be modified to incorporate additional incremental input constraints $\|u_r(t+1)-u_r(t)\|_{\infty}\leq \epsilon$.
Setpoints are included as a special case, with $\mathcal{R}(r)=r$ and the steady-state manifold $\mathcal{Z}_r$.
\end{remark}
\subsection{Terminal cost and terminal set}
Denote the tracking error by $e_r(t)=x(t)-x_r(t)$.
The control goal is to stabilize the tracking error $e_r(t)=0$ and achieve constraint satisfaction $(x(t),u(t))\in\mathcal{Z}$, $\forall t\geq 0$.
To this end we define the quadratic reference tracking stage cost
\begin{align}
\label{eq:stage}
\ell(x,u,r)=\|x-x_{r}\|_Q^2+\|u-u_{r}\|_R^2,
\end{align}
with positive definite weighting matrices $Q,~R$.
\begin{remark}
\label{remark:output_cost}
The extension to an tracking stage cost $\ell(x,u,r)=\|h(x,u)-h(x_r,u_r)\|_{S(r)}^2$ with some output $y=h(x,u)$ and a positive definite weighting matrix $S$ is discussed in the Appendix~\ref{sec:app_output}.
\end{remark}
As discussed in the introduction, we need suitable terminal ingredients to ensure stability and recursive feasibility for the closed-loop system.
\begin{assumption}
\label{ass:term}
There exist matrices $K_f(r)\in\mathbb{R}^{m\times n}$, $P_f(r)\in\mathbb{R}^{n\times n}$ with $c_l I_n\leq P_f(r)\leq c_u I_n$, a terminal set $\mathcal{X}_f(r)=\{x\in\mathbb{R}^n|~V_f(x,r)\leq \alpha\}$ with the terminal cost $V_f(x,r)=\|x-x_r\|_{P_f(r)}^2$, such that the following properties hold for any $r\in\mathcal{Z}_r$, any $x\in\mathcal{X}_f(r)$ and any $r^+\in\mathcal{R}(r)$
\begin{subequations}
\label{eq:term}
\begin{align}
\label{eq:term_dec}
V_f(x^+,r^+)\leq& V_f(x,r)-\ell(x,k_f(x,r),r),\\
\label{eq:term_con}
(x,k_f(x,r))\in&\mathcal{Z},
\end{align}
\end{subequations}
with $x^+=f(x,k_f(x,r))$, $k_f(x,r)=u_r+K_f(r)\cdot (x-x_r)$ and positive constants $c_l,~c_u,~\alpha$.
\end{assumption}
For $r=r^+=0$ this reduces to the standard conditions in~\cite{chen1998quasi}.
For a given trajectory $r$, this implies time-varying terminal ingredients, compare~\cite{faulwasser2011model,aydiner2016periodic}.
Designing suitable\footnote
In principle, this assumption can always be satisfied with a terminal equality constraint $\mathcal{X}_f(r)=x_r$.
However, this can lead to numerical problems, and decrease performance and robustness of the MPC scheme.
In addition, tracking schemes such as~\cite{limon2008mpc,limon2018nonlinear,kohler2018mpc}, typically require a non-vanishing terminal set size $\alpha$ to ensure exponential stability, compare Section~\ref{sec:ext}.
} terminal ingredients that satisfy this assumption is the main contribution of this paper and is discussed in more detail in the Section~\ref{sec:loc_stab}.
\begin{remark}
\label{rk:invarset}
Assumption~\ref{ass:ref} implies that the reference $r(t)$ is contained within a control invariant subset $\mathcal{Z}_{\infty}\subseteq\mathcal{Z}_r$.
Thus, Assumption~\ref{ass:term} could be relaxed, such that the conditions~\eqref{eq:term} only need to be satisfied for points $r\in\mathcal{Z}_{\infty}$.
The exact characterization of the set $\mathcal{Z}_{\infty}$ is, however, challenging and thus we consider the stricter\footnote{If there exists a fixed constant $T_0$, such that $r(t+k)\in\mathcal{Z}_r,~\forall k\in[0,T_0]$, implies $r(t)\in\mathcal{Z}_{\infty}$, then the conditions in Assumption~\ref{ass:term} are not stricter.
However, if we use a convex overapproximation (Prop.~\ref{prop:LMI_lpv}) and/or parameterize the matrices $P_f,~K_f$, then this may introduce additional conservatism.
} conditions as formulated in Assumption~\ref{ass:term}.
\end{remark}
\subsection{Preliminary results}
Denote the reference $r$ over the prediction horizon $N$ by ${r}(\cdot|t)\in\mathbb{R}^{(n+m)\times (N+1)}$ with $r(k|t)=r(t+k)$, $k=0,\dots,N$.
Given a predicted state and input sequence $x(\cdot|t)\in\mathbb{R}^{n\times N+1 },~u(\cdot|t)\in\mathbb{R}^{m\times N}$ the tracking cost with respect to the reference $r(\cdot|t)$ is given by
\begin{align*}
J_N(x(\cdot|t),u(\cdot|t),r(\cdot|t)):=&\sum_{k=0}^{N-1}\ell(x(k|t),u(k|t),r(k|t))\\
&+V_f(x(N|),r(N|t)).
\end{align*}
The MPC scheme is based on the following (standard) MPC optimization problem
\begin{subequations}
\label{eq:MPC}
\begin{align}
\label{eq:MPC_cost}
V(x(t),r(\cdot|t))=\min_{u(\cdot|t)}&J_N(x(\cdot|t),u(\cdot|t),r(\cdot|t))\\
\label{eq:MPC_dyn}
\text{s.t. }&x(k+1|t)=f(x(k|t),u(k|t)),\\
\label{eq:MPC_init}
&x(0|t)=x(t),\\
\label{eq:MPC_con}
&(x(k|t),u(k|t))\in\mathcal{Z},\\
\label{eq:MPC_term}
&x(N|t)\in\mathcal{X}_f({r}(N|t)).
\end{align}
\end{subequations}
The solution to this optimization problem are the value function $V$ and the optimal input trajectory $u^*(\cdot|t)$.
In closed-loop operation we apply the first part of the optimized input trajectory to the system, leading to the following closed loop
\begin{align}
\label{eq:close}
x(t+1)=f(x(t),u^*(0|t))=x^*(1|t),\quad t\geq 0.
\end{align}
The following theorem summarizes the standard theoretical properties of the closed-loop system~\eqref{eq:close}.
\begin{theorem}
\label{thm:MPC}
Let Assumptions~\ref{ass:ref} and \ref{ass:term} hold.
Assume that Problem~\eqref{eq:MPC} is feasible at $t=0$.
Then Problem~\eqref{eq:MPC} is recursively feasible and the tracking error $e_r=0$ is (uniformly) exponentially stable for the resulting closed-loop system~\eqref{eq:close}.
\end{theorem}
\begin{proof}
This theorem is a straight forward extension of standard MPC results in~\cite{rawlings2009model}, compare also~\cite{faulwasser2011model}.
Given the optimal solution $u^*(\cdot|t)$, the candidate sequence
\begin{align}
\label{eq:candidate_input}
u(k|t+1)= \begin{cases}
u^*(k+1|t)& k\leq N-2 \\
k_f(x^*(N|t),r(N|t))&k=N-1
\end{cases},
\end{align}
is a feasible solution to~\eqref{eq:MPC_cost} and implies
\begin{align}
\label{eq:V}
V(x(t+1),r(\cdot|t+1))\leq V(x(t),r(\cdot|t))-\ell(x(t),u(t),r(t)).
\end{align}
Compact constraints in combination with the quadratic terminal cost imply
\begin{align*}
\|x(t)-x_r(t)\|_Q^2\leq V(x(t),r(\cdot|t))\leq c_v \|x(t)-x_r(t)\|_Q^2,
\end{align*}
for some $c_v\geq 1$.
Uniform exponential stability follows from standard Lyapunov arguments using the value function $V$.
\end{proof}
This theorem shows that if we can design suitable terminal ingredients (Ass.~\ref{ass:term}), the closed-loop tracking MPC has all the (standard) desirable properties.
In Section~\ref{sec:ext} we discuss how this can be extended to more general tracking problems.
This scheme can be easily modified to ensure robust reference tracking using the method in~\cite{kohler2018novel}, for details see Appendix~\ref{sec:app_robust} and the numerical example in Section~\ref{sec:num}.
\begin{remark}
\label{rk:withoutterm}
A powerful alternative to the proposed quasi-infinite horizon reference tracking MPC scheme would be a reference tracking MPC scheme without terminal ingredients~\cite{kohlernonlinear19} ($V_f(x,r)=0$, $\mathcal{X}_f(r)=\mathcal{X}$).
If it is possible to design terminal ingredients (Ass.~\ref{ass:term}), the value function of such an MPC scheme without terminal constraints is locally bounded by $V(x(t),r(\cdot|t))\leq \gamma \ell(x,u,r)$, with a suitable constant $\gamma$, compare~\cite[Prop.~2]{kohlernonlinear19}.
Thus, an MPC scheme without terminal constraints enjoys similar closed-loop properties to Theorem~\ref{thm:MPC}, provided a sufficiently large prediction horizon $N$ is used, compare~\cite[Thm.~2]{kohlernonlinear19}.
One of the core advantages of including suitably designed terminal ingredients is that we can implement the MPC scheme with a short prediction horizon $N$.
On the other hand, if the reference is not reachable (Ass.~\ref{ass:ref}), MPC schemes without terminal constraints can still be successfully applied~\cite[Thm.~4]{kohlernonlinear19}, which is in general not the case for MPC schemes with terminal constraints.
\end{remark}
|
\section{Introduction}
This is the second of four papers in which we investigate the following conjecture of the second named author (see \cite[Conjecture 3.4]{Go2}). Recall that a {\it hyperbolic knot manifold} is a compact, connected, orientable $3$-manifold with torus boundary whose interior admits a complete, finite volume hyperbolic structure.
\begin{conj}\label{conj} {\rm (C. McA. Gordon)} Suppose that $M$ is a hyperbolic knot
manifold and $\alpha, \beta$ are slopes on $\partial M$ such that $M(\alpha)$ is Seifert fibred and $M(\beta)$ toroidal. If $\Delta(\alpha, \beta) > 5$, then $M$ is the figure eight knot exterior.
\end{conj}
Our first result reduces the verification of the conjecture to the case where the Seifert filling is atoroidal.
\begin{thm}\label{reduction2}
Suppose that $M$ is a hyperbolic knot
manifold and $\alpha, \beta$ are slopes on $\partial M$ such that
$M(\alpha)$ is a toroidal Seifert fibred manifold and $M(\beta)$
is toroidal. Then $\Delta (\alpha,\beta) \leq 4$.
Furthermore, if $\Delta (\alpha,\beta) =4 $ then $(M;\alpha,\beta)\cong
(N(-\frac12,-\frac12); -4,0)$ where $N$ is the exterior of the 3-chain link \cite{MP}.
\end{thm}
We have that $N(-\frac12,-\frac12,-4)$ is Seifert fibred with base orbifold
$P^2 (2,3)$, and $N(-\frac12,-\frac12,0)$ contains an incompressible
torus separating $N(-\frac12,-\frac12,0)$ into Seifert fibred manifolds
with base orbifolds $D^2 (2,2)$ and $D^2 (2,3)$.
(See \cite[Table 2]{MP}.)
A {\it small Seifert} manifold is a $3$-manifold which admits a Seifert structure with base orbifold of the form $S^2(a,b,c)$ where $a, b, c \geq 1$. For instance, a closed, atoroidal Seifert manifold is small Seifert.
A small Seifert manifold is a {\it prism manifold} if its base orbifold is $S^2(2,2,n)$ for some $n \geq 2$.
Since the distance between a toroidal filling slope and a reducible filling slope is at most $3$ (\cite{Oh}, \cite{Wu1}), Theorem \ref{reduction2} reduces our analysis of Conjecture \ref{conj} to understanding the case where the Seifert Dehn filling is irreducible and small Seifert.
In an earlier paper \cite{BGZ2} we verified the conjecture in the case where $M$ admits no essential punctured torus of boundary slope $\beta$ which is a fibre or semi-fibre, or which has fewer than three boundary components; more precisely, we showed that in this case $\Delta(\alpha, \beta) \leq 5$. Here we focus on the case where $M$ admits an essential punctured torus with one boundary component.
Let {\it $Wh$} denote the left-handed Whitehead link exterior (see Figure \ref{bgz4-whitehead}). We parameterise the slopes on a boundary component of {\it $Wh$} using the standard meridian-longitude coordinates.
\begin{thm} \label{once-punctured}
Let $M$ be a hyperbolic knot manifold and $\alpha$ a slope on $\partial M$ such that $M(\alpha)$ is small Seifert. If $M$ admits an essential, once-punctured torus $F$ of boundary slope $\beta$ then $\Delta(\alpha, \beta) \leq 5$. Further, if $\Delta(\alpha, \beta) > 3$, then $F$ is not a fibre and $\pi_1(M(\alpha))$ is finite. More precisely, \\
$(1)$ if $\Delta(\alpha, \beta) = 4$, then $(M;\alpha,\beta)\cong (Wh(\frac{-2n\pm1}{n}); -4,0)$
for some integer $n$ with $|n|>1$ and $M(\alpha)$ has base orbifold $S^2(2,2,|\mp 2n-1|)$, so
$M(\alpha)$ is a prism manifold; \\
$(2)$ if $\Delta(\alpha, \beta) = 5$, then $(M; \alpha, \beta) \cong (Wh(-3/2); -5, 0)$, and $M(\alpha)$ has base orbifold $S^2(2,3,3)$.
\end{thm}
Baker \cite{Ba} has proven Theorem \ref{once-punctured} in the case that $M(\alpha)$ is a lens space. We provide an alternate proof of his result.
Theorem \ref{once-punctured} is sharp; see the infinite family of examples in \S \ref{d=4} for (1) and \cite[Table A.3]{MP} for (2). Another family of examples is provided by hyperbolic twist knots. These are genus one knots in the $3$-sphere whose exteriors admit small Seifert filling slopes of distance $1, 2,$ and $3$ from the longitudinal slope. Finally, Baker \cite[Theorem 1.1(IV)]{Ba} has constructed an infinite family of non-fibred hyperbolic knot manifolds which admit a once-punctured essential torus whose boundary slope is of distance $3$ to a lens space filling slope.
Here is an outline of the proof of Theorem \ref{once-punctured}. We begin by showing that the result holds unless, perhaps, $M$ admits an orientation-preserving involution $\tau$ with non-empty branch set $L$ contained in the interior of the quotient $M/\tau$, which is a solid torus. The results of \cite{BGZ2} reduce us to the case that $L$ has a very particular form (see Figure \ref{fig3}). On the other hand, $\tau$ extends to an involution $\tau_\alpha$ of $M(\alpha)$ with branch set $L_\alpha$ contained in the lens space $M(\alpha)/\tau_\alpha$. The fundamental group of $M(\alpha)/\tau_\alpha$ is non-trivial if the distance between $\alpha$ and $\beta$ is at least $3$. Since the involutions on small Seifert manifolds with such quotients are well-understood, we can explicitly describe the branch set $L_\alpha$ of $\tau_\alpha$. Comparing this description with the constraints we have already deduced on $L$ leads to the proof of the theorem.
Recall that an {\it exceptional filling slope} on the boundary of a hyperbolic $3$-manifold is a slope $\gamma$ such that $M(\gamma)$ is not hyperbolic. Geometrisation of $3$-manifolds implies that a slope $\gamma$ is exceptional if and only if $M(\gamma)$ is either reducible, toroidal, or Seifert fibred. Theorem \ref{once-punctured} combines with \cite{Oh}, \cite{Wu1}, \cite{Go1}, \cite{GW}, and Proposition \ref{compresses} to yield the next result.
\begin{thm} \label{once-punctured-exceptional}
Let $M$ be a hyperbolic knot manifold which admits an essential,
once-punctured torus $F$ of boundary slope $\beta$ and let $\gamma$ be an exceptional filling slope on $\partial M$. \\
$(1)$ $\Delta(\gamma, \beta) \leq 7$. \\
$(2)$ If $\Delta(\gamma, \beta) > 3$, then $M(\gamma)$ is either toroidal or has a finite fundamental group. \\
$(3)$ If $\Delta(\gamma, \beta) > 3$ and $M(\gamma)$ is toroidal, then either \\
\indent \hspace{.3cm} $(a)$ $\Delta(\gamma, \beta) = 4$ and $(M; \gamma, \beta) \cong (Wh(\delta); -4, 0)$ for some slope $\delta$; or \\
\indent \hspace{.3cm} $(b)$ $\Delta(\gamma, \beta) = 5$ and $(M; \gamma, \beta) \cong (Wh(-4/3); -5, 0)$ or $(Wh(-7/2); -5/2, 0)$; or \\
\indent \hspace{.3cm} $(c)$ $\Delta(\gamma, \beta) = 7$ and $(M; \gamma, \beta) \cong (Wh(-5/2); -7/2, 0)$. \\
$(4)$ If $\Delta(\gamma, \beta) > 3$ and $\pi_1(M(\gamma))$ is finite, then either \\
\indent \hspace{.3cm} $(a)$ $\Delta(\gamma, \beta) = 4$, $(M;\gamma,\beta)\cong (Wh(\frac{-2n\pm1}{n}); -4,0)$
for some integer $n$ with $|n|>1$, and $M(\gamma)$ has base orbifold $S^2(2,2,|\mp 2n-1|)$; or
\\
\indent \hspace{.3cm} $(b)$ $\Delta(\gamma, \beta) = 5, (M; \gamma, \beta) \cong (Wh(-3/2); -5, 0)$, and $M(\gamma)$ has base orbifold $S^2(2,3,3)$.
\end{thm}
Next we specialize to the case where $M$ is the exterior of a hyperbolic knot in the $3$-sphere.
\begin{thm} \label{genusones3}
Let $K \subset S^3$ be a hyperbolic knot of genus one with exterior $M_K$ and suppose $p/q$ is an exceptional filling slope on $\partial M_K$. \\
$(1)$ $M_K(0)$ is toroidal but not Seifert. \\
$(2)$ $M_K(p/q)$ is either toroidal or small Seifert with hyperbolic base orbifold. \\
$(3)$ If $M_K(p/q)$ is small Seifert with hyperbolic base orbifold, then $0 < |p| \leq 3$. \\
$(4)$ If $M_K(p/q)$ is toroidal, then $|q| = 1$ and $|p| \leq 4$ with equality implying $K$ is a twist knot.
\end{thm}
Here is how the paper is organised. We prove Theorem \ref{reduction2} in \S \ref{sec: reduction2}. In \S \ref{background} we show that there are strong topological constraints on $M$ which must be satisfied if Theorem \ref{once-punctured} doesn't hold. These constraints will be applied later in the paper to construct an involution on $M$. In \S \ref{involutions} we describe the branching set of an orientation-preserving involution on a small Seifert manifold with quotient space a lens space with non-trivial fundamental group. Using this we reduce the proof of Theorem \ref{once-punctured} to five problems involving links in lens spaces in \S \ref{sec: once-punctured},
and a problem in which ${\Delta}(\alpha,\beta)=4$ and $M({\alpha})$ is a prism manifold. These problems are resolved in \S \ref{m=1 Seifert}, \S \ref{dist7}, \S \ref{lens space delta = 5}, \S \ref{sec6.3}, \S \ref{delta = 5} and \S \ref{prism-section} respectively. The infinite family of examples realising distance $4$ in Theorem \ref{once-punctured} is constructed in \S \ref{d=4}. Theorems \ref{once-punctured-exceptional} and \ref{genusones3} are dealt with in \S \ref{sec: genus 1}.
\section{The case where $M(\alpha)$ is toroidal} \label{sec: reduction2}
In this section we prove Theorem \ref{reduction2}. Recall from the introduction that $N$ denotes the exterior of the 3-chain link of \cite{MP}. Note that $N(-\frac12,-\frac12)$ is obtained by Dehn filling on $N(-\frac12)$,
which is the exterior of the rational link associated with the rational
number $10/3$.
To prove Theorem \ref{reduction2} we consider all $(M;\alpha,\beta)$ where
$M$ is hyperbolic, $M(\alpha)$ and $M(\beta)$ are toroidal and
$\Delta (\alpha,\beta)\ge 4$.
For $\Delta (\alpha,\beta)\ge 6$ there are only four such $(M;\alpha,\beta)$
\cite{Go1}, and in all four cases neither $M(\alpha)$ nor $M(\beta)$ is Seifert
fibred.
For $\Delta (\alpha,\beta) = 4$ or 5, the triples $(M;\alpha,\beta)$ are
determined in \cite{GW}:
there are 14 hyperbolic manifolds $M_i$, $1\le i\le 14$, each with a pair
of toroidal filling slopes $\alpha_i,\beta_i$ at distance 4 or 5, where
$M_1,M_2,M_3$ and $M_{14}$ have two (torus) boundary components, and the
others, one.
It is shown in \cite{GW} that a hyperbolic manifold $M$ has two toroidal
filling slopes $\alpha$ and $\beta$ at distance 4 or 5 if and only if
$(M;\alpha,\beta) \cong (M_i;\alpha_i,\beta_i)$ for some $1\le i\le 14$,
or $(M;\alpha,\beta) \cong (M_i(\gamma);\alpha_i,\beta_i)$ for $i= 1,2,3$ or
14 and some slope $\gamma$ on the second boundary component of $M_i$.
(We adopt the convention that in the above homeomorphisms either
$\alpha \mapsto \alpha_i$, $\beta\mapsto\beta_i$, or $\alpha\mapsto \beta_i$,
$\beta\mapsto \alpha_i$.)
We prove Theorem \ref{reduction2} by showing that firstly, for $i\ne 1,2,3$
or 14, neither of the toroidal manifolds $M_i (\alpha_i)$ or $M_i(\beta_i)$
is Seifert fibred, secondly, for $i = 1,3$ or 14, there is no hyperbolic
manifold of the form $M_i(\gamma)$ with either $M_i(\gamma)(\alpha_i)$
or $M_i (\gamma)(\beta_i)$ toroidal Seifert fibred, and thirdly, there is
a unique example $(M_2(\gamma);\alpha_2,\beta_2)$ (up to homeomorphism)
where $M_2(\gamma)$ is hyperbolic, $M_2(\gamma)(\alpha_2)$ and
$M_2(\gamma)(\beta_2)$ are toroidal, and one is Seifert fibred; this is the
example described in Theorem \ref{reduction2}.
We first consider the manifolds $M_i$, $6\le i\le 13$.
The toroidal fillings on $M_i$, $M_i(0)$ and $M_i(\beta_i)$, are
described in Lemma~22.2 of \cite{GW}.
We adopt the notation introduced in \cite[p.116]{GW}.
\begin{lemma}\label{lem1}
For $6 \le i\le 13$, $M_i(0)$ is not Seifert fibred.
\end{lemma}
\begin{proof}
$M_i(0)$ is of the form $X(p_1,q_1;p_2,q_2)$; it is the double branched
cover of the tangle $Q_i(0)$, which is of the form $T(p_1,q_1;p_2,q_2)$,
the union of two Montesinos tangles.
Assume the numbering is chosen so that $p_1,q_1$ are not both 2;
(actually this is only an issue when $i=8$).
Then the Seifert fibre $\varphi_1$ of $X(p_1,q_1)$ is unique.
Since $X(p_1,q_1)$ and $X(p_2,q_2)$ are not both twisted $I$-bundles, to
show that $M_i(0)$ is not Seifert fibred it suffices to show that, in the
gluing of $X(p_1,q_1)$ and $X(p_2,q_2)$, $\varphi_1$ is not identified
with the Seifert fibre $\varphi_2$ of $X(p_2,q_2)$.
(When $i=8$, $p_2 = q_2 =2$ and there are two possible choices for
$\varphi_2$.)
We do this by identifying the image of $\varphi_1$ in the boundary of the
tangle $T(p_1,q_1)$, and then capping off the tangle $T(p_2,q_2)$ with the
corresponding rational tangle; in the double branched cover this corresponds
to doing Dehn filling on $X(p_2,q_2)$ along the slope $\varphi_1$.
If $M_i(0)$ were Seifert fibred then this Dehn filling would be reducible,
and so the corresponding rational tangle filling on $T(p_2,q_2)$ would
give a link that is either composite or split.
One checks that this is not the case.
\end{proof}
\begin{lemma}\label{lem2}
For $6\le i\le 13$, $M_i(\beta_i)$ is not Seifert fibred.
\end{lemma}
\begin{proof}
First note that $M_7 (\beta_7)$ is of the form $X(2,3;2,2)$.
We check that this is not Seifert fibred in the same way as we did
for $M_8(0)$ in Lemma~\ref{lem1}.
When $i\ne 7$, $M_i (\beta_i)$ is the double branched cover of a
2-component link $L_i$;
see \cite[Lemma~22.2]{GW}.
More specifically,
for $i = 6,8,9$ or 12, $L_i$ is a cabled Hopf link $C(p_1,q_1;p_2,q_2)$
with $p_1,p_2 >1$,
for $i=10$ or 11, $L_i$ is the link $C(C;2,1)$ (see \cite[page 116]{GW}), and
for $i=13$, $L_i$ is the 2-string cable of the trefoil shown in
\cite[Figure~22.13(d)]{GW}.
In all cases, $L_i$ is {\em toroidal}, i.e. its exterior contains an
essential torus. Moreover the exterior of $L_i$ is not Seifert fibred.
Therefore if $M_i(\beta_i)$ were Seifert fibred then $L_i$ would be a Montesinos link.
But the only toroidal Montesinos links are (see \cite[Corollary~5]{Oe})
$K(\frac12,\frac12,-\frac12,-\frac12)$,
$K(\frac23,-\frac13,-\frac13)$,
$K(\frac12,-\frac14,-\frac14)$,
and $K(\frac12,-\frac13,-\frac16)$.
One easily checks that no $L_i$ is of this form.
\end{proof}
\begin{lemma}\label{lem3}
$M_4 (\alpha_4)$ and $M_4 (\beta_4)$ are not Seifert fibred.
\end{lemma}
\begin{proof}
$M_4 (\alpha_4)$ and $M_4(\beta_4)$ contain incompressible tori
$\widehat F_a$ and $\widehat F_b$; the corresponding punctured tori
$F_a$ and $F_b$ in $M_4$ have four and two boundary components, respectively.
The intersection of $F_a$ and $F_b$ is described by the intersection graphs
$\Gamma_a \subset \widehat F_a$ and $\Gamma_b\subset \widehat F_b$
depicted in Figures~11.9(a) and (b) of \cite{GW}, respectively.
Note that $\widehat F_a$ separates $M_4 (\alpha_4)$, into $M_B$ and $M_W$,
say, while $\widehat F_b$ is non-separating in $M_4(\beta_4)$.
The faces of the graph $\Gamma_b$ lie alternately in $M_B$ and $M_W$;
we choose the notation so that all the faces of $\Gamma_b$ that lie in
$M_B$ are bigons.
Let $f_1, f_2, f_3$, and $g_1, g_2, g_3$ be the faces of $\Gamma_b$ with edges
$G,H; J,K; A,B;$ and $D,E; K,P,R; A,G,L;$ respectively. Let $h_1, h_2, h_3$ be
the faces of $\Gamma_a$ with edges $E,N; H,E;$ and $B,G,N,R;$ respectively.
(The notation refers to the edges illustrated in Figure 11.9 of [GW].)
For computations in $\pi_1(M_B)$ and $\pi_1(M_W)$ we take as
``base-point'' the rectangle in $\widehat F_a$ shown in Figure~11.9(a)
of \cite{GW}.
Let $s,t$ be the pair of generators of $\pi_1 (\widehat F_a)$ determined
by the downward vertical and rightward horizontal edges of that rectangle,
respectively.
Let $x_1$ and $x_3$ be the elements of $\pi_1(M_B)$ corresponding to the
1-handles $H_{(12)}$ and $H_{(34)}$, in the usual way.
The faces $f_1,f_2$ and $f_3$ give the relations in
$\pi_1 (M_B)$:
\begin{align*}
&x_1^2 t =1\\
&x_3^2 t^{-1} =1\\
&s^{-1}x_3 x_1 =1
\end{align*}
It follows that $M_B$ is Seifert fibred with base orbifold $D^2 (2,2)$, and
that the classes in $\pi_1 (\widehat F_a)$ of the Seifert fibres in the
two Seifert fibrings of $M_B$ are $t$ and $s$.
Let $x_2$ and $x_4$ be the elements of $\pi_1 (M_W)$ corresponding to
$H_{(23)}$ and $H_{(41)}$.
Then the faces $g_1,g_2$ and $g_3$ give the relations in $\pi_1 (M_W)$:
\begin{align*}
&tx_4 x_2 =1\\
&x_2 x_4 t^{-1} x_2 st =1\\
&x_2 x_4^2 t^{-1} =1
\end{align*}
These show that $M_W$ is Seifert fibred with base orbifold $D^2 (2,3)$,
the class of the Seifert fibre in $\pi_1 (\widehat F_a)$ being $st^2$.
Since this is distinct from either of the Seifert fibres of $M_B$,
$M_4 (\alpha_4)$ is not Seifert fibred.
We now consider $M_4 (\beta_4)$.
Let $u,v$ be the pair of generators for $\pi_1 (\widehat F_b)$ given by
the downward vertical and leftward horizontal edges of the rectangle in
Figure~11.9(b) of \cite{GW}.
(We take this rectangle as ``base-point'' for computations in
$\pi_1 (M_4 (\beta_4))$.)
Let $x,y$ be the elements of $\pi_1 (M_4 (\beta_4))$ given by the
1-handles $H_{(12)}$ and $H_{(21)}$.
The faces $h_1,h_2,h_3$ give the relations in $\pi_1(M_4(\beta_4))$:
\begin{align*}
&x(uv) y^{-1} v^{-1} = 1\\
&yv x^{-1} =1\\
& x^{-1} u^{-1} xux^{-1} (vu)^{-1} y =1
\end{align*}
The second relation gives $x= yv$, and the first then gives
$$y^{-1} vy = uv^2$$
The third relation gives
$$(y^{-1} u^{-1} y) u (y^{-1} u^{-1} y) u^{-1}v^{-3} =1$$
Now if $M_4 (\beta_4)$ were Seifert fibred, the non-separating torus
$\widehat F_b$ would be horizontal and so $M_4 (\beta_4)$ would be a
torus bundle over the circle with fibre $\widehat F_b$.
Hence $y^{-1} u^{-1}y$ would belong to $\pi_1 (\widehat F_b)$.
But the last relation above shows that if this is the case then
$$(y^{-1} u^{-1}y)^2 = v^3$$
Since $v^3$ is not a square in $\pi_1 (\widehat F_b)$, this is a
contradiction.
\end{proof}
\begin{lemma}\label{lem4}
$M_5 (\alpha_5)$ and $M_5 (\beta_5)$ are not Seifert fibred.
\end{lemma}
\begin{proof}
This can be proved in a similar fashion to Lemma~\ref{lem3}, using
\cite[Figure~11.10]{GW}.
Another way to establish the result is to note that, according to
\cite[\S6]{L2}, $M_5 \cong N(1,-\frac13)$, the toroidal filling slopes
$\alpha_5,\beta_5$ being $-4$ and 1.
We see that $N(1,-\frac13,-4)$ and $N(1,-\frac13,1)$ are not Seifert fibred
from Tables~4 and 3 of \cite{MP}, respectively.
\end{proof}
We next consider the manifolds $M_1,M_2$ and $M_3$, namely the exteriors
of the Whitehead link, the $10/3$-rational link, and the Whitehead sister
(or $(-2,3,8)$-pretzel) link, respectively.
These are all obtained by Dehn filling on the 3-chain link:
$M_1\cong N(1)$, $M_2 \cong N(-\frac12)$, $M_3 \cong N(-4)$.
Furthermore, their exceptional slopes and toroidal slopes are as
follows (see \cite[Table~A.1]{MP}):
$$\vbox{\offinterlineskip
\halign{\strut
\vrule#&\enspace \hfil$#$\hfil\enspace
&\vrule#&\enspace$#$\hfil\enspace
&\vrule#&\enspace\hfil$#$\hfil\enspace
&\vrule#\cr
\noalign{\hrule}
&&&\hbox{exceptional slopes}&&\hbox{toroidal slopes}&\cr
\noalign{\hrule}
&N(1)&&\infty ,-3,-2,-1,0,1&&-3,1&\cr
\noalign{\hrule}
&N(-\frac12)&&\infty,-4,-3,-2,-1,0&&-4,0&\cr
\noalign{\hrule}
&N(-4)&&\infty,-3,-2,-1,-\frac12,0&&-\frac12,0&\cr
\noalign{\hrule}
}}$$
\begin{lemma}\label{lem5}
In each of the following cases, the manifold $N(\alpha,\beta,\gamma)$
is a toroidal Seifert fibre space if and only if $\gamma$ is one of
the values listed.
$(a)$ $N(1,-3,\gamma) :\hskip9pt $\quad $\gamma = -3,1$; \\
\indent \hspace{5mm} $N(1,1,\gamma) :\hskip17pt $\quad $\gamma = -3, -2,-1,0$.
$(b)$ $N(-\frac12,-4,\gamma) :$\quad $\gamma = -\frac12$;\\
\indent \hspace{5mm} $N(-\frac12,0,\gamma) :\hskip9pt $\quad $\gamma = -\frac72$.
$(c)$ $N(-4,-\frac12,\gamma) : $\quad $\gamma = -\frac12$;\\
\indent \hspace{5mm} $N(-4,0,\gamma) :\hskip9pt $\quad no $\gamma$.
\end{lemma}
\begin{proof}
This follows by inspecting Tables~2, 3 and 4 of \cite{MP}.
We see from these that the only toroidal Seifert fibre spaces
$N(\alpha,\beta,\gamma)$ are
(1) $N(-3,1,1),\ \ N(-3,-\tfrac53,-\tfrac53),\ \
N(-3,-3,t/u)
\text{ where } t/u \ne-1, -1 +\tfrac1m \text{ or }\infty,$ and
(2) $N(0,\frac12 +n ,-\frac92 -n)$,
$N(1,1,n)$ where $|n+1| \le 1$,
$N(-\frac32,-\frac52,0)$, and $N(-4,-\frac12,-\frac12)$.
\end{proof}
Note that the values of $\gamma$ listed in parts (a) and (c) of
Lemma~\ref{lem5} all belong to the set of exceptional slopes of
$N(1)$ and $N(-4)$ respectively.
It follows that for $i=1$ and 3, there is no $\gamma$ such that
$M_i (\gamma)$ is hyperbolic and one of $M_i(\gamma)(\alpha_i)$,
$M_i(\gamma)(\beta_i)$ is toroidal Seifert fibred.
In the case $i=2$, note that by \cite[Proposition 1.5 part (1.4)]{MP},
there is an automorphism of $N(-\frac12)$ inducing homeomorphisms
\begin{align*}
&N(-\tfrac12, -4,-\tfrac12) \cong N(-\tfrac12, 0,-\tfrac72)\\
\noalign{\vskip6pt}
&N( -\tfrac12, 0,-\tfrac12) \cong N(-\tfrac12,-4,-\tfrac72)
\end{align*}
Also, we see from \cite[Table 2]{MP} that $N(-\frac12,0,-\frac12)$ is toroidal.
Thus part~(b) of Lemma~\ref{lem5} gives rise to the single example described
in Theorem \ref{reduction2}.
Finally, we take care of $M_{14}$:
\begin{lemma}\label{lem56}
For no slope $\gamma$ on the second boundary component of $M_{14}$
is $M_{14}(\gamma)(\alpha_{14})$ or $M_{14}(\gamma)(\beta_{14})$ toroidal
Seifert fibred.
\end{lemma}
\begin{proof}
In \cite{L1} Lee describes a hyperbolic 3-manifold $Y$ with two torus
boundary components having (homeomorphic) Dehn fillings $Y(0)$ and $Y(4)$
that contain Klein bottles.
In fact $Y(0) \cong Y(4) \cong Q(2,2) \cup Wh$,
where $Q (2,2)$ is the Seifert fibre space with base orbifold $D^2 (2,2)$
and $Wh$ is the exterior of the Whitehead link.
Hence $Y(0) \cong Y(4)$ is toroidal.
It follows from the classification in \cite{GW} of the hyperbolic 3-manifolds
with toroidal fillings at distance~4 that $Y\cong M_{14}$.
(The only other manifolds with two boundary components having toroidal
fillings at distance~4 are $M_1$ and $M_2$, and there the toroidal fillings
are graph manifolds; see e.g. \cite[Table~A.1]{MP}.)
It therefore suffices to show that $M_{14}(\gamma)(\alpha_{14})$ is not
toroidal Seifert fibred for any slope $\gamma$.
The manifold $M = M_{14}(\alpha_{14}) \cong Q(2,2)\cup Wh$ is the double
branched cover of the tangle shown in \cite[Figure~22.14(b)]{GW}.
Thus $M(\gamma) \cong Q(2,2) \cup Wh(\gamma)$.
Hence if $M(\gamma)$ is toroidal Seifert fibred then $\gamma$ must be an
exceptional slope for $Wh$.
These slopes (with respect to the parametrization in
\cite[Table~A.1]{MP}) are $\infty,-3,-2,-1,0$ and $1$.
Now $Wh(-3)$ and $Wh(1)$ are toroidal non-Seifert,
$Wh(\infty) \cong D^2 \times S^1$, and $Wh(-2)$, $Wh(-1)$ and $Wh(0)$ are
Seifert fibred with base orbifold $D^2 (3,3)$, $D^2(2,4)$ and $D^2(2,3)$,
respectively.
So we need only consider $M(\gamma)$ for $\gamma =\infty,-2,-1$ and $0$;
we do this by examining the corresponding rational tangle filling on the
tangle shown in \cite[Figure~22.14(b)]{GW}.
For $\gamma =\infty$, this yields the pretzel knot
$K(-\frac12,-\frac12,\frac12)$, so $M(\infty)$ is atoroidal.
For $\gamma = -2,-1$ and $0$ we show that the Seifert fibre of $Wh(\gamma)$
does not match the Seifert fibre in either of the two Seifert fibrings
of $Q(2,2)$.
This is straightforward to check, for example using the same approach
as in the proof of Lemma~\ref{lem1}.
\end{proof}
\section{Background Results for the Proof of Theorem \ref{once-punctured}} \label{background}
We collect various results in this section and the next which will be used throughout this paper and its sequel \cite{BGZ3}. In what follows, $M$ will be a hyperbolic knot manifold and $b_1(M)$ will denote its first Betti number. In this section we assume that $F$ is an essential, punctured torus of slope $\beta$ which is properly embedded in $M$.
For a closed, essential surface $S$ in $M$ we define ${\mathcal{C}}(S)$ to be the set
of slopes $\delta$ on $\partial M$ such that $S$ compresses in $M(\delta)$. A
slope $\eta$ on $\partial M$ is called a {\it singular slope} for $S$ if $\eta
\in {\mathcal{C}}(S)$ and $\Delta(\delta, \eta) \leq 1$ for each $\delta \in \mathcal{C}(S)$. A result of Wu \cite{Wu2} states that if ${\mathcal{C}}(S) \ne \emptyset$,
then there is at least one singular slope for $S$.
\begin{prop} \label{compresses}
Suppose that $M$ admits a non-separating, essential, genus $1$ surface of boundary slope $\beta$ which caps-off to a compressible torus in $M(\beta)$. If $\gamma$ is a slope on $\partial M$ such that $M(\gamma)$ is not hyperbolic, then $\Delta(\gamma, \beta) \leq 3$. If $M(\gamma)$ is an irreducible, atoroidal, small Seifert manifold, then $\Delta(\gamma, \beta) \leq 1$.
\end{prop}
\begin{proof}
By hypothesis $M(\beta)$ admits a non-separating $2$-sphere and so is reducible with first Betti number at least $1$.
In the case that $b_1(M) \geq 2$, there is a closed essential surface $S \subset \hbox{int}(M)$ which is Thurston norm minimizing in $H_2(M)$. By \cite[Corollary]{Ga}, $S$ is essential and Thurston norm minimizing in $H_2(M(\delta))$ for all slopes $\delta \ne \beta$. By \cite[Proposition 5.1]{BGZ1}, $\Delta(\gamma, \beta) \leq 1$ for any slope $\gamma$ such that $M(\gamma)$ is not hyperbolic. Suppose then that $b_1(M) = 1$ and note that by hypothesis $\beta$ is a strict boundary slope. In this case \cite[Theorem 3.2]{BCSZ2} implies that $\beta$ is a singular slope and so the conclusions of the lemma follow from \cite[Theorem 1.5]{BGZ1}.
\end{proof}
\begin{cor} \label{stays incompressible}
Theorem \ref{once-punctured} holds if $M$ admits a non-separating, essential, genus $1$ surface of boundary slope $\beta$ which caps-off to a compressible torus in $M(\beta)$.
{\hspace{2mm}{\small $\diamondsuit$}}
\end{cor}
The torus in $M(\beta)$ obtained by capping-off $F$ with a meridional disks will be denoted $\widehat F$. We use $M_F$ to denote the compact manifold obtained by cutting $M$ open along $F$ and $M(\beta)_{\widehat F}$ the manifold obtained by cutting $M(\beta)$ open along $\widehat F$.
\begin{prop} \label{reduction}
Suppose that $M(\alpha)$ is a Seifert fibred manifold and $M(\beta)$ is toroidal. Then $\Delta(\alpha, \beta) \leq 3$ as long as one of the following conditions is satisfied:
\noindent $(a)$ $\alpha$ or $\beta$ is a singular slope of a closed essential surface in $M$.
\noindent $(b)$ $M(\alpha)$ or $M(\beta)$ is reducible.
\noindent $(c)$ $(i)$ $|\partial F| = 1$ and $M_F$ is not a genus $2$ handlebody.
\indent \hspace{.4cm} $(ii)$ $|\partial F| = 2$ and $M_F$ is neither connected nor a union of two genus $2$ handlebodies.
\end{prop}
\begin{proof} If $\alpha$ or $\beta$ is a singular slope of a closed essential surface in $M$, then \cite[Corollary 1.6]{BGZ1} shows that $\Delta(\alpha, \beta) \leq 3$, so we are done in case (a).
Assume next that $M(\gamma)$ is reducible where $\gamma$ is one of $\alpha$ or $\beta$. If $\gamma = \alpha$, then $\Delta(\alpha, \beta) \leq 3$ by \cite{Oh} and \cite{Wu1}. Assume then that $\gamma = \beta$. If $b_1(M) \geq 2$, then $\Delta(\gamma, \beta) \leq 1$ for any exceptional slope $\gamma$ as in the proof of Proposition \ref{compresses}. Assume then that $b_1(M) = 1$. Since $M(\beta)$ is toroidal, it is neither $S^1 \times S^2$ nor a connected sum of lens spaces. Hence \cite[Proposition 6.2]{BGZ1} implies that $\beta$ is a singular slope of a closed essential surface in $M$. Thus we are done by part (a).
Finally consider part (c) of the proposition. If $|\partial F| = 1$, any compression of $\partial M_F$ in $M_F$ yields one or two tori, so as $M$ is hyperbolic it is not hard to see that $M_F$ is a handlebody, contrary to hypothesis. Thus $\partial M_F$ is incompressible in $M_F$, and hence in $M$. Let $S \subset \hbox{int}(M)$ be the inner boundary component of a collar of $\partial M_F$ in $M_F$.
Then $S$ is incompressible in $M$, and by construction there is an annulus $A$ in $M$ with boundary components $\partial_1 A$ and $\partial_2 A$, say, where $A \cap S = \partial_1 A$ and $A \cap \partial M = \partial_2 A$ has slope $\beta$ on $\partial M$. It follows from \cite{Sh} that $S$ is incompressible in $M(\gamma)$ whenever $\Delta(\gamma,\beta) > 1$. Thus $\beta$ is a singular slope for $S$ and so part (a) of this proposition shows $\Delta(\alpha,\beta) \le 3$. Thus (i) holds.
If $|\partial F| = 2$ and $M_F$ is not connected, then $M = X_1 \cup_{F} X_2$ where $\partial X_j$ is a genus $2$ surface for $j = 1, 2$. If $\partial X_j$ compresses in $X_j$ for both $j$, then $X_1$ and $X_2$ are genus $2$ handlebodies as $M$ is hyperbolic. Since this possibility is excluded by our hypotheses, $\partial X_j$ is incompressible in $X_j$ for some $j$. Then it is essential in $M$ but compresses in $M(\beta)$, so as in the previous paragraph, $\beta$ is a singular slope for $\partial X_j$. Thus $\Delta(\alpha, \beta) \leq 3$. This completes the proof.
\end{proof}
Theorem \ref{reduction2} and Propositions \ref{compresses} and \ref{reduction} yield the following corollary.
\begin{cor} \label{smallseifert}
Conjecture \ref{conj} holds as long as it holds when $M(\alpha)$ is an irreducible, atoroidal, small Seifert manifold.
{\hspace{2mm}{\small $\diamondsuit$}}
\end{cor}
Here is a result from \cite{BGZ2}. Recall from \S 6 of that paper that $t_j^+$ is the number of {\it tight components} of $\breve{\Phi}_j^+$.
A $3$-manifold is {\it very small} if its fundamental group does not contain a non-abelian free group.
\begin{prop} \label{main1}
Suppose that $F$ is a once-punctured essential genus $1$ surface of boundary slope $\beta$ in a hyperbolic knot manifold $M$ which completes to an essential torus in $M(\beta)$ but is not a fibre in $M$. If $M(\alpha)$ is a small Seifert manifold, then
$$\Delta(\alpha, \beta) \leq \left\{ \begin{array}{ll} 6 & \hbox{if $M(\alpha)$ is very small} \\ 8 & \hbox{otherwise} \end{array} \right. $$
Moreover if $t_1^+ > 0$, then
$$\Delta(\alpha, \beta) \leq \left\{ \begin{array}{ll} 3 & \hbox{if $M(\alpha)$ is very small} \\ 4 & \hbox{otherwise} \end{array} \right. $$
\end{prop}
\begin{rem} \label{t1+ > 0}
{\rm When $t_1^+ = 0$, $M(\beta)_{\widehat F}$
is Seifert with base orbifold an annulus with one cone point \cite[Lemma 7.9]{BGZ2}. }
\end{rem}
\begin{proof}[Proof of Proposition \ref{main1}] The first inequality is the conclusion of \cite[Proposition 13.2]{BGZ2}. To deduce the second we use the notation and results of \cite{BGZ2}.
Suppose next that $t_1^+ > 0$. Since $t_1^+$ is even and the number of boundary components $F$ is bounded below by $\frac{1}{2} t_1^+$, we have $t_1^+ = 2$. Proposition 13.1 of \cite{BGZ2} then shows that $\Delta(\alpha, \beta) \leq 4$. Suppose that $M(\alpha)$ is very small.
The first paragraph of the proof of \cite[Proposition 13.1]{BGZ2} shows that $\Delta(\alpha, \beta) \leq 3$ if $\overline{\Gamma}_S$ has a vertex of valency $3$ or less while the second shows the same inequality holds if it doesn't. This completes the proposition's proof.
\end{proof}
\section{Involutions on small Seifert manifolds} \label{involutions}
We collect several results about involutions on small Seifert manifolds in this section.
\begin{lemma} \label{invariant}
Let $W$ be a small Seifert manifold and $\tau$ an orientation-preserving involution on $W$ with non-empty fixed point set. Then there is a $\tau$-invariant Seifert structure on $W$ with base orbifold of the form $S^2(a,b,c)$ where $1 \leq a \leq b \leq c$.
\end{lemma}
\begin{proof} If $W$ is a lens space, the result follows from \cite{HR}.
Assume then that this isn't the case and fix a Seifert structure on $W$ with base orbifold $S^2(a,b,c)$ where
$a \leq b \leq c$. The assumption that $\pi_1(W)$ is not cyclic implies
that $a \geq 2$ and $a, b, c$ are determined by $W$.
Let $L \subset W/\tau$ be the branch set of $\tau$. The orbifold theorem implies that the orbifold $W/ \tau$ is geometric and since $L$ is a link, $W/\tau$ admits a Seifert structure with a $2$-dimensional base orbifold \cite{Du}. Thus $W$ admits a $\tau$-invariant Seifert structure. We claim that we can assume this structure has base orbifold $S^2(a,b,c)$. If $b \ne 2$, all Seifert structures on $W$ have this form, so assume $a = b = 2 \leq c$. If the base orbifold of the $\tau$-invariant structure is not $S^2(a,b,c)$, it must be $P^2(d)$ for some integer $d \geq 1$. When $d > 1$, there is a unique singular fibre $\phi$ in this structure, and it must be invariant under $\tau$. Then $\tau$ leaves the exterior $E$ of this fibre invariant, which is a twisted $I$-bundle over the Klein bottle. By assumption, $\tau$ leaves the Seifert structure on $E$ with base orbifold a M\"{o}bius band invariant. There is exactly one other Seifert structure on $E$, up to isotopy, and its base orbifold is $D^2(2,2)$. Moreover, there is at least one such structure which is $\tau|E$-invariant. This structure can be extended across a fibred neighbourhood of $\phi$ in a $\tau$-invariant fashion yielding the desired $\tau$-invariant structure on $W$.
The argument is similar if $d = 1$, for $\tau$ induces an involution of the base orbifold $P^2$ of $W$, and since any self-map of $P^2$ has a fixed point, there is a $\tau$-invariant fibre $\phi$ in $W$. Now proceed as in the case $d > 1$.
\end{proof}
For our next three results we let $W$ denote a small Seifert manifold and $\tau$ an orientation-preserving involution on $W$ with non-empty fixed point set such that the quotient $W/\tau$ is a lens space $L(\bar p, \bar q) \not \cong S^3$. We use $L_\tau$ to denote the branch set of $\tau$ in $L(\bar p, \bar q)$.
Fix a $\tau$-invariant Seifert structure on $W$ with base orbifold of the form $S^2(a,b,c)$ where $1 \leq a \leq b \leq c$ (Lemma \ref{invariant}) and let $\bar \tau$ be the involution of $S^2(a,b,c)$ (possibly the identity) induced by $\tau$.
Since the $\tau$-invariant Seifert structure on $W$ has an orientable base orbifold, its fibres can be coherently oriented.
Hodgson and Rubinstein have classified orientation-preserving involutions on lens spaces with non-empty fixed point sets. In particular, their work yields the following result.
\begin{lemma} {\rm (\cite[\S 4.7]{HR})} \label{cyclic case}
Suppose that $W$ is the lens space $L(p,q)$ and $W / \tau = L(\bar p, \bar q) \not \cong S^3$.
$(1)$ If $p$ is odd, then $L_\tau$ is connected and is either
\vspace{-.35cm}
\begin{itemize}
\item[(a)] the core of a solid torus of a genus one Heegaard splitting of $L(\bar p, \bar q)$;
\item[(b)] the boundary of a M\"{o}bius band spine of a Heegaard solid torus of $L(\bar p, \bar q)$;
\end{itemize}
\vspace{-.35cm}
$(2)$ If $p$ is even, then $L_\tau$ has two components and is either
\vspace{-.35cm}
\begin{itemize}
\item[(a)] the union of the cores of the two solid tori of a genus one Heegaard splitting of $L(\bar p, \bar q)$;
\item[(b)] the boundary of an annular spine of a Heegaard solid torus of $L(\bar p, \bar q)$.
{\hspace{2mm}{\small $\diamondsuit$}}
\end{itemize}
\vspace{-.35cm}
\end{lemma}
Next we suppose that $W$ is not a lens space. In this case $2 \leq a \leq b \leq c$.
\begin{lemma} \label{+-quotient}
Suppose that $W$ is not a lens space and that $\tau$ preserves the orientations of the Seifert fibres of $W$. Then there is an induced Seifert structure on $W/\tau$ such that $L_\tau$ is a union of at most three Seifert fibres where at least one of the fibres is regular. Further, $\bar \tau$ is either the identity or has two fixed points and
$(1)$ if $\bar \tau$ is the identity then $a = 2$, $|L_\tau|$ is the number of cone points of $S^2(a,b,c)$ of even order, and the components of $L_\tau$ which are regular fibres correspond to the cone points of order $2$.
$(2)$ if $\bar \tau$ is not the identity then $L_\tau$ has at most two components. Exactly one of its components is a regular fibre.
\end{lemma}
\begin{proof} The hypotheses imply that there is an induced Seifert structure on $L(\bar p, \bar q)$ whose fibres are the images of the fibres of $W$. Since $W$ has three exceptional fibres, $\bar \tau$ fixes precisely one or three cone points. In the latter case, $\bar \tau$ is the identity.
Suppose first that $\bar \tau$ is the identity on $S^2(a,b,c)$. Since $\tau$ has a $1$-dimensional fixed point set, $\tau$ rotates the regular fibres of $W$ by $\pi$. Its fixed point set is the union of the fibres of even multiplicity and therefore $L_\tau$ is a union of Seifert fibres. The reader will verify that if a fibre of $W$ has multiplicity $k$, then its image in $L(\bar p, \bar q)$ has multiplicity $\bar k = \frac{k}{\gcd(k,2)}$. Hence as $L(\bar p, \bar q)$ has at most two exceptional fibres, $a = 2$.
Suppose next that $\bar \tau$ fixes precisely one cone point of $S^2(a,b,c)$. In this case its fixed point set consists of this cone point and a regular point. Thus the fixed point set of $\tau$ is contained in a union of two fibres, so $L_\tau$ has at most two components. The reader will verify that each exceptional fibre of $W$ is sent to an exceptional fibre of $L(\bar p, \bar q)$, two of them to the same fibre. Thus the $\tau$-invariant regular fibre of $W$ is sent to a regular fibre of $L(\bar p, \bar q)$. It follows that this fibre lies in the fixed-point set of $\tau$ and therefore $L_\tau$ contains a regular fibre of $L(\bar p, \bar q)$.
\end{proof}
\begin{lemma} \label{--quotient}
Suppose that $W$ is not a lens space and that $\tau$ reverses the orientations of the Seifert fibres of $W$.
If $W/\tau = L(\bar p, \bar q) \not \cong S^3$, then
\noindent $(1)$ $W$ has base orbifold $S^2(\bar p, \bar p, m)$ where $m \geq 2$ and the Seifert
invariants of the exceptional fibres of order $\bar p$ are the same. Hence if
$W$ is not a prism manifold, $\bar p \ne 2$.
\noindent $(2)$ There is an integer $n$ coprime with $m$ such that $L_\tau$ is isotopic
to the closure $K(m/n)$ of an $m/n$ rational tangle in a genus $1$ Heegaard
solid torus of $W / \tau$ as depicted in Figure \ref{bgz4-fig24}.
In particular,
$$|L_\tau| = \left\{ \begin{array}{ll}
1 & \hbox{ if } n \hbox{ is odd} \\
2 & \hbox{ if } n \hbox{ is even}
\end{array} \right. $$
\end{lemma}
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-fig24.eps}} \caption{}\label{bgz4-fig24}
\end{figure}
\begin{proof} The fixed point set of $\bar \tau$ is non-empty so as it reverses orientation, it is reflection in an equator of $S^2(a,b,c)$. This equator cannot contain all three cone points as otherwise $\tau$ would be the Montesinos involution on $W$ and therefore $L(\bar p, \bar q)$ would be $S^3$. Thus it contains exactly one cone point and $\bar \tau$ permutes the other two. It follows that up to relabeling, $(a,b,c) = (r,r,m)$ for some integers $r, m \geq 2$. Further, $S^2(r,r,m)/ \bar \tau = D^2(r; m)$, where $D^2(r; m)$ is the $2$-orbifold with underlying space a $2$-disk and singular set consisting of a cone point of order $r$, a corner-reflector point $x$ of order $m$, and a reflection line $\partial D^2 \setminus \{x\}$. Therefore $L(\bar p, \bar q) = W / \tau \cong L(r, t)$ for some integer $t$. Thus $r = \bar p$, which proves part (1).
A Montesinos-type analysis of the quotient of the $\tau$-invariant solid torus given by the inverse image in $W$ of a small annular neighbourhood of $\hbox{Fix}(\bar \tau)$ in $S^2(\bar p, \bar p, m)$ shows that the branch set of this quotient is of the form described in part (2). It is well known that this branch set has one component if $n$ is odd and two otherwise, so part (2) holds.
\end{proof}
\section{Beginning of the proof of Theorem \ref{once-punctured}} \label{sec: once-punctured}
\subsection{Assumptions} \label{assumptions 1}
We assume throughout the rest of the paper that $M$ is a hyperbolic knot manifold containing an essential once-punctured torus $F$ of boundary slope $\beta$ which caps off to an essential torus in $M(\beta)$ (cf. Corollary \ref{stays incompressible}) and that $M(\alpha)$ is an atoroidal, irreducible, small Seifert manifold (cf. Corollary \ref{smallseifert}). We assume as well that $\Delta(\alpha, \beta) > 3$, and (therefore) $M_F$ is a genus $2$ handlebody by Proposition \ref{reduction}.
We will show that under these assumptions, $\Delta(\alpha, \beta) \leq 5$, $F$ is not a fibre, $\pi_1(M(\alpha))$ is finite non-cyclic, and
(a) if $\Delta(\alpha, \beta) = 4$, $(M;\alpha,\beta)\cong (Wh(\frac{-2n\pm1}{n}); -4,0)$
for some integer $n$ with $|n|>1$ and $M(\alpha)$ has base orbifold $S^2(2,2,|\mp 2n-1|)$;
(b) if $\Delta(\alpha, \beta) = 5$, then $(M; \alpha, \beta) \cong (Wh(-3/2); -5, 0)$ and $M(\alpha)$ has base orbifold $S^2(2,3,3)$.
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-fig0.eps}} \caption{ }\label{bgz4-fig0}
\end{figure}
\subsection{An involution on $M$} \label{involution 1}
\noindent There is an involution $\tau_F$ on $F$ with exactly three fixed points whose
action on $\partial F$ is rotation by $\pi$. See Figure
\ref{bgz4-fig0}. Thus $F / \tau_F$ is the $2$-orbifold $D^2(2,2,2)$. Let $N \cong F \times I$
be a small neighbourhood of $F$ in $M$ and extend $\tau_F$ to an involution
$\tau_N$ in the obvious way. Then $\tau_N| F \times \partial I$ extends to a
hyperelliptic involution of $\partial M_F$. Since $M_F$ is a genus $2$ handlebody, the latter extends to an
involution $\tau_{M_F}$ of $M_F$. Piecing together $\tau_N$ and $\tau_{M_F}$ we
obtain an orientation-preserving involution $\tau:M \to M$ with non-empty
$1$-dimensional fixed point set $\widetilde L \subset \hbox{int}(M)$. Further,
$V := M/\tau$ is a solid torus containing the branch set $L$ of $\tau$. By construction, this is a hyperbolic link which
intersects some meridional disk of $V$ transversely and in three points. When $F$ is a fibre in $M$, $L$ is braided in $V$.
Note that $L$
cannot intersect any meridional disk in one point as $M$ is
$\partial$-irreducible.
The slopes on $\partial M$ can be identified with $\pm$-classes of primitive elements of $H_1(\partial M)$.
In particular we assume $\alpha, \beta \in H_1(\partial M)$. Let $\mu$ be any dual slope to $\beta$. This means that $1 = \Delta(\mu, \beta) = |\mu \cdot \beta|$. Hence $\{\mu, \beta\}$ form a basis for $H_1(\partial M)$. Write
\begin{equation}\label{alpha 1}
\text{\em $\alpha = p \mu + q \beta$}
\end{equation}
where $p, q$ are coprime. After possibly changing the signs of $\mu$ and $\beta$ we may assume that
\begin{equation}\label{p 1}
\text{\em $p = \Delta(\alpha, \beta)$}
\end{equation}
Without loss of generality we may suppose that $p \geq 1$.
The map $M \to V$ is a double cover when restricted to $\partial M$. It sends $\beta$ to a slope $\bar \beta$, a meridian of $V$, and sends $\mu$ to $\bar \mu$, a longitude of $V$.
For each slope $\gamma$ on $\partial M$, $\tau$ extends to an involution $\tau_\gamma: M(\gamma) \to M(\gamma)$. Moreover, if $\widetilde U_\gamma$ denotes the filling torus in $M(\gamma)$ and $\widetilde K_\gamma$ its core, then
\begin{equation}\label{fix 1}
\text{\em $\hbox{Fix}(\tau_\gamma) = \left\{ \begin{array}{ll}
\widetilde L & \hbox{ if } \Delta(\gamma, \beta) \hbox{ is odd} \\
\widetilde L \cup \widetilde K_\gamma & \hbox{ if } \Delta(\gamma, \beta) \hbox{ is even}
\end{array} \right.$}
\end{equation}
It is clear that $\widetilde U_\gamma/\tau_\gamma$ is a solid torus $U_\gamma$.
Denote its core $\widetilde K_\gamma / \tau_\gamma$ by $K_\gamma$. Thus
$M(\gamma) / \tau_\gamma = V \cup_{\bar {\gamma}} U_\gamma$ is a lens space. Indeed,
if $\gamma = r\mu + s \beta$, then under the double cover $\partial M \to
\partial V$ we have $\gamma \mapsto r \bar \mu + 2 s \bar \beta$. Let $\bar
\gamma = \frac{1}{\gcd(2,r)}(r \bar \mu + 2 s\bar \beta)$ denote the associated
slope and $L_\gamma$ the branch set in $M(\gamma) / \tau_\gamma$. Then
$$(M(\gamma) / \tau_\gamma, L_\gamma) = (V(\bar \gamma), L_{\gamma}) \cong \left\{ \begin{array}{ll}
(L(r, 2s), L) & \hbox{ if } r \hbox{ is odd} \\
(L(\frac{r}{2}, s), L \cup K_\gamma) & \hbox{ if } r \hbox{ is even}
\end{array} \right.$$
We are interested in the case $\gamma = \alpha$. Set
\begin{equation}\label{pbar 1}
\text{\em $\bar p = p/ \gcd(p,2)$ \;\;\;\;\; \hbox{ and } \;\;\;\;\;
$\bar q = 2q/\gcd(p,2)$}
\end{equation}
so that $\bar \alpha = \bar p \bar \mu + \bar q \bar
\beta$ and
$$M(\alpha) / \tau_\alpha \cong L(\bar p, \bar q)$$
From \ref{fix 1} we see that
\begin{equation}\label{comps 1}
\text{\em $|L_\alpha| = \left\{ \begin{array}{ll}
|L| & \hbox{ if } p \hbox{ is odd} \\
|L| + 1 & \hbox{ if } p \hbox{ is even}
\end{array} \right. $}
\end{equation}
Fix a $\tau_\alpha$-invariant Seifert structure on $M(\alpha)$ with base orbifold $S^2(a,b,c)$ where $1 \leq a \leq b \leq c$ (Lemma \ref{invariant}).
Let $\bar \tau_\alpha$ be the involution of $S^2(a,b,c)$ (possibly the identity) induced by $\tau_\alpha$.
\begin{lemma} \label{+-quotient 2}
Suppose that assumptions \ref{assumptions 1} hold. Suppose as well that $M(\alpha)$ is not a lens space and that $\tau_\alpha$ preserves the orientations of the Seifert fibres of $M(\alpha)$. Then there is a Seifert structure on $L(\bar p, \bar q)$ in which $L_\alpha$ is a union of at most three fibres, at least one of which is regular. Further, $L_\alpha = L$ so that $p = \Delta(\alpha, \beta)$ is odd.
\end{lemma}
\begin{proof} Lemma \ref{+-quotient} shows that $L$ is a union of fibres in the induced Seifert structure on $L(\bar p, \bar q)$ and at least one of these fibres is regular. This implies that $K_\alpha \not \subset L_\alpha$ as otherwise $L = L_\alpha \setminus K_\alpha$ would not be a hyperbolic link in $V$. Thus $L = L_\alpha$ so $p$ is odd by \ref{comps 1}.
\end{proof}
\begin{lemma} \label{--quotient 2}
Suppose that assumptions \ref{assumptions 1} hold. Suppose as well that $M(\alpha)$ is not a lens space and that $\tau_\alpha$ reverses the orientations of the Seifert fibres of $M(\alpha)$. Then
\noindent $(1)$ $M(\alpha)$ has base orbifold $S^2(\bar p, \bar p, m)$ where $m \geq 2$ and the Seifert
invariants of the exceptional fibres of order $\bar p$ are the same. Hence if
$M(\alpha)$ is not a prism manifold, $\Delta(\alpha, \beta) \ne 4$.
\noindent $(2)$ There is an integer $n$ coprime with $m$ such that $L_\alpha$ is isotopic
to the closure $K(m/n)$ of an $m/n$ rational tangle in a genus $1$ Heegaard
solid torus of $M(\alpha) / \tau_\alpha$ as depicted in Figure \ref{bgz4-fig24}.
In particular,
$$|L_\alpha| = \left\{ \begin{array}{ll}
1 & \hbox{ if } n \hbox{ is odd} \\
2 & \hbox{ if } n \hbox{ is even}
\end{array} \right. $$
\noindent $(3)$ $|L| = 1$, $m$ is odd, and $n \equiv p$ (mod $2$).
\end{lemma}
\begin{proof} Parts (1) and (2) follow from Lemma \ref{--quotient}.
In order to prove part (3), suppose that $|L| = 2$. Then part (2) shows that $L = L_\alpha$. In particular, $p$ is odd (\ref{comps 1}). Consideration of the form of $L_\alpha$ (cf. Figure \ref{bgz4-fig24}) shows that its two components are isotopic to one another. But since $L$ is transverse to a meridian disk of $V$ and intersects it in three points, the generator $\gamma$ of $H_1(V(\bar \alpha)) \cong \mathbb Z / \bar p$ carried by the core of $V$ satisfies $\gamma = \pm 2 \gamma$. Hence $\bar p = 3$. But $p$ is odd so $\Delta(\alpha, \beta) = p = \bar p = 3$, contrary to our hypotheses. Thus $|L| = 1$.
Next suppose that $m$ is even. Then $L_\alpha = K(m/n)$ is connected, so $L = L_\alpha$ and $p$ is odd, and $L$ is homotopically trivial in $L(\bar p, \bar q)$. But $L$ intersects a meridian disk of the Heegaard torus $V \subset L(\bar p, \bar q)$ transversely and in three points, so the only way it can be null homotopic is for $3 = \bar p$. Since $p$ is odd, $p = 3$, which contradicts our hypotheses. Thus $m$ is odd.
By (2), $|L_\alpha| \equiv n$ (mod $2$). Since $|L| = 1$ by (3), Identity \ref{comps 1} shows that $|L_\alpha| \equiv p$ (mod $2$).
\end{proof}
\subsection{Constraints on the branch set $L$}
Here we deduce strong constraints on the form of the branch set $L$ in $V$.
\begin{lemma} \label{cyclic-cover}
Suppose that assumptions \ref{assumptions 1} hold and that $\tau_\alpha$ reverses the orientation of the Seifert fibres of $M(\alpha)$. Let $k \geq 1$ be an integer dividing $\bar p$ and consider the $k$-fold cyclic cover $S^2(\frac{\bar p}{k}, \frac{\bar p}{k}, m, m, \ldots , m) \to S^2(\bar p, \bar p, m)$
obtained by the $k$-fold unwrapping of $S^2(\bar p, \bar p, m)$ about the two cone points labeled $\bar p$. Let $\widetilde {M(\alpha)}_k \to M(\alpha)$ be the associated $k$-fold cyclic cover where $\widetilde {M(\alpha)}_k$ is Seifert with base orbifold $S^2(\frac{\bar p}{k}, \frac{\bar p}{k}, m, m, \ldots , m)$ and the inclusion of a regular fibre of $M(\alpha)$ lifts to $\widetilde {M(\alpha)}_k$. Define $\widetilde{M}_k \to M$ to be the cover obtained by restricting $\widetilde {M(\alpha)}_k \to M(\alpha)$ to $M$. Then
\noindent $(1)$ $\partial \widetilde M_k$ is connected and $F$ lifts to $\widetilde M_k$. In particular, $\beta$ lifts to a slope $\widetilde \beta$ on $\partial \widetilde M_k$.
\noindent $(2)$ $\alpha$ lifts to a slope $\widetilde \alpha$ on $\partial \widetilde M_k$ such that $\widetilde {M(\alpha)}_k = \widetilde M_k(\widetilde \alpha)$. Further, $\Delta(\widetilde \alpha, \widetilde \beta) = \frac{p}{k}$.
\noindent $(3)$ $\widetilde \alpha$ is the singular slope of a closed essential surface in $\widetilde M_k$ if $S^2(\frac{\bar p}{k}, \frac{\bar p}{k}, m, m, \ldots , m)$ is hyperbolic with at least four cone points. If this is the case, $p/k \leq 3$.
\end{lemma}
\begin{proof} The cover $S^2(\frac{\bar p}{k}, \frac{\bar p}{k}, m, m, \ldots , m) \to S^2(\bar p, \bar p, m)$ is determined by the homomorphism $\varphi: H_1(S^2(\bar p , \bar p, m)) = \langle x,y : \bar p x = \bar p y = m(x+y) = 0 \rangle \to \mathbb Z/k$ where $\varphi(x) \equiv - \varphi(y) \equiv 1 \hbox{ (mod $k$)}$.
First note that the homomorphism $H_1(M(\alpha)) \to H_1(V(\bar \alpha)) \cong \mathbb Z / \bar p$ kills any class carried by a regular Seifert fibre of $M(\alpha)$ (i.e. there are regular fibres with image an interval). Thus it factors through a homomorphism $\psi: H_1(S^2(\bar p, \bar p, m)) \to H_1(V(\bar \alpha))$. Since $\tau_\alpha$ preserves the fibre of multiplicity $m$ in $M(\alpha)$, but reverses its orientation, $(\bar \tau_\alpha)_* (x + y) = -(x + y)$. Thus $2(x+y)$ is sent to zero in $H_1(V(\bar \alpha))$ while $x$ is sent to a generator. Since $m$ is odd and $m(x+y) = 0$, $x+y \mapsto 0 \in H_1(V(\bar \alpha))$. It follows that $\varphi$ factors as $H_1(S^2(\bar p, \bar p, m)) \stackrel{\psi}{\longrightarrow} H_1(V(\bar \alpha)) \stackrel{\cong}{\longrightarrow} \mathbb Z / \bar p \to \mathbb Z / k$. Since $H_1(F)$ lies in the kernel of $H_1(M) \to H_1(V)$ while $\mu$ is sent to a generator of $H_1(V)$, we conclude that $\partial \widetilde M_k$ is connected and $F$ lifts to $\widetilde M_k$. This proves (1).
For (2), note that by construction, there is a basis $\{\widetilde \mu, \widetilde \beta\}$ of $H_1(\partial \widetilde M_k)$ where $\widetilde \mu$ is sent to $k \mu$ and $\widetilde \beta$ is sent to $\beta$ in $H_1(\partial M)$. Then $\alpha = p \mu + q \beta$ lifts to $(\frac{p}{k}) \widetilde \mu + q \widetilde \beta$. Clearly $\Delta(\widetilde \alpha, \widetilde \beta) = \frac{p}{k}$.
Part (3) is a consequence of \cite[Theorems 1.5 and 1.7]{BGZ1}.
\end{proof}
\begin{lemma} \label{holds}
Suppose that assumptions \ref{assumptions 1} hold. Then $M$ is not a once-punctured torus bundle. In particular, Theorem \ref{once-punctured} holds when $F$ is a fibre.
\end{lemma}
{\noindent{\bf Proof.\hspace{2mm}}} We assume that $M$ is a once-punctured torus bundle in order to obtain a contradiction.
There is a $3$-braid $\sigma$ whose closure in $V$ is $L$. Altering $\sigma$ by conjugation in
$B_3 = \langle \sigma_1, \sigma_2 : \sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2 \rangle$
leaves its closure invariant. (Here $\sigma_1, \sigma_2$ are the standard generators of $B_3$.) There is an isomorphism $B_3 \cong \langle a, b : a^3 = b^2 \rangle$ where $a = \sigma_1 \sigma_2$ and $b = \sigma_1 \sigma_2 \sigma_1$. The center of $B_3$ is generated by $a^3$ with $B_3 / \langle a^3 \rangle \cong \mathbb Z/2 * \mathbb Z / 3$. We will use $\bar \sigma$ to denote the image of a braid $\sigma$ in $B_3 / \langle a^3 \rangle$. Thus $\bar a$ has order $3$ and $\bar b$ has order $2$. In particular,
$$\bar \sigma_1 = \bar a^{-1} \bar b$$
\vspace{-.7cm}
$$\bar \sigma_2 = \bar b \bar a^2$$
The inverse image $\widehat L$ of $L \subset V \subset L(\bar p, \bar q)$ under the universal cover $S^3 \to L(\bar p, \bar q)$ is the closure the braid $\sigma^{\bar p} a^{-3 \bar q}$.
\begin{claim} \label{nontrivial}
{\it $\widehat L$ is not the trivial knot.}
\end{claim}
\begin{proof}[Proof of Claim \ref{nontrivial}]
If $\widehat L$ is trivial then $\sigma^{\bar p} a^{-3 \bar q}$ is conjugate to $\sigma_1 \sigma_2, \sigma_1^{-1} \sigma_2^{-1}$, or $\sigma_1 \sigma_2^{-1}$ (\cite[Classification Theorem, page 27]{BiMe}). The first two cases can be ruled out since they would imply that the exterior of $\widehat L$ in the inverse image of $V$ in $S^3$ is not hyperbolic. On the other hand, in the third case we have $\bar \sigma^{\bar p} = \bar \sigma_1 \bar \sigma_2^{-1} = \bar a^2 \bar b \bar a \bar b \in B_3 / \langle a^3 \rangle \cong \mathbb Z/2 * \mathbb Z / 3$. But this is impossible since $\bar a^2 \bar b \bar a \bar b$ is not a proper power.
\end{proof}
\begin{claim} \label{+}
{\it $\tau_\alpha$ preserves the orientation of the Seifert fibres of $M(\alpha)$. In particular, $\widehat L$ is a union of fibres in some Seifert structure on $S^3$ and $p$ is odd.}
\end{claim}
\begin{proof}[Proof of Claim \ref{+}] Suppose otherwise and consider the $\bar p$-fold cyclic cover $\widetilde M_{\bar p} \to M$ constructed in Lemma \ref{cyclic-cover}. The base orbifold $S^2(m, m, \ldots , m)$ of $\widetilde{M(\alpha)}$ has $\bar p$ cone points, each of order $m \geq 3$ by Lemma \ref{--quotient 2}(3). If $\bar p \geq 4$, Lemma \ref{cyclic-cover} (3) implies that $\widetilde M_{\bar p}$ contains a closed essential surface, contrary to \cite{CJR} or \cite{FH}. Hence $\bar p$ is $2$ or $3$ and therefore as $p > 3$, $p$ is $4$ or $6$. Identity \ref{comps 1} then combines with parts (2) and (3) of Lemma \ref{--quotient 2} to show that $|L_\alpha| = 2$ and $m$ is odd. It follows that each component of $L_\alpha$ is isotopic to the core of a genus one Heegaard solid torus in $L(\bar p, \bar q)$ (cf. Figure \ref{bgz4-fig24}). In particular this is true of $L = L_\alpha \setminus K_\alpha$. It follows that $\widehat L$ is a trivial knot, contrary to the conclusion of Claim \ref{nontrivial}. Thus $\tau_\alpha$ preserves the orientation of the Seifert fibres of $M(\alpha)$. The remaining conclusions are a consequence of Lemma \ref{+-quotient 2}.
\end{proof}
Claim \ref{+} implies that $\bar p = p$ and $\bar q = 2q$.
Since $L$ is a hyperbolic link in $V$, $\widehat L$ is a hyperbolic link in the inverse image of $V$ in $S^3$. Thus the Schreier normal form for $\sigma^{p} a^{-6 q}$ is generic (cf. \cite[Theorem 5.2]{FKP}). On the other hand, by Claim \ref{+}, $\widehat L$ is not a hyperbolic link in $S^3$ so \cite[Theorem 5.5]{FKP} implies that $\sigma^{p} a^{-6 q}$ is conjugate in $B_3$ to a braid of the form $\sigma_1^c \sigma_2^d$ where $c, d \in \mathbb Z \setminus \{0\}$. We must have $\min\{|c|, |d|\} = 1$ as otherwise $\widehat L$ would be a connected sum of non-trivial torus links, contrary to the conclusion of Claim \ref{+}. Thus $\sigma^{p} a^{-6 q}$ is conjugate to $\sigma_1^c \sigma_2^\epsilon$ for some $\epsilon \in \{\pm 1\}$ and non-zero $c$. The following claim completes the proof of Lemma \ref{holds}.
\begin{claim} \label{2}
{\it If $p > 3$, $\sigma^{p} a^{-6 q}$ is not conjugate to $\sigma_1^c \sigma_2^\epsilon$ for any $\epsilon \in \{\pm 1\}$. }
\end{claim}
\begin{proof}[Proof of Claim \ref{2}]
Suppose that $\sigma^{p} a^{-6 q}$ is conjugate to $\sigma_1^c \sigma_2^\epsilon$ for some $\epsilon \in \{\pm 1\}$. Projecting into $B_3 / \langle a^3 \rangle$ shows that $\bar \sigma_1^c \bar \sigma_2^\epsilon$ is a $p^{th}$-power in that group. The latter condition is invariant under conjugation and taking inverse, so without loss of generality we can suppose that $\epsilon = 1$. Now
$$\bar \sigma_1^c \bar \sigma_2 = (\bar a^{-1} \bar b)^c (\bar b \bar a^{-1}) = \left\{
\begin{array}{ll}
(\bar b \bar a)^{|c|} (\bar b \bar a^{-1}) & \hbox{if } c \leq 0 \\
\bar a & \hbox{if } c = 1 \\
\bar a^{-1} \bar b \bar a & \hbox{if } c = 2 \\
(\bar a^{-1} \bar b) \bar a^{-1} (\bar a^{-1} \bar b)^{-1} & \hbox{if } c = 3 \\
(\bar a^{-1} \bar b \bar a)(\bar a \bar b) (\bar a^{-1} \bar b)^{c-4} (\bar a^{-1} \bar b \bar a)^{-1} & \hbox{if } c > 3
\end{array} \right. $$
Consideration of the normal form for elements of $\mathbb Z/2 * \mathbb Z / 3$ shows that the only values of $c$ which give proper powers in $B_3 / \langle a^3 \rangle$ are $c = 1, 2$, or $3$.
Say $c = 1$ or $3$. Then up to conjugation, $\bar \sigma^p = \bar a^{\pm 1}$ and therefore $\bar \sigma = \bar a ^{\pm 1}$. Hence $\sigma = a^{3k \pm 1}$ for some integer $k$. But then it is easy to see that $L$ is boundary-parallel in $V$, contrary to the fact that $V \setminus L$ is hyperbolic.
Next suppose that $c = 2$. Then $\bar \sigma^p = \bar b$ up to conjugation and therefore the same is true of $\bar \sigma$. As $a^3 = b^2$, $\sigma = b^{2n + 1}$ for some integer $n$. Then $L \subset \hbox{int}(V)$ has two components. One is a core curve $K_0$ of $V$ while the other is isotopic in $V \setminus K_0$ into $\partial V$. It follows that there is an essential annulus properly embedded in the exterior of $L$ in $\hbox{int}(V)$. But this contradicts the fact that $L$ is a hyperbolic link in $V$.
\end{proof}
\vspace{-.5cm} \hfill {\hspace{2mm}{\small $\diamondsuit$}} (of Lemma \ref{holds})
Recall that $t_1^+$ is the number of tight components of $\breve{\Phi}_1^+$ (cf. \cite[\S 6]{BGZ2}).
\begin{lemma} \label{secondreduction}
Suppose that assumptions \ref{assumptions 1} hold. Then $t_1^+ = 0$. In particular, $M(\beta)_{\widehat F}$ is Seifert with base
orbifold of the form $A(a)$ where $A$ is an annulus and $a \geq 2$.
\end{lemma}
\begin{proof} Lemma \ref{holds} implies that $F$ is not a fibre and so Proposition \ref{main1} and Remark \ref{t1+ > 0} show that the lemma holds as long as either $M(\alpha)$ is very small or $\Delta(\alpha, \beta) > 4$. Assume then that $M(\alpha)$ is not very small and that $\Delta(\alpha, \beta) = 4$. The latter equality combines with Lemma \ref{+-quotient 2} to show that $\tau_\alpha$ reverses the orientations of the fibres of $M(\alpha)$. But then Lemma \ref{--quotient 2}(1) implies that $M(\alpha)$ is a prism manifold, contradicting our assumption that $
|
M(\alpha)$ is not very small. Thus the lemma holds.
\end{proof}
\begin{lemma} \label{types}
Suppose that assumptions \ref{assumptions 1} hold. Then there are coprime integers
$a \geq 2$ and $b$ as well as a $3$-braid $\sigma$ such that $L$ is isotopic
to the link depicted in Figure \ref{fig3}.
\end{lemma}
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-fig3.eps}} \caption{ }\label{fig3}
\end{figure}
\begin{proof} By Lemma \ref{secondreduction}, $M(\beta)_{\widehat F}$ is Seifert with base
orbifold of the form $A(a)$ where $A$ is an annulus and $a \geq 2$.
Consider the involution $\widehat \tau: M(\beta)_{\widehat F} \to
M(\beta)_{\widehat F}$ induced by $\tau_\beta$.
Note that $M(\beta)_{\widehat F}/ \widehat \tau = V(\bar \beta)_{\widehat F/
\widehat\tau} \cong (S^2 \times S^1)_{S^2 \times \{x\}} \cong S^2 \times I$. Now
$M(\beta)_{\widehat F}$ has a unique Seifert structure which we can suppose is
$\widehat \tau$-invariant. Let $\overline{\widehat \tau}$ be the induced
involution on $A(a)$. Note $\overline{\widehat \tau}$ cannot preserve
orientation as otherwise $M(\beta)_{\widehat F}/ \widehat \tau \cong S^2 \times
I$ would admit a Seifert structure. Thus it reverses orientation and since it
fixes the cone point and leaves each boundary component invariant, it must be
reflection along a pair of disjoint properly embedded arcs, each of which runs
from one boundary component to the other. The quotient $A(a) /
\overline{\widehat \tau}$ is a disk whose boundary contains two disjoint,
compact arcs, each a reflector arc, one of which contains the $\mathbb Z/ a$ cone point.
It follows that the branch set in $M(\beta)_{\widehat F}/ \widehat \tau \cong
S^2 \times I$ consists of a $2$-braid and an $\frac{a}{b}$-rational tangle running from
one end to the other which are separated by a properly embedded vertical annulus. See
Figure \ref{fig1}.
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-fig1.eps}} \caption{ }\label{fig1}
\end{figure}
We claim that $K_\beta \cap M(\beta)_{\widehat F}$ is a component of the 2-braid. To see this, first note that by Lemma \ref{secondreduction}, $\breve{\Phi}_1^+$ has no tight components. Next we refer the reader to the final paragraph of the proof of \cite[Lemma 7.9]{BGZ2}. It is shown there that $M_{F} = X^+$ is obtained by attaching a solid torus $V$ to the product of an interval $I$ and a once-punctured annulus $A_*$ where $V \cap (A_* \times I)$ is a pair of annuli which have winding number $a$ in $V$ and components of $\partial A_* \times I$ in $A_* \times I$. This decomposition is invariant under the restriction of $\hat \tau$ to $M_F$ and it is easy to see that the quotient of $V$ contains the $\frac{a}{b}$-rational tangle. Since $(\partial M)_{\partial F} \subset A_* \times I$ is disjoint from $V$, it follows that $K_\beta \cap M(\beta)_{\widehat F}$ is a component of the 2-braid. Thus $L \cap M_F/ \tau$ is
as depicted in Figure \ref{fig2}, where ${\delta}$ is a $3$-braid. It follows that there is a $3$-braid $\sigma$ such that $L$ is as depicted
in Figure \ref{fig3}.
\end{proof}
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-fig2.eps}} \caption{ }\label{fig2}
\end{figure}
\subsection{The lens space case} \label{lens space case}
The methods of this paper can be used to give a new proof of Ken Baker's theorem: {\it if $M$ contains a once-punctured essential genus $1$ surface of boundary slope $\beta$ and $M(\alpha)$ is a lens space, then $\Delta(\alpha, \beta) \leq 3$} \cite{Ba}. We begin the proof here and complete it in \S \ref{lens space delta = 5}.
\begin{lemma} \label{once-punctured cyclic case}
Suppose that assumptions \ref{assumptions 1} hold. If $\pi_1(M(\alpha))$ is cyclic, then $p = 5$, $F$ is not a fibre, and $L_\alpha$ is either the core of a solid torus of a genus one Heegaard splitting of $L(5, 2q)$ or the boundary of a M\"{o}bius band spine of a Heegaard solid torus of $L(5, 2q)$.
\end{lemma}
\begin{proof} We know that $F$ is not a fibre (Lemma \ref{holds}), so $p = \Delta(\alpha, \beta) \leq 6$ by Proposition \ref{main1}.
As $\Delta(\alpha, \beta) = p \geq 4$, $M(\alpha) / \tau_\alpha \cong L(\bar p, \bar q)$ is not $S^3$.
Hence by Lemma \ref{cyclic case}, $L_\alpha$ is a union of Seifert fibres of some Seifert fibring of $L(\bar p, \bar q)$. Since $L$ is hyperbolic in $V$, $K_\alpha$ cannot be contained in $L_\alpha$. Thus $p$ is odd by \ref{fix 1}, so $p = \bar p = 5$, $\bar q = 2q$, and $L = L_\alpha$.
Lemma \ref{cyclic case}(1) then shows that $L_\alpha$ is either the core of a solid torus of a genus one Heegaard splitting of $L(5, 2q)$ or the boundary of a M\"{o}bius band spine of a Heegaard solid torus of $L(5, 2q)$.
\end{proof}
\begin{rem}
{\rm We can complete the proof of Baker's result mentioned above at this point by invoking a theorem of Sangyop Lee \cite{L3} which states that the distance between a toroidal filling slope and a lens space filling slope is at most $4$. Nevertheless, we give an independent proof that $\Delta(\alpha, \beta) \ne 5$ (and so $\Delta(\alpha, \beta) \leq 3$) in \S \ref{lens space delta = 5} below.}
\end{rem}
\subsection{Reduction of the proof of Theorem \ref{once-punctured}}
In this section we reduce the proof of Theorem \ref{once-punctured} to several problems concerning links. These will be solved in the subsequent sections of the paper. We begin with a slight sharpening of our upper bound for $\Delta(\alpha, \beta)$.
\begin{lemma} \label{at most 7}
If assumptions \ref{assumptions 1} hold, then $\Delta(\alpha, \beta) < 8$.
\end{lemma}
\begin{proof} By Lemma \ref{holds}, $F$ is not a fibre in $M$. Hence $\Delta(\alpha, \beta) \leq 8$ by Proposition \ref{main1} (or \cite{LM}). Suppose that $\Delta(\alpha, \beta) = 8$. Then $M(\alpha)$ is not very small by Proposition \ref{main1}. Further, Proposition \ref{reduction} implies that $M_F$ is a genus two handlebody, so we can construct an involution $\tau$ as above. Then Lemma \ref{+-quotient 2} implies that $\tau_\alpha$ reverses the orientations of the Seifert fibres of $M(\alpha)$. Parts (1) and (3) of Lemma \ref{--quotient 2} imply that $M(\alpha)$ has a Seifert structure with base orbifold $S^2(4,4, m)$ where $m \geq 3$ is odd. Let $\widetilde M_2 \to M$ be the $2$-fold cover constructed in Lemma \ref{cyclic-cover}. By part (2) of that lemma, $\widetilde M_2 (\widetilde \alpha)$ is Seifert with base orbifold $S^2(4, 4, m, m)$. But then Lemma \ref{cyclic-cover} (3) implies $4 = \frac{8}{2} \leq 3$, which is false. Thus $\Delta(\alpha, \beta) \ne 8$.
\end{proof}
\begin{lemma} \label{delta = 4 case}
Suppose that assumptions \ref{assumptions 1} hold and that $\Delta(\alpha, \beta) = 4$. Then $M(\alpha)$ is a prism manifold.
\end{lemma}
\begin{proof} Since $\Delta(\alpha, \beta)$ is even, $M(\alpha)$ is not a lens space (Lemma \ref{once-punctured cyclic case}) and so Lemma \ref{+-quotient 2} implies that $\tau_\alpha$ reverses the orientations of the fibres of $M(\alpha)$. Lemma \ref{--quotient 2}(1) now implies that $M(\alpha)$ is a prism manifold.
\end{proof}
Given the last two lemmas, to complete the proof of Theorem \ref{once-punctured} under assumptions \ref{assumptions 1}, we must consider the possibility that $\Delta(\alpha, \beta) \in \{5,6,7\}$ besides the case when ${\Delta}(\alpha,\beta)=4$
and $M({\alpha})$ is a prism manifold. We do this by comparing the constraints obtained above on the branch sets $L$ and $L_\alpha$:
\begin{itemize}
\item $L$ lies in $V$ as depicted in Figure \ref{fig3} (Lemma \ref{types});
\vspace{.3cm} \item when $M(\alpha)$ is not a lens space and $\tau_\alpha$ preserves the orientation of the Seifert fibres of $M(\alpha)$, then $\Delta(\alpha, \beta)$ is odd and $L_\alpha$ is the union of at most three fibres of some Seifert structure on $L(\bar p, \bar q)$ (Lemma \ref{+-quotient 2});
\vspace{.3cm} \item when $M(\alpha)$ is not a lens space and $\tau_\alpha$ reverses the orientation of the Seifert fibres of $M(\alpha)$, then $L_\alpha$ lies in some Heegaard solid torus of $L(\bar p, \bar q)$ as depicted in Figure \ref{bgz4-fig24} (Lemma \ref{--quotient});
\vspace{.3cm} \item when $M(\alpha)$ is a lens space, then $\Delta(\alpha, \beta) = 5$ and $L_\alpha$ is either the core of a Heegaard solid torus of $L(5, 2q)$ or the boundary of a M\"{o}bius band spine of a Heegaard solid torus of $L(5, 2q)$
(Lemma \ref{once-punctured cyclic case}).
\end{itemize}
The proof of Theorem \ref{once-punctured} therefore reduces to proving the following claims.
\begin{enumerate}
\item If $\tau_\alpha$ preserves the orientation of the Seifert fibres and $M(\alpha)$ is not a lens space, then $\Delta(\alpha, \beta) = 5$ and $(M; \alpha, \beta)$ is homeomorphic to $(Wh(-3/2); -5, 0)$.
\vspace{.3cm} \item The links contained in the universal cover $S^3$ of $L(7, \bar q)$ which are depicted in Figure \ref{7ab} and Figure \ref{7mn} are not equivalent when $\Delta(\alpha, \beta) = 7$, $|L| = 1$, $m$ is odd, and $n \equiv 1$ (mod 2).
\vspace{.3cm} \item the link depicted in Figure \ref{fig3} considered as lying in a Heegaard solid torus in $L(5, 2q)$ is not isotopic to either the core of a Heegaard solid torus or the boundary of a M\"{o}bius band spine of a Heegaard solid torus.
\vspace{.3cm} \item The links contained in a Heegaard solid torus in $L(3, \bar q)$ depicted in Figure \ref{bgz4-fig24} and Figure \ref{fig3} are not equivalent.
\vspace{.3cm} \item The links contained in the universal cover $S^3$ of $L(5, \bar q)$ which are depicted in Figure \ref{bgz4-5ab} and Figure \ref{bgz4-5mn} are not equivalent in the universal cover $S^3$ of $L(5, \bar q)$ when $\Delta(\alpha, \beta) = 5$, $|L| = 1$, $m$ is odd, and $n \equiv 1$ (mod 2).
\vspace{.3cm} \item ${\Delta}(\alpha,\beta)=4$ and $M({\alpha})$ is a prism manifold if and only if
$(M; \alpha,\beta)\cong (Wh(\frac{-2n\pm1}{n});-4,0)$ for some integer $n$ with $|n|>1$.
\end{enumerate}
These will be proved in \S \ref{m=1 Seifert}, \S \ref{dist7}, \S \ref{lens space delta = 5}, \S \ref{sec6.3}, \S \ref{delta = 5} and \S \ref{prism-section} respectively.
\section{The case that $\tau_\alpha$ preserves the orientation of the Seifert fibres, $M(\alpha)$ is not a lens space, and $\Delta(\alpha, \beta) \in \{5, 7\}$.} \label{m=1 Seifert}
In this section we suppose that assumptions \ref{assumptions 1} hold and show that if $\tau_\alpha$ preserves the orientation of the Seifert fibres, $M(\alpha)$ is not a lens space, and $\Delta(\alpha, \beta) \in \{5, 7\}$, then $\Delta(\alpha, \beta) = 5$ and $(M; \alpha, \beta)$ is homeomorphic to $(Wh(-3/2); -5, 0)$.
By hypothesis, $M(\alpha)$ is small Seifert with exactly three singular fibres. It is not a prism manifold
by \cite{L2} and so has a unique Seifert structure. Recall that $M(\alpha)/\tau_\alpha = V(\bar \alpha)$ is the lens space
$L(\bar p, \bar q) = L(p, 2q)$ and the branch set of $\tau_\alpha$ in $L(p, 2q)$ is a link denoted by $L_\alpha$. As $p$ is odd, $L_\alpha=L$ (cf. \ref{comps 1}).
Suppose that $L_\alpha$ is a Seifert link with respect to the induced Seifert
fibration on $L(p, 2q) = M(\alpha)/\tau_\alpha$. We need to show that
$p=5$ and $(M; \alpha, \beta)$ is homeomorphic to $(Wh(-3/2); -5, 0)$.
By Lemma \ref{+-quotient 2}, at least one component of $L$ is a regular fibre of $L(p, 2q)$.
Let $K$ be such a component and denote by $X$ the exterior of $L$ in $L(p, 2q)$.
Then $X$ has the induced Seifert fibration with $|\partial X|=|L|$ boundary components, each a torus.
Let $T_K$ be the component of $\partial X$ corresponding to the knot $K$.
\begin{lemma}\label{unique annulus for K}
There is an essential separating vertical annulus $(A,\partial A)\subset (X, T_K)$
which cuts $X$ into two components $X_1$ and $X_2$ such that each $X_i$ is
either a torus cross interval or a fibred solid torus
whose core is a singular fibre of $X$ of order larger than $2$.
\end{lemma}
\begin{proof} The lemma follows from Lemma \ref{+-quotient} and its proof.
Let $\bar \tau_\alpha$ be the induced map on the orbifold $S^2(a,b,c)$ of $M(\alpha)$
where each of $a, b, c$ is $\geq 2$.
Then $\bar \tau_\alpha$ is either the identity or an involution with two
fixed points. Let $\sigma_1,\sigma_2,\sigma_3$ denote the singular fibres of $M(\alpha)$ and let their orders be
$a,b,c$ respectively.
First assume that $\bar \tau_\alpha$ is the identity map. Then Lemma \ref{+-quotient}(a) implies that at least one of $a,b,c$, say $a$, is $2$ and the fixed point set of $\tau_\alpha$ in $M(\alpha)$ is the union of those $\sigma_i$ with even orders.
In particular $\sigma_1$ belongs to the fixed point set of $\tau_\alpha$
and its image in $L(p, 2q)$ is a regular fibre.
Note that if $\sigma_2$, respectively $\sigma_3$, does not belong to the fixed point
set of $\tau_\alpha$, then $b$, respectively $c$, is odd, and the image of $\sigma_2$,
respectively $\sigma_3$, in $L(p, 2q)$
is a fibre of $L(p, 2q)$ of order $b$, respectively $c$.
Hence the sum of $|\partial X|=|L|$ and the number of the singular fibres of $X$
equals $3$. Since the surface underlying the base orbifold of $X$ is planar, the lemma follows in this case.
Next assume that $\bar \tau_\alpha$ is an involution.
Then two of the singular fibres of $M(\alpha)$, say $\sigma_1$ and $\sigma_2$, have the same order $a = b$. Both are mapped to a common singular fibre in $L(p, 2q)$ of order $a$.
Since $M(\alpha)$ is not a prism manifold, $a = b > 2$.
By Lemma \ref{+-quotient}(b), the fixed point set of $\tau_\alpha$ in $M(\alpha)$ consists of a regular fibre
and possibly the remaining singular fibre $\sigma_3$.
If $\sigma_3$ does not belong to $\hbox{Fix}(\tau_\alpha)$, then its image in $L(p, 2q)$ is a singular fibre of order
$2c \geq 4$ and therefore the sum $|\partial X|=|L|$ and the number of the singular fibres of $X$
again equals $3$. As in the previous case, the lemma follows from this.
\end{proof}
Recall that $K_\alpha$ is the core circle of the filling solid torus in $V(\bar \alpha) = L(p, 2q)$.
The exterior $Y$ of $K_\alpha$ in $X$ is also the exterior of $L$ in $V$ and so is hyperbolic.
Let $T_V = \partial V \subset \partial Y$.
The solid torus $V$ has a meridian disk $D$ which intersects $L$ in three points
such that $P=D\cap Y$ is an essential thrice-punctured disk in $Y$.
Let $d_V = \partial P \cap T_V$ and let $c_1,c_2,c_3$
be the three components of $\partial P$ contained in $\partial Y\setminus T_V$.
Note $d_V$ has the slope $\bar {\beta}$ in $T_V$, and each $c_i$ is a meridian curve
of some component of $L$.
Among all annuli satisfying the conditions of Lemma \ref{unique annulus for K},
we choose one, denoted $A$, which intersects $T_V$ in the minimal number of components.
Since $Y$ is hyperbolic, $A\cap T_V$ is non-empty.
The surface $Q=A\cap Y$ is essential in $Y$.
Since $A$ is separating in $X$, $\partial Q\cap T_V$ consists of an even number, say $n$,
of simple essential loops in $T_V$ of slope $\bar \alpha$.
Let $a_1,a_2$ be the two components of $\partial Q$ in $T_K$, and let $b_1,...,b_{n}$ be
the components of $\partial Q$ in $T_V$ numbered so that they occur successively around $d_V$.
Each $a_i$ is a Seifert fibre of $X$, and each $b_j$ has slope $\bar \alpha$ on $T_V$. If $c_j$ is a meridian curve of $K$, then the distance between $c_j$ and $a_i$ is $1$ since $K$ is a regular fibre of $L(p,2q)$.
Now define the labeled intersection graphs $\Gamma_P$ and $\Gamma_Q$ as usual.
We may consider $d_V$, $c_1,c_2,c_3$, $a_1,a_2$, $b_1,...,b_{n}$ as
the boundaries of the fat vertices of these graphs.
Each $b_i$, $i=1,...,n$, has valency $p = \Delta(\bar \alpha,\bar \beta)=\Delta(\alpha,\beta)$, and the valency of $d_V$
is $np$.
Note that the valency of $a_1$ is equal to the valency of $a_2$ and is equal to
the number of $c_i$'s which are meridians of $K$. Further, the valency of $c_i$
is either $2$ or $0$ depending on whether $c_i$ is a meridian curve of $K$ or not.
We call the edges in $\Gamma_Q$ connecting some $b_i$ to some $b_j$
{\it B-edges}, and call the edges in $\Gamma_P$ connecting $d_V$ to itself {\it $D$-edges}.
Similarly we define $A$-edges, $C$-edges, $AB$-edges, and $CD$-edges.
Note that an arc in $P\cap Q$ is a $B$-edge in $\Gamma_Q$ if and only if it is a $D$-edge in $\Gamma_P$, is an $A$-edge in $\Gamma_Q$ if and only if it is an $C$-edge in $\Gamma_P$,
and is an $AB$-edge in $\Gamma_Q$ if and only if it is a $CD$-edge in $\Gamma_P$.
Every $D$-edge is positive, so by the parity rule, every $B$-edge is negative.
By construction, no $D$-edge in $\Gamma_P$ is boundary parallel
in $P$. Thus there are at most three different $D$-edges in the reduced graph $\overline{{\Gamma}}_P$ (cf. Figure \ref{d-edges}).
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-d-edges.eps}} \caption{The maximal possible $D$-edges in
$\overline{{\Gamma}}_P$ }\label{d-edges}
\end{figure}
\begin{lemma}\label{S cycle with D-edges} There can be no
$S$-cycle in $\Gamma_P$ consisting of $D$-edges.
\end{lemma}
\begin{proof} Suppose otherwise that
$\{e_1, e_2\}$ is an $S$-cycle in $\Gamma_P$ consisting of $D$-edges with
label pair $\{j,j+1\}$.
We may assume that the bigon face $E$ between $e_1$ and $e_2$
lies on the $X_1$-side of $A$.
Let $H$ be the portion of the filling solid torus of $L(p, 2q)$ lying in $X_1$ which contains
$\hat b_j$ and $\hat b_{j+1}$.
In $\Gamma_Q$, $e_1\cup b_j \cup e_2\cup b_{j+1}$ cannot be contained in a disk region $D_*$ of $A$ as otherwise a regular neighborhood of $D_*\cup E\cup H$ in $X_1$ would be a punctured projective space.
Thus $e_1\cup b_j \cup e_2\cup b_{j+1}$ contains a core circle of $A$ (cf. Figure \ref{S-cycle}).
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-S-cycle.eps}} \caption{The corresponding
cycle $\{e_1, e_2\}$ in
${\Gamma}_Q$}\label{S-cycle}
\end{figure}
Let $U$ be a regular neighborhood of $E\cup H\cup A$ in $X_1$. Then $U$ is a solid torus and the frontier
of $U$ in $X_1$ is an annulus $(A', \partial A')\subset (X, T_K)$ for which $\partial A'$ is parallel to $\partial A$ in $T_K$ and which intersects $T_V$ in $n - 2$ components. By construction, $A'$ is inessential in $X_1$ and therefore
$X_1$ cannot be a torus cross interval.
It follows that $X_1$ is a fibred solid torus of $X$.
Since $A'$ has winding number $2$ in the solid torus $U$, the singular fibre of $X_1$
has order $2$, contrary to Lemma \ref{unique annulus for K}. Thus the lemma holds.
\end{proof}
Note that ${\Gamma}_P$ has at most six $CD$-edges and thus
${\Gamma}_P$ has at least $(np-6)/2$ $D$-edges, so there is a family of at least $(np-6)/6$
mutually parallel $D$-edges.
By Lemma \ref{S cycle with D-edges} we have $(np-6)/6\leq n/2$. Hence $n\leq 6/(p-3)$
and therefore $p=5$ and $n=2$. If ${\Gamma}_P$ has a $C$-edge, it would have only one family of parallel
$D$-edges, and this family would have at least three edges, contrary to the fact that no two $D$-edges can be parallel in
$\Gamma_P$ by Lemma \ref{S cycle with D-edges}.
Also, ${\Gamma}_P$ has at least four $CD$-edges as otherwise there would be four $D$-edges
two of which would form an $S$-cycle.
Thus ${\Gamma}_P$ has either six or four $CD$-edges.
We first consider the case when there are exactly four $CD$-edges.
In this case we have three $D$-edges in ${\Gamma}_P$, no two of which can be parallel.
Hence ${\Gamma}_P$ may be assumed
to be as illustrated in Figure \ref{db not parallel}, i.e. $c_1$ and $c_2$ are contained in $T_K$ and $c_3$ is contained $\partial X\setminus T_K$.
Thus $|L|=|\partial X|=2$ and we may assume that $X_1$ is a solid torus and $X_2$ is a
torus cross interval.
In particular $c_3$ is contained in $X_2$.
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-db-not-parallel2.eps}} \caption{
${\Gamma}_P$ when $\Delta(\alpha, \beta)=5$, $n=2$ and $4$ $CD$-edges.}\label{db not parallel}
\end{figure}
Consider the face $f$ given in Figure \ref{db not parallel}.
From the figure we see that $f$ and $c_3$ are on the same side of $A$ (since $A$ is separating
in $X$) and thus
$f$ is contained in $X_2$.
Let $T_*$ be the component of $\partial X_2$ containing $A$ and $H$ that part of filling solid torus of $L(p, 2q)$ contained
in $X_2$. We use $\partial_0 H$ to denote $\partial H \cap T_V$. It is evident that the boundary $\partial f$ of $f$
is contained in $T_*\cup \partial_0 H$.
Also note that $\partial f\cap T_*$ cannot be contained in
a disk in $T_*$ as otherwise $X_2$ would contain a projective space as a summand.
Thus $\partial f\cap T_*$ is contained in an annulus $A_*$ of $T_*$.
A regular neighborhood $W$ of $H \cup f\cup T_*$ in $X_2$ is a Seifert fibred space whose base orbifold is an annulus with a cone point of order $2$. Since $X_2$ is a torus cross interval, the frontier of $W$ in $X_2$ is an incompressible torus in $X_2$.
But this torus cannot be parallel to $T_*$ in $X_2$, contradicting the fact that
$X_2$ is a torus cross interval. Thus the case when there are exactly four $CD$-edges does not arise.
We now know that ${\Gamma}_P$ must have six $CD$-edges.
Hence there are exactly two $D$-edges in ${\Gamma}_P$ and they are not parallel. It follows that ${\Gamma}_P$ is as illustrated in
Figure \ref{no-c-edges} (1) or (2). (Without
loss of generality, we may assume that the labels around $d_V$ are as shown in these figures and that the vertices $c_1$, $c_2$ and $c_3$
are numbered as given there.)
Therefore $L=K$ and both $X_1$ and $X_2$ are solid tori.
We are going to show that part (1) of Figure \ref{no-c-edges} cannot arise and that in case of part (2) of Figure \ref{no-c-edges}
the dual graph ${\Gamma}_Q$ may be assumed to be as shown in part (6) of
Figure \ref{dual-of-f}.
\begin{figure}[!ht]
\centerline{\includegraphics{bgz4-no-c-edges.eps}} \caption{${\Gamma}_P$ when $p=5$, $n=2$ with $6$ $CD$-edges}
\label{no-c-edges}
\end{figure}
\begin{lemma}\label{part (1) impossible} The graph ${\Gamma}_P$ cannot be as shown in part (1) of
Figure \ref{no-c-edges}.
\end{lemma}
\begin{proof}
Suppose otherwise that ${\Gamma}_P$ is given by part (1) of Figure \ref{no-c-edges}.
Since $A$ is a separating annulus, the faces $f_1, f_2$ of ${\Gamma}_P$ lie
on the same side of $A$, say in $X_1$, and the faces $g_1, g_2$ lie in
$X_2$.
Let $H$ be the part of the filling solid torus of $L(p, 2q)$ contained
in $X_1$ and set $\partial_0 H = \partial H \cap T_V$.
The boundary edges of $f_1$ consist of two $CD$-edges $e_1$, $e_2$ and one $D$-edge $e_3$.
Without loss of generality, we may assume that the label of the edge $e_1$ at the vertex $c_1$ is $2$.
In ${\Gamma}_Q$, the boundary edges of $f_1$ may be assumed to be as illustrated
in part (1) of Figure \ref{dual-of-f}.
Note that the boundary $\partial f_1$ of $f_1$, including the corners,
lies in $\partial X_1 \cup \partial_0 H$.
Further, $\partial f_1\cap \partial X_1$ is contained in an annulus $A_*$ of $\partial X_1$ whose slope has
distance $1$ from that of $\partial A$. Note as well that $\partial f_1\cap (\partial X_1\setminus A)$ is an essential arc in the annulus
$(\partial X_1 \setminus A)$.
A regular neighborhood $U$ of $H \cup f_1 \cup A_*$ in $X_1$ is a solid torus whose frontier in $X_1$ is an annulus $A_\#$
of winding number $2$ in $U$.
Thus $A_\#$ must be parallel to $\partial X_1\setminus A_*$ through $X_1\setminus U$.
It follows that the fundamental group of $X_1$ is carried by $U$ and thus
has presentation
$$<x,t; x^2t=1>$$
where we take a fat base point in $A$ containing $b_1\cup b_2\cup (\partial f_1 \cap A)\cup (\mbox{all $AB$-edges})$,
$x$ is a based loop formed by a cocore arc of $\partial_0 H$, and $t$ is a based loop formed by a cocore arc of $\partial X_1\setminus A$.
Now consider the face $f_2$.
We claim that the label of the edge $e_4$ at the vertex $c_3$ cannot be $2$.
Otherwise in ${\Gamma}_Q$, the boundary edges of $f_2$, $e_4$ and $e_5$ would be as depicted in part (2) or part (3) of Figure \ref{dual-of-f}. In either case, the face $f_2$ would add the relation
$xts=1$ to the presentation for $\pi_1(X_1)$ above, where $s$ is the element represented by a core circle of the annulus $A$. Thus the fundamental group of the solid torus $X_1$ would be generated by $s=x$.
But $s$ can be considered as a regular fibre of
$X$. So the singular fiber of $X_1$ would have order one, which contradicts Lemma \ref{unique annulus
|
psilon(n)$, $SNR_{in0}$, and for the reconstructed signal $x_R(n),$ $SNR_{out}$, for every realization, is given in Fig. \ref{RANSAC_CS_SNRoutD}.
The statistical results averaged over all considered realizations are given in Table \ref{tableI}.
\begin{table}[htbp]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& $N_{it}$ & $SNR_{in} $ & $SNR_{in0}$ & $SNR_{out0}$ & $SNR_{out}$ & $D$ \\ \hline
$I=8$ & 2.94 & -8.81 & 13.01 & 16.23 & 27.07 & 119.46 \\ \hline
$I=16$ & 11.99 & -16.73 & 13.00 & 14.39 & 26.46 & 111.33 \\ \hline
$I=24$ & 57.66 & -17.53 & 13.01 & 13.29 & 25.91 & 103.56 \\ \hline
\end{tabular}
\smallskip
\caption{Results for the case with $N=128$, $M=32$, $K=5$.}
\label{tableI}
\end{table}
The theoretically expected improvement (Section \ref{SNRD}) in the SNR (omitting the impulsive noise in the input signal) for the case $I=8$, in Table \ref{tableI}, is
$$SNR_{out}-SNR_{in0}=10\log\bigg(\frac{D}{K}\bigg)=10\log\bigg(\frac{119.46}{5}\bigg)=13.77 \text{ dB}$$
For other two cases, $I=16$ and $I=24$, we get $$SNR_{out}-SNR_{in0}=13.46 \text{ dB}$$
and $SNR_{out}-SNR_{in0}=13.14$ dB, respectively. This is in high agreement with the statistical data in Table \ref{tableI}.
\smallskip
\noindent\textbf{Example 2:}
Since the impulsive Cauchy noise may take some small values as well (some of the assumed $I$ outliers may happen to be inliers, in reality), the expected number of iterations is smaller than the theoretically obtained result given by (\ref{vjerovatnoca}). In order to check the expected number of iterations, $N_{it}$, against its theoretical value, avoiding the ambiguity of possible Cauchy noise inliers, we will provide that all $I$ signal samples are certainly outliers. The impulsive noise is modified as $\nu(n) \to \nu(n)+100$, to be sure that all of these samples are the outliers. Then, we have repeated the same experiment with $100$ realizations and obtained $N_{it}=10.74$, while the theory in (\ref{vjerovatnoca}) predicts $P=0.0927$ with $N_{it}=1/P=10.78$, for $I=8$. The same numerical experiment as in Example 1 is performed for $I=16$, and we get $P=0.0071$ with $N_{it}=1/P=140.95$, while the statistics for this case produced $N_{it}=139.78$. The same holds for $I=24$. The complete results of the experiment, with the modified impulsive noise, are given in Table \ref{TableII}.
\begin{table}[htbp]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& $N_{it}$ & $SNR_{in} $ & $SNR_{in0}$ & $SNR_{out0}$ & $SNR_{out}$ & $D$ \\ \hline
$I=8$ & 10.74 & -22.45 & 13.01 & 21.02 & 27.20 & 119.96 \\ \hline
$I=16$ & 139.78 & -26.75 & 13.00 & 21.06 & 27.05 & 111.94 \\ \hline
$I=24$ & 2335.67 & -27.21 & 13.01 & 20.71 & 26.31 & 103.56 \\ \hline
\end{tabular}
\smallskip
\caption{Results for the case with $N=128$, $M=32$, $K=5$, with highly impulsive noise so that all its values are outliers.}
\label{TableII}
\end{table}
In this case (of the modified impulsive noise), we can also check the result for the SNR ratio in the final RANSAC mid-result, when the consensus is detected on a small subset with $M$ samples. Then, $$SNR_{out0}-SNR_{in0}=10\log\bigg(\frac{M}{K}\bigg)=10\log\bigg(\frac{32}{5}\bigg)=8.06 \ \mathrm{ dB}$$
for all three considered cases, $I=8$, $I=16$, and $I=24$.
|
This result is in complete agreement with the statistical results for these SNR values in Table \ref{TableII}.
\smallskip
\noindent\textbf{Example 3:}
The experiment form Example 1 is repeated with some other numbers of the available samples, $N$, sparsites, $K$, the number of impulsive disturbances, $I$, and the samples used in the RANSAC-based CS reconstruction. The results are given in Tables \ref{TableIII} and \ref{TableIV} and further prove the efficiency of the proposed method and accuracy of the proposed SNR descriptors.
\begin{table}[htbp]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& $N_{it}$ & $SNR_{in} $ & $SNR_{in0}$ & $SNR_{out0}$ & $SNR_{out}$ & $D$ \\ \hline
$I=8$ & 5.40 & -5.24 & 16.79 & 14.90 & 26.30 & 107.52 \\ \hline
$I=12$ &
15.06 & -10.13 & 16.76 & 13.33 & 25.79 & 104.85 \\ \hline
$I=16$ &
99.47 & -11.42 & 16.80 & 13.10 & 25.41 & 101.41
\\ \hline
\end{tabular}
\smallskip
\caption{Results for the case with $N=128$, $M=64$, $K=12$.}
\label{TableIII}
\end{table}
\begin{table}[htbp]
\centering
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& $N_{it}$ & $SNR_{in} $ & $SNR_{in0}$ & $SNR_{out0}$ & $SNR_{out}$ & $D$ \\ \hline
$I=16$ &
5.36 & -8.63 & 16.83 & 15.79 & 29.14 & 231.65 \\ \hline
$I=32$ &
61.84 & -14.55 & 16.84 & 14.15 & 28.30 & 214.07 \\ \hline
$I=40$ &
261.30 & -16.90 & 16.81 & 14.42 & 28.25 & 208.48
\\ \hline
\end{tabular}
\smallskip
\caption{Results for the case with $N=256$, $M=64$, $K=12$.}
\label{TableIV}
\end{table}
\smallskip
\noindent\textbf{Example 4:}
The experiment from Example 1 is repeated with the impulsive noise only, that is, when $y(n)=x(n)+\nu(n)$. As expected, for all considered numbers of outliers, the obtained results are within the computer precision accuracy. They are given in Table \ref{TableV}. In this case, the value of $d$ should very small. We used $d=10^{-6}$, for this experiment.
\begin{table}[htbp]
\centering
\begin{footnotesize}
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
& $N_{it}$ & $SNR_{in} $ & $SNR_{in0}$ & $SNR_{out0}$ & $SNR_{out}$ & $D$ \\ \hline
$I=8$ & 10.68 & -22.44 & 316.64 & 276.25 & 282.12 & 120 \\ \hline
$I=16$ & 138.21 & -26.75 & 316.67 & 271.86 & 277.52 & 112 \\ \hline
$I=24$ & 2402.23 & -27.47 & 316.67 & 270.51 & 275.40 & 104 \\ \hline
\end{tabular}
\end{footnotesize}
\smallskip
\caption{Results for the case with $N=128$, $M=32$, $K=5$, with highly impulsive noise in outliers and without noise in inliers.}
\label{TableV}
\end{table}
\section{Conclusion}
Inspired by recent advances in compressive sensing and sparse signal processing, we have developed a RANSAC-based methodology for the detection of disturbances. Upon detecting disturbance-free samples, a compressive sensing algorithm is used for the recovery of the disturbed samples, which are considered as unavailable. The presented methodology is general -- no specific assumptions have been made regarding the range of values or statistical behavior of the disturbance. The presented approach exploits the fact that disturbances degrade signal sparsity. It has been only assumed that the signals of interest exhibit sparsity in a known transformation domain. The theory has been verified on numerical examples.
\section*{Acknowledgment}
This work was supported by the Croatian Science Foundation under the project IP-2020-02-4358.
|
\section{Conclusion}
We proposed four self-supervised methods to enhance semantic understanding of conversational text by multiple interlocutors for abstractive dialogue summarization. The methods, switching utterance, switching interlocutor, inserting utterance, and masking interlocutor, specifically strengthen the learning of structural and component information in dialogues. By enhancing an off-the-shelf pre-trained BERT with our methods, we build an abstractive summarizer in a shared encoder-decoder architecture for sequence-to-sequence training. Our experimental results on the SAMSum corpus indicate a substantive improvement measured in ROUGE scores. Through a careful ablation study, we provide a practical insight for the crucial hyperparameter settings of the proposed self-supervised methods.
We believe there is still much room for improvements in our setting. We used only `bert-base-uncased'~version to prove our concept, "Can intuitively enhanced representation of encoder produce better performance in the sequence generation task?". Nevertheless, this base model has already achieved promising results so that it will inspire more investigation into engaging large language models with more sophisticated experimental setup.
\section{Experiments}
\subsection{Dataset}
We evaluate our approach on the SAMSum corpus~\cite{gliwa-etal-2019-samsum} that is constructed by linguists fluent in English, which contains over 16k chat dialogues consists of over 182K utterances (see Table~\ref{tab:samsum} for statistics on dataset).
Each utterance has the specified format that a colon at the beginning of an utterance separates the interlocutor's name and their speaking content.
And also, we append 311 facial emojis into our vocab that appear most often in the train dataset in order to avoid including too many \emph{[unk]} in the input sequence. The effect of appended tokens is on Table~\ref{tab:result}.
\begin{table}[t!]
\caption{The number of each components of SAMSum corpus, where "Dial." stands for "Dialogue.", "Utter." for "Utterance", "Inter." for "Interlocutor's name" appearing in dialogues. "OOV." is short for the number (proportion) of utterance that contains at least one out of vocabulary token which is mostly facial emojis such as in "Nadine: caaaaaat \emojicat\emojicat\emojicat\emojicat\emojicat\emojicat\emojicat" , and "OOV. w/ FE" for the case "with facial emoji appended vocabulary".}
\label{tab:samsum}
\centering
\resizebox{1.00\columnwidth}{!}{
\begin{tabular}{l|ccccc}
\toprule
\textbf{Type} & \textbf{\# Dial.} & \textbf{\# Utter.} & \textbf{\# Inter.} & \textbf{\# OOV. } & \textbf{\# OOV. w/ FE} \\
\midrule
Train & 14,732 & 164,505 & 4,289 & 3,315 (2.0\%) & 1,165 (0.7\%)\\
Valid & 819 & 8,860 & 894 & 179 (2.0\%)& 44 (0.4\%)\\
Test & 818 & 9,212 & 912 & 179 (1.9\%)& 82 (0.9\%)\\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Implementation and training details}
All our experiment start from publicly available `bert-base-uncased'~version.
For self-supervised tasks, we preprocess the dialogue datasets for our experiments following steps: a) add \emph{[SEP]} token at every end of utterance which improves structural information of dialogue~\cite{zhao-etal-2020-improving} and is used as the utterance representation in our self-supervised methods, b) replace each name with a single token included in our vocabulary to simplify the problem so that model can learn the semantic relations between the subject and action ("Who did What"), rather than distribution of name tokens.
Then, we add a linear layer with dropout (0.1) on top of BERT, where \emph{[SEP]} and \emph{[MASK]} tokens are fed into for classification task.
Throughout abstractive summarization task, we use the shared encoder-decoder architecture (so called {\sc{bertShare}}~ in \cite{rothe-etal-2020-leveraging}) where both model's weights are shared and initialized with BERT trained by one of proposed self-supervised methods. Note that weight sharing between encoder and decoder is necessary rather than optional in that it reduces model parameters and improves the final performance as well in our experimental setup.
The hyper-parameters of self-supervised learning and summarization are almost identical except training epochs.
We use the AdamW optimizer~\cite{adamw} for both self-supervised and summarization task, with batch size 128, input sequence length 512 and learning rate from 2e-5 to 5e-5, warmup steps are 500.
For self-supervised learning, we train the model until the train loss converged (upper bounded by 5K steps).
After that, we fine-tune the model for summarization task until the validation loss converges.
\begin{table}[t!]
\caption{Results in terms of ROUGE metric on the SAMSum corpus test set.}
\label{tab:result}
\centering
\resizebox{1.00\columnwidth}{!}{
\begin{tabular}{lcccc}
\toprule
\textbf{Model} & \textbf{R-1} & \textbf{R-2} & \textbf{R-L} & \textbf{R-AVG} \\
\midrule
LONGEST-3 & 32.46 & 10.27 & 29.92 & 24.22 \\
Transformer~\cite{46201} & 37.27 & 10.76 & 32.73 & 26.92 \\
Fast Abs RL Enhanced~\cite{chen-bansal-2018-fast} & 41.95 & 18.06 & 39.23 & 33.08\\
D-HGN~\cite{feng2020incorporating} & 42.03 & 18.07 & 39.56 & 33.22 \\
TGDGA~\cite{zhao-etal-2020-improving} & 43.11 & \bf 19.15 & 40.49 & 34.25 \\
\midrule
{\sc{bertShare}}~ & 39.07 & 12.74 & 36.05 & 29.29 \\
+ Facial Emoji & 39.97 & 13.7 & 36.42 & 30.03 \\
w/ Masking Interlocutor & 40.29 & 13.91 & 37.46 & 30.55 \\
w/ Inserting Utterance & 44.17 & 18.76 & 41.68 & 34.87 \\
w/ Switching Interlocutor & 44.01 & 18.03 & 41.42 & 34.49 \\
w/ Switching Utterance & \bf 44.78 & 19.12 & \bf 42.21 & \bf 35.37 \\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Evaluation Metrics}
ROUGE~\cite{lin-2004-rouge} is one of standard measures to evaluate machine generated text over many natural language processing fields.
However, the metric based on only n-gram overlapping may not be the best choice for abstractive dialogue summarization~\cite{gliwa-etal-2019-samsum}. Such measure is lacking aspects of fluency, intelligibility, and repetition~\cite{Savelieva2020AbstractiveSO}.
In order to address this issue, we also report consine-similarity between model's prediction (i.e. generated summary) and ground truth using the sentence encoder~\cite{reimers-2019-sentence-bert}, `stsb-roberta-large (available at https://www.sbert.net/)' fine-tuned on Semantic Text Similarity dataset.
More precisely, we use only the longest 100 samples for calculating cosine-similarity in order to uncover the differences between good and poor quality summaries more clearly.
\section{Introduction}
In natural language processing, abstractive summarization generates a concise summary for lengthy source text using words that do not necessarily appear in the source. Such creative aspect (in comparison with extractive summarization) makes abstractive summarization one of the most challenging tasks in computational linguistics. In speech, dialogue summarization enables a useful capability to capture salient information scattered in a dialogue containing the utterances by multiple interlocutors and rewrite them into simplified, easy-to-grasp text. With the rapid growth of online communications, providing dialogue summaries is becoming one of the most important features in a speech system. The recent outbreak of the coronavirus pandemic (COVID-19) and other on-going global incidents demand a crisp summary of long speech conversations more useful and appealing than ever.
Self-supervised learning has been used widely to complement or replace entirely human-annotated datasets in training deep networks of language, speech, and visual models. There are numerous approaches for neural dialogue summarization, yet not many have aimed to improve semantic and structural understanding of dialogue with a specifically-designed self-supervised learning method.
In this paper, we propose self-supervised methods for training a neural abstractive dialogue summarization model. We have designed pretext tasks that require the model under training predict whether there are incorrect ordering and irrelevant information in utterances or not. There are also tasks to predict switched and masked interlocutor names. When pre-training with our self-supervised methods, the model seems to learn a better understanding of the semantic relevance between interlocutor and utterance. This can be crucial to capture the essence of a whole dialogue in much shortened text while preserving the most salient information.
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{figure2.pdf}
\caption{An overview of proposed self-supervised methods. They are \textbf{a) switching utterance}, \textbf{b) switching interlocutor}, \textbf{c) inserting utterance}, \textbf{d) masking interlocutor} methods. Each method is carried out separately. (Note that some methods can be combined to run beneficially.)}
\label{fig:methods}
\end{figure}
In Figure~\ref{fig:methods}, we describe an illustrative example about how our self-supervised methods take place during the pre-training of a neural language model upon which abstractive summarization task can be fine-tuned. There are four self-supervised methods, namely switching utterance, switching interlocutor, inserting utterance, and masking interlocutor. Notice that three out of the four self-supervised methods are set up as a simple binary classification problem (\emph{i.e.}, corrupted or not). In masking interlocutor method, mapping to a correct interlocutor's name would be required instead. We remark that our proposed method is fully compatible with a publicly available pre-trained language model like BERT~\cite{devlin-etal-2019-bert}. This makes our approach more appealing and retrofitting in the popular paradigm comprising the pre-training and fine-tuning stages for building contemporary NLP applications.
To construct a neural abstractive dialogue summarizer, we enhance a pre-trained BERT with our self-supervised methods and use it as both encoder and decoder in sequence-to
|
-sequence model~\cite{Bahdanau2015NeuralMT, rothe-etal-2020-leveraging} while sharing the weights between the encoder and the decoder (see Figure~\ref{fig:archi}). We fine-tune and evaluate empirically our BERT-based summarizer using the SAMSum corpus~\cite{gliwa-etal-2019-samsum}. Our self-supervised methods indicate a significantly improved performance compared to the baseline ({\sc{bertShare}}~), which is using the pre-trained BERT as is (without applying the proposed self-supervised methods).
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figure1.pdf}
\end{center}
\caption{Illustration of our approach on abstractive summarization task. First, we enhance dialogue context understanding of BERT via (a) proposed self-supervised methods. Then, we initialize the traditional encoder-decoder model with enhanced BERT and fine-tune on abstractive summarization task.}
\label{fig:archi}
\end{figure}
\section{Methods}
In this section, we describe how to enhance BERT's semantic understanding of dialogue in self-supervised fashion.
\begin{itemize}
\item \textbf{Switching Utterance}: Similar with previous works~\cite{wu-etal-2019-self, Logeswaran_Lee_Radev_2018} of predicting switched sentences, this task is to predict whether each utterance (not sentence) is switched or not. We switch some utterances selected with the probability \emph{P$_u$}. Hence, the number of switched utterances changes dynamically at each training step.
At the same time, we mask some interlocutor's name with the probabilty \emph{P$_n$}. This optional constraint of input utterances makes the task more challenging.
\item \textbf{Switching Interlocutor}: For this task, we switch interlocutor's name in utterances instead of utterances with the probabilty \emph{P$_i$} and we did not mask the name of interlocutors. Additionally, we concatenate corresponding reference summary with the utterances for each input sequence, so it can help the model find a mismatch between interlocutor and what they said.
\item \textbf{Inserting Utterance}: For this task, we insert \emph{K} utterances from other dialogues selected with a pre-defined probability into randomly selected \emph{K} positions from inter-utterances and the model predict whether each utterance is from other dialogue or not. If interlocutor's name in inserted utterances remains same as the original, which look obviously unfamiliar, the task has a little worth to solve. Hence, we replace the name of them properly to camouflage where they come from.
\item \textbf{Masking Interlocutor}: This task is similar to masked language modeling~\cite{devlin-etal-2019-bert} task which a model needs to recover masked token from a vocabulary. In our work, we only masked interlocutor's name appeared in reference summary. Then, the model predicts masked names using the information of utterances.
\end{itemize}
\section{Related Work}
\subsection{Self-supervised pre-training for text summarization}
In recent years, self-supervised learning has pushed the performance of a wide range of natural language processing (NLP) tasks to new state-of-the-art and becoming a dominant paradigm in NLP. Numerous pre-training methods based on self-supervised learning for text summarization have been introduced. A common approach lets a model predict the original input tokens from randomly masked tokens (or sentences) in a document to resemble a target downstream task, \emph{i.e.}, abstractive summarization~\cite{lewis-etal-2020-bart, pmlr-v119-zhang20ae, song2019mass}. Zhang \emph{et~al.}~\cite{zhang-etal-2019-hibert} adopt masked sentence prediction in pre-training stage and employed it as a sentence encoder for extractive summarization on large-scale news corpus. The most similar approach to ours is Wang \emph{et~al.}~\cite{wang-etal-2019-self}, where they introduced three self-supervised pre-training tasks for extractive summarization with CNN/DM datasets~\cite{see-etal-2017-get}.
Compared to previous work, our approach focuses more on how to incorporate the heterogeneous attributes of a dialogue to self-supervised methods in order to overcome the challenge that fine-tuning is often unstable on small datasets and causes performance degradation~\cite{rothe-etal-2020-leveraging}.
\subsection{Leveraging pre-trained model for text summarization}
Liu \& Lapata~\cite{liu-lapata-2019-text} has shown that BERT can beneficially be applied to both extractive and abstractive document summarization. For abstractive task, their model consists of a BERT pre-trained on extractive summarization task as the encoder and randomly initialized the 6-layer Transformer blocks for the decoder. On the contrary, we aim to leverage the full power of the proposed self-supervised methods by employing BERT as both an encoder and a decoder, which is pre-trained on the task destined to enhance semantic understanding of a dialogue. In recent empirical studies by Google Research~\cite{rothe-etal-2020-leveraging, goodman2019multistage}, it is possible to achieve state-of-the art results on text summarization without any auxiliary task with the encoder-decoder network utilizing pre-trained BERT, RoBERTa, GPT-2 (so-called {\sc{bert2bert}}, {\sc{bert2gpt2}}).
Our implementation of {\sc{bert2bert}}~architecture, however, could not have reached higher performance than baselines because the dataset of our choice is much smaller (about 10 times smaller) than previous work. This means that {\sc{bert2bert}}~architecture has a great potential for a nice warm-starting model but may not be sufficient for low-resource datasets to achieve a better performance.
\subsection{Abstractive dialogue summarization}
We have chosen a dialogue dataset of messenger-like natural conversations that have very distinct features from formally-written styled documents like a news article. Most salient pieces of conversations are scattered across the utterances by multiple interlocutors, and it makes difficult to decide what the key point of a dialogue is. Moreover, There are no large enough annotated datasets for abstractive dialogue summarization to train deep neural generative models. To address these problems, Ganesh and Dingliwal~\cite{ganesh2020restructuring} propose a two-phase pipeline method that uses discourse labels and an existing document summarizer in the zero-shot learning perspectives.
Feng~\emph{et~al.}~\cite{feng2020incorporating} propose the first to incorporate commonsense knowledge into abstractive dialogue summarization with a graph neural net that includes both utterance and knowledge nodes. Another approach using the graph structure for dialogue summarization is in Zhao~\emph{et~al.}~\cite{zhao-etal-2020-improving}. They have tackled the problem of previous sequence-to-sequence models~\cite{Bahdanau2015NeuralMT} about not paying an attention to handle the sentence-level long-distance dependency and capture the cross-sentence relations by proposing a method that can construct the whole dialogue as a graph for abastractive dialogue summarization. These two approaches constitute the most recent work on the SAMSum corpus. In this work, we choose them as the baselines to compare our model's score with.
\section{Results}
\subsection{Evaluation of Self-supervised Learning}
Table \ref{tab:result} shows the result of the proposed self-supervised learning methods on SAMSum dataset.
The upper part of Table \ref{tab:result} is the scores reported in each paper or \cite{gliwa-etal-2019-samsum}.
We use the official PyRouge package (https://pypi.org/project/py-rouge) to compute the ROUGE score.
In contrast to {\sc{bertShare}}~'s promising results~\cite{rothe-etal-2020-leveraging} on a large corpus of news like CNN/DM, it shows the relatively lower performance on our dataset compared to recent strong baselines. However, all combinations of {\sc{bertShare}}~and our methods improve performance dramatically by up to 6.08\% (Switching Utterance) in averaged ROUGE score.
On the other hand, Masking Interlocutor methods shows the poor result compared to other methods.
This is because the generative training objective that predicts masked tokens from thousands of candidates usually requires much longer training steps with a larger corpus than other binary decision task (i.e. switched or not), and we will investigate proper training setup in future work.
In addition, we confirm that the inclusion of facial emojis in the vocabulary is advantageous for abstractive summarization task, even though they do not appear in the reference summary at all. This is also evidence that if the encoder is enhanced and is employed as decoder too, it leads to better performance on sequence generation task.
\subsection{Ablation Test}
\subsubsection{Adding reference summary or elimination of interlocutor's name}
In the three among our proposed methods, the reference summary (gold label) and interlocutor's name can be added or removed freely.
For example, a model can detect the inconsistent utterance order (Switch Utterance) or mismatch of $\langle$interlocutor, utterance$\rangle$ pair (Switch Interlocutor) by only using the information flows in utterances without summary and interlocutor's name.
We conduct ablation studies to investigate how the model incorporates these additional information and the result is on Table \ref{tab:ablation}.
From the result, adding these two factors does not lead to better results in most cases in terms of ROUGE score.
Note that the highest ROUGE score achieved without any factors.
Interestingly, in most cases, using the concatenation of summary and dialogue as the input sequence for our three methods helps achieve higher scores in terms of cosine-similarity of reference summaries and generated ones.
It implies that there exists inconsistency between ROUGE and cosine-similarity metric and reference summary can help a model learn useful semantic information to generate better summary.
\subsubsection{Probability of Switching and Masking Name}
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{ablation.png}
\end{center}
\caption{Ablation results of Switching Utterance method according to combinations of two probabilities in terms of average ROUGE scores.}
\label{fig:heatmap}
\end{figure}
We also investigate the effects of probability of switching, \emph{P$_u$} and masking name, \emph{P$_n$} for Switching Utterance task.
We set the \emph{P$_u$} a range of [0.33, 1] and (0, 1] for \emph{P$_n$}.
Even if \emph{P$_u$} is 1.0, that does not mean all utterances are switched, because there are chances to restore the original order of utterances while re-ordering utterances randomly.
The numbers in Figure \ref{fig:heatmap} refer to the average score of ROUGE-1, 2 and L measure, obtained from the model pre-trained by Switching Utterance method with each probability and fine-tuned on SAMSum dataset for abstractive summarization.
As shown in Figure \ref{fig:heatmap}, we found that combination of two probabilities on the edges in the matrix, i.e. (1.0, 1.0), (1.0, 0.0), (0.33, 0), tends to show better performance.
On the other hand, the combination of 0.5 and 0.5 yields the worst result.
|
\section{The Models}\label{sect-models}
The general Lagrangian for a single-field model with second-order field equation is an arbitrary function $p(X,\phi)$ of the scalar field $\phi$ and its kinetic energy $X \equiv \frac{1}{2} \partial_\mu \phi \partial^\mu \phi$. In addition to the general case, in this paper we will consider four specific Lagrangians within this class:
\begin{enumerate}
\item Canonical field with potential $V(\phi)$:
\begin{equation}
\label{m:canonical}
p(X,\phi) = BX - V(\phi) \,,
\end{equation}
where we add the coefficient $B$ denoting the coupling strength of the kinetic energy, though traditionally it is set to 1.
\item Non-canonical inflation (NCI) model: This features an arbitrary power on the kinetic term and was studied in Refs.~\cite{Mukhanov:2005bu,Li:2012vt,Unnikrishnan:2012zu,Unnikrishnan2}, the Lagrangian being
\begin{equation}
p(X,\phi) = BX^n - V(\phi) \,,
\end{equation}
where $n$ is a positive integer (equal to 1 in the canonical case).
\item {Tachyon model}: The Tachyon model was introduced in Refs.~\cite{Sen:2002nu,Sen:2002in}, and later studied in Refs.~\cite{Gibbons:2002md,Fairbairn:2002yp,Kofman:2002rh,Piao:2002vf}. Its Lagrangian is
\begin{equation}
\label{m:tachyon}
p(X,\phi) = -V(\phi) \sqrt{1-2 \lambda_{\rm s} X} \,,
\end{equation}
where the warp factor $\lambda_{\rm s}$ is a constant.
\item {DBI inflation model}: Its Lagrangian is given by
\begin{equation}
\label{m:DBI}
p(X,\phi) = - \frac{1}{f(\phi)} \sqrt{1-2 f(\phi) X} - V(\phi) \,,
\end{equation}
where we follow Ref.~\cite{Alishahiha:2004eh} and take the warp factor $f(\phi) \simeq \lambda_{\rm s}/\phi^4$ with $\lambda_{\rm s}$ constant. Some authors, e.g.\ Ref.~\cite{Silverstein:2003hf}, include an additional term $1/f(\phi)$ to cancel the leading-order term from expanding the square root. We have absorbed such a term into $V(\phi)$. Investigations of this model include Refs.~\cite{Peiris:2007gz,Lorenz:2008et,Powell:2008bi}; in particular Ref.~\cite{Bean:2007eh} contains a detailed study of one particular regime of this model.
\end{enumerate}
The sound speed is defined as
\begin{equation}
\css = \frac{\delta p}{\delta \rho} = \frac{\partial{p}/\partial X}{\partial\rho/\partial X}\,,
\end{equation}
and one can show that for both non-canonical inflation models it is $\cs = \sqrt{1 - 2f(\phi)X}$, where the warp factor takes the unified form as $f=f(\phi)$. This factor is constant in Tachyon models, and a function of the scalar field in the DBI inflation models.
We investigate the observables of interest, being the power spectrum $\Ps$, its spectral index $n_{{\rm s}}$, and the tensor--scalar ratio $r$, within the slow-roll approximation. Following Refs.~\cite{ArmendarizPicon:1999rj,Garriga:1999vw} (see also Refs.~\cite{Lorenz:2008et} and \cite{MRV} for more precise computations of the spectra), these are given by
\begin{eqnarray}} \def\ea{\end{eqnarray}
\Ps &=& \frac{1}{8\pi^2} \frac{H^2}{\cs\epsilon} \label{mod-ps}\,,\\
\ns &=& \frac{d\ln\Ps}{d\ln k} = - (2\epsilon + \eta + s) \label{mod-ns}\,, \\
r & = & 16 \cs \epsilon \label{mod-r} \,,
\ea
where the various small parameters are defined by
\begin{equation}} \def\ee{\end{equation}
\epsilon = -\frac{d\ln H}{Hdt} \spc \eta = \frac{d\ln\epsilon}{Hdt} \spc s = \frac{d\ln\cs}{Hdt} \label{mod-pms} \,.
\ee
In this paper, for each case we will focus on power-law potentials, either with general exponent $m$ or the simplest cases $V= \frac{1}{2} M^2 \phi^2$ and $V = \frac{1}{4}\lambda \phi^4$. Also we take the reduced Planck mass $M_{\rm Pl}^{2}} %Squared Reduced Planck Mass = (8\pi G)^{-1=1$ for convenience, for instance in the expression for $\Ps$.
We do not make an extensive discussion of non-gaussianity in this article, as the theory of non-gaussianity in these models is already well developed, and for instance has been applied to DBI models in the recent {\it Planck} analysis of Ref.~\cite{Ade:2013ng}. It is well known that non-canonical models are a way of generating detectable non-gaussianity, of equilateral shape. But this by no means implies that typical non-canonical models do; it takes quite some effort to obtain a non-gaussianity parameter above unity. By contrast, if one analyses simple potentials and kinetic terms, as we do in this article, typically the non-gaussianity remains small due to slow-roll suppression. This is true of all but one of our models, the exception being the DBI in the relativistic case where the constraint has already been provided in Ref.~\cite{Ade:2013ng}. Hence accurate calculations of the scalar and tensor power spectra, as provided here, are the only way to constrain these models at present.
\sect{The General Systematic Method}\label{sect-methods}
Here we propose a systematic method in the slow-roll regime to obtain and express the final solutions for observables, such as the power spectrum $\Ps$ and spectral index $n_{\rm s}$, as a function of e-folds $N$.
\subs{General formula for the spectral index $n_{{\rm s}}$}\label{sect3}
We continue with the general Lagrangian $\L=p(\phi,X)$, noting for later use that any field redefinition to a new field $\varphi$, that is a function of $\phi$ and whose kinetic energy density is $\td X} \def\tY{\td Y = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi$, still leaves the Lagrangian as a general function $\tp(\varphi,\td X} \def\tY{\td Y)$ and hence results in the same field equations in the new variables.
We write down the energy density of the universe, $\rho$, as usual \cite{ArmendarizPicon:1999rj,Garriga:1999vw}
\begin{equation}} \def\ee{\end{equation}\label{dens-eq}
\rho = 2 X p_{,X} - p \,.
\ee
The Friedmann equations are
\begin{equation}} \def\ee{\end{equation}\label{frw-eq}
H^{2} = \frac{\rho}{3} \spc \dot{H} = -X p_{,X} \,,
\ee
where we continue to use the convention $M_{\rm Pl}^{2}} %Squared Reduced Planck Mass = (8\pi G)^{-1 = 1$.
We now define a variable $u$ by
\begin{equation}} \def\ee{\end{equation}\label{def-u}
u \vcentcolon= \frac{1}{\epsilon}= \frac{H^{2}}{X p_{,X}} \,.
\ee
We will use Eqs.~(\ref{dens-eq}), (\ref{frw-eq}), (\ref{def-u}) and their derivatives to obtain the observables for the considered inflationary scenarios in the slow-roll regime.
Eq.~(\ref{frw-eq}) gives the continuity equation
\begin{equation}} \def\ee{\end{equation}\label{conti}
\rho' = 3(\rho + p) = 6Xp_{,X} \,,
\ee
where $^{\prime}$ indicates derivative w.r.t.\ the e-folds $N$, which as usual are counted backwards from the end of inflation. According to Eqs.~(\ref{def-u}) and (\ref{conti}) we can write
\begin{equation}} \def\ee{\end{equation}\label{u-drr}
\frac{2}{u} = \frac{\rho'}{\rho} \,.
\ee
This compact form suggests that typically $\epsilon = 1/u \propto 1/2N$, in view of dimensional analysis.
We wish to write the density of the Universe as
\begin{equation}} \def\ee{\end{equation}\label{rho-eps}
\rho =\rho(V(\phi),u) \,,
\ee
so that $X$ and $\dot{X}$ will play the role of simplification in recursion relations for $u$ and $u^{\prime}} \def\Xp{\tX{^{\prime}}$, etc. This form is general but useful, since it indicates that the considered quantities, such as the power spectrum, are determined by the potential and a small parameter $1/u$. The final result can be obtained by the perturbation method in terms of the small parameters coming from $u$ and its derivatives (here $\epsilon$, $\eta$, $\cs$, $s$). Once we find the solution for the quantity $u$ we can then obtain the potential via Eq.~(\ref{u-drr}), where it is a first derivative w.r.t.\ the scalar field $\phi$.
To find an expression for $u$ without assuming a particular form of the Lagrangian, we differentiate Eq.~(\ref{def-u}) w.r.t.\ $N$ to obtain
\begin{eqnarray}} \def\ea{\end{eqnarray}\label{duu-def}
\frac{u^{\prime}} \def\Xp{\tX{^{\prime}}}{u} &=& \frac{2}{u} - \frac{X^{\prime}}{X}\frac{1+\css}{2\css} - \frac{p_{,X\phi}}{p_{,X}}\phi^{\prime} \,.
\ea
In deriving this equation, we have applied the relation $1/\css = 1 + 2 Xp_{,XX}/p_{,X}$.
Differentiating Eq.~(\ref{u-drr}) w.r.t.\ $N$,
\begin{equation}} \def\ee{\end{equation}\label{duu-drr}
-\frac{u^{\prime}} \def\Xp{\tX{^{\prime}}}{u} = \frac{\rho''}{\rho'} - \frac{\rho'}{\rho} = \frac{2}{u} {\left(} \def\rrb{\right) \frac{\rho''\rho}{{\rho'}^{2}} - 1 \rrb} \,,
\ee
and the relation $X=\frac{1}{2}H^{2}\phi^{\prime 2}$ gives
\begin{equation}} \def\ee{\end{equation}\label{dxx}
\frac{X'}{X} = 2 {\left(} \def\rrb{\right) \frac{\phi^{\prime\prime}}{\phi^{\prime}} + \frac{1}{u} \rrb} \,.
\ee
Equation~(\ref{dxx}) can be used to eliminate the field-dependent terms, such as $X'/X$ and $\phi^{\prime}/\phi$, which are present in Eqs.~(\ref{duu-def}) and (\ref{duu-drr}).
The quantity $u^{\prime}} \def\Xp{\tX{^{\prime}}$ has a clear meaning, namely
\begin{equation}} \def\ee{\end{equation}\label{eta-eps-sr}
u^{\prime}} \def\Xp{\tX{^{\prime}} = \frac{\eta}{\epsilon} \,.
\ee
Once $u$ is obtained, we will have an explicit relation between $\eta$ and $\epsilon$.
\ssubs{Exact formula for $u(c_{\rm s})$}
We now use Eq.~(\ref{conti}) to reformulate Eq.~(\ref{duu-def}) as,
\begin{equation}} \def\ee{\end{equation}\label{du-cs-v}
u^{\prime}} \def\Xp{\tX{^{\prime}} = 2 \left(1 - \frac{u}{2}\frac{p_{,X\phi} \phi^{\prime}}{p_{,X}}\right) - 3(1+\css)\left(1 - \frac{u}{2} \frac{\rho_{\phi}\phi^{\prime}}{\rho}\right)u \,.
\ee
Equation (\ref{du-cs-v}) is derived from the field equation without any assumptions or model specification. It is general and has a quite symmetric form, from which we can get some descriptive results by inserting or approximating Eq.~(\ref{u-drr}). For example, qualitatively, if one approximates $\rho \sim V$, then for the following two models we will have
\begin{itemize}} \def\ei{\end{itemize}
\item Canonical inflation\\
This type of inflation model has Lagrangian $\L = X - V(\phi)$. Then according to the equation above, we will have $\epsilon = 1/2N$, and $\eta = 2 \epsilon = 1/N$, due to
\begin{equation}} \def\ee{\end{equation}\label{two-cases-i}
p_{,X\phi} \equiv 0 \spc \frac{2}{u} \simeq \frac{\rho_{\phi}\phi^{\prime}}{\rho} = \frac{V_{,\phi}\phi^{\prime}}{V}\,.
\ee
\item Tachyon models\\
The Lagrangian for this model is $\L = - V(\phi) \sqrt{1 - 2 \lambda_{\rm s} X}$, where $\lambda_{\rm s}$ is a constant. As in the analysis above, we find $\epsilon = {\rm const.}$ due to,
\begin{equation}} \def\ee{\end{equation}\label{two-cases-ii}
\frac{p_{,X\phi}\phi^{\prime}}{p_{,X}} \equiv \frac{V_{,\phi}\phi^{\prime}}{V} \equiv \frac{\rho_{\phi}\phi^{\prime}}{\rho} \simeq \frac{2}{u} \,.
\ee
\ei
The outcomes are concise, but the approximate results of Eqs.~(\ref{two-cases-i}) and (\ref{two-cases-ii}) are only for qualitative understanding. We cannot make this approximation to eliminate any term in Eq.~(\ref{du-cs-v}). The evolution equation for $u$ is derived from the exact equation of motion, and if one wants to make an assumption on any term in Eq.~(\ref{du-cs-v}), one must also rederive the evolution equation for $u$. In the next subsection we address this issue. Therefore, although we have obtained this equation, it does not yet lead to a clear understanding for the observables for given model.
\ssubs{Predictions within the slow-roll approximation}
We need to derive a suitable equation for $u$. In the following, we will apply a well-defined approximation scheme to derive results based on Eqs.~(\ref{def-u}), (\ref{conti}), (\ref{duu-def}), and (\ref{duu-drr}). In obtaining the observables, we use the variable $u$ and its derivatives.
In the slow-roll regime, the following equation is obtained after expanding Eq.~(\ref{conti})
\begin{equation}} \def\ee{\end{equation}\label{dxp-sr}
\rho_{,\phi} \phi^{\prime} \simeq 6Xp_{,X} \,.
\ee
Then Eq.~(\ref{u-drr}) can be written as
\begin{equation}} \def\ee{\end{equation}\label{rho-eps-loa}
\frac{2}{u} = \frac{\rho_{,\phi} \phi^{\prime}}{\rho} \,,
\ee
Then the full Eq.~(\ref{du-cs-v}) will not applicable because it is derived from the full equation of motion for scalar field. We need to find the derivative of $X'$ from Eq.~(\ref{dxp-sr}) in order to get a similar equation for $\epsilon$ and $\cs$, by eliminating $X'/X$ in Eq.~(\ref{duu-def}). In view of this, we obtain the following equation,
\begin{equation}} \def\ee{\end{equation}
\frac{2}{u} \frac{\rho_{,\phi\phi} \rho}{\rho_{,\phi}^{2}} + \frac{\phi^{ \prime\prime}}{ \phi^{\prime}} = \frac{X'}{X} \frac{1 + \css}{2\css} + \frac{p_{,X\phi}\phi^{\prime}}{p_{,X}} \label{dpp-sr} \,.
\ee
We also define a variable $\delta$,
\begin{equation}} \def\ee{\end{equation}
\delta \equiv \left( \frac{\rho_{,\phi\phi} \rho}{\rho_{,\phi}^{2}} - \frac{1}{2} \right) \label{drr-sr} \,,
\ee
for later convenience. Assembling Eqs.~(\ref{duu-def}), (\ref{dxx}), (\ref{dxp-sr}), (\ref{dpp-sr}), and (\ref{drr-sr}) we eliminate $Xp_{,X}$, $\phi^{ \prime\prime}/ \phi^{\prime}$ and $X'/X$, but we keep the sound speed $\cs$ as it can be related to the variable $u$. After some effort, we can obtain the final result for Eq.~(\ref{duu-def}),
\begin{equation}} \def\ee{\end{equation}\label{ode-u-sr}
u^{\prime}} \def\Xp{\tX{^{\prime}} = 2 \left[} \def\rsb{\right] 1 - \delta(1+\css) \rsb + \frac{p_{,X\phi}\phi^{\prime}}{p_{,X}} u \css \,.
\ee
As we have noted in Eqs.~(\ref{two-cases-i}) and (\ref{two-cases-ii}), the last term in the above equation can now be approximated and then the whole equation can be simplified and solved.
\subs{Predictions for two classes of models}
Using Eq.~(\ref{ode-u-sr}) we consider some classes of inflation models. Unless explicitly stated, the following discussion will consider a monomial potential of the form $V \propto \phi^{m}$. This will provide a constant $\delta =1/2 - 1/m$, independent of the field value itself.
\ssubs{Variable separable class}
This class of inflation models includes two subclasses: Sum-Separable Models ({\bf SSMs}) and Product-Separable Models ({\bf PSMs}).
\begin{itemize}} \def\ei{\end{itemize}
\item[$\star$] \underline{{\bf SSMs}:} Here we have separate terms for $\phi$ and $X$ which are added together, for example $\mathcal{L} = BX^{n} - A\phi^{m}$.\footnote{Actually when the kinetic term consists of a single monomial term, any constant prefactor can readily be set to unity without loss of generality by a field rescaling.} In this class, similarly to Eq.~(\ref{two-cases-i}), we have the following useful relations
\begin{equation}} \def\ee{\end{equation}
\rho(V(\phi),u) = \frac{V(\varphi)}{1-1/3u} \spc p_{,X\phi} \equiv 0 \spc {\css} \equiv \frac{1}{2n-1} \,,
\ee
which can be substituted into Eq.~(\ref{ode-u-sr}). Then we can obtain a compact form, recovering the results we found for this model in Ref.~\cite{Li:2012vt}:
\begin{equation}} \def\ee{\end{equation}\label{can-du-N-sr}
u^{\prime}} \def\Xp{\tX{^{\prime}} = 2 \frac{\beta}{m} \spc \beta = \frac{(n-1)m + 2n}{2n-1} \,.
\ee
{\it Solution for $u$:} According to the equation above, the solution is explicitly obtained as
\begin{equation}} \def\ee{\end{equation}\label{can-eps-N-sr}
\epsilon = \frac{m}{\beta} \frac{1}{2N} \,.
\ee
We have a model independent $\eta$ according to Eqs.~(\ref{eta-eps-sr}), (\ref{can-du-N-sr}) and (\ref{can-eps-N-sr}),
\begin{equation}} \def\ee{\end{equation}\label{eta-N-sr}
\eta = 2 \frac{\beta}{m} \epsilon \equiv \frac{1}{N} \,,
\ee
under the linear assumption by considering only the leading-order small parameter, such as $\epsilon$. We see that the parameter $\epsilon$ depends on the exponents $m$ and $n$ of the given model, while the parameter $\eta$ does not.
{\it Power spectrum $\Ps$, spectral index $n_{{\rm s}}$, and tensor-to-scalar ratio $r$:} Now that we have obtained the solution for $u$ expressed by Eq.~(\ref{can-eps-N-sr}), we can use Eqs.~(\ref{dxp-sr}), (\ref{rho-eps-loa}), and (\ref{can-du-N-sr}) to find the scalar potential $V=A\phi^m$ in terms of e-folds $N$. We recall that Eq.~(\ref{dxp-sr}) gives the slow-roll prediction
\begin{equation}} \def\ee{\end{equation}
V^{\prime}} \def\Vpp{V^{\prime\prime} = 6^{1-n}n B \frac{V^{n-1}}{(\beta N)^{2n}} {\left(} \def\rrb{\right) \frac{V}{A} \rrb}^{2n/m} \,,
\ee
and Eq.~(\ref{rho-eps-loa}) gives
\begin{equation}} \def\ee{\end{equation}
\frac{\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}}^{\prime}}{\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}}} \simeq \frac{V^{\prime}} \def\Vpp{V^{\prime\prime}}{V} = \frac{2}{u} \,.
\ee
Finally we obtain the potential $V$ as
\begin{equation}} \def\ee{\end{equation}\label{can-V-N-sr}
V = {\left(} \def\rrb{\right) \frac{m6^{n-1}}{n} \, \frac{A^{\frac{2n}{m}}} {B} \rrb}^{\frac{m}{\beta}\frac{1}{2n-1}} {(\beta N)}^{m/\beta} \,.
\ee
Therefore we can now write down the power spectrum from Eq.~(\ref{mod-ps}) and its spectral index from the Eq.~(\ref{mod-ns}),
\begin{eqnarray}} \def\ea{\end{eqnarray}
\Ps &=& \frac{\sqrt{2n-1}}{12\pi^{2}} \frac{\beta}{m} \times {\left(} \def\rrb{\right) \frac{m\beta^{2n-1}6^{n-1}}{n} \, \frac{A^{\frac{2n}{m}}} {B} \rrb}^{\frac{m}{\beta}\frac{1}{2n-1}} \times N^{\frac{m}{\beta}+1} \label{ps-ssm} \,,\\
\ns &=& -2 \left(1+\frac{m}{\beta} \right) \times \frac{1}{2N} \label{ns-ssm}\,,\\
r &=& 16\frac{m}{\beta} \frac{1}{2N} \label{r-ssm}\,.
\ea
\item[$\star$] \underline{{\bf PSMs}:} These take the form $\mathcal{L} = -K(X)V(\phi)$ where both $K(X), V(\phi)>0$, the Tachyon being an example. We will still have the following relations, analogous to the case in Eq.~(\ref{two-cases-ii}),
\begin{equation}} \def\ee{\end{equation}
\rho_{,\phi} \propto V_{,\phi} \spc \frac{p_{,X\phi}\phi^{\prime}}{p_{,X}} \equiv \frac{V_{,\phi}\phi^{\prime}}{V} = \frac{2}{u} \label{tachyon-u-phi} \,,
\ee
which leads Eq.~(\ref{ode-u-sr}) to be,
\begin{equation}} \def\ee{\end{equation}\label{tachyon-du-N-css-sr}
u^{\prime}} \def\Xp{\tX{^{\prime}} = 2(1 - \delta) (1+\css) \,.
\ee
Here we have an implicit function, the sound speed $\css = \css(u)$. To get an explicit result for model observables, such as $\Ps$ and $\ns$ via this differential equation (\ref{tachyon-du-N-css-sr}), we first need to specify the particular form of $\css$ in terms of $u$. In general it is not straightforward to solve this equation due to the undetermined sound speed, but this is possible for the Tachyon Models as presented in Section~\ref{app-to-sect-tachyon}.
\ei
\ssubs{A more general ansatz}
Not all Lagrangians are sum or product separable, of course; the Tachyon is product-separable but the DBI case is neither. To set up formalism to deal with the latter, we consider the more general Lagrangian
\begin{equation}} \def\ee{\end{equation}
\label{e:330}
\L = - \left( W(\phi) K(X) + U(\phi) \right) \,.
\ee
This contains the SSM and PSM as special cases, respectively $W$ constant and $U \propto W$. In its conventional form, Eq.~(\ref{m:DBI}), the DBI model does not take this form, but we will show below that it can be written in this form via a field redefinition. Although we will not undertake a general study for the above ansatz, due to the complexity of the analysis, we will use the DBI case to illustrate our procedure on Lagrangians of this class.
\sect{Application of the Systematic Method to Two Models}\label{sect-twomods}
\subs{Tachyon models}\label{app-to-sect-tachyon}
We now consider the Tachyon model given by Eq.~(\ref{m:tachyon}) and apply Eq.~(\ref{tachyon-du-N-css-sr}). At this point we don't have to impose any simplification since the Lagrangian already takes the product-separable form. However, we will implement a field-redefinition approach to rederive these results in a different way in Section~\ref{canon-tach}.
For this type of model, the relation between $u$ and the sound speed $\cs$ is
\begin{equation}} \def\ee{\end{equation}\label{tachyon-css-u}
\epsilon = \frac{1}{u} = \frac{3}{2}(1-\css) \,,
\ee
while the energy density $\rho$ is
\begin{equation}} \def\ee{\end{equation}
\rho(V(\phi),u) = \frac{V(\phi)}{\sqrt{1-2/3u}} \simeq V(\phi) \,.
\ee
Then we can solve for $u$ according to Eq.~(\ref{tachyon-du-N-css-sr}), which can be written as
\begin{equation}} \def\ee{\end{equation}
u^{\prime}} \def\Xp{\tX{^{\prime}} = 4\mu \left(1 - \frac{1}{3u}\right) \spc \mu = 1 - \delta = \frac{1}{2} + \frac{1}{m} \,.
\ee
The solution for $\epsilon = 1/u$ is therefore
\begin{equation}} \def\ee{\end{equation}\label{tachyon-u-N-sr}
N = \frac{1}{4\mu} {\left[ \frac{1}{\epsilon} + \frac{1}{3}\ln{\left(\frac{3}{\epsilon} - 1\right)} \right]} \,.
\ee
The solution can be inverted using the Lambert $\mathcal W$ function\footnote{Two useful properties of the Lambert $\mathcal W$ function \cite{lambert-wiki} are $\mathcal{W}(e) = 1 \label{W-e-1}$
and its asymptotic behaviour for any real $x \geq e$ \cite{lambertw-HH},
$$
L_1 - L_2 + \frac{1}{2}L_3\leq \mathcal W(x) \leq L_1 - L_2 + \frac{e}{e-1}L_3
$$
where $L_1 = \ln{x} ,\; L_2 = \ln\ln{x} ,\; L_3 = L_2/L_1$.}
so we will have,
\begin{equation}} \def\ee{\end{equation}
u = \frac{1}{\epsilon} = 1 + \mathcal W(e^{x-1}) \spc x = 12\mu N \label{tachyon-u-eps-W} \,.
\ee
[The Lambert function is not really necessary here, but we use this method as it is required in later parts of this article.]
As $\mu$ is of order one, $x \gg 1$ for any $N$ of interest, which means that $\epsilon \ll 1$ will always hold. Then we can obtain the following relations
\begin{eqnarray}} \def\ea{\end{eqnarray}
\epsilon &=& \frac{1}{2\mu}\frac{1}{2N} \,, \label{tachyon-u-N-sr-close}\\
\eta = 4\mu \epsilon \left(1 - \frac{\epsilon}{3}\right) &=& \frac{1}{N} {\left(} \def\rrb{\right) 1 - \frac{1}{6\mu} \frac{1}{2N} \rrb} \simeq \frac{1}{N}\,.
\ea
To find the power spectrum, we need to find the relation of $\phi$ to $N$. To achieve this, we combine Eq.~(\ref{dxp-sr}) with Eqs.~(\ref{tachyon-u-phi}), (\ref{tachyon-css-u}), and (\ref{tachyon-u-N-sr-close}) and find the potential $V$ in terms of $N$ and the model parameters $A,\, \lambda_{\rm s}$,
\begin{equation}} \def\ee{\end{equation}
V = \kappa N^{1/2\mu} \spc \kappa = {\left(} \def\rrb{\right) 2m^{2}\mu \frac{\cs A^{2\mu-1}}{\lambda_{\rm s}} \rrb}^{1/2\mu} \,.
\ee
According to Eq.~(\ref{mod-ps}) the scalar power spectrum is
\begin{equation}} \def\ee{\end{equation}\label{tachyon-ps-N}
\Ps = \frac{1}{12\pi^{2}}\cs^{1/2\mu -1} \times{\left(} \def\rrb{\right) m^2 (2\mu)^{1+2\mu} \frac{A^{2/m}}{\lambda_{\rm s}} \rrb}^{1/2\mu} \times N^{1+1/2\mu}\,.
\ee
The spectral index receives contributions from the last term and also from the time variation of the sound speed $\cs$, but we will now see that the latter term does not contribute at lowest order in slow-roll. To check this, we need $s$ which is
\begin{equation}} \def\ee{\end{equation}
s = -\frac{\cs^{\prime}}{\cs} = \frac{1-\css}{2\css} \frac{X^\prime}{X} \,.
\ee
Combining with $u^{\prime}} \def\Xp{\tX{^{\prime}}$ from Eq.~(\ref{ode-u-sr}), we have
\begin{equation}} \def\ee{\end{equation}
u^{\prime}} \def\Xp{\tX{^{\prime}} = -\frac{1+\css}{2\css} \frac{X^\prime}{X} u \,,
\ee
and therefore
\begin{equation}} \def\ee{\end{equation}\label{tachyon-s-N-sr}
s = \eta \frac{- \epsilon/3}{1 - \epsilon/3} = -\frac{4}{3} \mu \epsilon^{2} \spc \frac{X^\prime}{X} = 4(\delta-1)\epsilon\css \,.
\ee
The spectral index is then
\begin{equation}} \def\ee{\end{equation}\label{tachyon-ns-sr}
\ns = -(2\epsilon + \eta + s) = - (2+4\mu)\epsilon + \frac{8}{3}\mu\epsilon^{2} \,.
\ee
Retaining only the lowest-order terms in slow-roll, as required for consistency as only those have been included throughout, we have the spectral index $n_s$ and $r$ as
\begin{eqnarray}} \def\ea{\end{eqnarray}
\ns &=& -\frac{2m+2}{m+2}\frac{1}{N} \label{tachyon-ns-N-est}\,,\\
r &=& \frac{8\cs}{\mu}\frac{1}{2N} \simeq \frac{8m}{m+2}\frac{1}{N} \label{tachyon-r-N-est}\,.
\ea
This indicates that the spectral index is always red tilted, $n_{{\rm s}} < 1$, in Tachyon models. The number of e-folds between observable perturbation generation and the end of inflation, usually taken to be $N \simeq 50$ \cite{Liddle:2003as}. So take an example for quadratic potential where $m=2$, we will have $n_s=0.97$ and $r=0.08$ at pivot $N_*=50$.
\subs{DBI inflation models}\label{app-to-sect-dbi}
The DBI action, as already given in Eq.~(\ref{m:DBI}), is
\begin{equation}} \def\ee{\end{equation}\label{m:DBI-2}
p(\phi,X) = -\frac{1}{f(\phi)} \sqrt{1 - 2f(\phi) X} - V(\phi) \,,
\ee
where the sound speed $\cs = \sqrt{1 - 2f(\phi) X}$ and the warp factor is $f(\phi) = \lambda_{\rm s}/\phi^4$. If the last term $V$ is zero the model reduces to the Tachyon model with a constant potential, but we are not interested in this case here; instead we are going to discuss a more general case in the following subsections.
\ssubs{Field redefinition}
To proceed with our investigation using the method of the previous sections, and in view of simplifying the later calculations, we apply a field redefinition $\varphi = 1/\phi$ to the DBI action~(\ref{m:DBI}). A variant of this technique will also be used in studying the Tachyon model in Section~\ref{sect-implement}. However, we will now focus on the current case, where the Lagrangian for DBI inflation with a potential $V = A\phi^m$ becomes
\begin{eqnarray}} \def\ea{\end{eqnarray}
\varphi &=& \frac{1}{\phi} ,\;W(\varphi) = \frac{1}{\varphi^{4}} ,\; \tilde{V}} \def\tVp{\tilde{V^{\prime}} \def\Vpp{V^{\prime\prime}}(\varphi) = A\varphi^{4-m}\,, \label{dbi-phi-vpahi}\\
\L &=& - W(\varphi) {\left(} \def\rrb{\right) \frac{1}{\lambda_{\rm s}}\tilde{\cs} + \tilde{V}} \def\tVp{\tilde{V^{\prime}} \def\Vpp{V^{\prime\prime}}(\varphi) \rrb} \label{dbi-vphi-L} ,\, \tilde{\cs} = \sqrt{1 - 2\lambda_{\rm s} \td X} \def\tY{\td Y} \label{dbi-cs-red-tX} \,.
\ea
The notation $\td X} \def\tY{\td Y$ stands for the kinetic term after applying the field redefinition. With this new field definition, the DBI action falls in the class defined by Eq.~(\ref{e:330}).
The energy density $\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}}$ is
\begin{equation}} \def\ee{\end{equation}
\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}} = W(\varphi) {\left(} \def\rrb{\right) \frac{1}{\lambda_{\rm s} \cs} + \tilde{V}(\varphi) \rrb} = \frac{3}{2}\frac{W}{\lambda_{\rm s}} \frac{1-\css}{\cs} u \,.
\ee
We have a sound speed (for later convenience, we use the same notation $\cs$ for sound speed instead of $\tilde\cs$) which is of the same form as for the Tachyon. The above action in Eq.~(\ref{dbi-cs-red-tX}) will induce an equation of motion,
\begin{equation}} \def\ee{\end{equation}\label{dbi-vphi}
\frac{\ddot\varphi}{\cs^{2}} + 3H \dot\varphi + \left[\frac{W^{\prime}}{W} \frac{1}{\lambda_{\rm s}} + \cs\tilde{V}\left(\frac{W^{\prime}} \def\Wpp{W^{\prime\prime}}{W} + \frac{\tildeV^{\prime}} \def\Vpp{V^{\prime\prime}}{\tilde{V}} \right) \right] = 0 \,.
\ee
Rearranging Eq.~(\ref{dbi-vphi}) it can be reformulated as
\begin{equation}} \def\ee{\end{equation}\label{dbi-vphi2}
\frac{\ddot\varphi}{\cs^{2}} + 3H \dot\varphi + \frac{W^{\prime}}{\lambda_{\rm s} W} \left[ 1 + {\lambda_{\rm s}\cs}\tilde{V}\left(1+ \frac{\tildeV^{\prime}} \def\Vpp{V^{\prime\prime}}{\tilde{V}}\frac{W}{W^{\prime}} \def\Wpp{W^{\prime\prime}} \right) \right] = 0\,,
\ee
where the Hubble rate is
\begin{equation}} \def\ee{\end{equation}\label{dbi-vphi-H}
H^{2} = \frac{u}{2}\frac{W}{\lambda_{\rm s}}\frac{1-\css}{\cs} \,.
\ee
Also Eq.~(\ref{dbi-vphi2}) suggests the same form as the Tachyon, if the potential $V$ is constant.
\ssubs{Predictions for the quartic potential}\label{sect-eval-dbi-lp4}
In this section we will consider the quartic potential $V = \frac{\lambda}{4}\phi^4$, which will provide a simple form of differential relation in Eq.~(\ref{tachyon-du-N-css-sr}) for $u = 1/\epsilon$ and its first derivative. We can write
\begin{eqnarray}} \def\ea{\end{eqnarray}
\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}} = \frac{W(\varphi)}{\lambda_{\rm s}\cs} ( 1 + \alpha \cs ) &\spc& \alpha = A\lambda_{\rm s} = \frac{\lambda_{\rm s}\lambda}{4} = {\rm const} \,, \\
\delta = \frac{\Wff W}{W_{,\varphi}} \def\Wff{W_{,\varphi\varphi}^{2}} - \frac{1}{2} \equiv \frac{3}{4} &\spc& \epsilon = \frac{3}{2} \frac{1 - \css}{1 + \alpha \cs} \label{dbi-eps-cs} \,.
\ea
Unlike the case of Tachyon models, for the quartic potential model in DBI inflation we have a constant $\delta$. We now have the approximate equation for relation (\ref{dxp-sr}),
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-lp4}
\frac{2}{u} = \frac{\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}}_{,\varphi}\varphi^{\prime}}{\rho} \def\rp{\tr{^{\prime}}} \def\rpp{\tr{^{\prime\prime}}} \equiv \frac{W_{,\varphi}\varphi^{\prime}}{W} = \frac{p_{,\td X} \def\tY{\td Y\varphi} \varphi^{\prime}}{p_{,\td X} \def\tY{\td Y}}\,.
\ee
Also, we can readily solve Eq.~(\ref{dbi-sr-lp4}) so that we can write down the power spectrum $\Ps$ and its spectral index $n_{{\rm s}}$. This is simplified because of the field redefinition, in contrast to the conventional treatment where one could not get a slow-roll solution for $\Ps$ etc. According to Eq.~(\ref{dxp-sr}) and Eqs.~(\ref{dbi-vphi2}) and (\ref{dbi-sr-lp4}) we have\footnote{One can just expand Eq.~(\ref{dbi-eps-cs}) to obtain this relation, but for consistency of our treatment we present the same procedure as followed in the previous sections.}
\begin{equation}} \def\ee{\end{equation}
3X = \epsilon \frac{1+\alpha\cs}{\lambda_{\rm s}}\,,
\ee
which then solves $\varphi$ and the redefined warp factor $W(\varphi)$,
\begin{equation}} \def\ee{\end{equation}
W = 64\css \frac{1}{\epsilon^2} \label{dbi-W-cs-eps-sr} \,.
\ee
Then we have $\Ps$ as
\begin{equation}} \def\ee{\end{equation}
\Ps = \frac{8}{3\pi^2} \frac{1+\alpha\cs}{\lambda_{\rm s}} \frac{1}{\epsilon^3} \label{dbi-Ps-cs-eps-ls-1-sr}\,.
\ee
Now we notice that there is another relation for $\epsilon$ in Eq.~(\ref{dbi-eps-cs}), so the above expression can be written in another equivalent form,
\begin{equation}} \def\ee{\end{equation}
\Ps = \frac{4}{\pi^2} \frac{1-\css}{\lambda_{\rm s}} \frac{1}{\epsilon^4} \label{dbi-Ps-cs-eps-ls-2-sr} \,.
\ee
These formulae may be used interchangeably. Considering the two asymptotic limits for the sound speed $\cs=0$ or $1$, either of the above leads to the prediction for the power spectrum. In the limit $\cs\sim1$, Eq.~(\ref{dbi-Ps-cs-eps-ls-1-sr}) straightforwardly indicates the leading dependence for $\Ps$ of order $1/\lambda_{\rm s}\epsilon^3$, while in the case $\cs \raw 0$ we may use the second expression for the power spectrum $\Ps$ by Eq.~(\ref{dbi-Ps-cs-eps-ls-2-sr}). This alternative equation is more useful in some cases. For example, providing no knowledge about the model parameter degeneracy which is encoded in $\alpha$, we cannot obtain good insight from Eq.~(\ref{dbi-Ps-cs-eps-ls-1-sr}) in the cases where the limits $\cs \raw 0$ or $\alpha \raw 0$ are approached. On the contrary, Eq.~(\ref{dbi-Ps-cs-eps-ls-2-sr}) will tell us the prediction in spite of $\cs$ or the degenerate combination $\alpha$. And in the limit of $\cs \sim 0$, we will obtain the usual prediction for the power spectrum $\Ps = 4/\pi^2 \lambda_{\rm s} \epsilon^4$ when the inflaton field moves relativistically in DBI inflation.
We would however like to go further in obtaining the expression for $\Ps$ for the reason that in either of the expressions above, which have two or three small parameters, it remains unclear where the model degeneracies lie. Therefore, as in previous sections, we are going to find an expression for $u$ in terms of e-folds $N$. To do this, following Eq.~(\ref{tachyon-du-N-css-sr}), we obtain a similar equation for $u$
\begin{equation}} \def\ee{\end{equation}\label{dbi-du-cs-sr}
u^{\prime}} \def\Xp{\tX{^{\prime}} = \frac{1}{2} (1 + \css) \,.
\ee
However, as $u$ is a rational function of $\cs$ (see Eq.~(\ref{dbi-eps-cs})), we will use another approach to obtain the solution for $u$.
Assembling all of Eqs.~(\ref{duu-def}), (\ref{dbi-cs-red-tX}), and (\ref{dbi-sr-lp4}), we will have the ODEs below
\begin{eqnarray}} \def\ea{\end{eqnarray}
\frac{u^{\prime}} \def\Xp{\tX{^{\prime}}}{u} &=& - \frac{{\td X} \def\tY{\td Y}^{\prime}}{2\td X} \def\tY{\td Y} \frac{1+\css}{\css} - {\left(} \def\rrb{\right) \frac{2}{u} - \frac{p_{,\td X} \def\tY{\td Y\varphi}\varphi^{\prime}}{p_{,\td X} \def\tY{\td Y}} \rrb} \,,\label{d1} \\
\frac{\cs^{\prime}}{\cs} &=& -\frac{1-\css}{\css}\frac{\tilde{X}^{\prime}}{2\tilde X} = -s \label{dbi-sr-dcs}\,.
\ea
We turn to investigate the evolution equation (\ref{dbi-sr-dcs}) for the sound speed, whose evolution is very simple due to the field redefinition to $\varphi$-field space.
We can solve the relation for $\tilde{X}^{\prime}/\tilde X$ by using Eqs.~(\ref{dbi-sr-lp4}), (\ref{dbi-du-cs-sr}), and (\ref{d1})
\begin{equation}} \def\ee{\end{equation}
\frac{\tilde{X}^{\prime}}{\tilde X} = -\frac{\css}{u} \,,
\ee
then we get the evolution equation for sound speed $\cs$,
\begin{equation}} \def\ee{\end{equation}\label{dbi-cs-N-ode}
\cs^{\prime} = \frac{3}{4} \frac{(1-\css)^{2}}{1 + \alpha \cs} \cs \,.
\ee
The solution is obtained as follows
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-N-eps-cs}
N = \frac{1}{\epsilon} + \frac{2}{3} {\left(} \def\rrb{\right) \ln{\frac{\css}{1 - \css}} + \frac{\alpha}{2} \ln{\frac{1 + \cs}{1 - \cs}} \rrb} \,.
\ee
We are interested in the predictions in the different limits of the sound speed $\css$, although we cannot explicitly invert the general results.
\begin{itemize}} \def\ei{\end{itemize}
\item[$\star$] {\it Non-relativistic case}\\
It is not generally possible to invert the relation for $\epsilon$ and the e-folds $N$ since $\alpha$ is an important parameter. Then we will not have the full expression for the scalar power spectrum. However, in the non-relativistic case the condition $\css \sim 1$ can be applied such that the latter terms are negligible. So we will have the approximate solution for $\epsilon \simeq 1/N$. Then by this relation, we obtain the redefined scalar field $\varphi$,
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-W-N-cs}
W(\varphi) = \tau N^{2} \spc \tau = 64\css \,.
\ee
Finally we have the scalar power spectrum by Eq.~(\ref{mod-ps}) as\footnote{To determine $\tau$, we applied Eq.~(\ref{rho-eps-loa}). It is a bit tricky to determine the constant $\tau$ in this case. This relation is for $\css \raw 1$; one should replace $1-\css$ with (1 + $\alpha\cs$) via the definition of $\epsilon$. Then one finds this proportionality coefficient is exactly $64\css$.}
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-ps}
\Ps \simeq \frac{8}{3\pi^2} \frac{1+\alpha}{\lambda_{\rm s}} N^3 \propto {\left(} \def\rrb{\right) A + \frac{1}{\lambda_{\rm s}} \rrb} N^3 \,.
\ee
Therefore we can obtain the spectral index
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-ns}
\ns \simeq -\frac{3}{N} \,.
\ee
This relation can also be derived from the differential system. We can evaluate the two small parameters according to Eqs.~(\ref{eta-eps-sr}), (\ref{dbi-du-cs-sr}), and (\ref{dbi-sr-dcs}),
\begin{equation}} \def\ee{\end{equation}
\eta = \frac{1+\css}{2}\epsilon \spc s = -\frac{1-\css}{2} \epsilon \,,
\ee
to obtain the spectral index which is then presented as,
\begin{equation}} \def\ee{\end{equation}\label{dbi-sr-ns-eps-cs}
\ns = - (2\epsilon + \eta + s) = - (2 + \css) \epsilon \,,
\ee
in terms of $\epsilon$ and $\cs$. From Eq.~(\ref{dbi-sr-ns-eps-cs}) we find that for a quartic potential the spectrum is always red tilted, since $2+\css>0$ is always satisfied regardless of the warp strength or the mass of the scalar field. Also if the DBI scalar field rolls asymptotically in the manner of a canonical field in which $\css \raw 1$, then we have $n_{s} \raw 1 - 3/N_{*} = 0.94$ at $N_{*}=50$. This result is obviously recovered due to the canonical-like inflation with a quartic potential.
One should note that the relation in Eq.~(\ref{dbi-sr-ns-eps-cs}) does {\em not} require any limit for $\cs$. It can also be applied when $\css \ll 1$, but if that is the case we should modify the value for $\epsilon \simeq 2/N$. We will see this below.
\item[$\star$] {\it Relativistic case}\\
The behaviour of DBI inflaton can be relativistic, which corresponds to the limit $\css \ll 1$. In this case, we can also use the method from this section. Unlike the case of $\css \raw 1$ we only need to approximate $\epsilon \sim 2/N$.\footnote{This relation can be obtained either from Eq.~(\ref{dbi-cs-N-ode}) by inserting $\css\raw0$, or using limiting properties for the third term in Eq.~(\ref{dbi-sr-N-eps-cs}) which is $\log{\frac{1+\cs}{1-\cs}} \raw 2\cs$. In this limit, we will infer another relation for $\alpha\cs\epsilon \simeq 3/2$.}
Therefore, we can write down the relation for the redefined scalar field $\varphi$, the scalar power spectrum and its spectral index as follows
\begin{eqnarray}} \def\ea{\end{eqnarray}
W = \xi N^4 &\spc& \xi \simeq {\left(} \def\rrb{\right) \frac{3}{\alpha} \rrb}^2 \label{dbi-rel-w-N} \,, \\
\Ps &=& \frac{1}{4\pi^2} \frac{1}{\lambda_{\rm s}} N^4 \label{dbi-rel-Ps-N} \,, \\
\ns &=& -\frac{4}{N} \label{dbi-rel-ns-N}\,.
\ea
To derive the proportionality constant in Eq.~(\ref{dbi-rel-w-N}), we have used the speed limit relation $\lambda_{\rm s}{\dot\varphi}^2 \sim 1$. While deriving Eqs.~(\ref{dbi-rel-w-N}) and (\ref{dbi-rel-Ps-N}), we have also applied the relation $\alpha\cs \epsilon \simeq 3/2$ for this relativistic case. These relations can be obtained by inserting $\css \ll 1$ into Eq.~(\ref{dbi-eps-cs}). The sound speed in this scenario is constrained by non-gaussianity to be greater than about 0.07 \cite{Ade:2013ng}.
\ei
Since we can have slow-roll solutions for $\epsilon$ in each case, we can just substitute either solution to the power spectrum into Eq.~(\ref{dbi-Ps-cs-eps-ls-1-sr}) or Eq~(\ref{dbi-Ps-cs-eps-ls-2-sr}). Eventually we will have the predictions as above. We can see both predictions for the power spectrum for DBI inflation with quartic potential are valid. However, the first one (\ref{dbi-Ps-cs-eps-ls-1-sr}) will reduce to the conventional canonical inflation with quartic potential in the non-relativistic limit, while Eq.~(\ref{dbi-Ps-cs-eps-ls-2-sr}) will give the prediction for it in the relativistic limit.
\begin{figure}} \def\efg{\end{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{all-3-nsr.pdf}
\caption{Slow-roll predictions for several inflation models alongside constraints from {\it Planck} and other probes. [Based on an image from Ref.~\cite{Ade:2013uln}, original image credit ESA/{\it Planck} Collaboration.]}
\label{fig:all-3-nsr}
\efg
\subs{Comparison to {\it Planck} constraints}
We have studied several models and derived their predictions within the slow-roll approximation. Now we present these predictions by visualising them against the latest constraints on inflation models compiled by the {\it Planck} collaboration \cite{Ade:2013uln}.
The {\it Planck}+WP observational data favours a concave potential for viable canonical inflation models and limits the tensor-to-scalar ratio below $r=0.11$ at 95\% confidence. In Figure \underline{\ref{fig:all-3-nsr}} we can see that for canonical inflation models with the polynomial potential $V\propto \phi^m$, the $m=3,4$ cases are ruled out by the observational requirements, while potentials with $m=2/3$, $1$, $4/3$ and $2$ are within 95\% confidence, though at $N_*=50$ the quadratic potential ($m=2$) lies outside the 95\% confidence region \cite{Ade:2013uln}.
This plot shows two of our models. The first is the NCI model where the Lagrangian has the form $\L = BX^2 - V(\phi)$, where both quadratic potential (red diamond) and quartic potential (green diamond) are located within the observationally-permitted region. The second is Tachyon inflation; the figure shows that the quadratic potential (red triangle) is well within the permitted region, while the quartic potential is marginal with only the large $e$-foldings case $N_* = 60$ lying within the 95\% confidence region.
The polynomial potential in NCI-$X^2$ models and Tachyon inflation model can be considered as reshaped by the non-canonical term, and these reshaped potentials can correspond to standard inflation models. In $X^2$ models the quadratic potential is reshaped to $m\in(2/3,1)$ and the quartic potential to $m\in(1,4/3)$. In Tachyon models, the quadratic potential is reshaped to $m=1$ and the quartic potential to $m=4/3$. This reshaping in Tachyon models can be found in the following section.
One may infer from the figure that if the inflaton rolls in a polynomial potential $V\propto \phi^m/m$ with higher $m$, the higher-order kinetic term $X^n$ in NCI models may be supported by current observational datasets such as {\it Planck} and BAO. These models can potentially give significant non-gaussianity while satisfying power spectrum constraints, since large non-gaussianity occurs via a small sound speed $\cs$, which the NCI-$X^n$ models can provide with $\css = 1/(2n-1)$.
\sect{Implementation Methods for Particular Cases}\label{sect-implement}
For some particular models, there may be more direct treatments. For example for Tachyon models, we can take advantage of a field redefinition, along with the assumption $\epsilon \ll 1$ which in turn implies that $\cs \raw 1$. This method leads the same prediction as those obtained in Section \ref{sect-twomods}, as we show next. For DBI inflation with a quadratic potential, the methods in Section \ref{sect-twomods} require a lot of effort in order to obtain the observables, but initially applying a slow-roll assumption gives results for the model predictions. The following methods can be considered as an implementation of the systematic approach in Section \ref{sect-methods}.
\subs{Field redefinition in Tachyon models}\label{canon-tach}
A more direct approach to the observables is possible for Tachyon models, as a combination of the slow-roll approximation and a field redefinition allows the theory to be approximated by a canonical Lagrangian.
Starting from
\begin{equation}} \def\ee{\end{equation} \label{tachyon}
P(X,\phi) = -V\sqrt{1-2\lambda_{\rm s} X},
\ee
and making the slow-roll assumption $\lambda_{\rm s} X \ll 1$, we can approximate it by a different non-canonical model with Lagrangian
\begin{equation}} \def\ee{\end{equation} \label{sim-tachyon}
P(X,\phi) \simeq V\lambda_{\rm s} X - V.
\ee
This can be transformed to a canonical action if we can find a new field $\varphi$, with corresponding kinetic energy $\tilde X = \frac{1}{2} \partial_\mu \varphi \partial^\mu \varphi$, such that $\tilde X= V\lambda_{\rm s} X$. This can be done in principle for any well-defined potential, though not always analytically. Again focussing on the monomial potential $V(\phi)=A\phi^m$, the required transformation is
\begin{equation}} \def\ee{\end{equation}
\varphi = \frac{2\sqrt{A\lambda_{\rm s}}}{m+2}\,\phi^{1+m/2} \,,
\ee
so that we have $V(\varphi)$ as,
\begin{equation}} \def\ee{\end{equation} \label{norm-non}
V(\varphi) = \varphi^{\frac{2m}{m+2}} \left(\frac{m+2}{2}A^{\frac{1}{m}} \lambda_{\rm s}^{-\frac{1}{2}} \right)^{\frac{2m}{m+2}}\,.
\ee
By completing the above transformation, then the original Lagrangian (\ref{tachyon}) can be rewritten within this approximation as
\begin{equation}} \def\ee{\end{equation}
P(X,\phi) \raw \tilde P(\tilde X, \varphi) = \tilde X - \tilde A \varphi^{\frac{2m}{m+2}} \,,
\ee
where the normalisation coefficient became
\begin{equation}} \def\ee{\end{equation}
\tilde A = \left(\frac{m+2}{2} \right)^{\frac{2m}{m+2}} \left(A^{1/m} \lambda_{\rm s}^{-1/2} \right)^{\frac{2m}{m+2}} \,.
\ee
The usual results for canonical inflation with $V(\varphi) \propto \varphi^\alpha$ then apply, giving a spectral index and tensor--scalar ratio of \cite{LL92}
\begin{eqnarray}
n_{\rm s} -1 & = & - \frac{2+\alpha}{2N} = - \frac{2m+2}{m+2} \frac{1}{N} \,,\\
r & = & \frac{4\alpha}{N} = \frac{8m}{m+2} \frac{1}{N} \,.
\end{eqnarray}
These match the results we found in Section~\ref{app-to-sect-tachyon}.
We therefore conclude that, provided our slow-roll assumption holds, a power-law potential in a Tachyon model behaves as if it were a canonical model but with a different power which is smaller (and indeed never bigger than 2). For example, the quadratic potential $m=2$ rescales to a linear potential $V(\varphi) = 2\sqrt{A/\lambda_{\rm s}}\,\varphi$, while the quartic potential $m=4$ rescales to
\begin{equation}} \def\ee{\end{equation}
V(\varphi)=\left(\frac{3A^{1/4}}{\sqrt{\lambda_{\rm s}}} \right)^{4/3}\varphi^{4/3}\,.
\ee
The upper limit for the rescaled potential is the quadratic type $V(\varphi)= \hat A \varphi^{2}$, which in canonical inflation is well explored and permitted by present data including that from the {\it Planck} satellite \cite{Ade:2013uln}. Consequently, as long as the slow-roll assumption made at the start is valid, we expect this potential to match the observational data for any value of the power-law, unlike in the canonical case.
These results explain the degeneracy between the potential normalization and $\lambda_{\rm s}$ found numerically in Ref.~\cite{Devi:2011qm}, and also confirmed in some numerical analysis we have undertaken ourselves that we display later. As the spectral index and tensor--scalar ratio match observations, the only parameter tightly constrained by observations is the potential normalization in the $\varphi$ representation. Hence we predict a perfect degeneracy $A \propto \lambda_{\rm s}$ in the quadratic case and $A \propto \lambda_{\rm s}^2$ in the quartic case.
We can also now check the validity of the slow-roll assumption made to obtain these solutions. Within the canonical frame, it is well known that the slow-roll approximation $\tilde{X} \ll V(\varphi)$ is valid. Hence $V\lambda_{\rm s} X \ll V$, and hence $\lambda_{\rm s} X \ll 1$ as required for our original approximation in the Lagrangian. This shows the self-consistency of the slow-roll approximation we have deployed. Nevertheless, it remains possible that there are other observationally-valid solutions that do not obey the slow-roll condition; investigation of this would require a numerical analysis.
\subs{DBI inflation with a quadratic potential}\label{sect-dbi-m2p2-sra}
We now carry out a similar procedure for DBI inflation with a quadratic potential. According to Eq.~(\ref{dbi-vphi}), in which the first term is negligible compared to the second term (see Appendix~\ref{dbi-sra-validity}), we can obtain an approximate slow-roll equation,
\begin{equation}
3H \dot\varphi \simeq - \frac{W^{\prime}}{W} \frac{1}{\lambda_{\rm s}} \left( 1 + \frac{m}{4} \alpha c_s \varphi^{4-m} \right) \spc \alpha = A\lambda_{\rm s} \,. \label{sr-eom-dbi-m2p2}
\end{equation}
With the quadratic potential $\tilde{V} = A \varphi^{2}$ and the warp factor in form of $W(\varphi)$, the Hubble rate in Eq.~(\ref{dbi-vphi-H}) and the slow-roll equation in Eq.~(\ref{sr-eom-dbi-m2p2}) for scalar field $\varphi$ simplifies to
\begin{eqnarray}} \def\ea{\end{eqnarray}
3H^2 &=& \frac{1}{\lambda_{\rm s}\cs\varphi^4} y \label{dbi-H-y} \,,\\
3H\dot\varphi &\simeq& \frac{2}{\lambda_{\rm s}\varphi} (1+y) \label{eom-dbi-m2p2-y} \,,\\
y &=& 1 + \alpha \cs \varphi^2 \label{dbi-m2p2-y} \,.
\ea
We note that $y>1$ since $\alpha>0$ always holds, so we can immediately obtain the parameters $\delta$ and $\epsilon$ as
\begin{eqnarray}} \def\ea{\end{eqnarray}
\delta &=& \frac{\dot\varphi}{H\varphi} = \frac{2}{\alpha} {\left(} \def\rrb{\right)\frac{y+1}{y}\rrb} (y-1) \label{sr-dbi-m2p2-delta} \,,\\
\epsilon &=& \frac{\dot\varphi^2}{2} \frac{1}{H^2} \frac{1}{\cs\varphi^4} = \frac{2}{\alpha} {\left(} \def\rrb{\right) \frac{y+1}{y} \rrb}^2 (y-1) \label{sr-dbi-m2p2-eps} \,.
\ea
We wish to find the relation between the field $\varphi$ and the e-folds $N$ via the relation
\begin{equation}} \def\ee{\end{equation}
N = -\int H dt = -\int \frac{d\varphi}{\varphi} \frac{1}{\frac{\dot\varphi}{H\varphi}} = -\int \frac{d\varphi}{\varphi} \frac{1}{\delta} \label{dbi-efold-vphi} \,.
\ee
According to Eq.~(\ref{dbi-m2p2-y}) we can derive the relation for $dy$ and $d\varphi$ as,
\begin{equation}\label{sr-dbi-m2p2-dy-deps}
\frac{dy}{y-1} = 2\frac{d\varphi}{\varphi} {\left(} \def\rrb{\right) 1 + \frac{1}{2} \frac{s}{\delta} \rrb} \sim 2\frac{d\varphi}{\varphi} \,,
\end{equation}
where $s$ is the small parameter defined in Eq.~(\ref{mod-pms}). To approximate the last term in Eq.~(\ref{sr-dbi-m2p2-dy-deps}), we have used the relation\footnote{The approximation $s\ll\epsilon$ cannot tell us $\css\sim1$ even though $s\sim0$. However, in Appendix \ref{dbi-sr-quad-sra} we show that we indeed have $\css\sim O(1)$ if using the relation (\ref{sr-dbi-m2p2-dy-deps}) for approximation.}
\begin{equation}} \def\ee{\end{equation}
\frac{s}{2\delta} = \frac{s}{\epsilon} \frac{y+1}{2y} < \frac{s}{\epsilon} \ll 1 \label{sr-dbi-appro} \,.
\ee
Now we can obtain $N$ under this approximation. Substituting Eq.~(\ref{sr-dbi-m2p2-delta}) into Eq.~(\ref{dbi-efold-vphi}) and applying Eq.~(\ref{sr-dbi-m2p2-dy-deps}), we have
\begin{equation}} \def\ee{\end{equation}
N = -\frac{\alpha}{4}\int \frac{ydy}{(y-1)^2(y+1)} \,,
\ee
and its solution is,
\begin{equation}} \def\ee{\end{equation}
\frac{8N}{\alpha} = \frac{1}{y-1} + \frac{1}{2}\ln\frac{y+1}{y-1} \spc\quad (y>1, \cs\simeq1) \label{sr-dbi-m2p2-N-y}\,.
\ee
For the quadratic potential $V\propto \phi^2$ we will recover a relation for $\varphi^{-2}= \phi^2 = 4N = 2mN$ when $y\gg1$ according to the above equation. This is the slow-roll prediction for the quadratic potential in canonical inflation, where the sound speed is exactly $\css=1$, and hence applies if DBI inflation with a quadratic potential is approximated by canonical inflation.
Equation~(\ref{sr-dbi-m2p2-N-y}) is the general relation between the scalar field $\varphi$ (or $\phi$) and $N$. Note that $y$ can be either of order 1 or much greater, according to the calculation in Appendix \ref{dbi-sr-quad-sra}, even when we impose the constraint $\css \sim O(1)$.
We can write the Eq.~(\ref{sr-dbi-m2p2-N-y}) in terms of the Lambert $\mathcal W$ function as follows:
\begin{equation}} \def\ee{\end{equation}
y = 1+\frac{2}{\theta(x)} = 1 + \frac{2}{\mathcal W(e^{x + 1}) - 1} ,\quad\quad\mbox{(at $\cs\simeq 1$)} \label{sr-dbi-alpha-N} \,,
\ee
where $x={16N}/{\alpha}$. Therefore we can write the $\delta$ and $\epsilon$ according to Eqs.~(\ref{sr-dbi-m2p2-delta}) and (\ref{sr-dbi-m2p2-eps})
\begin{eqnarray}} \def\ea{\end{eqnarray}
\delta &=& \frac{8}{\alpha} \frac{\mathcal W}{\mathcal W^2-1} \label{sr-dbi-m2p2-delta-N-theta}\,,\\
\epsilon &=& \frac{16}{\alpha} \frac{\mathcal W}{\mathcal W^2-1} \frac{\mathcal W}{\mathcal W+1} \label{
|
sr-dbi-m2p2-eps-N-theta}\,, \\
N &=& \frac{\alpha}{16} (1+\mathcal W+\ln{\mathcal W}) \label{sr-dbi-m2p2-N-theta} \,.
\ea
They imply $\epsilon \sim 1/N$. We now can treat Eqs.~(\ref{sr-dbi-m2p2-eps-N-theta}) and (\ref{sr-dbi-m2p2-N-theta}) as parametric equations for $\delta$, $\epsilon$ and $1/N$, respectively, with parameter $\theta$.
By denoting $\beta = \left(1+ 1/{\mathcal W} \right)^2$ we can also write
\begin{numcases}{\beta=}
1 \spc & \; $\mathcal W \gg 1 \Longleftrightarrow \alpha \ll 16N$ \,,\\
4 \spc &\; $\mathcal W \raw 1 \Longleftrightarrow \alpha \gg 16N$ \,.
\end{numcases}
Since this quantity (ratio) is monotonically increasing to 4 with $\alpha$, it will never contribute a term like $1/N$, so we can regard this ratio $\beta$ as a `pseudo-constant'. Therefore according to Eqs.~(\ref{sr-dbi-m2p2-eps-N-theta}) and (\ref{sr-dbi-m2p2-N-theta}) we represent the parameter $\theta$ as
\begin{equation}} \def\ee{\end{equation}
\theta = \beta \frac{16}{\alpha\epsilon} \quad,\quad \theta\alpha \sim 16\beta{N} \label{sr-dbi-m2p2-theta} \,,
\ee
where the second relation is approximated by setting $\epsilon \sim 1/N$ for later convenience in discussion. The parameters $\theta$ and $\alpha$, due to the relation in Eq.~(\ref{sr-dbi-m2p2-theta}), will be treated interchangeably when we discuss the parameter constraints in a later section.
We will use these relations to generalise our conclusions below. So far we have completed the slow-roll calculation for DBI inflation with a quadratic potential. Now we need to evaluate the power spectrum $\Ps$ and its spectral index. According to Eq.~(\ref{mod-ps}) and the variables which have been derived in Eqs.~(\ref{dbi-H-y}), (\ref{dbi-m2p2-y}), (\ref{sr-dbi-m2p2-delta-N-theta}), (\ref{sr-dbi-m2p2-eps-N-theta}) and (\ref{sr-dbi-m2p2-theta}), the power spectrum $\Ps$ is written as,
\begin{equation}} \def\ee{\end{equation}
\Ps = \frac{1}{48\pi^2}\frac{1}{\epsilon} \frac{\alpha^2}{\lambda_{\rm s}} \theta \left(1+\frac{\theta}{2}\right) = \frac{1}{96\pi^2} \frac{\alpha^3}{\lambda_{\rm s}} \frac{(\mathcal W^2-1)^2(\mathcal W+1)}{W^2} \label{sr-dbi-m2p2-ps-N} \,,
\ee
where $N_*$ is the e-folds number at the end of the DBI inflation. According to Eq.~(\ref{sr-dbi-m2p2-theta}), the power spectrum $\Ps$ in Eq.~(\ref{sr-dbi-m2p2-ps-N}) provided from the quadratic potential in DBI inflation can also be rewritten in terms of $\epsilon$ together with model parameters,
\begin{equation}} \def\ee{\end{equation}
\Ps = \frac{1}{3\pi^2} {\left(} \def\rrb{\right) \beta \frac{A}{\epsilon^2} + 8\beta^2 \frac{1}{\lambda_{\rm s}\epsilon^3} \rrb} \label{sr-dbi-m2p2-ps-eps-2}\,.
\ee
We can see that the slow-roll prediction for the power spectrum has terms due to the scalar potential denoted by its scale $A$, and the warp geometry denoted by the strength $\lambda_{\rm s}$. This is a new output from our slow-roll calculation, in comparison to the conventional approach where the first term appears only. Since in that case the model has been always studied at $\cs \sim 1$ initially, the DBI action is reduced to the canonical type, which no doubt will only present a limited prediction that is the first term in our generalised equation in Eq.~(\ref{sr-dbi-m2p2-ps-eps-2}).
According to Eqs.~(\ref{mod-ns}) and (\ref{sr-dbi-m2p2-ps-N}), we obtain the spectral index as
\begin{equation}} \def\ee{\end{equation}
\ns = - (2\epsilon + \eta) \label{sr-dbi-m2p2-ns-N} \,.
\ee
The spectral index here is exactly derived from Eq.~(\ref{sr-dbi-m2p2-ps-N}). There is no significant contribution from the derivatives of the sound speed, since $\cs \sim 1$ or from Eq.~(\ref{sr-dbi-appro}) which tells us that the third parameter $s$ is much less than $\epsilon$. We only need to work out the second parameter $\eta$ via the result for $\epsilon$, so that,
\begin{equation}} \def\ee{\end{equation}
\eta = \frac{\mathcal W^2 - \mathcal W + 2}{\mathcal W^2} \epsilon \label{sr-dbi-m2p2-eta-theta}\,.
\ee
Then by means of Eq.~(\ref{mod-ns}), we have the result for the spectral index,
\begin{equation}} \def\ee{\end{equation}
\ns = -3\epsilon \frac{\mathcal W^2 - \mathcal W/3 + 2/3}{\mathcal W^2}, \quad (\mathcal W>1)\label{sr-dbi-m2p2-ns-eps-theta}\,.
\ee
However, this formula does not give a clear understanding of the spectral index, unlike the one in Eq.~(\ref{sr-dbi-m2p2-ns-N}). However, we can check that the relations for the spectral index will have the same asymptotic behaviour (at $\cs \sim 1$)
\begin{numcases}{\ns=}
-\frac{2}{N_*} & \; $\mathcal W\raw1 \Longleftrightarrow \alpha\gg16N$ \,, \\
-\frac{3}{N_*} & \; $\mathcal W \gg 1 \Longleftrightarrow \alpha\ll16N$ \,,
\end{numcases}
where one can derive the $\epsilon$ as
\begin{numcases}{\epsilon=}
\frac{1}{2N_*} & \; $\mathcal W\raw1 \Longleftrightarrow \alpha\gg16N$ \,, \\
\frac{1}{N_*} & \; $\mathcal W \gg 1 \Longleftrightarrow \alpha\ll16N$ \,.
\end{numcases}
Finally, in terms of $(N,x;\mathcal W(e^{1+x}))$, we present both the spectral index $n_s$ and the tensor-to-scalar ratio $r$ as,
\begin{eqnarray}} \def\ea{\end{eqnarray}
n_s &=& 1 - \frac{1}{N} \frac{x}{\mathcal W-1}\frac{3\mathcal W^2-\mathcal W+2}{(\mathcal W+1)^2} ,\quad (\mathcal W>1) \label{sr-dbi-m2p2-r-W} \,,\\
r &=& \frac{16\cs}{N} \frac{x}{\mathcal W-1} \frac{\mathcal W^2}{(\mathcal W+1)^2} ,\quad (\mathcal W>1, \cs\sim1) \label{sr-dbi-m2p2-r-W2}\,,
\ea
where $x=16N/\alpha$, and the sound speed can be obtained from $\css = 1 - 2\epsilon/[3(\mathcal W-1)]$. Since $x>0$ and $\mathcal W>1$, the spectral index is always red tilted. We can see that the prediction for both $n_s$ and $r$ are model parameter dependent, in view of $\alpha=M^2\lambda_{\rm s}/2$. DBI inflation, however, does not provide a simple form for the spectral index $n_s$ and the tensor-to-scalar ratio $r$ as in previous models, where these quantities can be expressed as a function of the variable set $(n,m,N)$. This model has another variable $\alpha$ which can play an important role in determining the value of $n_s$ and $r$.
These results, for the power spectrum in Eq.~(\ref{sr-dbi-m2p2-ps-eps-2}) and spectral index in Eq.~(\ref{sr-dbi-m2p2-r-W}), are derived in the non-relativistic limit $\cs \sim 1$. Meanwhile, we can evaluate the following relation
\begin{equation}} \def\ee{\end{equation}
\frac{23}{24} < \frac{\mathcal W^2 - \mathcal W/3 + 2/3}{\mathcal W^2} < \frac{4}{3} \qquad (\mathcal W > 1)\,,
\ee
which appeared in Eq.~(\ref{sr-dbi-m2p2-ns-eps-theta}). Therefore we will have a leading contribution for the spectral index $\ns = O(\epsilon)$ when the DBI inflaton field moves with $\cs\sim1$.
\begin{figure}} \def\efg{\end{figure}[t]
\centering
\includegraphics[width=0.9\textwidth]{sr-dbi-m2p2-nsr.pdf}
\caption{Slow-roll predictions for the DBI inflation with quadratic potential and the {\it Planck} constraints on inflation models. [Based on an image from Ref.~\cite{Ade:2013uln}, original image credit ESA/{\it Planck} Collaboration.]}
\label{fig:sr-dbi-m2p2-nsr}
\efg
Figure~\underline{\ref{fig:sr-dbi-m2p2-nsr}} shows the predictions against data. We see that for any available model parameter $\alpha$ DBI inflation has difficulties in providing the required observables, as the results lie above the $m=2$ canonical model which is coming under observational pressure, particularly with $N_* = 50$. But for larger pivot e-folds, for example $N_*=60$, the DBI model with this quadratic potential has the possibility to match observations. Note incidentally that our predictions meet those of natural inflation (the purple shaded region), which approach the canonical quadratic case from below.
Our result is different from the result that $\ns = O(\epsilon^2)$ in Ref.~\cite{Alishahiha:2004eh}. The difference between these results is that ours is in the $\cs \rightarrow 1$ limit, while theirs applies in the warp-factor dominated regime of $\cs \rightarrow 0$, a limit in which we are able to reproduce their result. To show this we start with the the Hubble parameter $H^2$ and the sound speed $\css$ given by
\begin{equation}} \def\ee{\end{equation}
3H^2 = \frac{1}{\varphi^4} \frac{y}{\lambda_{\rm s}\cs} \quad\spc\quad \epsilon = \frac{3}{2} \frac{1-\css}{y} \,.
\ee
So at $\cs\raw0$, which also means $y\gg1$ if we focus on $\epsilon\ll1$, according to Eq.~(\ref{sr-dbi-m2p2-eps}) we have the following approximate relations
\begin{equation}} \def\ee{\end{equation}
3H^2 \simeq \frac{3}{2\lambda_{\rm s}\epsilon} \frac{1}{\cs\varphi^4}\quad\spc\quad\epsilon \simeq \frac{3}{2} \frac{1}{y} \quad\spc\quad \alpha = \frac{3}{\epsilon^2} \label{ecs0} \,.
\ee
Therefore, according to Eqs.~(\ref{mod-ps}) and (\ref{mod-ns}) we have
\begin{eqnarray}} \def\ea{\end{eqnarray}
\Ps &=& \frac{1}{16\pi^2} \frac{1}{\lambda_{\rm s}\epsilon^2} \frac{1}{\css\varphi^4} = \frac{1}{16\pi^2} \frac{1}{\lambda_{\rm s}\epsilon^2} \frac{\alpha^2}{(y-1)^2} \label{ps-0}\,,\\
\ns &=& -\left(-2\frac{\epsilon^\prime}{\epsilon} -2\frac{y^\prime}{y-1}\right) = 2\frac{y^\prime}{y} \frac{1}{y-1} \label{ns-0}\,.
\ea
Using the relation in Eq.~(\ref{ecs0}), we can simplify them to be
\begin{eqnarray}} \def\ea{\end{eqnarray}
\Ps &=& \frac{1}{4\pi^2}\frac{1}{\lambda_{\rm s}\epsilon^4} \,,\\
\ns &=& \frac{4}{3}\epsilon\eta \propto O(\epsilon^2)\,,
\ea
since $\epsilon,\, \eta=-\epsilon^\prime/\epsilon=y^\prime/y$ are of the same order. As this regime predicts that $\ns$ is very close to one, it is under strong pressure from observations.
To summarise our results in the slow-roll approximation, we obtained formulae for the power spectrum $\Ps$ in Eq.~(\ref{sr-dbi-m2p2-ps-N}) or (\ref{sr-dbi-m2p2-ps-eps-2}) and its spectral index $n_{{\rm s}}$ in Eq.~(\ref{sr-dbi-m2p2-ns-N}). We present all the relations in each limit in Table \underline{\ref{sr-dbi-m2p2-obs-theta}}.
\begin{table}} \def\etb{\end{table}[t]
\centering
\def1.2{1.}
\begin{tabular}} \def\et{\end{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\theta$ or $\alpha$ & $\frac{1}{N_*}$& $\epsilon(\theta)$ &$\epsilon(N_*)$ & $\Ps(\epsilon)$ & $\Ps(N_*)$ & $\ns$ & Domination \\
\hline
$\theta \raw 0$ or $\alpha \gg 16N_*$ & $\frac{8}{\alpha\theta}$ & $\frac{4}{\alpha\theta}$& $\frac{1}{2N_*}$ & $ \frac{A}{\epsilon^2}$ & $\propto AN_*^2$& $-\frac{4}{2N_*}$ & Scalar potential \\
$\theta \gg 1$ or $\alpha \ll 16N_*$ & $\frac{16}{\alpha\theta}$ & $\frac{16}{\alpha\theta}$& $\frac{1}{N_*}$ & $ \frac{1}{\lambda_{\rm s}\epsilon^3}$ & $\propto \frac{N_*^3}{\lambda_{\rm s}}$ & $-\frac{6}{2N_*}$ & Warp geometry \\
\hline
\et\caption{Predictions for the observables in DBI inflation with a quadratic potential, in each limit for $\theta$. At other values of $\theta$, the outcome is determined by both the scalar potential and the warp geometry (see Eq.~(\ref{sr-dbi-m2p2-ps-N}) or Eq.~(\ref{sr-dbi-m2p2-ps-eps-2})). These results are obtained under the assumption in Eq.~(\ref{sr-dbi-m2p2-dy-deps}), which requires the sound speed $\css \sim O(1)$. Also note that $\theta = \mathcal W - 1$.}\label{sr-dbi-m2p2-obs-theta}
\etb
Finally, it is worth mentioning that we can also make predictions for the case where the sound speed $\css \ll 1$ in the same manner that we have just applied. We can obtain the solution in this relativistic case, but the relation $\epsilon=\epsilon(N)$ is not obviously applicable from the e-folds $2/(y^2-1)\propto\mathcal{W}(N)$ which, according to Eq.~(\ref{dbi-z-css-0}), can be obtained from Eq.~(\ref{dbi-efold-vphi}). These models are subject to the non-gaussianity constraint that the sound speed be greater than about 0.07 \cite{Ade:2013ng}.
\sect{Model Parameter Estimation}\label{sect-mpe}
We now discuss parameter estimation for the models investigated in the previous sections. Throughout this paper, the potential considered possesses only one free parameter. Since the amplitude of the scalar power spectrum is accurately determined to be about $2.5\times 10^{-9}$ from observational data, we can now estimate the model parameters for each case, for example taking the pivot e-folds to be $N_{*}=50$.
\subs{Sum-separable models}
We take the expression of the scalar power spectrum from Eq.~(\ref{ps-ssm}) and apply a (base 10) logarithm, giving
\begin{eqnarray}} \def\ea{\end{eqnarray}
\log{A} &=& \frac{m}{2n} \left[ - \frac{6.53 + (1+\frac{m}{\beta})\log{N} + \log(\frac{\beta}{m}\sqrt{2n-1})}{\frac{m}{\beta}\frac{1}{2n-1}} - \log \left(\frac{m6^{n-1}}{n} \beta^{2n-1} \right) \right] \,.\label{ssm-mpe}
\ea
Note that the model parameter $A$ denotes the amplitude appearing in the potential $V(\phi) = A\phi^{m} = \lambda_p\phi^{m}/m$ after the coefficient of the kinetic term has been rescaled to unity. For example, for the quadratic potential $A = M^{2}/2$ where $M^{2}$ is the mass of field, while for the quartic potential $A = \lambda/4$.
As an application of this formula, we consider the canonical inflation cases where $n=1$ and $m=2,4$. Then we will obtain an estimate for the mass scale for the quadratic and quartic potential respectively. Then
\begin{eqnarray}} \def\ea{\end{eqnarray}
\log M^2 &\simeq& -10.2 \,,\\
\log\lambda &\simeq& -12.5 \,.
\ea
For non-canonical inflation models, a similar analysis can be performed.
\subs{Tachyon models}
For the Tachyon model, we consider the results in Section~\ref{app-to-sect-tachyon}. From Eq.~(\ref{tachyon-ps-N}) we find the relation between $A$ and $\lambda_{\rm s}$ to be
\begin{eqnarray}} \def\ea{\end{eqnarray}
\log\lambda_{\rm s} &=& \frac{2}{m}\log{A} + 2(\mathbf{A} + \mathbf{B} + \mathbf{C}) + {(1+2\mu)\log{(2N)}} \\
\mathbf A &=& -\mu \log{(12\pi^{2} \times 2.5\times10^{-9})} = 6.53\mu \nonumber ,\,
\mathbf B = \log{m}+ \left(\frac{1}{2}+\mu\right)\log{\mu} ,\,
\mathbf C = -\frac{1}{m}\log{\cs} \,.
\ea
The equation for $\mathbf{C} \sim 0$ above is approximate due to the condition $1/3< \cs < 1$, and particularly $\cs \sim 1$ if $\epsilon \ll 1$. Therefore, for the quadratic potential $A = M^{2}/2\,, m=2 \,, \mu = 1$, we can evaluate the parameters as,
\begin{equation}} \def\ee{\end{equation}\label{tachyon-num-m2p2}
\log\lambda_{\rm s} = \log{\frac{M^{2}}{2}} + 19 + 2\log{2} \simeq \log{M^{2}} + 19.3 \,.
\ee
For the quartic potential, $A = \lambda/4 \,, m=4 \,, \mu=3/4$, the parameters are
\begin{equation}} \def\ee{\end{equation}\label{tachyon-num-lp4}
\log\lambda_{\rm s} = \frac{1}{2}\log{\frac{\lambda}{4}} + 18.2 \simeq \frac{1}{2}\log{\lambda} + 17.9 \,.
\ee
We have carried out a Monte Carlo Markov chain fit to the data through an extension of our analysis in Ref.~\cite{Li:2012vt} and found results which are in agreement, shown in Figure~\ref{f:tachnum}. Similar results can be found in Ref.~\cite{Devi:2011qm}.
\begin{figure}} \def\efg{\end{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{m2p2_nov_jan_1_2D_needed_single_ls_16} \hspace*{1cm}
\includegraphics[width=0.4\textwidth]{lp4_nov_feb_2_2D_needed_single_ls_16}
\caption{MCMC constraints on model parameters for the quadratic (left) and quartic (right) tachyon cases. The exact degeneracy between the potential normalisation and the tachyon parameter predicted by our analytic analysis is clearly seen.}
\label{f:tachnum}
\efg
\subs{DBI models}
\ssubs{Quadratic potential}
We only study the parameter estimation for the case of $\css \sim 1$, as in Section~\ref{sect-dbi-m2p2-sra}, as otherwise that it is hard to find the required function $\epsilon = \epsilon(N)$. For this reason, for the quadratic potential we present the results in the {\it non-relativistic case} only.
According to the results in Section \ref{sect-dbi-m2p2-sra}, though the formulae for the power spectrum are complicated, we can still approximate the value for $\alpha$ which encodes the scale of the potential ($A$) and the strength of the warped geometry ($\lambda_{\rm s}$). According to Eq.~(\ref{sr-dbi-m2p2-ps-N}), the combined contribution from these indicates that
\begin{equation}} \def\ee{\end{equation}\label{sr-dbi-m2p2-mpe-aes}
\theta \sim 2 \quad{{\rm or}}\quad \alpha \gtrsim 8N \,.
\ee
Taking the logarithm of this relation (\ref{sr-dbi-m2p2-mpe-aes}) at the pivot scale $N_*=50$ we have the linear equation
\begin{equation}} \def\ee{\end{equation}
l \equiv \log M^2 + \log\lambda_{\rm s} \simeq 2.9 \label{sr-dbi-log-alpha-N} \,.
\ee
The relation in Eq.~(\ref{sr-dbi-m2p2-mpe-aes}) gives us two possibilities for the locations or choices for the model parameters $A \,, \lambda_{\rm s}$ at some critical value such as $\alpha \sim 8N_*$. We will see according to Eq.~(\ref{sr-dbi-log-alpha-N}) and Eq.~(\ref{dbi-vphi-H}) that larger $\alpha$ implies a strong contribution from the scalar field potential. In the other case, $\alpha$ smaller than the critical $8N_*$, inflation will be dominated by the warped geometry.
Recalling Table~\underline{\ref{sr-dbi-m2p2-obs-theta}}, we already have the relation between $\theta$ and e-folds $N$. Therefore we can approximate the model parameter range, as presented in Table~\underline{\ref{sr-dbi-m2p2-mod-pms-bound}}. This table also indicates the value $\log m^2 + \log \lambda_{\rm s} \sim 3$ (or similar) should be found in parameter space. Above this critical value DBI inflation will be dominated by the scalar field potential, while on the other side the warp geometry will dominate.
\begin{table}} \def\etb{\end{table}[t]
\centering
\def1.2{1.2}
\begin{tabular}} \def\et{\end{tabular}{|c|c|c|}
\hline
$\alpha$ & $\log M^2 + \log \lambda_{\rm s}$ & Domination\\
\hline
$\alpha \gg 8N_*$ & $>2.9$ & Scalar potential\\
$\alpha \ll 8N_*$ & $<2.9$ & Warp geometry \\
\hline
\et\caption{Constraints on the model parameters in DBI inflation with quadratic potential. The model parameters, $A$ and $\lambda_{\rm s}$, are correlated along the line $l \vcentcolon= \log M^2 + \log \lambda_{\rm s}$. The parameter $\alpha = A\lambda_{\rm s} = M^2\lambda_{\rm s}/2$, where $M^2$ is the mass scale of the scalar potential and $\lambda_{\rm s}$ the strength of the warp factor.}\label{sr-dbi-m2p2-mod-pms-bound}
\etb
\ssubs{Quartic potential}
The power spectrum was obtained in Section \ref{sect-eval-dbi-lp4}. Following the discussion there, we can approximate the prediction for model parameters at the $N_* = 50$. We note that in this case, although we can still constrain model parameters by requiring the correct amplitude of density perturbations at the pivot scale, the
models typically do not lie in the within the 95\% confidence region of the $n_s$--$r$ plane.
\begin{itemize}} \def\ei{\end{itemize}
\item[$\star$] {\it Non-relativistic case}\\
According to Eq.~(\ref{dbi-sr-ps}) in the non-relativistic limit where $\cs \sim 1$
\begin{equation}} \def\ee{\end{equation}\label{kmc-dbi-sr-m2p2-mpe-nonrel}
\log \lambda_{\rm s} - \log(1+\alpha) \simeq 13.1 \,,
\ee
for the parameter correlation. We can estimate the bound for each parameter with different $\alpha$. If $\alpha \ll 1$ then we can expect the low bound value for the strength $\lambda_{\rm s}$ for warp factor,
\begin{equation}} \def\ee{\end{equation}
\log \lambda_{\rm s} \gtrsim 13.2 \label{sr-dbi-lp4-nrel-lambda-1} \,,
\ee
while in the limit $\alpha \gg 1$, we will have the upper bound for strength $\lambda$ of the scalar potential,
\begin{equation}} \def\ee{\end{equation}
\log\lambda < - 13.1 + \log4 \simeq -12.5 \label{sr-dbi-lp4-nrel-lambda-2} \,.
\ee
This is the observational value for canonical inflation with a quartic potential.
\item[$\star$] {\it Relativistic case}\\
Similarly we consider the case $\cs \ll 1$ from Eq.~(\ref{dbi-rel-Ps-N}), obtaining
\begin{equation}} \def\ee{\end{equation}\label{kmc-dbi-sr-lp4-mpe-rel}
\log\lambda_{\rm s} \simeq 13.8\,.
\ee
Hence to have relativistic motion during inflation, the strength of the warp geometry must take this value. We cannot see a relation for the parameter $A$ for the potential if we just consider the power spectrum in Eq.~(\ref{dbi-rel-Ps-N}), but if we recall the footnote in Section \ref{sect-eval-dbi-lp4} when deriving Eq.~(\ref{dbi-rel-Ps-N}), we have an asymptotic condition relevant to both parameters $A$ and $\lambda_{\rm s}$:
\begin{equation}} \def\ee{\end{equation}
\alpha \epsilon \cs \sim 1.5 \,,
\ee
where $\epsilon = 2/N$ in the relativistic case. So the asymptotic relation between $\lambda$ and $\lambda_{\rm s}$ is
\begin{equation}} \def\ee{\end{equation}\label{kmc-dbi-sr-lp4-mpe-rel-2}
\log \lambda_{\rm s} + \log \lambda \sim \log\frac{3N_*}{\cs} \simeq 2.2 - \log\cs \,.
\ee
We can evaluate this equation at some values of $\cs$. For example, for $\cs \sim 0.01$, which is of the same order as $\epsilon$, we will have $\log \lambda_{\rm s} + \log \lambda \sim 4.2$.
\ei
All these parameter estimation cases have also been examined using MCMC methods, and the results from our CosmoMC \cite{Lewis:2002ah} exploration are presented in our companion paper II.
\sect{Conclusions}\label{sect-conclusion}
In this paper we have developed and applied a systematic method for deriving observational predictions in non-canonical single-field inflation models, using the slow-roll approximation encoded as a differential equation for $u = 1/\epsilon$ rather than as a set of slow-roll parameters. We have given explicit calculations for several such models, including the Tachyon and DBI cases, deriving observables such as the power spectrum $\Ps$ and its spectral index $n_s$ in terms of e-folds $N$. For some models we also present the results in terms of slow-varying parameters rather than $N$, when we are unable to explicitly solve the transcendental equation for $\epsilon$ and $\cs$, for example in DBI inflation with quartic potential.
The use of field redefinition is another key methodology in this paper. It can simplify the process of finding the solution for $u$, and in terms of the redefined scalar field the reshaped potential can reveal the degeneracy of model parameters which includes both the strength of the kinetic energy and the strength of the potential energy. By this method, in Tachyon models we have obtained an explicit correlation of $f$ and $\lambda$ which has previously been found only via numerical calculation. For the DBI inflation models, we have also obtained for the first time a similar formula describing the correlations between its two model parameters, though they are not as explicit as the relation in the Tachyon model.
While this method cannot give a full exact solution, as obtained either via non-slow-roll approaches or by numerical exploration, it can nevertheless offer some advantages for modelling inflationary cosmology. For one thing it may suggest to reconstruct the Lagrangian which will potentially give an explicitly solvable relation for $\css=\css(u)$. For another, it can provide a quick parameter estimation since the power spectrum, which is accurately determined from the observational data, is also formulated via slow-varying parameters. In other words, given the Lagrangian or potential, by finding the $p_{,X\varphi}$ and then the variable $u$, we can solve the power spectrum by Eq.~(\ref{mod-ps}) which constrains the model parameters for the considered model. For some classes of inflation model, this process will be quite straightforward, such as the sum-separable class where $p_{,X\varphi}=0$ and the product-separable class where $p_{,X\varphi}\varphi^{\prime}/p_{,X}=-2/u$.
For DBI inflation models with different potentials, we adopted different approaches due to their individual complexities. The DBI inflation with a quadratic potential, if not immediately using limiting cases of $\cs$, is the least tractable model in the current paper. However we still found predictions for this case, though we have only presented in detail the non-relativistic predictions for this potential. Additionally, in the relativistic case we have obtained predictions matching those derived in the conventional manner using the field speed limit $\cs=0$ \cite{Alishahiha:2004eh}. Due to the irreversible relation for $\epsilon=\epsilon(N)$ from the e-folds integration $2/(y^2-1)=\mathcal{W}(N)$ we have not presented the details, but they can readily be computed if one is interested in the application of the method. The case of DBI inflation with a quartic potential is examined in both the non-relativistic and relativistic cases. For both models, not only do we recover the same results as conventional treatments, but our analysis gives a novel formula for the power spectrum, Eq.~(\ref{sr-dbi-m2p2-ps-eps-2}).
The proposed approach is not only able to address monomial potentials, but can also be applied to models with other potentials. For future applications, reconstruction of the inflation model will be an important direction. For the models we have described in this paper, we hope the method will be useful in that more general context.
A benefit of considering non-canonical Lagrangians is that model predictions can be in better agreement with current data than the canonical case, where even the quadratic potential is starting to come under pressure. We have affirmed previous results \cite{Li:2012vt,Unnikrishnan:2012zu,Unnikrishnan2} showing that the non-canonical $X^n$ models give predictions that, for a given potential, are in better agreement with data. In this article we have extended that conclusion to the Tachyon models, most explicitly via the field redefinition approach which shows that under slow-roll they are equivalent to canonical models with reshaped potentials of shallower slope, as favoured by the data. For DBI models the situation is less promising; with a quadratic potential the predictions in the non-relativistic limit, seen in Fig.~\ref{fig:sr-dbi-m2p2-nsr}, are further from scale-invariance than the canonical case, while in the relativistic limit the cancellation of the leading-order slow-roll correction noted in Ref.~\cite{Alishahiha:2004eh} places the model too close to scale-invariance. In the quartic case the situation is even less promising. We conclude therefore that DBI models with simple potentials are in significant tension with current data.
\acknowledgments
S.L.\ was supported by a Sussex International Research Scholarship, and A.R.L.\ by the Science and Technology Facilities Council [grant numbers ST/I000976/1 and ST/K006606/1] and a Royal Society--Wolfson Research Merit Award. We thank Antony Lewis, Hiranya Peiris, and David Seery for useful discussions.
\input{appendix}
|
\section{Introduction}
Foukal et al. (1974) introduced the notation ``sunspot plumes'' to
describe areas above sunspot umbrae that are ``the brightest features
in an active region by an order of magnitude''.
This led to the idea that sunspot plumes are regions within large magnetic
loops, extending to altitudes of several thousand kilometers above the
photosphere, in which the temperature is one to two orders of magnitude
lower than in the corona of the surrounding active region
(Noyes et al. 1985).
In contrast, based on numerous sunspot observations with the Ultraviolet
Spectrometer and Polarimeter (UVSP) on the {\it Solar Maximum Mission (SMM)}
Gurman (1993) found that sunspot plumes were nearly nonexistent.
Most recently Maltby et al. (1998) observed sunspot plumes in five out
of nine sunspot regions with the Coronal Diagnostic Spectrometer
(CDS; Harrison et al. 1995) on the {\it Solar and Heliospheric
Observatory (SOHO)} and discussed briefly previous conflicting results.
The CDS observations showed that sunspot plumes exist in
the upper part of the transition region, occur both in magnetic
unipolar and bipolar regions, and may extend outside the umbra and
into the penumbra.
>From the energy requirements in sunspot loops Foukal (1976) suggested
that rapid downflows occur in the plumes.
Strong downflows over sunspots were reported by Brueckner, Bartoe, \&
VanHoosier (1977) and studied by Nicolas et al. (1982), while Kjeldseth-Moe
et al. (1988) found that both upflows and downflows occurred. Other
investigations have confirmed and extended these observations, for a
review see Maltby (1997). None of the observations above referred
specifically to plumes and to our knowledge the velocity in sunspot
plumes is not known. An investigation by Brynildsen et al. (1998) on
the connection between line emission and wavelength shift in sunspot
regions may, however, hold some relevance to this.
In this paper we extend the CDS material to twelve sunspot regions
and present the first measurements of velocities in sunspot plumes.
\section{Observations and Data Reduction}
Observations of twelve sunspot regions, listed in Table~1, were obtained
with the Normal Incidence Spectrometer (NIS) of the CDS instrument
as part of a joint observing program on {\it SOHO}.
Since NOAA 7981 recurs as NOAA 7986 eleven different regions were observed.
A large fraction of the observing time was used to raster an area of
120$\arcsec \times$ 120$\arcsec$, moving the narrow 2.0$\arcsec$
spectrometer slit perpendicular to the slit direction in steps of 2.0$\arcsec$.
The exposure time was 20~s; each raster required 25 min and contains
information from 60 adjacent slit locations for ten emission lines.
The observations contain spectra with spectral resolution,
$\lambda / \delta \lambda$, in the range 3635 -- 4500 within ten narrow
spectral windows centered on the selected lines, which cover a wide range of
ionization temperatures, see Table~2.
White light sunspot contours are taken from the MDI instrument
(Scherrer et al. 1995) on SOHO and the National Solar Observatory.
To compensate for the influence of solar rotation on the alignment,
the images were re-aligned by moving the frequently observed MDI magnetograms
artificially to the same observing time as the CDS recordings.
The data acquisition and detector characteristics that are relevant
for this study were described by Harrison et al. (1995).
Briefly, the CDS data are corrected for geometrical distortions,
the CCD readout bias is removed, the non-wavelength-dependent
calibration parameters peculiar to the detector are applied, including
the exposure time, the amplification of the microchannel plate, and
a flat-field correction.
The final step in the calibration is to convert the photon events into
absolute intensity units.
A careful approach to the line-of-sight velocity determination
is required to avoid the possible influence of other lines within
the spectral windows, particularly in areas where the line of primary
interest is weak.
Detailed investigations of the line profiles show that
most of them may be represented with a single Gaussian profile.
The profile parameters for He~{\scriptsize I} $\lambda$584
and O~{\scriptsize III} $\lambda$599 are determined in this way
since no disturbing lines are found.
We find that it is possible to improve the line-of-sight determination
for the other lines by representing the observations by a composite
line profile, comprised of two Gaussian components, where one Gaussian
is adjusted to fit the line of primary interest and the other
accounts for the second most intense line within the spectral window,
see Table~2.
This approach is used even in cases where the distance between the
lines is too large to consider the second most intense line as a
blend.
After exploring different ways of reducing the data, we decided to use
two, slightly different methods.
In the first method the line profile parameters for both lines
are determined by a least squares fit to the observations.
This requires good signal to noise ratio both for the line and the blend
and is applied to the Mg~{\scriptsize IX} $\lambda$368 line.
In the second method we locate parts of the rastered area where
the line of primary interest is weak and use this location to determine
the wavelength position and line width of the second line.
Keeping these parameters constant,
we are able to fit the observations with a composite line profile.
This method is tested by applying it to the He~{\scriptsize I}
$\lambda$522 line and compare the results with those obtained
for the strong He~{\scriptsize I} $\lambda$584 line, which is not
influenced by other lines.
We estimate the accuracy in the line-of-sight velocity determinations
to be 5 km~s$^{-1}$ for He~{\scriptsize I} $\lambda$584 and
O~{\scriptsize V}, 10 km~s$^{-1}$ for Mg~{\scriptsize IX}, and
15 km~s$^{-1}$ for O~{\scriptsize III}, O~{\scriptsize IV},
Ne~{\scriptsize VI} and Fe~{\scriptsize XVI}.
\section{Results}
Following Maltby et al. (1998) we set the criterion for the presence
of a sunspot plume by requiring that the contours for peak line intensity
$I \ge 5 \times \overline I$ are located (1) above the umbra or part
thereof and (2) with most of the emission inside the white light sunspot.
Figure~1 (Plate L00) shows the observed spatial distribution of
Ne~{\scriptsize VI} $\lambda$562 peak line intensity in twelve sunspots.
The brightest features with peak line intensity, $I$ larger than 5 times
the average intensity, $\overline I$, are encircled by yellow contours,
whereas medium bright features with $I > 2.5 \times \overline I$ are
encircled by green contours.
NOAA 7973, 7986, 8073, 8083, 8085, 8113 and 8123 satisfy the adopted
|
criterion for containing a sunspot plume.
We also regard NOAA 8011 as containing a sunspot plume since the
O~{\scriptsize V} $\lambda$629 peak line intensity exceeds
5 $\times$ $\overline I$ in a small region above the umbra.
It should be remarked that sunspots without plumes,
NOAA 7981, 7999, 8076 and 8108, also
show Ne~{\scriptsize VI} $\lambda$562 peak line intensity
$I > 5 \times \overline I$ both above and outside the sunspot.
Since NOAA 7981 recurs as 7986 we note that Figure~1
(Plate L00) illustrates a finding of previous observers
({\frenchspacing e.g.} Foukal 1976), that plumes may be absent
during part of the sunspot's lifetime.
A remarkable feature in Figure~1 (Plate L00) is that the enhanced
Ne~{\scriptsize VI} emission appears to outline one or a few thin emission
structures, extending from the sunspot to the surrounding regions.
In NOAA 8076 the extended feature resembles a magnetic loop, but in
other regions, such as NOAA 7986, the medium bright emission
features look like footpoints of magnetic loops.
Similar features emitting strongly in the transition region lines
O~{\scriptsize IV} $\lambda$554 and O~{\scriptsize V} $\lambda$629
are observed in almost the same locations.
We note that Foukal (1976) remarked that strong cool emission was
often found near both foot points of a loop, even if only
one foot point could be confidently traced to the umbra.
This suggests that Foukal (1976) observed similar emission structures
to those seen in Figure~1 (Plate L00).
We now consider the line-of-sight velocity in bright areas above sunspots,
{\frenchspacing i.e.} areas encircled with yellow contours in Figure~1
(Plate L00) both in sunspot plumes and in equally bright areas above the
other sunspots.
Figure~2 shows the relative line-of-sight velocity, $v$, versus line
formation temperature, $T$, in these bright areas above sunspots.
Since most of the sunspots are observed close to the disk
centre we tend to use the words upflow and downflow,
even though contributions from horizontal velocities cannot be excluded.
The CDS spectra contain few chromospheric lines and therefore
the line-of-sight velocity is measured relative to the average
line-of-sight velocity in the rastered area,
120$\arcsec \times$~120~$\arcsec$.
No corrections for the differential redshift between transition region-
and chromospheric lines are applied.
Figure~2 shows that the relative line-of-sight velocity is
directed away from the observer and increases with increasing temperature,
reaches a maximum between 15 and 41 km~s$^{-1}$ close
to log T $\approx$ 5.5 and then decreases abruptly.
For nine out of twelve sunspots the maximum relative velocity exceeds
26 km~s$^{-1}$.
The result is valid both for sunspot plumes and equally bright regions
above the other sunspots.
No connection between line-of-sight velocity and heliocentric angle,
$\theta$, is apparent.
For the coronal lines with log T $\approx$ 6.0 (6.4) the velocity is
below 10 km~s$^{-1}$ in eleven out of twelve sunspots.
This implies a marked change from low velocities in the corona to
strong downflows in the sunspot transition region.
To clarify the problem let us study the spatial distribution of the
relative line-of-sight velocity in O~{\scriptsize V} $\lambda$629,
formed in the transition region and compare the results with those
obtained for the low corona line Mg~{\scriptsize IX} $\lambda$368,
see Figures~3 and 4 (Plates L00 and L00), respectively.
Figures~3 and 4 (Plates L00 and L00) confirm the results presented
in Figure~2.
Almost the entire area encircled with a yellow contour in the sunspot
transition region is strongly redshifted, see Figure~3 (Plate L00),
whereas the corresponding areas in Figure~4 (Plate L00) show little or
no wavelength shift.
This fascinating result deserves a few comments.
The observations are obtained with small to moderate heliocentric angles
and show that the vertical flow in the corona is too small
to maintain a strong flow in the sunspot transition region.
This suggests that the gas has to be supplied from regions surrounding the
sunspot.
Let us examine Figures~3 and 4 (Plates L00 and L00).
Almost all sunspot regions in Figure~3 (Plate L00) contain one or a few
prominent, strongly redshifted velocity channels, several of which extend
from inside the sunspot to considerable distances from the sunspot.
We note that plasmas at transition region temperatures
may occur at considerable heights within the active region
({\frenchspacing e.g.} Brekke, Kjeldseth-Moe, \& Harrison 1997).
Since the gas is moving away from the observer in the velocity channels
that end in the sunspot, it seems likely that the gas is moving from
regions located at a greater height outside the sunspots and towards the
sunspot plume region.
This interpretation is compatible with the results presented in Figure~3
(Plate L00) and with the low velocities observed in Figure~4 (Plate L00),
where only NOAA 8076 shows a prominent, redshifted velocity channel.
The present observations, interpreted in terms of gas at transition
region temperature moving from greater height towards the sunspot
plume, may be of interest in future studies of the suggestion that
transition region structures are not physically connected to the
coronal structures (Feldman 1983).
It is interesting to compare corresponding images in Figures~1 and 3
(Plates L00 and L00).
We find that the enhanced line emission regions tend to be redshifted.
For a few sunspots, such as NOAA 8076, there is good correspondence between
redshift and peak line intensity, whereas in others, such as NOAA 8113,
there is a marked difference between the location of the enhanced line
emission and the most prominent, redshifted channels.
Above we have tacitly assumed that the wavelength shift is caused by
material flow.
We note that the observations may be compatible with an alternative
interpretation of redshifts in the transition region lines,
proposed by Hansteen, Maltby, and Malagoli (1996).
Based on numerical simulations they find that episodic, magneto-hydrodynamic
disturbances that originate in the corona and become nonlinear as they
propagate towards the transition region may produce an overabundance of
redshifts in the transition region lines, combined with small to moderate
wavelength shifts in the coronal lines.
\acknowledgements
We would like to thank all the members of the large international CDS
team for their extreme dedication in developing and operating this
excellent instrument, the Michelson Doppler Imager team
and T. Rimmele at the National Solar Observatory for permission
to use their data and the Research Council of Norway for financial support.
SOHO is a mission of international cooperation between ESA and NASA.
\clearpage
|
\section{\label{intro} Introduction}
\allowdisplaybreaks
The lepton sector is one of the most interesting object for experiments to search for new physics (NP) beyond the prediction of the standard model (SM). For example, the evidence of neutrino oscillation confirms that the SM must be extended. Recently, the experimental data of anomalous magnetic moments (AMM) of charged leptons $(g-2)_{e_a}/2\equiv a_{e_a}$ has been updated, where the deviation between SM prediction and the lasted experiment data for muon is \cite{Muong-2:2021ojo}
\begin{equation}
\label{eq_damu}
\Delta a^{\mathrm{NP}}_{\mu}\equiv a^{\mathrm{exp}}_{\mu} -a^{\mathrm{SM}}_{\mu} =\left(251 \pm 59\right) \times 10^{-11},
\end{equation}
corresponding to the $4.2\sigma$ deviation from standard model (SM) prediction \cite{Aoyama:2020ynm} combined from various contributions \cite{Davier:2010nc, Davier:2017zfy, Keshavarzi:2018mgv, Colangelo:2018mtw, Hoferichter:2019mqg, Davier:2019can, Keshavarzi:2019abf, Kurz:2014wya, Melnikov:2003xd, Masjuan:2017tvw, Colangelo:2017fiz, Hoferichter:2018kwz, Gerardin:2019vio, Bijnens:2019ghy, Colangelo:2019uex, Colangelo:2014qya, Blum:2019ugy, Aoyama:2012wk, Aoyama:2019ryr, Czarnecki:2002nt, Gnendiger:2013pva}.
For the electron anomaly, the deviation between SM and experiment is $1.6\sigma$ discrepancy \cite{Morel:2020dww}.
On the other hand, $\Delta a_{e,\mu}$ are strongly constrained by the experimental data obtained from searching for the charged lepton flavor violating (cLFV) decays $e_b\rightarrow e_a\gamma$ are~\cite{MEG:2016leq, BaBar:2009hkt}:
\begin{align}
\label{eq_ebagaex}
\mathrm{Br}(\tau\rightarrow \mu\gamma)&<4.4\times 10^{-8}, \;
\mathrm{Br}(\tau\rightarrow e\gamma) <3.3\times 10^{-8}, \;
\mathrm{Br}(\mu\rightarrow e\gamma) < 4.2\times 10^{-13}.
\end{align}
This important property was discussed previously, for example see discussions for a general estimation in Ref. \cite{Crivellin:2018qmi}, and many particular models beyond the standard model (BSM) \cite{Lindner:2016bgg, Dorsner:2020aaz, Hue:2021xap, Hue:2021xzl, Hong:2022xjg, Li:2022zap}. General formulas expressing simultaneously both one-loop contributions to AMM and cLFV amplitudes were introduced in the limits of new heavy scalar and/or gauge boson exchanges $m_B^2 \gg m^2_{a}$ with $m_a$ being the mass of a charged lepton $e_a=e,\mu,\tau$ \cite{Crivellin:2018qmi}. Other calculations in the unitary gauge were discussed \cite{Yu:2021suw, Leveille:1977rc} for the one-loop contributions to $a_{e_a}$ with $m_{a}\neq0$, without the relations with the cLFV amplitudes. The analytic one-loop formulas for cLFV amplitudes calculated in the 't Hooft Feynman (HF) gauge were also shown in Ref. \cite{Lavoura:2003xp}, using the notations of the Passarino-Veltman (PV) functions \cite{Passarino:1978jh, tHooft:1978jhc} with $m_{a}\neq m_{b}$. The approximate formulas with $m_{a}=m_{b}=0$ were introduced and consistent with those given in Ref. \cite{Crivellin:2018qmi}, as shown particularly in Ref. \cite{Hue:2017lak} for 3-3-1 models. The general analytic formulas of these PV functions were introduced for numerical investigations. They are consistent with the results generated by LoopTools \cite{Hahn:1998yk}, which can be transformed into other PV notations implemented in the Fortran numerical package \textit{Collier} \cite{Denner:2016kdg}, used to investigate cLFV decays in a two Higgs doublet model (2HDM) \cite{Jurciukonis:2021izn}. Many particular expressions to compute the AMM and/or cLFV decay amplitudes predicted by different particular BSM were constructed \cite{Lindner:2016bgg}. The relations among them can be checked by using suitable transformations, starting from the set of particular PV notations in this work. On the other hand, in a discussion on analytic formulas for one-loop contributions to AMM, a class of fermion-scalar-vector ($FSV$) diagrams consisting of a photon coupling with two different physical particles, namely one scalar and one gauge boson, were considered even in the unitary gauge \cite{Yu:2021suw}. It leads us to a question whether the Ward identity (WI) for the external photon is still valid with the presence of this diagram type. We emphasize that the general results for one-loop contributions to decays $e_b\to e_a \gamma$ and AMM of leptons introduced in many previous works do not include this $FSV$ diagrams. Moreover, they imply the existence of the triple photon coupling with two distinguishable physical particles that has never been mentioned previously.
In particular, many works introducing general one-loop contributions for AMM of charged leptons \cite{Leveille:1977rc, Lindner:2016bgg, Crivellin:2018qmi}, or decays relating with photon such as cLFV decays $e_b\to e_a\gamma$ \cite{Lavoura:2003xp, Lindner:2016bgg, Crivellin:2018qmi}, loop-induced Higgs decays $h\to \gamma \gamma$ \cite{Gunion:1989we, Bunk:2013uea}, $h\to Z\gamma, f\bar{f} \gamma$ \cite{Bunk:2013uea, Hue:2017cph, Phan:2021xwc, VanOn:2021myp}, quark decays $q\to q'\gamma$, $\dots$. Excluding the $FSV$ vertex type will reduce a huge number of related one- and two-loop diagrams as well as confirm the validation of general one-loop calculation introduced previously.
In this work, we will show precisely the important steps to derive the one-loop contributions to both AMM and cLFV decays. The calculation is performed by hand, which is consistent with another cross-checking using FORM package \cite{Vermaseren:2000nd}. The final formulas are expressed exactly in terms of the PV functions defined by LoopTools. The results are then easily to change into all of the other available forms using suitable transformations. The convention of the PV-functions are very convenient to derive the exact formulas before solving particular pure mathematical problems. We also determine contributions arising from a new form of photon coupling with vector bosons such as leptoquarks and confirm the consistence between our results and those introduced in Refs. \cite{Bunk:2013uea, Barbieri:2015yvd, Biggio:2016wyy}.
Our paper is organized as follows. Section \ref{intro} explains our aim of this work. Section \ref{sec_formulas} introduces notations and important formulas to establish the relations between AMM and cLFV amplitudes. Section \ref{sec_discuss} shows discussions to confirm the consistence of our results and previous works, and the validation of the WI for the relevant analyitic formulas. Section \ref{sec_conclusion} summarizes main features of our work. Finally, we provide many appendices showing precisely many intermediate steps and notations to derive the final results mentioned in this work, including the analytic formulas of the PV functions consistent with LoopTools given in appendix~\ref{app_PVLT}.
\section{ \label{sec_formulas} General amplitudes and notations}
It is well-known that analytic formulas of one-loop contributions to the cLFV amplitudes $e_b(p_2)\rightarrow e_a(p_1)\gamma(q)$ and AMM of SM charged leptons $e_a$ can be presented in the same expressions, see for example Ref. \cite{Crivellin:2018qmi} corresponding to the presence of new heavy particles in BSM. Possible one-loop Feynman diagrams contributing to $a_{e_a}$ and cLFV decay amplitudes $e_b\to e_a \gamma$ in BSM are shown in Fig.~\ref{fig_eab}, where $F$ is a fermion coupling with the SM charged lepton $e_a=e,\mu,\tau$; and the boson $B=h, V$ is a scalar or gauge boson, respectively.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{ebaOneLoop}
\caption{Feynman diagrams for one-loop contribution to $a_{e_a}$ and cLFV amplitudes $e_b\to e_a \gamma$ in the unitary gauge.
\label{fig_eab}}
\end{figure}
We note here that Ref. \cite{Yu:2021suw} argues another type of $FSV$ one-loop diagrams giving new contributions to the AMM. They will be discussed in details in this work.
Firstly, we adopt the Lagrangian generating one-loop diagrams in Fig. \ref{fig_eab}, namely \cite{Crivellin:2018qmi}
\begin{align}
\mathcal{L}_h &=\overline{F}(g_{a, Fh}^{L}P_L +g_{a,Fh}^{R}P_R) e_a h +\mathrm{h.c.}, \label{eq_LFh}
\\
\mathcal{L}_V& =\overline{F}\gamma^{\mu} ( g_{a,FV}^{L}P_L +g_{a, FV}^{R} P_R)e_aV_{\mu} +\mathrm{h.c.} \label{eq_LFV},
\end{align}
where the fermion $F$ and the boson $B=V_{\mu},h$ have electric charges $Q_F$ and $Q_B$, and masses $m_F$ and $m_B$, respectively. These Lagrangians \eqref{eq_LFh} and \eqref{eq_LFV} are consistent with those in Ref. \cite{Lavoura:2003xp}. Moreover, the photon couplings with all physical particles should be mentioned clearly, as given in Ref. \cite{Lavoura:2003xp}, i.e., we will adopt the Feynman rules that the photon alway couples with two identical physical particles, as given in table \ref{t_AXX},
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Vertex & Coupling & Vertex & Couplings &Vertex & Couplings\\
\hline
$A^{\mu}(p_0)V^{\nu}(p_+)V^{*\lambda}(p_-)$&$-ieQ_V\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-) $&$A^{\mu}h(p_+)h^*(p_-)$&$ ieQ_h(p_+-p_-)_{\mu}$ & $A^{\mu}\overline{F}F$& $ieQ_F\gamma_{\mu}$\\
\hline
\end{tabular}
\caption{Feynman rules for cubic couplings of photon $A^{\mu}$, where $p_{0,\pm}$ are incoming momenta into the relevant vertex.
\label{t_AXX}}
\end{table}
where $\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-) = g_{\mu\nu} (p_0 -p_+)_{\lambda} +g_{\nu \lambda} (p_+ -p_-)_{\mu} +g_{ \lambda \mu} (p_- -p_0)_{\nu}$ is the standard form. The more general form of $\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-)$ introduced in Refs. \cite{Bunk:2013uea, Barbieri:2015yvd, Biggio:2016wyy} will be discussed in details
later.
All couplings listed in Lagrangians \eqref{eq_LFh}, \eqref{eq_LFV}, and table \ref{t_AXX} result in the following form factors relevant with one-loop contributions:
\begin{align} \label{eq_Boson}
c_{R B}^{ab}= &\dfrac{e}{16\pi^2}g_{a,FB}^{L*} g_{b,FB}^{R} m_F \times \dfrac{f_B(x_B) +Q_Fg_B(x_B)}{m_B^2}
%
\nonumber \\
&+ \dfrac{e}{16\pi^2}\left(m_{b} g_{a,FB}^{L*} g_{b, FB}^{L} +m_{a} g_{a,FB}^{R*} g_{b, FB}^{R}\right) \times \dfrac{\tilde{f}_B (x_B) +Q_F \tilde{g}_B(x_B)}{m_B^2},
\end{align}
where $x_B \equiv m_F^2/m_B^2$. The four scalar functions $f_B(x)$, $g_B(x)$, $\tilde{f}_B(x)$, and $\tilde{g}_B(x)$ are listed in Eq. \eqref{eq_fgx} of appendix \ref{app_PVLT}, as the approximate formulas in the limit $m_a,m_b\ll m_B$. The formula in Eq. \eqref{eq_Boson} does not contain contributions from the $FSV$ diagrams mentioned in Ref. \cite{Yu:2021suw}, because of the absence of photon coupling $AVh$. The corresponding formulas of AMM and cLFV decay rates are:
\begin{align}
a_{e_a} &\equiv -\dfrac{2m_{a}}{e}\left(c^{aa}_R + c^{aa*}_R\right) = -\dfrac{4m_{a}}{e}\mathrm{Re}[ c^{aa}_R], \label{eq_aea1}
\\ \mathrm{Br} (e_b\to e_a \gamma)&= \frac{m^3_b}{4\pi \Gamma_b}\left( \left|c^{ab}_R\right|^2 + \left|c^{ba}_R\right|^2\right),
\end{align}
where $m_a$, $m_b$, and $\Gamma_b$ are the masses and total decay width of the leptons $e_a$, $e_b$, and
\begin{align}
\label{eq_cabR}
c^{ab}_R &\equiv \sum_{B,F} c^{ab}_{RB}.
\end{align}
The amplitude for a vertex $\bar{e}_a e_aA_{\mu}$ in Ref. \cite{Peskin:1995ev} is consistent with the following form presenting both AMM and cLFV amplitudes \cite{Escribano:1996wp,Eidelman:2016aih}
\begin{equation}\label{eq_eeAeff}
i\mathcal{M}=-ie \overline{u_a}(p_1)\left[ \gamma^{\mu}F_1 -\frac{\sigma^{\mu\nu}q_{\nu}}{2m_{a}} \left(iF_2 + \gamma^5 F_3\right)\right]u_b(p_2)\varepsilon^*_{\mu},
\end{equation}
where $\sigma^{\mu\nu}\equiv \frac{i}{2}\left[ \gamma^{\mu} \gamma^{\nu} - \gamma^{\nu} \gamma^{\mu}\right]$; $F_{1,2,3}$ are scalar form factors; $\varepsilon^*_{\mu}$ and $q_{\nu}$ is the polarized vector of the external photon. The form factor $F_{2,3}$ gets contribution only from loop corrections. They relate with the well-known experimental quantities called the anomalous magnetic moment $a_{e_a}$ and electric dipole moment $d_{e_a}$ for $b=a$, respectively. Specifically we have
\begin{equation}\label{eq_ga}
F_{1}=1;\quad a_{e_a}=F_2; \quad d_{e_a}=-\frac{e}{2m_{a}}F_3.
\end{equation}
Regarding to the LFV decay $ e_b\rightarrow e_a \gamma$ the amplitude can also be written in the same form \cite{Cheng:1984vwu, Lavoura:2003xp}, suggesting that $F_2$ can be calculated based on the one-loop corrections to LFV decays. In particular, the second term of the amplitude \eqref{eq_eeAeff} can be expanded as follows \cite{Hue:2017lak}
\begin{align}
\mathcal{M}&=(2p_1.\varepsilon^*)\overline{u_a} \left( C_{(ab)L} P_L +C_{(ab)R} P_R\right)u_b
+\overline{u_a} \left[D_{(ab)L} \slashed{\varepsilon}^* P_L +D_{(ab)R} \slashed{\varepsilon}^* P_R\right]u_b,
\end{align}
where $m_{a}=m_{b}$ and we can prove that $C_{(ab)L}P_L +C_{(ab)R} P_R =\frac{e}{2m_{a}}(F_2 -i\gamma^5 F_3)$. The WI for the external photon gives
\begin{equation}\label{eq_DLR}
D_{(ab)L}= -(m_{b}C_{(ab)R} +m_{a}C_{(ab)L}), \; D_{(ab)R} = -(m_{b} C_{(ab)L} +m_{a}C_{(ab)R}).
\end{equation}
The hermiticity that $C_{(aa)R}=C^*_{(aa)L}$ \cite{Eidelman:2016aih} gives
\begin{align}
\label{eq_FCLR}
a_{e_a}&= \frac{m_{a}(C_{(aa)L} + C_{(aa)R})}{e} =\frac{2m_{a} \mathrm{Re}{[C_{(aa)L,R}]} }{e},\nonumber \\
%
d_{e_a}&=i(C_{(aa)R} -C_{(aa)L})=\mathrm{Im}{[C_{(aa)L}]}= -\mathrm{Im}{[C_{(aa)R}]}.
\end{align}
Hence, the following relations between two different notations must be satisfied:
\begin{equation}\label{eq_relationCbaR}
c^{ab}_{R}= -\frac{1}{2} C_{(ab)R} \; \mathrm{and} \; c^{ba}_{R}= -\frac{1}{2} C_{(ab)L}.
\end{equation}
From the above discussion, we see that one-loop contributions to the $a_{e_a}$ and $d_{e_a}$ can be written in terms of well-known PV functions, see detailed discussions in Ref. \cite{Hue:2017lak} or general formula introduced for calculations the LFV decay rates of charged leptons \cite{Lavoura:2003xp}, with the identification that $\sigma_{L,R}\equiv -C_{(ab)L,R}$. In the limit of $0\simeq m_{a},m_{b}\ll m_B$, the numerical values of $a_{e_a}$ can be evaluated using the numerical packages such as LoopTools \cite{Hahn:1998yk} or Collier \cite{Denner:2016kdg}. Although the exact analytic formulas of one-loop three point functions presented in Ref.~\cite{Hue:2017lak} can not be applied to calculate $a_{e_a}$, but the limit of $m_b\to m_a$ can be used to solve this problem. The analytic formulas
of $a_{e_a}$ were introduced completely in Ref. \cite{Yu:2021suw}.
Because of the relations in Eq. \eqref{eq_DLR}, only $C_{(ab)L,R}$ is needed to determine $a_{e_a}$ and Br$(e_b\to e_a \gamma)$. Because all two-point diagrams give contributions to just $D_{(ab)L,R}$, $C_{(ab)L,R}$ are calculated by considering only three-point diagrams. In this work, the analytic formulas of $D_{(ab)L,R}$ will be determined directly from all diagrams in Fig. \ref{fig_eab} to check the validation of the WI in the presence of the $FSV$.
The analytic formulas for one-loop contributions to the cLFV decay amplitudes presented
in this work are more general than the results introduced in Ref.~\cite{Hue:2017lak} for general 3-3-1 models. Many important steps in our calculations were shown in appendix~\ref{app_detailedStep}. Using this unitary gauge, the assumption for a particular form of the Goldstone boson couplings given in Ref. \cite{Lavoura:2003xp} is unnecessary. In contrast, we use the same photon couplings to other physical particles in an arbitrary BSM, as given in table \ref{t_AXX}. Namely, a tree-level photon coupling always contains two identical physical particles. This implies that the contributions from the $FSV$ diagrams are not included.
Using the notations of PV-functions defined in appendix \ref{app_PVLT}, the $Fhh$ contributions from diagram (1) in Fig. \ref{fig_eab} are:
\begin{align}\label{eq_C_Lfhh}
C^{Fhh}_{(ab)L} =& \frac{-eQ_h}{16\pi^2} \left[ m_{a}g_{a, Fh}^{L*} g_{b, Fh}^{L} X_1^{f}
%
+ m_{b}g_{a, Fh}^{R*} g_{b,Fh}^{R} X_2^{f} -m_F g_{a,Fh}^{R*} g_{b,Fh}^L X_0^{f} \right],
%
\nonumber \\ C^{Fhh}_{(ab)R} =& \frac{-eQ_h}{16\pi^2}\left[m_{a}g_{a,Fh}^{R*} g_{Fh}^{bR}X_1^{f}
%
+ m_{b}g_{a, Fh}^{L*} g_{b,Fh}^{L}X_2^{f} -m_F g_{a, Fh}^{L*}g_{b, Fh}^{R}X_0^{f}\right],
%
\end{align}
where $X^f_{0},X^f_1,\dots$ are linear combinations of the PV-functions $C_{0, 00, i, ij}$ defined precisely in appendix \ref{app_PVLT}.
The diagram (2) in Fig. \ref{fig_eab} gives $hFF$ contributions as follows:
\begin{align}
\label{eq_CLhff}
C^{hFF}_{(ab)L} =& \frac{-eQ_F}{16\pi^2}\left[ m_{a}g_{a,Fh}^{L*} g_{b,Fh}^{bL} X_1^{h}
%
+ m_{b}g_{a,Fh}^{R*} g_{b,Fh}^{R}X_2^{h} +m_F g_{a,Fh}^{R*} g_{b,Fh}^L X_3^{h} \right],
%
\nonumber \\ C^{hFF}_{(ab)R} =& \frac{-eQ_F}{16\pi^2} \left[m_{a}g_{a,Fh}^{R*} g_{b,Fh}^{R}X_1^{h}
+ m_{b}g_{a,Fh}^{L*} g_{b, Fh}^{L} X_2^{h} +m_F g_{a, Fh}^{L^*}g_{b,Fh}^{R}X_3^{h}\right].
%
\end{align}
where $X^h_{1,2,3}$ are linear combinations of $C_{0,i,ij}(m_h^2, m_F^2, m_F^2)$. The above result are completely consistent with the results introduced in Ref. \cite{Lavoura:2003xp}, except an overall sign and the signs before the PV-functions $\bar{c}_{1,2}$, arising from the different definitions of the external momenta $p_i$ in the denominators of the one-loop integrals. We also give the analytic formulas of $D^{Fhh}_{(ab)L,R}$ and $D^{hFF}_{(ab)L,R}$, used to confirm the WI given in Eq. \eqref{eq_DLR} for the only-scalar contributions. The PV-functions derived from the diagram (2) defined as $X^h_{i}$ are different from $X^f_{i}$ defined for three diagrams (1), (3) and (4). In contrast, the equal functions are denoted as follows:
$$ B^{(i)}_0 \equiv B^{(i)f}_0=B^{(i)h}_0, \;X_0\equiv X_0^f=X_0^h, \; i=1,2. $$
The form factors $D_{(ab)L,R}$ originated from scalar contributions are:
\begin{align}
\label{eq_Hformfactor}
D_{(ab)L}^{Fhh}=& \frac{-eQ_H}{16\pi^2} \left\{g^{L*}_{a,Fh}g^{L}_{b,Fh} \times 2 C^f_{00} \right\}
\nonumber \\ &
+ \frac{-eQ_e}{16\pi^2(m_a^2 -m_b^2)} \left\{ \left(m_b g^{L*}_{a,Fh}g^{R}_{b,Fh} + m_a g^{R*}_{a,Fh} g^{L}_{b,Fh} \right)m_F \left( B^{(1)}_0 -B^{(2)}_0 \right)
\right. \nonumber \\ &- \left. g^{L*}_{a,Fh} g^{L}_{b,Fh}\left( m^2_a B^{(1)f}_1- m^2_b B^{(2)f}_1 \right) - m_a m_b g^{R*}_{a,Fh} g^{R}_{b,Fh} \left( B^{(1)f}_1- B^{(2)f}_1\right) \right\},
\nonumber \\ D_{(ab)R}^{Fhh} =& D_{(ab)L}^{FHH} \left[ g^L_{a,Fh} \leftrightarrow g^R_{a,Fh}, g^L_{b,Fh} \leftrightarrow g^R_{b,Fh} \right],
\nonumber \\ D_{(ab)L}^{hFF} =& -\frac{eQ_F}{16 \pi^2} \left\{ g^{L*}_{a,Fh} g^{L}_{b,Fh} \left[m_F^2 C^h_0 +(2-d) C^h_{00} -m_a^2X_1^h - m_b^2 X_2^h \right]
\right.\nonumber \\&\left. + g^{R*}_{a,Fh} g^{R}_{b,Fh} m_a m_bX_0 + \left[ g^{R*}_{a,Fh} g^{L}_{b,Fh} m_a + g^{L*}_{a,Fh} g^{R}_{b,Fh} m_b\right] m_F C^h_0 \right\},
\nonumber \\ D_{(ab)R}^{hFF} =& D_{(ab)L}^{hFF}\left[ g^L_{a,Fh} \leftrightarrow g^R_{a,Fh}, g^L_{b,Fh} \leftrightarrow g^R_{b,Fh} \right].
\end{align}
It is noted that the $Fhh$ contributions are the sum of three diagrams (1), (3), and (4), while the $hFF$ contributions are from the only diagram (2). We emphasize that the electric charge conversation $Q_F=Q_h+Q_e$ is one of the necessary requirements to guarantee the WI given in Eq. \eqref{eq_DLR}, see a detailed proof in appendix \ref{app_detailedStep}. We can see this crudely from the necessary condition that div$[D^{hFF}_{(ab)L}]+ \mathrm{div}[D^{Fhh}_{(ab)L}]\sim g_{a}^{L*}g^L_b (Q_e+Q_h-Q_F)=0$ and div$[D^{hFF}_{(ab)R}]+ \mathrm{div}[D^{Fhh}_{(ab)R}] \sim g_{a}^{R*}g^R_b(Q_e+Q_h-Q_F)=0$. This conclusion supports completely the only case of electric conversation among the remaining ones mentioned in Ref. \cite{Lavoura:2003xp}.
Regarding Lagrangian \eqref{eq_LFV}, which results in four diagrams in the second line of Fig. \ref{fig_eab}, diagram (5) gives the following $FVV$ contributions:
\begin{align}
\label{eq_CLFVV}
C_{(ab)L}^{FVV}
= -\frac{eQ_V}{16\pi^2} &
\left\{ g_{a,FV}^{R*}g_{b,FV}^{L} m_F \left[ 3X_3^f +\frac{1}{2m_V^2} \right]
%
- g_{a,FV}^{L*}g_{b, FV}^{R} m_F\times \frac{ m_{a}m_{b}}{ m_V^2} X^f_{012}
\right. \nonumber \\ & + g_{a, FV}^{L*}g_{b,FV}^{L}m_{a} \left[2 (X^f_1 -X^f_3) + \frac{m_F^2 X^f_{01} +m_{b}^2X^f_2}{m_V^2} \right]
%
\nonumber \\ & \left. + g_{a, FV}^{R*}g_{b, FV}^{R}m_{b} \left[ 2 (X^f_2 -X^f_3) + \frac{m_F^2 X^f_{02} +m_{a}^2X^f_1}{m_V^2} \right] \right\},
\end{align}
where $X^{f}_{i}=X_{i}(m_F^2, m_V^2, m_V^2)$, and
\begin{align}
\label{eq_CRFVV}
C_{(ab)R}^{FVV}
= -\frac{eQ_V}{16\pi^2} &
\left\{ g_{a, FV}^{L*}g_{b, FV}^{R} m_F \left[ 3X^f_3 +\frac{1}{2m_V^2} \right] - g_{a, FV}^{R*}g_{b, FV}^{L} m_F\times \frac{ m_{a}m_{b}}{ m_V^2} X^f_{012}
\right. \nonumber \\ & +g_{a, FV}^{R*}g_{b, FV}^{R}m_{a} \left[ 2 (X^f_1 -X^f_3) +\frac{m_F^2 X^f_{01}+ m_{b}^2 X^f_2}{m_V^2} \right]
%
\nonumber \\ & \left. +g_{a, FV}^{L*}g_{b,FV}^{L}m_{b} \left[2 (X^f_2 -X^f_3) +\frac{m_F^2X^f_{02}+ m_{a}^2 X^f_1}{m_V^2} \right] \right\}.
\end{align}
Diagram (6) gives $VFF$ contributions:
\begin{align}
\label{eq_VFFCL}
C_{(ab)L}^{VFF}= -\frac{eQ_F}{16\pi^2} &\left\{ m_{a} g_{a, FV}^{L*} g_{b, FV}^{L} \left[ \frac{}{} 2 X^v_{01}
+\frac{m_F^2\left( X^v_1 -X^v_3\right)+ m_{b}^2X^v_2 }{m_V^2} \right]
%
\right. \nonumber \\&\; + m_{b}g_{a, FV}^{R*} g_{b,FV}^{R} \left[ \frac{}{}2 X^v_{02}
%
+\frac{m_F^2\left( X^v_2 -X^v_3\right)+ m_{a}^2 X^v_1}{m_V^2} \right]
%
\nonumber \\&\; - g_{a,FV}^{R*} g_{b,FV}^{L} m_F \left[ 4 X_0
%
+ \frac{m_{a}^2X^v_1 + m_{b}^2X^v_2- m^2_{F}X^v_3}{m_V^2} \right]
%
\nonumber \\&\left. \;-g_{a, FV}^{L*}g_{b, FV}^{R} \frac{m_{a} m_{b}}{m_V^2}\times m_F (X^v_{12} -X^v_3) \right\},
%
\end{align}
where all $X_{i}^v$ are expressed in terms of PV functions $C^{VFF}_{0,i,ij}=C_{0,i,ij}(m_V^2, m_F^2, m_F^2)$, and
\begin{align}
\label{eq_VFFCR}
C_{(ab)R}^{VFF}= -\frac{eQ_F}{16\pi^2} &\left\{ m_{a} g_{a, FV}^{R*} g_{b,FV}^{R} \left[ 2 X^v_{01}
+\frac{m_F^2\left( X^v_1 -X^v_3\right)+ m_{b}^2X^v_2}{m_V^2} \right]
%
\right. \nonumber \\&\; + m_{b}g_{a, FV}^{L*} g_{b, FV}^{L} \left[ 2 X^v_{02}
%
+\frac{m_F^2\left( X^v_2 -X^v_3\right)+ m_{a}^2X^v_1}{m_V^2} \right]
%
\nonumber \\&\; - g_{a, FV}^{L*} g_{b, FV}^{R} m_F \left[ 4 X^v_0
%
+ \frac{m_{a}^2X^v_1 + m_{b}^2X^v_2 - m^2_{F}X^v_3}{m_V^2} \right]
%
\nonumber \\&\left. \;-g_{a,FV}^{R*}g_{b, FV}^{L} \frac{m_{b} m_{a}}{m_V^2}\times m_F (X^v_{12} -X^v_3) \right\}.
%
\end{align}
Finally, using the simple notations $g^{L,R}_a\equiv g^{L,R}_{a,FV}$, the formulas of $D_{(ab)L}$ and $D_{(ab)R}$ are
\begin{align}
D^{(78)}_{(ab)L}=& D^{(7)}_{(ab)L}+ D^{(8)}_{(ab)L}
\nonumber \\ = & \frac{e Q_e}{ 16\pi^2 (m_a^2 -m_b^2)} \left\{ \frac{}{} \left( g^{L*}_{a} g^{R}_{b} m_b + g^{R*}_{a} g^{L}_{b} m_a\right) 3m_F \left[ B^{(1)}_0 -B^{(2)}_0\right]
\right. \nonumber \\ &- \left. m_b \left( m_a g^{R*}_{a} g^{R}_{b} + m_b g^{L*}_{a} g^{L}_{b}\right) \left[ \left(2+ \frac{m_F^2 +m_b^2}{m_V^2}\right) B^{(2)v}_1 +\frac{A_0(m_V^2) +2m_F^2B^{(1)}_0}{m_V^2} +1 \right]
\right. \nonumber \\ &+\left. m_a \left( m_b g^{R*}_{a} g^{R}_{b} + m_a g^{L*}_{a} g^{L}_{b}\right) \left[ \left(2+ \frac{m_F^2 +m_a^2}{m_V^2}\right) B^{(1) v}_1 +\frac{A_0(m_V^2) +2m_F^2B^{(2)}_0}{m_V^2} +1 \right]\right\}, \label{eq_DabLR78}
\\ D^{(78)}_{(ab)R}=& D^{(78)}_{(ab)L} \left[ g^L_{a} \leftrightarrow g^R_{a}, \; g^L_{b} \leftrightarrow g^R_{b}\right].
\nonumber \\ D_{(ab)L}^{FVV}=& -\frac{eQ_V}{16\pi^2} \left\{\frac{}{} g_{a}^{L*} g_{b}^{L} \left[\frac{}{} 2(d-2)C^f_{00} +2(m_a^2+m_b^2)X^f_3
\right.\right.\nonumber \\&\left. \left. -\frac{1}{m_V^2} \left( m_F^2(B^{(1)}_0 +B^{(2)}_0 -2C^f_{00} ) +A_0(m_V^2) +m_a^2 B^{(1) f}_1+m_b^2 B^{(2) f}_1 \right)\right]
\right. \nonumber \\&\; + g_{a}^{R*} g_{b}^{R} m_a m_{b} \left[4X^f_3 +\frac{2 C^f_{00}}{m_V^2} \right]
+ g_{a}^{R*} g_{b}^{L} \times m_a m_F \left[ 3C^f_0 -\frac{1}{2m_V^2} + \frac{m_b^2 X^f_{012}}{m_V^2}\right]
\nonumber \\&\left. \; +g_{a}^{L*}g_{b}^{R} \times m_b m_F \left[ 3C^f_0 -\frac{1}{2m_V^2} + \frac{m_a^2 X^f_{012}}{m_V^2}\right] \right\},
\nonumber \\ D_{(ab)R}^{FVV}=& C_{(ab)L}^{FVV} \left[ g^{L}_{a} \leftrightarrow g^{R}_{a}, g^{L}_{b} \leftrightarrow g^{R}_{a}\right] \label{eq_FVVDLhand}.
\end{align}
The remaining formulas of $D_{(ab)L,R}$ from diagram (6) of Fig. \ref{fig_eab} are
\begin{align}
\label{eq_DLR6}
D^{VFF}_{(ab)L}=& \frac{eQ_F}{16\pi^2} \left\{ g^{L*}_{a} g^{L}_{b} \left[\frac{}{} -2m_F^2 C_{0} +(d-2)^2 C_{00}^v +2m_a^2 X^v_{01}+2m_b^2 X^v_{02}
\right.\right.\nonumber \\&\left.\left. \qquad \qquad \qquad \quad -\frac{1}{m_V^2} \left[(2-d)m_F^2 C^v_{00} +A_0(m_V^2) +m_F^2 \left( B^{(1)}_0 +B^{(2)}_0 \right)
\right.\right. \right. \nonumber \\&\left.\left. \left. \qquad \qquad \qquad \qquad \qquad - m_a^2\left(B^{(1) v}_0 +B^{(1)}_1 \right) -m_b^2 \left( B^{(2) v}_0 +B^{(2)v}_1\right) +m_a^2m_b^2 X_0
\right.\right. \right. \nonumber \\&\left.\left. \left. \qquad \qquad \qquad \qquad \qquad
-m_F^2 \left( (m_a^2 +m_b^2-m_F^2)C_0 +m_a^2 X^v_1 +m_b^2X^v_2\right) \frac{}{} \right] \right]
\right.\nonumber \\&\left. \qquad \quad + g^{R*}_ag^{R}_b m_am_b \left[ 2X_0 -\frac{1}{m_V^2}\left((2-d) C^v_{00} +m_F^2 X^v_3 -m_a^2 X^v_1 -m_b^2 X^v_2 \right)\right]
\right.\nonumber \\ &\left.\qquad \quad + \frac{g^{R*}_ag^{L}_b m_am_F}{m_V^2} \left[ -B^{(1)v}_1 +(2-d)C_{00} -m_a^2 X^v_1 +m_b^2(X^v_3 -X^v_2) \right]
\right.\nonumber \\ &\left.\qquad \quad + \frac{g^{L*}_ag^{R}_b m_bm_F}{m_V^2} \left[ -B^{(2) v}_1 +(2-d)C^v_{00} -m_b^2 X^v_2 +m_a^2 (X^v_3 -X^v_1) \right] \right\},
\nonumber \\ D^{VFF}_{(ab)R}= & D^{VFF}_{(ab)L} \left[ g_a^L\leftrightarrow g_a^R,\; g_b^L\leftrightarrow g_b^R\right].
\end{align}
We note that all results presented here are crosschecked by FORM package \cite{Vermaseren:2000nd}, using intermediate steps given in appendix \ref{app_detailedStep}. There is a property that $C^X_{(ab)R}=C^X_{(ab)L}\left[ g_a^L\leftrightarrow g_a^R,\; g_b^L\leftrightarrow g_b^R\right]$ for all $X=Fhh,hFF,FVV,VFF$. The above results of one-loop contribution to $C_{(ab)L,R}$ are totally consistent with those introduced in Ref. \cite{Lavoura:2003xp}, after some
transformations of notations presented in appendix \ref{app_special}. In the limit of $m_h^2,m_V^2\gg m^2_a,m^2_b$, i.e., $m^2_a/m_B^2,m^2_b/m_h^2\simeq 0$ with $B=h,V$, we get consistent results with those given in Refs. \cite{Crivellin:2018qmi, Freitas:2014pua, Stockinger:2006zn}. To derive the above results for gauge boson exchanges, we start with many important features different from those mentioned in Ref. \cite{Lavoura:2003xp}, namely: i) we do
not use the typical form of couplings relating with Goldstone bosons going along with the presence of new gauge bosons, ii) we have to use the massless property of the on-shell photon $q^2=0$, iii) to confirm the WI for all diagrams given in Fig. \ref{fig_eab}, we need the charge conversation law corresponding to the Lagrangian \eqref{fig_eab}: $Q_F=Q_V +Q_e$. Therefore, our calculation is another independent approach to confirm the result given in Ref. \cite{Lavoura:2003xp}. The details of the calculation to confirm the WI for all one-loop contributions are given in appendix \ref{app_detailedStep}. We remind that our results are derived from the photon couplings listed in the table \ref{t_AXX}, and do not contain the contributions from the FSV diagrams. In the following, we pay attention to the possibility of adding the FSV diagrams or the new forms of the photon couplings.
\section{\label{sec_discuss} Discussion on WI and previous results}
\subsection{WI to constrain the form of photon couplings}
Now we focus on the feature that the WI of the on-shell photon will constrain strongly the forms of the cubic photon couplings with two physical particles in a renormalized Lagrangian. Now we consider the existence of the photon couplings type at tree level:
\begin{align}
\label{eq_AX12}
\mathcal{L}^{\gamma XX} =& eQ_FA^{\mu}\left[ \overline{F_1}\gamma^{\mu}F_2 +\mathrm{h.c.} \right] + eQ_hA^{\mu} \left[ \left( h_1^* \partial_{\mu}h_2 -h_2 \partial_{\mu}h_1^*\right) +\mathrm{h.c.}\right]
%
\nonumber \\ & -\left[ eQ_VA^{\mu}V^{\nu}_1 V^{\lambda*}_2 \Gamma_{\mu \nu \lambda}(p_0,p_+p_-) + \mathrm{h.c.}\right] +\left[g_{\gamma hV} g_{\mu \nu}h^{-Q} A^{\mu}V^{Q\nu}+ \mathrm{h.c.}\right],
\end{align}
where all couplings are more general than those well-known as the standard forms given in Table \ref{t_AXX}. The last term corresponds to the photon couplings with a scalar $h$ and a gauge boson $V$. The above Lagrangian results in the following decays from the heavy particle to lighter one: i) $F_2\to F_1 \gamma$, ii) $h_2\to h_1 \gamma$, iii) $V_2\to V_1 \gamma$, and iv) $V\to h\gamma$. The WI for these decay amplitudes at tree level is $\mathcal{M}^{\mu}(X_1\to X_2 \gamma)p_{0\mu}=0$ with $p_{0\mu}$ being the external photon momentum. It can be derived that:
\begin{itemize}
\item Using the same convention of external momenta given in Fig. \ref{fig_eab}\Binh{,} we have $\mathcal{M}^{\mu}(F_2\to F_1 \gamma)q_{\mu}\sim (m_{F_2}-m_{F_1}) \overline{u}_{F_2}(p_2)u_{F_1}(p_1)=0$, where $p_0\equiv -q$. Therefore, $m_{F_2}=m_{F_1}$. This case is automatically satisfied for the tree level AMM amplitude.
\item $\mathcal{M}^{\mu}(h_2\to h_1 \gamma)p_{0\mu}\sim(p_2-p_1).(p_2+p_1)= (m^2_{h_2}-m^2_{h_1}) =0$, where all on-shell momenta are incoming the vertex $A^{\mu}h^*_1h_2$, implying that $p_0=-(p_1+p_2)$ and $p_{1,2}^2=m^2_{h_{1,2}}$. The consequence is $m_{h_1}=m_{h_2}$.
%
\item $\mathcal{M}^{\mu}(V \to h \gamma)p_{0\mu}\sim\varepsilon_v.p_{0} =0$, where $\varepsilon_v$ and $p_0$ are the polarization of gauge boson $V$ and the external momentum of the photon $A_{\mu}$. Hence the presence of a $AhV$ vertex does not automatically satisfy the WI. One-loop contributions for all diagrams arising from this vertex must be checked for the validation of WI.
%
\item $\mathcal{M}_{\mu}(V_1 \to V_2 \gamma)p^{\mu}_0 \sim\varepsilon^{\nu}_1\varepsilon^{\lambda*}_2p^{\mu}_0 \Gamma_{\mu \nu \lambda}(p_0,p_1,p_2) =0$, where $\varepsilon_{1,2}$, and $p_{1, 2, 0}$ are the polarization of the gauge boson $V_{1,2}$ and the external momentum of the gauge bosons $V_{1,2}$ and photon $A_{\mu}$, respectively. We will use the following properties of the external gauge bosons $V_i (i=1,2)$ and photon: $\varepsilon_i.p_{i}=0$, $p_0^2=0$, $p_i^2=m^2_{V_i}$, and the momentum conversation $p_0 +p_1 +p_2=0$ following notations in table \ref{t_AXX}. After some intermediate steps of calculation, we have:
\begin{align}
\label{eq_WIV12ga}
\mathcal{M}_{\mu}(V_1 \to V_2 \gamma)p^{\mu}_0 \sim& (p_0.\varepsilon_1) \left[ (p_0 -p_1). \varepsilon^*_2\right] + (\varepsilon_1.\varepsilon^*_2) \left[ (p_1 -p_2).p_0 \right] +(p_0.\varepsilon^*_2) \left[ (p_2 -p_0).\varepsilon_1 \right]
%
\nonumber \\ =& (\varepsilon_1.\varepsilon^*_2) \left[ m^2_{V_2} -m^2_{V_1} \right]=0.
\end{align}
Hence, $m_{V_1}= m_{V_2}$ is necessary. From this, we consider the more general photon coupling with a gauge boson \cite{Biggio:2016wyy} describing the couplings of a leptoquark field \cite{Barbieri:2015yvd}
\begin{align}
\label{eq_Gamma1}
\Gamma'_{\mu\nu\lambda}(p_0,p_1,p_2) =& g_{\mu\nu} (k_v p_0 -p_1)_{\lambda} +g_{\nu \lambda} (p_1 -p_2)_{\mu} +g_{ \lambda \mu} (p_2 - k_vp_0)_{\nu}
\nonumber \\ =&\Gamma_{\mu\nu\lambda}(p_0,p_1,p_2) +\delta k_v \left(g_{\mu \nu} p_{0\lambda} -g_{\lambda \mu} p_{0\nu}\right),
\end{align}
with $\delta k_v=k_v-1$ showing the deviation from the standard vertex listed in table \ref{t_AXX}. This may change the one-loop contributions of the diagram (5) in Fig. \ref{fig_eab}, hence change the formulas of $C^{FVV}_{(ab)L,R}$ given in Eqs. \eqref{eq_CLFVV} and \eqref{eq_CRFVV}, respectively. One can prove immediately that the vertex deviation
\begin{align}
\label{eq_dGamma}
\delta \Gamma_{\mu\nu\lambda}(p_0,p_1,p_2)\equiv \Gamma'_{\mu\nu\lambda}(p_0,p_1,p_2) -\Gamma_{\mu\nu\lambda}(p_0,p_1,p_2)= \delta k_v \left(g_{\mu \nu} p_{0\lambda} -g_{\lambda \mu} p_{0\nu}\right)
\end{align}
guarantees the WI. The new one-loop contributions arising from $\delta \Gamma$ are also satisfied the WI, see analytic formulas given in Eq. \eqref{eq_dCLR}.
\end{itemize}
Now we start from the point that all results of one loop contributions given from Eq. \eqref{eq_C_Lfhh} to Eq. \eqref{eq_DLR6} based on the standard forms of photon couplings given in table \ref{t_AXX}, where a photon always couples with two identical physical fields. On the other hand, a recent work \cite{Yu:2021suw} assumed the existence of a new photon coupling kind $ASV$, which may appear in some BSM, in which the photon couples with one gauge boson $V$ and one scalar $S$. The appearance of a boson $V$ or $S$ will generate by itself the one-loop contributions that always guarantee the WI by the respective set of four diagrams given in Fig. \ref{fig_eab}. Hence, the two FSV diagrams must give contributions satisfying the WI themselves, namely
\begin{align}
\label{eqFSVWi}
D^{FSV}_{(ab)_L} +m_a C^{FSV}_{(ab)_L} +m_b C^{FSV}_{(ab)_R}= D^{FSV}_{(ab)_R} +m_a C^{FSV}_{(ab)_R} +m_b C^{FSV}_{(ab)_L}=0.
\end{align}
As a result, the divergent parts for both $L$ and $R$ parts give:
\begin{align}
%
\label{eq_divFSV}
0=& g_{\gamma SV} \left[ 2g^{L*}_{ah}g^{L}_{bV} m_F -g^{L*}_{ah} g^{R}_{bV} m_b -g^{R*}_{ah}g^{L}_{bV} m_a \right]
\nonumber \\ =& g_{\gamma SV} \left[ 2g^{R*}_{ah}g^{R}_{bV} m_F -g^{R*}_{ah}g^{L}_{bV} m_b -g^{L*}_{ah}g^{R}_{bV} m_a \right].
\end{align}
Considering the case of $g_{\gamma SV}\neq0$. Then, all quantities $g^L_{ah}, g^R_{bh}$, $g^L_{bV}$, and $g^R_{bV}$ are zero if at least one of them is zero. More strictly, we require that the two Eqs. \eqref{eqFSVWi} must be hold for both divergent and finite parts arising from $D_{(ab)L,R}$ and $C_{(ab)L,R}$ given in appendix \ref{app_WiFSV}. Consequently, $g_{\gamma SV}=0$, i.e., the $FSV$ diagram type does not satisfy the WI.
Regarding to the vertex deviation of the $AVV$ couplings defined in Eq. \eqref{eq_dGamma}, the new one-loop contributions relating to $C^{FVV}_{(ab)L,R}$ and $D^{FVV}_{(ab)L,R}$ are shown in Eq. \eqref{eq_dCLR} of appendix \ref{app_detailedStep}. Our results are consistent with previous works \cite{Barbieri:2015yvd, Biggio:2016wyy}. Although they satisfy the WI, they contain divergent parts, for example
\begin{align}
\label{eq_divdCLR}
\mathrm{div}\left[ -\delta C^{FVV}_{L}\right] =& \frac{\delta k_v eQ_V}{32\pi^2 m_V^2} \left[g^{L*}_a g^{L}_bm_a + g^{R*}_a g^{R}_b m_b -2g^{R*}_a g^{L}_b m_F \right].
\end{align}
Hence, $\delta k_v=0$ is equivalent to the renormalizable condition of the theory, see a more detailed explanation in Ref. \cite{Biggio:2016wyy}. This confirms that the $AVV$ coupling listed in table \ref{t_AXX} is still valid for a general UV-complete model. Consequently, $\delta C^{FVV}_{L}=0$, implying that the results of $C^{FVV}_{(ab)L,R}$ given in Eqs. \eqref{eq_CLFVV} and \eqref{eq_CRFVV} are unchanged for many renormalizable theories.
\subsection{Discussions on previous results }
It is easy to derive that $C_{(ab)L,R} = \sigma_{L,R}$ corresponding to the notations given in Ref. \cite{Lavoura:2003xp}, see a detailed explanation in appendix \ref{app_special}. This confirms a perfect consistence of the two results obtaining from different original assumptions that we have indicted above. In addition, these results are also consistent with those given in Ref. \cite{Crivellin:2018qmi} in the limit of heavy boson masses in the loops, which are very useful for studying the correlations of AMM and cLFV decays.
In some BSM, SM light quark may play role of the light fermions $u,d\equiv F$ in the Yukawa couplings \cite{Dorsner:2020aaz}, hence the condition $m_F^2\gg m^2_{a},m^2_{b}$ is not held. But numerical illustrations \cite{Hue:2017lak} to investigate cLFV decays $e_b\to e_a \gamma$ with very light neutrinos show that the case of $m^2_{F}\ll m^2_{a}$ are also valid for approximation formulas with $m^2_{a}=m^2_{b}=0$, provided $m^2_{a}, m^2_{b}\ll m^2_h,m^2_{V}$. An analytic approximation to explain this result was given in, for example Ref. \cite{Hue:2015fbb}.
For analytic formulas of cLFV and $a_{e_a}$ introduced in Ref. \cite{Lindner:2016bgg}, They can be changed into the form of PV-functions consistent with our results. An exceptional case mentioned there is the couplings of a double charged boson with two identical leptons. For example, the Lagrangian containing couplings of a doubly charged Higgs boson is \cite{Lindner:2016bgg}:
\begin{align}
\label{eq_phieab0}
\mathcal{L}_{\mathrm{int}}= g^{ij}_{s3}\phi^{++} \overline{\ell^C_i}\ell_j + g^{ij}_{p3}\phi^{++} \gamma^5 \overline{\ell^C_i}\ell_j +\mathrm{h.c.},
\end{align}
where we can identify that $ g^R_{a,Fh}= g^{ij}_{s3} +g^{ij}_{p3}$ and $g^R_{a,Fh}= g^{ij}_{s3} -g^{ij}_{p3}$. But the Feynman rules for the a vertex $\overline{\ell^C_i}\ell_j \phi^{++}$ containing two identical leptons gives an extra factor 2, implying that $C_{(ab)L,R}$ given in Eqs. \eqref{eq_C_Lfhh} and \eqref{eq_CLhff} must be added a factor 4. Instead of many particular formulas to calculate one-loop contributions relating to different charged particles, the one-loop results for $(g-2)_{e_a}$ and $e_b\to e_a \gamma$ decays can be generalized for $a_{e_a}$ with an arbitrary electric charge $Q_F$ of a new fermion and the boson with $Q_B=Q_F-Q_e$ with $B=h,V$. Namely, the $a_{e_a}$ formulas are
\begin{align}
a_{e_a}(h) =& \frac{Q_hm_a }{16\pi^2} \int_0^1 dx \times \frac{x(x-1) \left[ 2\mathrm{Re}[g^{RL}] m_F + (g^{LL} +g^{RR})m_ax\right]}{(1-x) m_F^2 + x\left[ m_h^2 +m_a^2(x-1)\right]}
%
\nonumber \\& +\frac{Q_Fm_a }{16\pi^2}\int_0^1 dx \times \frac{x^2 \left[ -2g^{RL}[g^{RL}] m_F + (g^{LL} +g^{RR}) m_a (x-1) \right]}{(1-x) m_h^2 + x\left[ m_F^2 +m_a^2(x-1)\right]},
\label{eq_amh}
%
\\ a_{e_a}(V) =& -\frac{Q_Vm_a }{16\pi^2 m_V^2}\int_0^1 dx \times \left[ \frac{\mathrm{Re}[g^{RL}]m_F\left[m_F^2(x-1) + m_V^2 x(6x-1) +m_a^2 x(3-5x+2x^2)\right]}{(1-x) m_F^2 + x\left[ m_V^2 +m_a^2(x-1)\right]} \right.
%
\nonumber \\&\left. - \frac{m_a(g^{LL} +g^{RR})\left[m_F^2(2-3x +x^2) + m_V^2 2x(x+1) +m_a^2 x(x-1)\right]}{(1-x) m_F^2 + x\left[ m_V^2 +m_a^2(x-1)\right]} \right]
%
\nonumber \\ &+ \frac{Q_Fm_a }{16\pi^2 m_V^2}\int_0^1 dx\left[ \frac{2g^{RL}[g^{RL}]m_Fx\left[m_F^2x - 4m_V^2 (1-x) +m_a^2 x(2x -1 )\right]}{(1-x) m_V^2 + x\left[ m_F^2 +m_a^2(x-1)\right]} \right.
%
\nonumber \\ &\left.+ \frac{(g^{LL} +g^{RR}) m_a x\left[m_F^2 x(1+x) +2 m_V^2 (2 -3x +x^2) +m_a^2 x(x-1)\right]}{(1-x) m_V^2 + x\left[ m_F^2 +m_a^2(x-1)\right]} \right], \label{eq_amV}
\end{align}
where $g^{RL}=g^{R*}_{a,FB}g^{L}_{a,FB}$, $g^{LL}=g^{L*}_{a,FB}g^{L}_{a,FB}$, and $g^{RR}=g^{R*}_{a,FB}g^{R}_{a,FB}$ with $B=h,V$. The coupling identifications are $ g^R_{a,Fh}= g^{aa}_{s k} +g^{aa}_{pk}$ and $g^R_{a,Fh}= g^{aa}_{sk} -g^{aa}_{pk}$ for $k=1,2,3$ relating to neutral, singly, and doubly charged Higgs bosons. Similarly for the gauge bosons, $ g^R_{a,FV}= g^{aa}_{v k} +g^{aa}_{ak}$ and $g^R_{a,FV}= g^{aa}_{vk} -g^{aa}_{ak}$ for $Q_{V}=1,0,-1,2$ corresponding $k=1,2,3,4$.
The two formulas \eqref{eq_amh} and \eqref{eq_amV} are derived by inserting the PV functions given in appendix \ref{app_PVLT} in the limit $p_1^2=p_2^2=m_a^2$ into $C_{(ab)L,R}$. We have checked that our results are consistent with all $HFF$, $FHH$, and $VFF$ contributions relating to the diagrams (1), (2), and (6) in Fig. \ref{fig_eab}, respectively. For the one-loop $FVV$ contributions arising from the diagram (5), there is a difference between our result and that in Ref. \cite{Lindner:2016bgg}, namely
$$ \delta (a_{e_a})(FVV)= \frac{Q_Vm_a m_F}{16\pi^2 m_V^2} (|g^{aa}_{vk}|^2 - |g^{aa}_{ak}|^2)\int_0^1 dx(2x+1) = \frac{Q_Vm_a m_F}{8\pi^2 m_V^2} (|g^{aa}_{vk}|^2 -|g^{aa}_{ak}|^2). $$
It shows that the two results are consistent if $g^{aa}_{vk}=\pm g^{aa}_{ak}$,i.e., $g^L_{a,FB}g^R_{a,FB}=0$, which appears in many BSM such as the SM, 3-3-1 models,... We also see that the $FVV$ contribution to $a_{e_a}$ of the doubly gauge boson given in Ref. \cite{Lindner:2016bgg} has an opposite sign with our result.
We note that our results are also valid as the exact solutions for studying the AMM and $e_b\to e_a\gamma$ decay in BSM consisting of very light bosons $m_B \ll m^2_{a},m^2_{b}$ such as an axion-like particle (ALP) \cite{Bauer:2019gfk, Cornella:2019uxs}, or a new scalar singlet \cite{Liu:2018xkx}.
\section{\label{sec_conclusion} Conclusion}
Using the unitary gauge, we confirm the exact results of analytic formulas in terms of PV functions for one-loop contributions to the cLFV decay rates $e_b\to e_a\gamma$ given in Ref. \cite{Lavoura:2003xp}, which are also applicable to compute the AMM of charged leptons. These results are consistent with those given in Ref. \cite{Crivellin:2018qmi} in the limit of heavy bosons $m_B\gg m_a,m_b$. The general expressions in terms of PV-functions are very convenient to change into available forms. Our calculations here are in many new feature as follows. Our calculation is independent with the Goldstone boson couplings of the new gauge bosons. The Ward Identity of the external photon constrains allows only the couplings of photon with two identical physical particles, as given in table \ref{t_AXX}. At tree-level, the $ASV$ couplings do not satisfy the WI if $\varepsilon_v.p_{0} \neq 0$, where $\varepsilon_v$ and $p_0$ are the polarization of gauge boson $V$ and the external momentum of the photon, respectively. The one-loop $FSV$ contributions arising from this vertex type to cLFV amplitudes and AMM do violate the WI. Therefore, the results given in Refs. \cite{Lavoura:2003xp, Crivellin:2018qmi} are valid in all renormalizable BSM respecting the WI. They are still applied for other similar decays of quarks $q\to q'\gamma$. The photon-scalar-vector $ASV$ vertex does not appear in BSM satisfying the WI. Our conclusion is very useful for constructing loop calculations relating to photon couplings, where only the vertex types listed in Table \ref{t_AXX} are valid.
\section*{Acknowledgments}
This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 103.01-2019.387. L. T. Hue is thankful to Van Lang University.
\section{\label{intro} Introduction}
\allowdisplaybreaks
The lepton sector is one of the most interesting object for experiments to search for new physics (NP) beyond the prediction of the standard model (SM). For example, the evidence of neutrino oscillation confirms that the SM must be extended. Recently, the experimental data of anomalous magnetic moments (AMM) of charged leptons $(g-2)_{e_a}/2\equiv a_{e_a}$ has been updated, where the deviation between SM prediction and the lasted experiment data for muon is \cite{Muong-2:2021ojo}
\begin{equation}
\label{eq_damu}
\Delta a^{\mathrm{NP}}_{\mu}\equiv a^{\mathrm{exp}}_{\mu} -a^{\mathrm{SM}}_{\mu} =\left(251 \pm 59\right) \times 10^{-11},
\end{equation}
corresponding to the $4.2\sigma$ deviation from standard model (SM) prediction \cite{Aoyama:2020ynm} combined from various contributions \cite{Davier:2010nc, Davier:2017zfy, Keshavarzi:2018mgv, Colangelo:2018mtw, Hoferichter:2019mqg, Davier:2019can, Keshavarzi:2019abf, Kurz:2014wya, Melnikov:2003xd, Masjuan:2017tvw, Colangelo:2017fiz, Hoferichter:2018kwz, Gerardin:2019vio, Bijnens:2019ghy, Colangelo:2019uex, Colangelo:2014qya, Blum:2019ugy, Aoyama:2012wk, Aoyama:2019ryr, Czarnecki:2002nt, Gnendiger:2013pva}.
For the electron anomaly, the deviation between SM and experiment is $1.6\sigma$ discrepancy \cite{Morel:2020dww}.
On the other hand, $\Delta a_{e,\mu}$ are strongly constrained by the experimental data obtained from searching for the charged lepton flavor violating (cLFV) decays $e_b\rightarrow e_a\gamma$ are~\cite{MEG:2016leq, BaBar:2009hkt}:
\begin{align}
\label{eq_ebagaex}
\mathrm{Br}(\tau\rightarrow \mu\gamma)&<4.4\times 10^{-8}, \;
\mathrm{Br}(\tau\rightarrow e\gamma) <3.3\times 10^{-8}, \;
\mathrm{Br}(\mu\rightarrow e\gamma) < 4.2\times 10^{-13}.
\end{align}
This important property was discussed previously, for example see discussions for a general estimation in Ref. \cite{Crivellin:2018qmi}, and many particular models beyond the standard model (BSM) \cite{Lindner:2016bgg, Dorsner:2020aaz, Hue:2021xap, Hue:2021xzl, Hong:2022xjg, Li:2022zap}. General formulas expressing simultaneously both one-loop contributions to AMM and cLFV amplitudes were introduced in the limits of new heavy scalar and/or gauge boson exchanges $m_B^2 \gg m^2_{a}$ with $m_a$ being the mass of a charged lepton $e_a=e,\mu,\tau$ \cite{Crivellin:2018qmi}. Other calculations in the unitary gauge were discussed \cite{Yu:2021suw, Leveille:1977rc} for the one-loop contributions to $a_{e_a}$ with $m_{a}\neq0$, without the relations with the cLFV amplitudes. The analytic one-loop formulas for cLFV amplitudes calculated in the 't Hooft Feynman (HF) gauge were also shown in Ref. \cite{Lavoura:2003xp}, using the notations of the Passarino-Veltman (PV) functions \cite{Passarino:1978jh, tHooft:1978jhc} with $m_{a}\neq m_{b}$. The approximate formulas with $m_{a}=m_{b}=0$ were introduced and consistent with those given in Ref. \cite{Crivellin:2018qmi}, as shown particularly in Ref. \cite{Hue:2017lak} for 3-3-1 models. The general analytic formulas of these PV functions were introduced for numerical investigations. They are consistent with the results generated by LoopTools \cite{Hahn:1998yk}, which can be transformed into other PV notations implemented in the Fortran numerical package \textit{Collier} \cite{Denner:2016kdg}, used to investigate cLFV decays in a two Higgs doublet model (2HDM) \cite{Jurciukonis:2021izn}. Many particular expressions to compute the AMM and/or cLFV decay amplitudes predicted by different particular BSM were constructed \cite{Lindner:2016bgg}. The relations among them can be checked by using suitable transformations, starting from the set of particular PV notations in this work. On the other hand, in a discussion on analytic formulas for one-loop contributions to AMM, a class of fermion-scalar-vector ($FSV$) diagrams consisting of a photon coupling with two different physical particles, namely one scalar and one gauge boson, were considered even in the unitary gauge \cite{Yu:2021suw}. It leads us to a question whether the Ward identity (WI) for the external photon is still valid with the presence of this diagram type. We emphasize that the general results for one-loop contributions to decays $e_b\to e_a \gamma$ and AMM of leptons introduced in many previous works do not include this $FSV$ diagrams. Moreover, they imply the existence of the triple photon coupling with two distinguishable physical particles that has never been mentioned previously.
In particular, many works introducing general one-loop contributions for AMM of charged leptons \cite{Leveille:1977rc, Lindner:2016bgg, Crivellin:2018qmi}, or decays relating with photon such as cLFV decays $e_b\to e_a\gamma$ \cite{Lavoura:2003xp, Lindner:2016bgg, Crivellin:2018qmi}, loop-induced Higgs decays $h\to \gamma \gamma$ \cite{Gunion:1989we, Bunk:2013uea}, $h\to Z\gamma, f\bar{f} \gamma$ \cite{Bunk:2013uea, Hue:2017cph, Phan:2021xwc, VanOn:2021myp}, quark decays $q\to q'\gamma$, $\dots$. Excluding the $FSV$ vertex type will reduce a huge number of related one- and two-loop diagrams as well as confirm the validation of general one-loop calculation introduced previously.
In this work, we will show precisely the important steps to derive the one-loop contributions to both AMM and cLFV decays. The calculation is performed by hand, which is consistent with another cross-checking using FORM package \cite{Vermaseren:2000nd}. The final formulas are expressed exactly in terms of the PV functions defined by LoopTools. The results are then easily to change into all of the other available forms using suitable transformations. The convention of the PV-functions are very convenient to derive the exact formulas before solving particular pure mathematical problems. We also determine contributions arising from a new form of photon coupling with vector bosons such as leptoquarks and confirm the consistence between our results and those introduced in Refs. \cite{Bunk:2013uea, Barbieri:2015yvd, Biggio:2016wyy}.
Our paper is organized as follows. Section \ref{intro} explains our aim of this work. Section \ref{sec_formulas} introduces notations and important formulas to establish the relations between AMM and cLFV amplitudes. Section \ref{sec_discuss} shows discussions to confirm the consistence of our results and previous works, and the validation of the WI for the relevant analyitic formulas. Section \ref{sec_conclusion} summarizes main features of our work. Finally, we provide many appendices showing precisely many intermediate steps and notations to derive the final results mentioned in this work, including the analytic formulas of the PV functions consistent with LoopTools given in appendix~\ref{app_PVLT}.
\section{ \label{sec_formulas} General amplitudes and notations}
It is well-known that analytic formulas of one-loop contributions to the cLFV amplitudes $e_b(p_2)\rightarrow e_a(p_1)\gamma(q)$ and AMM of SM charged leptons $e_a$ can be presented in the same expressions, see for example Ref. \cite{Crivellin:2018qmi} corresponding to the presence of new heavy particles in BSM. Possible one-loop Feynman diagrams contributing to $a_{e_a}$ and cLFV decay amplitudes $e_b\to e_a \gamma$ in BSM are shown in Fig.~\ref{fig_eab}, where $F$ is a fermion coupling with the SM charged lepton $e_a=e,\mu,\tau$; and the boson $B=h, V$ is a scalar or gauge boson, respectively.
\begin{figure}[ht]
\centering
\includegraphics[width=14cm]{ebaOneLoop}
\caption{Feynman diagrams for one-loop contribution to $a_{e_a}$ and cLFV amplitudes $e_b\to e_a \gamma$ in the unitary gauge.
\label{fig_eab}}
\end{figure}
We note here that Ref. \cite{Yu:2021suw} argues another type of $FSV$ one-loop diagrams giving new contributions to the AMM. They will be discussed in details in this work.
Firstly, we adopt the Lagrangian generating one-loop diagrams in Fig. \ref{fig_eab}, namely \cite{Crivellin:2018qmi}
\begin{align}
\mathcal{L}_h &=\overline{F}(g_{a, Fh}^{L}P_L +g_{a,Fh}^{R}P_R) e_a h +\mathrm{h.c.}, \label{eq_LFh}
\\
\mathcal{L}_V& =\overline{F}\gamma^{\mu} ( g_{a,FV}^{L}P_L +g_{a, FV}^{R} P_R)e_aV_{\mu} +\mathrm{h.c.} \label{eq_LFV},
\end{align}
where the fermion $F$ and the boson $B=V_{\mu},h$ have electric charges $Q_F$ and $Q_B$, and masses $m_F$ and $m_B$, respectively. These Lagrangians \eqref{eq_LFh} and \eqref{eq_LFV} are consistent with those in Ref. \cite{Lavoura:2003xp}. Moreover, the photon couplings with all physical particles should be mentioned clearly, as given in Ref. \cite{Lavoura:2003xp}, i.e., we will adopt the Feynman rules that the photon alway couples with two identical physical particles, as given in table \ref{t_AXX},
\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Vertex & Coupling & Vertex & Couplings &Vertex & Couplings\\
\hline
$A^{\mu}(p_0)V^{\nu}(p_+)V^{*\lambda}(p_-)$&$-ieQ_V\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-) $&$A^{\mu}h(p_+)h^*(p_-)$&$ ieQ_h(p_+-p_-)_{\mu}$ & $A^{\mu}\overline{F}F$& $ieQ_F\gamma_{\mu}$\\
\hline
\end{tabular}
\caption{Feynman rules for cubic couplings of photon $A^{\mu}$, where $p_{0,\pm}$ are incoming momenta into the relevant vertex.
\label{t_AXX}}
\end{table}
where $\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-) = g_{\mu\nu} (p_0 -p_+)_{\lambda} +g_{\nu \lambda} (p_+ -p_-)_{\mu} +g_{ \lambda \mu} (p_- -p_0)_{\nu}$ is the standard form. The more general form of $\Gamma_{\mu \nu \lambda}(p_0,p_+,p_-)$ introduced in Refs. \cite{Bunk:2013uea, Barbieri:2015yvd, Biggio:2016wyy} will be discussed in details
later.
All couplings listed in Lagrangians \eqref{eq_LFh}, \eqref{eq_LFV}, and table \ref{t_AXX} result in the following form factors relevant with one-loop contributions:
\begin{align} \label{eq_Boson}
c_{R B}^{ab}= &\dfrac{e}{16\pi^2}g_{a,FB}^{L*} g_{b,FB}^{R} m_F \times \dfrac{f_B(x_B) +Q_Fg_B(x_B)}{m_B^2}
%
\nonumber \\
&+ \dfrac{e}{16\pi^2}\left(m_{b} g_{a,FB}^{L*} g_{b, FB}^{L} +m_{a} g_{a,FB}^{R*} g_{b, FB}^{R}\right) \times \dfrac{\tilde{f}_B (x_B) +Q_F \tilde{g}_B(x_B)}{m_B^2},
\end{align}
where $x_B \equiv m_F^2/m_B^2$. The four scalar functions $f_B(x)$, $g_B(x)$, $\tilde{f}_B(x)$, and $\tilde{g}_B(x)$ are listed in Eq. \eqref{eq_fgx} of appendix \ref{app_PVLT}, as the approximate formulas in the limit $m_a,m_b\ll m_B$. The formula in Eq. \eqref{eq_Boson} does not contain contributions from the $FSV$ diagrams mentioned in Ref. \cite{Yu:2021suw}, because of the absence of photon coupling $AVh$. The corresponding formulas of AMM and cLFV decay rates are:
\begin{align}
a_{e_a} &\equiv -\dfrac{2m_{a}}{e}\left(c^{aa}_R + c^{aa*}_R\right) = -\dfrac{4m_{a}}{e}\mathrm{Re}[ c^{aa}_R], \label{eq_aea1}
\\ \mathrm{Br} (e_b\to e_a \gamma)&= \frac{m^3_b}{4\pi \Gamma_b}\left( \left|c^{ab}_R\right|^2 + \left|c^{ba}_R\right|^2\right),
\end{align}
where $m_a$, $m_b$, and $\Gamma_b$ are the masses and total decay width of the leptons $e_a$, $e_b$, and
\begin{align}
\label{eq_cabR}
c^{ab}_R &\equiv \sum_{B,F} c^{ab}_{RB}.
\end{align}
The amplitude for a vertex $\bar{e}_a e_aA_{\mu}$ in Ref. \cite{Peskin:1995ev} is consistent with the following form presenting both AMM and cLFV amplitudes \cite{Escribano:1996wp,Eidelman:2016aih}
\begin{equation}\label{eq_eeAeff}
i\mathcal{M}=-ie \overline{u_a}(p_1)\left[ \gamma^{\mu}F_1 -\frac{\sigma^{\mu\nu}q_{\nu}}{2m_{a}} \left(iF_2 + \gamma^5 F_3\right)\right]u_b(p_2)\varepsilon^*_{\mu},
\end{equation}
where $\sigma^{\mu\nu}\equiv \frac{i}{2}\left[ \gamma^{\mu} \gamma^{\nu} - \gamma^{\nu} \gamma^{\mu}\right]$; $F_{1,2,3}$ are scalar form factors; $\varepsilon^*_{\mu}$ and $q_{\nu}$ is the polarized vector of the external photon. The form factor $F_{2,3}$ gets contribution only from loop corrections. They relate with the well-known experimental quantities called the anomalous magnetic moment $a_{e_a}$ and electric dipole moment $d_{e_a}$ for $b=a$, respectively. Specifically we have
\begin{equation}\label{eq_ga}
F_{1}=1;\quad a_{e_a}=F_2; \quad d_{e_a}=-\frac{e}{2m_{a}}F_3.
\end{equation}
Regarding to the LFV decay $ e_b\rightarrow e_a \gamma$ the amplitude can also be written in the same form \cite{Cheng:1984vwu, Lavoura:2003xp}, suggesting that $F_2$ can be calculated based on the one-loop corrections to LFV decays. In particular, the second term of the amplitude \eqref{eq_eeAeff} can be expanded as follows \cite{Hue:2017lak}
\begin{align}
\mathcal{M}&=(2p_1.\varepsilon^*)\overline{u_a} \left( C_{(ab)L} P_L +C_{(ab)R} P_R\right)u_b
+\overline{u_a} \left[D_{(ab)L} \slashed{\varepsilon}^* P_L +D_{(ab)R} \slashed{\varepsilon}^* P_R\right]u_b,
\end
|
{align}
where $m_{a}=m_{b}$ and we can prove that $C_{(ab)L}P_L +C_{(ab)R} P_R =\frac{e}{2m_{a}}(F_2 -i\gamma^5 F_3)$. The WI for the external photon gives
\begin{equation}\label{eq_DLR}
D_{(ab)L}= -(m_{b}C_{(ab)R} +m_{a}C_{(ab)L}), \; D_{(ab)R} = -(m_{b} C_{(ab)L} +m_{a}C_{(ab)R}).
\end{equation}
The hermiticity that $C_{(aa)R}=C^*_{(aa)L}$ \cite{Eidelman:2016aih} gives
\begin{align}
\label{eq_FCLR}
a_{e_a}&= \frac{m_{a}(C_{(aa)L} + C_{(aa)R})}{e} =\frac{2m_{a} \mathrm{Re}{[C_{(aa)L,R}]} }{e},\nonumber \\
%
d_{e_a}&=i(C_{(aa)R} -C_{(aa)L})=\mathrm{Im}{[C_{(aa)L}]}= -\mathrm{Im}{[C_{(aa)R}]}.
\end{align}
Hence, the following relations between two different notations must be satisfied:
\begin{equation}\label{eq_relationCbaR}
c^{ab}_{R}= -\frac{1}{2} C_{(ab)R} \; \mathrm{and} \; c^{ba}_{R}= -\frac{1}{2} C_{(ab)L}.
\end{equation}
From the above discussion, we see that one-loop contributions to the $a_{e_a}$ and $d_{e_a}$ can be written in terms of well-known PV functions, see detailed discussions in Ref. \cite{Hue:2017lak} or general formula introduced for calculations the LFV decay rates of charged leptons \cite{Lavoura:2003xp}, with the identification that $\sigma_{L,R}\equiv -C_{(ab)L,R}$. In the limit of $0\simeq m_{a},m_{b}\ll m_B$, the numerical values of $a_{e_a}$ can be evaluated using the numerical packages such as LoopTools \cite{Hahn:1998yk} or Collier \cite{Denner:2016kdg}. Although the exact analytic formulas of one-loop three point functions presented in Ref.~\cite{Hue:2017lak} can not be applied to calculate $a_{e_a}$, but the limit of $m_b\to m_a$ can be used to solve this problem. The analytic formulas
of $a_{e_a}$ were introduced completely in Ref. \cite{Yu:2021suw}.
Because of the relations in Eq. \eqref{eq_DLR}, only $C_{(ab)L,R}$ is needed to determine $a_{e_a}$ and Br$(e_b\to e_a \gamma)$. Because all two-point diagrams give contributions to just $D_{(ab)L,R}$, $C_{(ab)L,R}$ are calculated by considering only three-point diagrams. In this work, the analytic formulas of $D_{(ab)L,R}$ will be determined directly from all diagrams in Fig. \ref{fig_eab} to check the validation of the WI in the presence of the $FSV$.
The analytic formulas for one-loop contributions to the cLFV decay amplitudes presented
in this work are more general than the results introduced in Ref.~\cite{Hue:2017lak} for general 3-3-1 models. Many important steps in our calculations were shown in appendix~\ref{app_detailedStep}. Using this unitary gauge, the assumption for a particular form of the Goldstone boson couplings given in Ref. \cite{Lavoura:2003xp} is unnecessary. In contrast, we use the same photon couplings to other physical particles in an arbitrary BSM, as given in table \ref{t_AXX}. Namely, a tree-level photon coupling always contains two identical physical particles. This implies that the contributions from the $FSV$ diagrams are not included.
Using the notations of PV-functions defined in appendix \ref{app_PVLT}, the $Fhh$ contributions from diagram (1) in Fig. \ref{fig_eab} are:
\begin{align}\label{eq_C_Lfhh}
C^{Fhh}_{(ab)L} =& \frac{-eQ_h}{16\pi^2} \left[ m_{a}g_{a, Fh}^{L*} g_{b, Fh}^{L} X_1^{f}
%
+ m_{b}g_{a, Fh}^{R*} g_{b,Fh}^{R} X_2^{f} -m_F g_{a,Fh}^{R*} g_{b,Fh}^L X_0^{f} \right],
%
\nonumber \\ C^{Fhh}_{(ab)R} =& \frac{-eQ_h}{16\pi^2}\left[m_{a}g_{a,Fh}^{R*} g_{Fh}^{bR}X_1^{f}
%
+ m_{b}g_{a, Fh}^{L*} g_{b,Fh}^{L}X_2^{f} -m_F g_{a, Fh}^{L*}g_{b, Fh}^{R}X_0^{f}\right],
%
\end{align}
where $X^f_{0},X^f_1,\dots$ are linear combinations of the PV-functions $C_{0, 00, i, ij}$ defined precisely in appendix \ref{app_PVLT}.
The diagram (2) in Fig. \ref{fig_eab} gives $hFF$ contributions as follows:
\begin{align}
\label{eq_CLhff}
C^{hFF}_{(ab)L} =& \frac{-eQ_F}{16\pi^2}\left[ m_{a}g_{a,Fh}^{L*} g_{b,Fh}^{bL} X_1^{h}
%
+ m_{b}g_{a,Fh}^{R*} g_{b,Fh}^{R}X_2^{h} +m_F g_{a,Fh}^{R*} g_{b,Fh}^L X_3^{h} \right],
%
\nonumber \\ C^{hFF}_{(ab)R} =& \frac{-eQ_F}{16\pi^2} \left[m_{a}g_{a,Fh}^{R*} g_{b,Fh}^{R}X_1^{h}
+ m_{b}g_{a,Fh}^{L*} g_{b, Fh}^{L} X_2^{h} +m_F g_{a, Fh}^{L^*}g_{b,Fh}^{R}X_3^{h}\right].
%
\end{align}
where $X^h_{1,2,3}$ are linear combinations of $C_{0,i,ij}(m_h^2, m_F^2, m_F^2)$. The above result are completely consistent with the results introduced in Ref. \cite{Lavoura:2003xp}, except an overall sign and the signs before the PV-functions $\bar{c}_{1,2}$, arising from the different definitions of the external momenta $p_i$ in the denominators of the one-loop integrals. We also give the analytic formulas of $D^{Fhh}_{(ab)L,R}$ and $D^{hFF}_{(ab)L,R}$, used to confirm the WI given in Eq. \eqref{eq_DLR} for the only-scalar contributions. The PV-functions derived from the diagram (2) defined as $X^h_{i}$ are different from $X^f_{i}$ defined for three diagrams (1), (3) and (4). In contrast, the equal functions are denoted as follows:
$$ B^{(i)}_0 \equiv B^{(i)f}_0=B^{(i)h}_0, \;X_0\equiv X_0^f=X_0^h, \; i=1,2. $$
The form factors $D_{(ab)L,R}$ originated from scalar contributions are:
\begin{align}
\label{eq_Hformfactor}
D_{(ab)L}^{Fhh}=& \frac{-eQ_H}{16\pi^2} \left\{g^{L*}_{a,Fh}g^{L}_{b,Fh} \times 2 C^f_{00} \right\}
\nonumber \\ &
+ \frac{-eQ_e}{16\pi^2(m_a^2 -m_b^2)} \left\{ \left(m_b g^{L*}_{a,Fh}g^{R}_{b,Fh} + m_a g^{R*}_{a,Fh} g^{L}_{b,Fh} \right)m_F \left( B^{(1)}_0 -B^{(2)}_0 \right)
\right. \nonumber \\ &- \left. g^{L*}_{a,Fh} g^{L}_{b,Fh}\left( m^2_a B^{(1)f}_1- m^2_b B^{(2)f}_1 \right) - m_a m_b g^{R*}_{a,Fh} g^{R}_{b,Fh} \left( B^{(1)f}_1- B^{(2)f}_1\right) \right\},
\nonumber \\ D_{(ab)R}^{Fhh} =& D_{(ab)L}^{FHH} \left[ g^L_{a,Fh} \leftrightarrow g^R_{a,Fh}, g^L_{b,Fh} \leftrightarrow g^R_{b,Fh} \right],
\nonumber \\ D_{(ab)L}^{hFF} =& -\frac{eQ_F}{16 \pi^2} \left\{ g^{L*}_{a,Fh} g^{L}_{b,Fh} \left[m_F^2 C^h_0 +(2-d) C^h_{00} -m_a^2X_1^h - m_b^2 X_2^h \right]
\right.\nonumber \\&\left. + g^{R*}_{a,Fh} g^{R}_{b,Fh} m_a m_bX_0 + \left[ g^{R*}_{a,Fh} g^{L}_{b,Fh} m_a + g^{L*}_{a,Fh} g^{R}_{b,Fh} m_b\right] m_F C^h_0 \right\},
\nonumber \\ D_{(ab)R}^{hFF} =& D_{(ab)L}^{hFF}\left[ g^L_{a,Fh} \leftrightarrow g^R_{a,Fh}, g^L_{b,Fh} \leftrightarrow g^R_{b,Fh} \right].
\end{align}
It is noted that the $Fhh$ contributions are the sum of three diagrams (1), (3), and (4), while the $hFF$ contributions are from the only diagram (2). We emphasize that the electric charge conversation $Q_F=Q_h+Q_e$ is one of the necessary requirements to guarantee the WI given in Eq. \eqref{eq_DLR}, see a detailed proof in appendix \ref{app_detailedStep}. We can see this crudely from the necessary condition that div$[D^{hFF}_{(ab)L}]+ \mathrm{div}[D^{Fhh}_{(ab)L}]\sim g_{a}^{L*}g^L_b (Q_e+Q_h-Q_F)=0$ and div$[D^{hFF}_{(ab)R}]+ \mathrm{div}[D^{Fhh}_{(ab)R}] \sim g_{a}^{R*}g^R_b(Q_e+Q_h-Q_F)=0$. This conclusion supports completely the only case of electric conversation among the remaining ones mentioned in Ref. \cite{Lavoura:2003xp}.
Regarding Lagrangian \eqref{eq_LFV}, which results in four diagrams in the second line of Fig. \ref{fig_eab}, diagram (5) gives the following $FVV$ contributions:
\begin{align}
\label{eq_CLFVV}
C_{(ab)L}^{FVV}
= -\frac{eQ_V}{16\pi^2} &
\left\{ g_{a,FV}^{R*}g_{b,FV}^{L} m_F \left[ 3X_3^f +\frac{1}{2m_V^2} \right]
%
- g_{a,FV}^{L*}g_{b, FV}^{R} m_F\times \frac{ m_{a}m_{b}}{ m_V^2} X^f_{012}
\right. \nonumber \\ & + g_{a, FV}^{L*}g_{b,FV}^{L}m_{a} \left[2 (X^f_1 -X^f_3) + \frac{m_F^2 X^f_{01} +m_{b}^2X^f_2}{m_V^2} \right]
%
\nonumber \\ & \left. + g_{a, FV}^{R*}g_{b, FV}^{R}m_{b} \left[ 2 (X^f_2 -X^f_3) + \frac{m_F^2 X^f_{02} +m_{a}^2X^f_1}{m_V^2} \right] \right\},
\end{align}
where $X^{f}_{i}=X_{i}(m_F^2, m_V^2, m_V^2)$, and
\begin{align}
\label{eq_CRFVV}
C_{(ab)R}^{FVV}
= -\frac{eQ_V}{16\pi^2} &
\left\{ g_{a, FV}^{L*}g_{b, FV}^{R} m_F \left[ 3X^f_3 +\frac{1}{2m_V^2} \right] - g_{a, FV}^{R*}g_{b, FV}^{L} m_F\times \frac{ m_{a}m_{b}}{ m_V^2} X^f_{012}
\right. \nonumber \\ & +g_{a, FV}^{R*}g_{b, FV}^{R}m_{a} \left[ 2 (X^f_1 -X^f_3) +\frac{m_F^2 X^f_{01}+ m_{b}^2 X^f_2}{m_V^2} \right]
%
\nonumber \\ & \left. +g_{a, FV}^{L*}g_{b,FV}^{L}m_{b} \left[2 (X^f_2 -X^f_3) +\frac{m_F^2X^f_{02}+ m_{a}^2 X^f_1}{m_V^2} \right] \right\}.
\end{align}
Diagram (6) gives $VFF$ contributions:
\begin{align}
\label{eq_VFFCL}
C_{(ab)L}^{VFF}= -\frac{eQ_F}{16\pi^2} &\left\{ m_{a} g_{a, FV}^{L*} g_{b, FV}^{L} \left[ \frac{}{} 2 X^v_{01}
+\frac{m_F^2\left( X^v_1 -X^v_3\right)+ m_{b}^2X^v_2 }{m_V^2} \right]
%
\right. \nonumber \\&\; + m_{b}g_{a, FV}^{R*} g_{b,FV}^{R} \left[ \frac{}{}2 X^v_{02}
%
+\frac{m_F^2\left( X^v_2 -X^v_3\right)+ m_{a}^2 X^v_1}{m_V^2} \right]
%
\nonumber \\&\; - g_{a,FV}^{R*} g_{b,FV}^{L} m_F \left[ 4 X_0
%
+ \frac{m_{a}^2X^v_1 + m_{b}^2X^v_2- m^2_{F}X^v_3}{m_V^2} \right]
%
\nonumber \\&\left. \;-g_{a, FV}^{L*}g_{b, FV}^{R} \frac{m_{a} m_{b}}{m_V^2}\times m_F (X^v_{12} -X^v_3) \right\},
%
\end{align}
where all $X_{i}^v$ are expressed in terms of PV functions $C^{VFF}_{0,i,ij}=C_{0,i,ij}(m_V^2, m_F^2, m_F^2)$, and
\begin{align}
\label{eq_VFFCR}
C_{(ab)R}^{VFF}= -\frac{eQ_F}{16\pi^2} &\left\{ m_{a} g_{a, FV}^{R*} g_{b,FV}^{R} \left[ 2 X^v_{01}
+\frac{m_F^2\left( X^v_1 -X^v_3\right)+ m_{b}^2X^v_2}{m_V^2} \right]
%
\right. \nonumber \\&\; + m_{b}g_{a, FV}^{L*} g_{b, FV}^{L} \left[ 2 X^v_{02}
%
+\frac{m_F^2\left( X^v_2 -X^v_3\right)+ m_{a}^2X^v_1}{m_V^2} \right]
%
\nonumber \\&\; - g_{a, FV}^{L*} g_{b, FV}^{R} m_F \left[ 4 X^v_0
%
+ \frac{m_{a}^2X^v_1 + m_{b}^2X^v_2 - m^2_{F}X^v_3}{m_V^2} \right]
%
\nonumber \\&\left. \;-g_{a,FV}^{R*}g_{b, FV}^{L} \frac{m_{b} m_{a}}{m_V^2}\times m_F (X^v_{12} -X^v_3) \right\}.
%
\end{align}
Finally, using the simple notations $g^{L,R}_a\equiv g^{L,R}_{a,FV}$, the formulas of $D_{(ab)L}$ and $D_{(ab)R}$ are
\begin{align}
D^{(78)}_{(ab)L}=& D^{(7)}_{(ab)L}+ D^{(8)}_{(ab)L}
\nonumber \\ = & \frac{e Q_e}{ 16\pi^2 (m_a^2 -m_b^2)} \left\{ \frac{}{} \left( g^{L*}_{a} g^{R}_{b} m_b + g^{R*}_{a} g^{L}_{b} m_a\right) 3m_F \left[ B^{(1)}_0 -B^{(2)}_0\right]
\right. \nonumber \\ &- \left. m_b \left( m_a g^{R*}_{a} g^{R}_{b} + m_b g^{L*}_{a} g^{L}_{b}\right) \left[ \left(2+ \frac{m_F^2 +m_b^2}{m_V^2}\right) B^{(2)v}_1 +\frac{A_0(m_V^2) +2m_F^2B^{(1)}_0}{m_V^2} +1 \right]
\right. \nonumber \\ &+\left. m_a \left( m_b g^{R*}_{a} g^{R}_{b} + m_a g^{L*}_{a} g^{L}_{b}\right) \left[ \left(2+ \frac{m_F^2 +m_a^2}{m_V^2}\right) B^{(1) v}_1 +\frac{A_0(m_V^2) +2m_F^2B^{(2)}_0}{m_V^2} +1 \right]\right\}, \label{eq_DabLR78}
\\ D^{(78)}_{(ab)R}=& D^{(78)}_{(ab)L} \left[ g^L_{a} \leftrightarrow g^R_{a}, \; g^L_{b} \leftrightarrow g^R_{b}\right].
\nonumber \\ D_{(ab)L}^{FVV}=& -\frac{eQ_V}{16\pi^2} \left\{\frac{}{} g_{a}^{L*} g_{b}^{L} \left[\frac{}{} 2(d-2)C^f_{00} +2(m_a^2+m_b^2)X^f_3
\right.\right.\nonumber \\&\left. \left. -\frac{1}{m_V^2} \left( m_F^2(B^{(1)}_0 +B^{(2)}_0 -2C^f_{00} ) +A_0(m_V^2) +m_a^2 B^{(1) f}_1+m_b^2 B^{(2) f}_1 \right)\right]
\right. \nonumber \\&\; + g_{a}^{R*} g_{b}^{R} m_a m_{b} \left[4X^f_3 +\frac{2 C^f_{00}}{m_V^2} \right]
+ g_{a}^{R*} g_{b}^{L} \times m_a m_F \left[ 3C^f_0 -\frac{1}{2m_V^2} + \frac{m_b^2 X^f_{012}}{m_V^2}\right]
\nonumber \\&\left. \; +g_{a}^{L*}g_{b}^{R} \times m_b m_F \left[ 3C^f_0 -\frac{1}{2m_V^2} + \frac{m_a^2 X^f_{012}}{m_V^2}\right] \right\},
\nonumber \\ D_{(ab)R}^{FVV}=& C_{(ab)L}^{FVV} \left[ g^{L}_{a} \leftrightarrow g^{R}_{a}, g^{L}_{b} \leftrightarrow g^{R}_{a}\right] \label{eq_FVVDLhand}.
\end{align}
The remaining formulas of $D_{(ab)L,R}$ from diagram (6) of Fig. \ref{fig_eab} are
\begin{align}
\label{eq_DLR6}
D^{VFF}_{(ab)L}=& \frac{eQ_F}{16\pi^2} \left\{ g^{L*}_{a} g^{L}_{b} \left[\frac{}{} -2m_F^2 C_{0} +(d-2)^2 C_{00}^v +2m_a^2 X^v_{01}+2m_b^2 X^v_{02}
\right.\right.\nonumber \\&\left.\left. \qquad \qquad \qquad \quad -\frac{1}{m_V^2} \left[(2-d)m_F^2 C^v_{00} +A_0(m_V^2) +m_F^2 \left( B^{(1)}_0 +B^{(2)}_0 \right)
\right.\right. \right. \nonumber \\&\left.\left. \left. \qquad \qquad \qquad \qquad \qquad - m_a^2\left(B^{(1) v}_0 +B^{(1)}_1 \right) -m_b^2 \left( B^{(2) v}_0 +B^{(2)v}_1\right) +m_a^2m_b^2 X_0
\right.\right. \right. \nonumber \\&\left.\left. \left. \qquad \qquad \qquad \qquad \qquad
-m_F^2 \left( (m_a^2 +m_b^2-m_F^2)C_0 +m_a^2 X^v_1 +m_b^2X^v_2\right) \frac{}{} \right] \right]
\right.\nonumber \\&\left. \qquad \quad + g^{R*}_ag^{R}_b m_am_b \left[ 2X_0 -\frac{1}{m_V^2}\left((2-d) C^v_{00} +m_F^2 X^v_3 -m_a^2 X^v_1 -m_b^2 X^v_2 \right)\right]
\right.\nonumber \\ &\left.\qquad \quad + \frac{g^{R*}_ag^{L}_b m_am_F}{m_V^2} \left[ -B^{(1)v}_1 +(2-d)C_{00} -m_a^2 X^v_1 +m_b^2(X^v_3 -X^v_2) \right]
\right.\nonumber \\ &\left.\qquad \quad + \frac{g^{L*}_ag^{R}_b m_bm_F}{m_V^2} \left[ -B^{(2) v}_1 +(2-d)C^v_{00} -m_b^2 X^v_2 +m_a^2 (X^v_3 -X^v_1) \right] \right\},
\nonumber \\ D^{VFF}_{(ab)R}= & D^{VFF}_{(ab)L} \left[ g_a^L\leftrightarrow g_a^R,\; g_b^L\leftrightarrow g_b^R\right].
\end{align}
We note that all results presented here are crosschecked by FORM package \cite{Vermaseren:2000nd}, using intermediate steps given in appendix \ref{app_detailedStep}. There is a property that $C^X_{(ab)R}=C^X_{(ab)L}\left[ g_a^L\leftrightarrow g_a^R,\; g_b^L\leftrightarrow g_b^R\right]$ for all $X=Fhh,hFF,FVV,VFF$. The above results of one-loop contribution to $C_{(ab)L,R}$ are totally consistent with those introduced in Ref. \cite{Lavoura:2003xp}, after some
transformations of notations presented in appendix \ref{app_special}. In the limit of $m_h^2,m_V^2\gg m^2_a,m^2_b$, i.e., $m^2_a/m_B^2,m^2_b/m_h^2\simeq 0$ with $B=h,V$, we get consistent results with those given in Refs. \cite{Crivellin:2018qmi, Freitas:2014pua, Stockinger:2006zn}. To derive the above results for gauge boson exchanges, we start with many important features different from those mentioned in Ref. \cite{Lavoura:2003xp}, namely: i) we do
not use the typical form of couplings relating with Goldstone bosons going along with the presence of new gauge bosons, ii) we have to use the massless property of the on-shell photon $q^2=0$, iii) to confirm the WI for all diagrams given in Fig. \ref{fig_eab}, we need the charge conversation law corresponding to the Lagrangian \eqref{fig_eab}: $Q_F=Q_V +Q_e$. Therefore, our calculation is another independent approach to confirm the result given in Ref. \cite{Lavoura:2003xp}. The details of the calculation to confirm the WI for all one-loop contributions are given in appendix \ref{app_detailedStep}. We remind that our results are derived from the photon couplings listed in the table \ref{t_AXX}, and do not contain the contributions from the FSV diagrams. In the following, we pay attention to the possibility of adding the FSV diagrams or the new forms of the photon couplings.
\section{\label{sec_discuss} Discussion on WI and previous results}
\subsection{WI to constrain the form of photon couplings}
Now we focus on the feature that the WI of the on-shell photon will constrain strongly the forms of the cubic photon couplings with two physical particles in a renormalized Lagrangian. Now we consider the existence of the photon couplings type at tree level:
\begin{align}
\label{eq_AX12}
\mathcal{L}^{\gamma XX} =& eQ_FA^{\mu}\left[ \overline{F_1}\gamma^{\mu}F_2 +\mathrm{h.c.} \right] + eQ_hA^{\mu} \left[ \left( h_1^* \partial_{\mu}h_2 -h_2 \partial_{\mu}h_1^*\right) +\mathrm{h.c.}\right]
%
\nonumber \\ & -\left[ eQ_VA^{\mu}V^{\nu}_1 V^{\lambda*}_2 \Gamma_{\mu \nu \lambda}(p_0,p_+p_-) + \mathrm{h.c.}\right] +\left[g_{\gamma hV} g_{\mu \nu}h^{-Q} A^{\mu}V^{Q\nu}+ \mathrm{h.c.}\right],
\end{align}
where all couplings are more general than those well-known as the standard forms given in Table \ref{t_AXX}. The last term corresponds to the photon couplings with a scalar $h$ and a gauge boson $V$. The above Lagrangian results in the following decays from the heavy particle to lighter one: i) $F_2\to F_1 \gamma$, ii) $h_2\to h_1 \gamma$, iii) $V_2\to V_1 \gamma$, and iv) $V\to h\gamma$. The WI for these decay amplitudes at tree level is $\mathcal{M}^{\mu}(X_1\to X_2 \gamma)p_{0\mu}=0$ with $p_{0\mu}$ being the external photon momentum. It can be derived that:
\begin{itemize}
\item Using the same convention of external momenta given in Fig. \ref{fig_eab}\Binh{,} we have $\mathcal{M}^{\mu}(F_2\to F_1 \gamma)q_{\mu}\sim (m_{F_2}-m_{F_1}) \overline{u}_{F_2}(p_2)u_{F_1}(p_1)=0$, where $p_0\equiv -q$. Therefore, $m_{F_2}=m_{F_1}$. This case is automatically satisfied for the tree level AMM amplitude.
\item $\mathcal{M}^{\mu}(h_2\to h_1 \gamma)p_{0\mu}\sim(p_2-p_1).(p_2+p_1)= (m^2_{h_2}-m^2_{h_1}) =0$, where all on-shell momenta are incoming the vertex $A^{\mu}h^*_1h_2$, implying that $p_0=-(p_1+p_2)$ and $p_{1,2}^2=m^2_{h_{1,2}}$. The consequence is $m_{h_1}=m_{h_2}$.
%
\item $\mathcal{M}^{\mu}(V \to h \gamma)p_{0\mu}\sim\varepsilon_v.p_{0} =0$, where $\varepsilon_v$ and $p_0$ are the polarization of gauge boson $V$ and the external momentum of the photon $A_{\mu}$. Hence the presence of a $AhV$ vertex does not automatically satisfy the WI. One-loop contributions for all diagrams arising from this vertex must be checked for the validation of WI.
%
\item $\mathcal{M}_{\mu}(V_1 \to V_2 \gamma)p^{\mu}_0 \sim\varepsilon^{\nu}_1\varepsilon^{\lambda*}_2p^{\mu}_0 \Gamma_{\mu \nu \lambda}(p_0,p_1,p_2) =0$, where $\varepsilon_{1,2}$, and $p_{1, 2, 0}$ are the polarization of the gauge boson $V_{1,2}$ and the external momentum of the gauge bosons $V_{1,2}$ and photon $A_{\mu}$, respectively. We will use the following properties of the external gauge bosons $V_i (i=1,2)$ and photon: $\varepsilon_i.p_{i}=0$, $p_0^2=0$, $p_i^2=m^2_{V_i}$, and the momentum conversation $p_0 +p_1 +p_2=0$ following notations in table \ref{t_AXX}. After some intermediate steps of calculation, we have:
\begin{align}
\label{eq_WIV12ga}
\mathcal{M}_{\mu}(V_1 \to V_2 \gamma)p^{\mu}_0 \sim& (p_0.\varepsilon_1) \left[ (p_0 -p_1). \varepsilon^*_2\right] + (\varepsilon_1.\varepsilon^*_2) \left[ (p_1 -p_2).p_0 \right] +(p_0.\varepsilon^*_2) \left[ (p_2 -p_0).\varepsilon_1 \right]
%
\nonumber \\ =& (\varepsilon_1.\varepsilon^*_2) \left[ m^2_{V_2} -m^2_{V_1} \right]=0.
\end{align}
Hence, $m_{V_1}= m_{V_2}$ is necessary. From this, we consider the more general photon coupling with a gauge boson \cite{Biggio:2016wyy} describing the couplings of a leptoquark field \cite{Barbieri:2015yvd}
\begin{align}
\label{eq_Gamma1}
\Gamma'_{\mu\nu\lambda}(p_0,p_1,p_2) =& g_{\mu
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.